Combine 3050 Info

Introduction to InvestmentsFINAN 3050

Week 7:

Managing Bond Portfolios (Chp. 10.1-10.3)

Slide 2Week 7

Michael HallingUniversity of Utah

YTM vs. Spot Rates

� Definition of Yield to Maturity (YTM): the yield (constant across time) that gives the observed price; denoted by r (without any subscript)

� Definition of spot rates: the spot rate st is the rate of interest for money held from the present time (0) until time t (measured in years); YTM of zero-

coupon bonds

( ) ( )∑= +

++

=T

tTt

tB

r

FV

r

CP

1 11

( ) ( )∑= +

++

=T

tT

T

t

t

tB

s

FV

s

CP

1 11

Slide 3Week 7


YTM vs. Spot Rates: Example

� Consider a 3-year bond with a coupon rate (annual) of 7%.� Determine the price of the bond if the YTM is 5.48%:

� Determine the price of the bond if the 1-year spot rate s1=5%, the 2-year spot rate s2=5.2% and the 3-year spot rate s3=5.5%

( ) ( ) ( )11.104

055.01

107

052.01

7

05.01

7321=

++

++

+

( ) ( ) ( )11.104

0548.01

107

0548.01

7

0548.01

7321=

++

++

+

Slide 4Week 7


Extracting Spot Rates from Bond Prices

� Assume the following information:

─ There is a 1-year zero-coupon bond trading for $98.─ There is a 2-year, 4%-coupon (annual) bond trading for $102.

� Determine the 1-year and 2-year spot rate (i.e., s1 and s2).

Slide 5Week 7


Interest Rate Risk: Bond Pricing Relationships (1)

� Inverse relationship between price and yield (in the following we refer to the

yield to maturity)� An increase in a bond’s yield to maturity results in a smaller price decline than

the price gain associated with a decrease in yield � relationship between price an yield is non-linear (convex)

� The price of long-term bonds are more sensitive to yield changes than prices of

short-term bonds� Price sensitivity is inversely related to a bond’s coupon rate: higher coupon rate

means lower price sensitivity� Price sensitivity is inversely related to the yield to maturity at which the bond is

currently selling

Slide 6Week 7


Interest Rate Risk: Bond Pricing Relationships (2)

Slide 7Week 7


Interest Rate Risk: Duration (1)

� A measure of the effective maturity of a bond

� The weighted average of the times until each payment is received, with the

weights proportional to the present value of the payment

� Duration is shorter than maturity for all bonds except zero coupon bonds

� Duration is equal to maturity for zero coupon bonds

� A measure of a bond’s sensitivity to interest rate risk: higher duration means higher sensitivity to interest rate risk

Slide 8Week 7



Slide 9Week 7



� Calculation:

─D…Duration─ T…Maturity of the bond

─ r...yield to maturity

─Ct…Coupon in period t in $─ FV…Face Value in $

( ) ( )PriceBond

1

PriceBond

1

1

TT

t

tt

rFV

Tr

C

tD+

×++

×=∑=

Slide 10Week 7


Interest Rate Risk: Duration - Example

8% Bond Time years Payment PV of CF (10%) Weight C1 X C4 1 80 72.727 .0765 .0765 2 80 66.116 .0690 .1392 Sum

3

1080 811.420 950.263

.8539 1.0000

2.5617 2.7774

� Calculation of the duration for a 3-year 8% coupon bond. The YTM is 10%.

Slide 11Week 7


Interest Rate Risk: Duration - Exercise

� Calculate the price and the duration for a 3-year zero-coupon bond. The YTM

is 10%.

� Calculate the price and the duration for a 3-year 8% coupon bond. The YTM is 9%.

� Calculate the price and the duration for a 3-year zero-coupon bond. The YTM

is 9%.

Slide 12Week 7


Interest Rate Risk: Duration Price Relationship

� The duration of a bond can be used to approximate its price change if interest rates change.

� Calculation:

─∆P…Change in bond price (i.e., New Price minus Old Price)─∆P/P…Relative change in bond price

─∆r…Change in YTM

( )r

r

D

P

P

DurationModified

∆×+

−=∆

3211 ( )

Prr

DP

DurationModified

×∆×+

−=∆321

1

Slide 13Week 7


Interest Rate Risk: Duration Price Relationship - Example

� Consider a bond with maturity of 30 years, coupon rate of 8% (paid annually)

and a YTM of 9%. Its price is $897.26 and its duration is 11.37 years.

� Question: What will happen to the bond price if the bond’s yield to maturity

increases to 9.1%?

� Calculate the change in YTM: ∆r=9.1%-9%=0.1%

� Change in price equals -$9.36

( )36.9$26.897$

09.01

37.11−=××

+−=∆

0 . 0 0 10 . 0 0 10 . 0 0 10 . 0 0 1P

Slide 14Week 7


Interest Rate Risk: Duration Price Relationship - Exercise

� Consider a 3-year bond with a coupon rate of 8% and a duration of 2.7774. Thebond has a price of $95.026 and a YTM of 10%.

─Consider an increase in YTM by one basis point (i.e., by 0.001 or 0.1%) to 10.1%

• Calculate the resulting percentage change in the bond price

• Approximate the percentage change in the bond price using the bond’s duration

─Repeat the exercise from before for the following situations:• A decrease in YTM by one basis point

• An increase in YTM by 1%

• A decrease in YTM by 1%

� How precise is the approximation using the duration equation in each case?

Slide 15Week 7


Interest Rate Risk: Some simple duration rules

� Rule 1: The duration of a zero-coupon bond equals its maturity.

� Rule 2: Holding time to maturity and YTM constant, a bond’s duration is higher when the coupon rate is lower.

� Rule 3: Holding the coupon rate constant, a bond’s duration increases with

time to maturity.

Slide 16Week 7


Application of Duration: Passive Bond Management

� Passive managers take bond prices as fairly set and seek to control only the risk

of their fixed-income portfolios.� Immunization:

─ strategies that investors use to shield their fixed-income portfolios from

exposure to interest rate fluctuations─ especially important for banks, insurance companies and pension funds

that have assets and liabilities, i.e., fixed-income products with different characteristics and different sensitivities to interest rate on both sides of

the balance sheets (inflows and outflows) ─ in this case, immunization means to match the duration of assets and liabilities

Slide 17Week 7


Application of Duration: Immunization – Example (1)

� Consider the following problem:

─ an insurance company faces an obligation (liability) of $19,487 in seven years; the market interest rate equals 10% p.a.

─How can you immunize this liability using a combination of 3-year zero

coupon bonds and a 15-year bond with coupon rate of 2.7%?

� Step 1: calculate the duration of the liability� Step 2: calculate the duration of the asset portfolio

� Step 3: find the combination of bonds on the asset side such that the duration of liabilities matches the duration of the assets

� Step 4: Fully fund the obligation

Slide 18Week 7



� Step 1: calculate the duration of the liability─Duration of the liability equals 7.

� Step 2: calculate the duration of the asset portfolio

─Duration of 3-year zero-coupon equals 3 years─Duration of coupon-paying bond equals 11 years

─ Portfolio duration is the weighted average of duration of each component asset

─w…fraction of the portfolio invested in the zero

─Asset duration = w × 3years + (1-w) × 11years

Slide 19Week 7



� Step 3: find the combination of bonds on the asset side such that the duration of

liabilities matches the duration of the assets─Asset duration = Liability Duration

─w × 3years + (1-w) × 11years = 7 years─w=50%

� Step 4: Fully fund the obligation─ Present value of the liabilities equals $10,000 = 19,478/(1+0.1)7

─We have to invest $10,000 into the asset side• $5,000 into the 3-year zero-coupon bond (we buy 66.5 of these bonds with a face

value of $100 each at a price of $75.13 each)

• $5,000 into the coupon bond (we buy 112.4 of these bonds with a face value of

$100 each at a price of $44.48 each)

Slide 20Week 7



� Check if immunization works: consider a drop in interest rates to 9% on the next day (i.e., there are still 7 years left until liability matures)

─ Present value of liability: $10,660.06 = $19,478/(1+0.09)7

─ Present value of asset portfolio: • Zero-coupon bond: P (FV of 100) = $77.2

• Coupon-paying bond: P (FV of 100) = $49.2

• Portfolio Value = $77.2×66.5 + $49.2×112.4 = $10,671.89

─Gap of -$10,660.06+$10,671.89=$11.4

� This strategy ensures that the average duration of assets and liabilities is the

same � if interest rates change in the short run, both – liabilities and assets –change to the same extent and the assets will be sufficient to cover the liabilities

Slide 21Week 7



� Check if immunization works: consider a drop in interest rates to 9% in one year (i.e., there are only 6 years left until liability matures)

─ Present value of liability: $11,619.5 = $19,478/(1+0.09)6

─ Present value of asset portfolio: • Zero-coupon bond: P (FV of 100; 2 years left until mat.) = $84.2

• Coupon-paying bond: P (FV of 100; 14 years left until mat.) = $50.95

• Portfolio Value = $84.2×66.5 + $50.95×112.4 = $11,328.83

─Gap of -$11,619.5+$11,328.83=-$290.63

� This strategy requires rebalancing, as over one year

─ durations of liabilities and assets change─ present values of liabilities and assets change

Slide 22Week 7


Interest Rate Risk: Convexity (1)

� Remember the non-linear

(convex) relationship

between yields and bond

prices.

� Approximating bond price

changes by duration

assumes a linear relationship

� Convexity considers

curvature.

� Do investor like or dislike convexity?

Slide 23Week 7


Interest Rate Risk: Convexity (2)

� Remember the approximation of bond price changes using duration (it is linear in r∆):

� The same relationship looks as follows if convexity is considered:

( )r

r

D

P

P

DurationModified

∆×+

−=∆

3211

( )( )2

2

1

1rCXr

r

D

P

P

DurationModified

∆××+∆×+

−=∆

321

Slide 24Week 7


Interest Rate Risk: Calculating Convexity

� In the spirit of the duration equation – similar structure.

� Example: convexity for a 3-year 8% coupon bond. The YTM is 10%.

( )( ) ( ) ( ) ( )

+

×+×++

×+×+

= ∑= PriceBond

11

PriceBond

11

1

1

1

2

tT

t

tt

rFV

TTr

C

ttr

CX

8% Bond Time years Payment PV of CF (10%) Weight t × (t+1) × C4 1 80 72.727 .0765 .153 2 80 66.116 .0690 .4176 Sum

3

1080 811.420 950.263

.8539 1.0000

10.2468

10.82 ×××× 1/1.12=8.94

Slide 25Week 7


Interest Rate Risk: Convexity - Exercise

� Consider a 3-year bond with a coupon rate of 8%, a duration of 2.7774 and a convexity of 8.94. The bond has a price of $950.263 and a YTM of 10%.

─Consider an increase in YTM by one basis point (i.e., by 0.001 or 0.1%) to 10.1%

• Approximate the percentage change in the bond price using the bond’s duration

AND convexity

─Repeat the exercise from before for the following situations:• A decrease in YTM by one basis point

• An increase in YTM by 1%

• A decrease in YTM by 1%

� Compare your results to the ones from the exercise on slide 14 (approximation

using only duration).

Slide 26Week 7


Interest Rate Risk: Duration Only and Duration-Convexity Approximations – Summary of Results

Change

New

Yield Old Price New Price

Relative

Change

Duration

Appr.

Change

Rel. Dev.

from

Approx.

D-CX

Appr.

Chan.

Rel. Dev.

from

Approx.

0.001 0.101 95.026 94.787 -0.00252 -0.003 -0.18% -0.003 0.00%

-0.001 0.099 95.026 95.267 0.002529 0.003 0.18% 0.003 0.00%

0.01 0.11 95.026 92.669 -0.02481 -0.025 -1.78% -0.025 0.02%

-0.01 0.09 95.026 97.469 0.025702 0.025 1.76% 0.026 0.02%

� This table summarizes results related to the exercises on slide 14 and slide 25.� (Comment: it would be a good exercise to replicate all the results in detail)

Hm

k 1a

1. Y

ou purchased a share of stock for $20. One year later you received $1 as dividend and sold

the share for $24. Your holding-period return w

as __________. 2.

An investor invests 40%

of his wealth in a risky asset w

ith an expected rate of return of 15%

and a variance of 4% and 60%

in a treasury bill that pays 6%. H

er portfolio's expected rate of return and standard deviation are __________ and __________

3. C

onsider the following tw

o investment alternatives. First, a risky portfolio that pays 15%

rate of return w

ith a probability of 60% or 5%

with a probability of 40%

. Second, a treasury bill that pays 6%

. The risk premium

on the risky investment is __________.

4. Y

ou have $500,000 available to invest. The risk-free rate as well as your borrow

ing rate is 8%

. The return on the risky portfolio is 16%. If you w

ish to earn a 22% return, you should

__________. 5.

Risk that can be elim

inated through diversification is called ______ risk 6.

Asset A

has an expected return of 20% and a standard deviation of 25%

. The risk free rate is 10%

. What is the rew

ard-to-variability ratio? 7.

A portfolio is com

posed of two stocks, A

and B. Stock A

has a standard deviation of return of 25%

while stock B

has a standard deviation of return of 5%. Stock A

comprises 20%

of the portfolio w

hile stock B com

prises 80% of the portfolio. If the variance of return on the

portfolio is .0050, the correlation coefficient between the returns on A

and B is __________.

8. The standard deviation of return on investm

ent A is .10 w

hile the standard deviation of return on investm

ent B is .05. If the covariance of returns on A

and B is .0030, the correlation

coefficient between the returns on A

and B is __________.

9. A

portfolio is composed of tw

o stocks, A and B

. Stock A has a standard deviation of return

of 5% w

hile stock B has a standard deviation of return of 15%

. The correlation coefficient betw

een the returns on A and B

is .5. Stock A com

prises 40% of the portfolio w

hile stock B

comprises 60%

of the portfolio. The variance of return on the portfolio is __________. 10. Stocks A

, B, C

and D have betas of 1.5, 0.4, 0.9 and 1.7 respectively. W

hat is the beta of an equally w

eighted portfolio of A, B

and C?

11. Consider the C

APM

. The risk-free rate is 6% and the expected return on the m

arket is 18%.

What is the expected return on a stock w

ith a beta of 1.3? 12. C

onsider the CA

PM. The risk-free rate is 5%

and the expected return on the market is 15%

. W

hat is the beta on a stock with an expected return of 12%

? 13. C

onsider the CA

PM. The expected return on the m

arket is 18%. The expected return on a

stock with a beta of 1.2 is 20%

. What is the risk-free rate?

14. Consider the single factor A

PT. Portfolio A has a beta of 1.3 and an expected return of 21%

. Portfolio B

has a beta of 0.7 and an expected return of 17%. The risk-free rate of return is

8%. If you w

anted to take advantage of an arbitrage opportunity, you should take a short position in portfolio __________ and a long position in portfolio __________.

15. Consider the m

ulti-factor APT w

ith two factors. Portfolio A

has a beta of 0. 5 on factor 1 and a beta of 1.25 on factor 2. The risk prem

iums on the factors 1 and 2 portfolios are 1%

and 7%

respectively. The risk-free rate of return is 7%. The expected return on portfolio A

is __________ if no arbitrage opportunities exist.

16. Security X has an expected rate of return of 13%

and a beta of 1.15. The risk-free rate is 5%

and the market expected rate of return is 15%

. According to the capital asset pricing m

odel, security X

is __________. 17. Y

ou invest $600 in security A w

ith a beta of 1.5 and $400 in security B w

ith a beta of .90. The beta of this form

ed portfolio is __________. 18. Security A

has an expected rate of return of 12% and a beta of 1.10. The m

arket expected rate of return is 8%

and the risk-free rate is 5%. The alpha of the stock is __________.

19. Use the follow

ing to answer questions a-d:

SML

beta

return

15%

10%

5%

01 2

a. W

hat is the expected return on the market?

b. What is the beta for a portfolio w

ith an expected return of 15%

c. W

hat is the expected return for a portfolio with a beta of 0.5?

d. W

hat is the alpha of a portfolio with a beta of 2 and actual return of 15%

? 20. A

ccording to the CA

PM, w

hat is the market risk prem

ium given an expected return on a

security of 13.6%, a stock beta of 1.2, and a risk free interest rate of 4.0%

? 21. U

sing the index model, the alpha of a stock is 4.0%

, the beta if 0.9 and the market return is

10%. W

hat is the residual given an actual return of 15%?

22. The risk premium

for exposure to exchange rates is 5% and the firm

has a beta relative to exchanges rates of 0.4. The risk prem

ium for exposure to the consum

er price index is -6%

and the firm has a beta relative to the C

PI of 0.8. If the risk free rate is 3.0%, w

hat is the expected return on this stock?

23. The small firm

in January effect is strongest ________. 24. Evidence suggests that there m

ay be _______ mom

entum and ________ reversal patterns in

stock price behavior. 25. In a study of investm

ent behavior of men and w

omen, B

arber and Odean find ___

26. Psychological regret theory says ____. 27. M

ental accounting is a form of fram

ing which is consistent w

ith ___ . Solutions 1. 25 percent

2. (

)(

)(

)(

).0800

.04.4

.0960.06

.6.15

.4r

E.5

p p=

=σ

=+

=

3. (

)(

)[

].05

.06-

.05.4

.15.6

=+

=Prem

ium

Risk

4.

()

375,0001-

1.75500,000

751

0816

0822

==

=− −

=

Borrowing

..

..

.y

5. unique, firm

-specific, diversifiable 6.

.40

7. C

orr=.225 C

orr)

05)(.

25)(.

8)(.

2(.2

)05

(.)8

(.)

25(.

)2(.

0050.

22

22

++

=8.

.60(.05)]

.0030/[.10n

Correlatio

==

9.

)5)(.

15)(.

05)(.

6)(.

4(.2

)15

(.)6

(.)

05(.

)4(.

22

22

2+

+=

σ0103

.2=

σ10. .93 11. 21.6%

12. .7

13. 8%

14. short A, long B

; hint: compute the factor’s exp return for each stock

15. 1625.

)07

(.25.1

)01

(.5.

07.

)(

=+

+=

A rE

=16.25%

16. 16.5%

.05)-

1.15(.15.05

benorm

ally

would

)(

=+

x rE

=> overpriced

17. 26.1

)90

(.000,1 400

)5.1

(000,1 600

=+

=p

β

18. 3.7%

19. a. 10% b. ? c. 7.5%

d. ? 20. 8%

21. R

esidual = 15 – (4 + 0.9x10) = 2%

22. 0.2%

23. early in the month

24. short-run, long run 25. that m

en trade more actively than w

omen

26. contends that people have more regret (blam

e themselves m

ore) when a decision that turned

out badly was m

ore unconventional, is consistent with the firm

size anomaly, and is

consistent with the book-to-m

arket anomaly

27. taking a very risky position with one investm

ent account and a very conservative position w

ith another investment account, investors' irrational preference for stocks w

ith high cash dividends, a tendency to ride losing stock positions for too long

Hom

ework 1b – From

BK

M

Ch. 5:

9, 10, 11, C

h. 6: 6, 7, 18, 19

Ch. 7:

2, 3, 4, 6, 7, 28 C

h. 8: 6, 16, 24.

Solutions to book problems from

the Sol Manual:

CH

. 5. 9.

E(rX ) = [0.2 × (–20%)] + [0.5 × 18%

] + [0.3 × 50%)] = 20%

E(rY ) = [0.2 × (–15%

)] + [0.5 × 20%] + [0.3 × 10%

)] = 10%

10. σ

X2 = [0.2 × (–20 – 20) 2] + [0.5 × (18 – 20) 2] + [0.3 × (50 – 20) 2] = 592

σ

X = 24.33%

σY

2 = [0.2 × (–15 – 10) 2] + [0.5 × (20 – 10) 2] + [0.3 × (10 – 10) 2] = 175

σY = 13.23%

11.

E(r) = (0.9 × 20%) + (0.1 × 10%

) = 19%

CH

. 6. Think of your C

APM

project when solving problem

s 6 and 7, we w

ill compute m

eans and st deviation for a variety of portfolios, here w

ith diff proportions in stocks and bonds, they will form

our frontier. Then w

e’ll use some extra form

ulas (that apply only in the case of two-asset portfolios) to help

us identify the min variance portfolio and the tangent one, but the overall objective is to

construct the frontier, and the tangent Capital M

arket Line. 6.

The parameters of the opportunity set are:

E(rS ) = 15%, E(rB ) = 9%

, σB

S = 32%, σ

BB = 23%

, ρ = 0.15, rf = 5.5%

-------------- Y

ou don’t need to know the cov or m

in-var weight calculations but here they are:

From the standard deviations and the correlation coefficient w

e generate the covariance matrix [note

that Cov(rS , rB ) = ρσ

B

S σB

B]:

Bonds

Stocks B

onds529.0

110.4 Stocks

110.4 1024.0

The min-var portf proportions are:

)r,

r(C

ov2

)r,

r(C

ov)S

(w

BS

2B2S

BS

2BM

in−

σ+

σ−

σ=

3142.0

)4.

1102(

5291024

4.110

529=

×−

+−

=

w

Min (B

) = 0.6858

This calc will help us add an extra point to the table below

: 31.42% stocks/68.58%

bond portfolio

-----------------

The mean and standard deviation of the m

inimum

variance portfolio are:

E(rM

in ) = (0.3142 × 15%) + (0.6858 × 9%

) = 10.89%

[

]2 1

BS

BS

2B2B

2S2S

Min

)r,

r(C

ovw

w2w

w+

σ+

σ=

σ

= [(0.3142

2 × 1024) + (0.68582 × 529) + (2 × 0.3142 × 0.6858 × 110.4)] 1/2 = 19.94%

%

in stocks %

in bonds E

xp. return Std dev.

00.00

100.00 9.00

23.00

20.00 80.00

10.20 20.37

31.42

68.58 10.89

19.94 M

inimum

variance 40.00

60.00 11.40

20.18

60.00 40.00

12.60 22.50

70.75

29.25 13.25

24.57 T

angency portfolio 80.00

20.00 13.80

26.68

100.00 00.00

15.00 32.00

7.

Investment opportunity set

for stocks and bonds

min var

B

SC

AL

0 2 4 6 8

10 12 14 16 18

010

2030

Standard Deviation (%

)

40

The graph approximates the points:

E(r)

σ M

inimum

Variance Portfolio

10.89%

19.94%

Tangency Portfolio 13.25%

24.57%

18.

The expected rate of return on the stock will change by beta tim

es the unanticipated change in the m

arket return: 1.2 × (8% – 10%

) = – 2.4%

Therefore, the expected rate of return on the stock should be revised to: 12% – 2.4%

= 9.6%

19. a.

The risk of the diversified portfolio consists primarily of system

atic risk. Beta m

easures systematic

risk, which is the slope of the security characteristic line (SC

L). The two figures depict the stocks'

SCLs. Stock B

's SCL is steeper, and hence Stock B

's systematic risk is greater. The slope of the SC

L, and hence the system

atic risk, of Stock A is low

er. Thus, for this investor, stock B is the riskiest.

b.

The undiversified investor is exposed primarily to firm

-specific risk. Stock A has higher firm

-specific risk because the deviations of the observations from

the SCL are larger for Stock A

than for Stock B

. Deviations are m

easured by the vertical distance of each observation from the SC

L. Stock A

is therefore riskiest to this investor. C

h. 7 2.

a. E(rX ) = 5%

+ 0.8(14% – 5%

) = 12.2%

"X = 14%

– 12.2% = 1.8%

E(rY ) = 5% + 1.5(17%

– 5%) = 18.5%

"Y = 17%

– 18.5% = –1.5%

b. i. For an investor w

ho wants to add this stock to a w

ell-diversified equity portfolio, Kay should

recomm

end Stock X because of its positive alpha, w

hile Stock Y has a negative alpha. In

graphical terms, Stock X

’s expected return/risk profile plots above the SML, w

hile Stock Y’s

profile plots below the SM

L. Also, depending on the individual risk preferences of K

ay’s clients, Stock X

’s lower beta m

ay have a beneficial impact on overall portfolio risk.

ii. For an investor who w

ants to hold this stock as a single-stock portfolio, Kay should

recomm

end Stock Y, because it has higher forecasted return and low

er standard deviation than Stock X

. Stock Y’s Sharpe ratio is: (0.17 – 0.05)/0.25 = 0.48

Stock X’s Sharpe ratio is only: (0.14 – 0.05)/0.36 = 0.25

The market index has an even m

ore attractive Sharpe ratio: (0.14 – 0.05)/0.15 = 0.60

How

ever, given the choice between Stock X

and Y, Y

is superior. When a stock is held in

isolation, standard deviation is the relevant risk measure. For assets held in isolation, beta as a

measure of risk is irrelevant. A

lthough holding a single asset in isolation is not typically a recom

mended investm

ent strategy, some investors m

ay hold what is essentially a single-asset

portfolio (e.g., the stock of their employer com

pany). For such investors, the relevance of standard deviation versus beta is an im

portant issue. 3.

E(rP ) = rf + β[E(rM ) – rf ]

20% = 5%

+ β(15% – 5%

) ⇒ β = 15/10 = 1.5

4. If the beta of the security doubles, then so w

ill its risk premium

. The current risk premium

for the stock is: (13%

- 7%) = 6%

, so the new risk prem

ium w

ould be 12%, and the new

discount rate for the security w

ould be: 12% + 7%

= 19%

If the stock pays a constant dividend in perpetuity, then w

e know from

the original data that the dividend (D

) must satisfy the equation for a perpetuity:

Price = D

ividend/Discount rate

40 = D/0.13 ⇒

D = 40 × 0.13 = $5.20

At the new

discount rate of 19%, the stock w

ould be worth: $5.20/0.19 = $27.37

The increase in stock risk has lowered the value of the stock by 31.58%

. 6.

a. False. β = 0 im

plies E(r) = rf , not zero. b. False. Investors require a risk prem

ium for bearing system

atic (i.e., undiversifiable) risk. c.

False. You should invest 0.75 of your portfolio in the m

arket portfolio, and the remainder in T-

bills. Then: βP = (0.75 × 1) + (0.25 × 0) = 0.75

7. a.

The beta is the sensitivity of the stock's return to the market return. C

all the aggressive stock A and the defensive stock D

. Then beta is the change in the stock return per unit change in the m

arket return. We com

pute each stock's beta by calculating the difference in its return across the tw

o scenarios divided by the difference in market return.

00

.220

532

2A

=− −

=β

70

.020

514

5.3

D=

− −=

β

b. With the tw

o scenarios equal likely, the expected rate of return is an average of the two possible

outcomes: E(rA ) = 0.5 × (2%

+ 32%) = 17%

E(rB ) = 0.5 × (3.5% + 14%

) = 8.75%

B

c. The SM

L is determined by the follow

ing: T-bill rate = 8% w

ith a beta equal to zero, beta for the m

arket is 1.0, and the expected rate of return for the market is:

0.5 × (20% + 5%

) = 12.5%

E(r)

β

8%

12.5%

1.0 2.0 A

SML

M

.7

αD

D

The equation for the security market line is:

E(r) = 8% + β(12.5%

– 8%)

d.

The aggressive stock has a fair expected rate of return of:

E(rA ) = 8% + 2.0(12.5%

– 8%) = 17%

The security analyst’s estimate of the expected rate of return is also 17%

. Thus the alpha for the aggressive stock is zero. Sim

ilarly, the required return for the defensive stock is:

E(rD ) = 8%

+ 0.7(12.5% – 8%

) = 11.15%

The security analyst’s estim

ate of the expected return for D is only 8.75%

, and hence:

α

D = actual expected return – required return predicted by CA

PM

= 8.75%

– 11.15% = –2.4%

The points for each stock are plotted on the graph above.

e. The hurdle rate is determ

ined by the project beta (i.e., 0.7), not by the firm’s beta. The

correct discount rate is therefore 11.15%, the fair rate of return on stock D

. 28. Equation 7.11 applies here:

E(rP ) = rf + βP1 [E(r1 ) − rf ] + β

P2 [E(r2 ) – rf ] W

e need to find the risk premium

for these two factors: γ1 = [E(r1 ) − rf ] and γ2 = [E(r2 ) − rf ]

To find these values, we solve the follow

ing two equations w

ith two unknow

ns:

40% = 7%

+ 1.8γ1 + 2.1γ2

10% = 7%

+ 2.0γ1 + (−0.5)γ2The solutions are: γ1 = 4.47%

and γ2 = 11.86%

Thus, the expected return-beta relationship is: E(rP ) = 7% + 4.47β

P1 + 11.86βP2

Ch. 8.

6. d.

16. a.

The grandson is recomm

ending taking advantage of (i) the small firm

anomaly and (ii) the January

anomaly. In fact, this seem

s to be one anomaly: the sm

all-firm-in-January anom

aly.

b. (i) C

oncentration of one’s portfolio in stocks having very similar attributes m

ay expose the portfolio to m

ore risk than is desirable. The strategy limits the potential for diversification.

(ii) Even if the study results are correct as described, each such study covers a specific tim

e period. There is no assurance that future tim

e periods would yield sim

ilar results.

(iii) A

fter the results of the studies became publicly know

n, investment decisions m

ight nullify these relationships. If these firm

s in fact offered investment bargains, their prices m

ay be bid up to reflect the now

-known opportunity.

24. i.

Mental accounting is best illustrated by Statem

ent #3. Sampson’s requirem

ent that his income

needs be met via interest incom

e and stock dividends is an example of m

ental accounting. M

ental accounting holds that investors segregate funds into mental accounts (e.g., dividends and

capital gains), maintain a set of separate m

ental accounts, and do not combine outcom

es; a loss in one account is treated separately from

a loss in another account. Mental accounting leads to

an investor preference for dividends over capital gains and to an inability or failure to consider total return.

ii. Overconfidence (illusion of control) is best illustrated by Statem

ent #6. Sampson’s desire to select

investments that are inconsistent w

ith his overall strategy indicates overconfidence. O

verconfident individuals often exhibit risk-seeking behavior. People are also more confident

in the validity of their conclusions than is justified by their success rate. Causes of

overconfidence include the illusion of control, self-enhancement tendencies, insensitivity to

predictive accuracy, and misconceptions of chance processes.

iii.

Reference dependence is best illustrated by Statem

ent #5. Sampson’s desire to retain poor

performing investm

ents and to take quick profits on successful investments suggests reference

dependence. Reference dependence holds that investm

ent decisions are critically dependent on the decision-m

aker’s reference point. In this case, the reference point is the original purchase price. A

lternatives are evaluated not in terms of final outcom

es but rather in terms of gains and

losses relative to this reference point. Thus, preferences are susceptible to manipulation sim

ply by changing the reference point.

Extra Questions(no solutions):

4. The Fama-French (1996) m

odel has three factors what are they?

6. What are the day of the w

eek effect, the January effect and the small-size firm

effect? 7. Provide at least one exam

ple for each of the “mental errors” (forecasting errors, overconfidence,

conservatism, representativeness) and “behavioral biases” (fram

ing, mental accounting, regret, loss

aversion)

Hm

k 2 C

h. 3: 2, 3, 4, 6, 8, 12, 13, 14, 20, 21 C

h. 4: 13 C

h. 12: 3, 5, 8, 18, 19 C

h. 17: 6, 11abcd C

h. 3. 2.

a. In principle, potential losses are unbounded, grow

ing directly with increases in the price of

IBM

.

b. If the stop-buy order can be filled at $128, the m

aximum

possible loss per share is $8. If the price of IB

M shares go above $128, then the stop-buy order w

ould be executed, limiting the

losses from the short sale.

3. a.

The stock is purchased for: 300 × $40 = $12,000 The am

ount borrowed is $4,000. Therefore, the investor put up equity, or m

argin, of $8,000. b. If the share price falls to $30, then the value of the stock falls to $9,000. B

y the end of the year, the am

ount of the loan owed to the broker grow

s to:

$4,000 × 1.08 = $4,320

Therefore, the rem

aining margin in the investor’s account is:

$9,000 − $4,320 = $4,680

The percentage m

argin is now: $4,680/$9,000 = 0.52 = 52%

Therefore, the investor will not receive a m

argin call.

c. The rate of return on the investm

ent over the year is:

(Ending equity in the account − Initial equity)/Initial equity

= ($4,680 − $8,000)/$8,000 = −0.415 = −41.5%

4. a.

The initial margin w

as: 0.50 × 1,000 × $40 = $20,000 A

s a result of the increase in the stock price Old Econom

y Traders loses:

$10 × 1,000 = $10,000

Therefore, margin decreases by $10,000. M

oreover, Old Econom

y Traders must pay the

dividend of $2 per share to the lender of the shares, so that the margin in the account

decreases by an additional $2,000. Therefore, the remaining m

argin is:

$20,000 – $10,000 – $2,000 = $8,000

b.

The percentage margin is: $8,000/$50,000 = 0.16 = 16%

So there w

ill be a margin call.

c.

The equity in the account decreased from $20,000 to $8,000 in one year, for a rate of return of:

(−$12,000/$20,000) = −0.60 = −60%

6. a.

The buy order will be filled at the best lim

it-sell order price: $50.25

b. The next m

arket buy order will be filled at the next-best lim

it-sell order price: $51.50

c.

You w

ould want to increase your inventory. There is considerable buying dem

and at prices just below

$50, indicating that downside risk is lim

ited. In contrast, limit sell orders are sparse,

indicating that a moderate buy order could result in a substantial price increase.

8. a.

Initial margin is 50%

of $5,000 or $2,500.

b. Total assets are $7,500 ($5,000 from

the sale of the stock and $2,500 put up for margin).

Liabilities are 100P. Therefore, net worth is ($7,500 – 100P). A

margin call w

ill be issued w

hen:

P

100P

100500,7

$−

= 0.30 ⇒ w

hen P = $57.69 or higher

12. The broker is instructed to attempt to sell your M

arriott stock as soon as the Marriott stock trades

at a bid price of $38 or less. Here, the broker w

ill attempt to execute, but m

ay not be able to sell at $38, since the bid price is now

$37.85. The price at which you sell m

ay be more or less than $38

because the stop-loss becomes a m

arket order to sell at current market prices.

13. a.

55.50

b. 55.25

c.

The trade will not be executed because the bid price is low

er than the price specified in the limit

sell order.

d. The trade w

ill not be executed because the asked price is greater than the price specified in the lim

it buy order. 14.

a. In an exchange m

arket, there can be price improvem

ent in the two m

arket orders. Brokers for

each of the market orders (i.e., the buy and the sell orders) can agree to execute a trade inside the

quoted spread. For example, they can trade at $55.37, thus im

proving the price for both custom

ers by $0.12 or $0.13 relative to the quoted bid and asked prices. The buyer gets the stock for $0.13 less than the quoted asked price, and the seller receives $0.12 m

ore for the stock than the quoted bid price.

b.

Whereas the lim

it order to buy at $55.37 would not be executed in a dealer m

arket (since the asked price is $55.50), it could be executed in an exchange m

arket. A broker for another

customer w

ith an order to sell at market w

ould view the lim

it buy order as the best bid price; the tw

o brokers could agree to the trade and bring it to the specialist, who w

ould then execute the trade.

20. (d)

The broker will sell, at current m

arket price, after the first transaction at $55 or less. 21.

(b) C

h.4. 13.

Start of year NA

V = $20

D

ividends per share = $0.20

End of year N

AV

is based on the 8% price gain, less the 1%

12b-1 fee:

End of year N

AV

= $20 × 1.08 × (1 – 0.01) = $21.384

R

ate of return = 20

$20.0

$20

$384.

21$

+−

= 0.0792 = 7.92%

Ch. 12

3. a.

gk D

P1

0−

=

g16

.02$

50$

−=

⇒

%12

12.0

50$

2$16.0

g=

=−

=

b.

18.

18$

05.0

16.0

2$g

k DP

10

=−

=−

=

The price falls in response to the m

ore pessimistic forecast of dividend grow

th. The forecast for current earnings, how

ever, is unchanged. Therefore, the P/E ratio decreases. The lower P/E

ratio is evidence of the diminished optim

ism concerning the firm

's growth prospects.

5. a.

g = RO

E × b = 0.20 × 0.30 = 0.06 = 6.0%

D

1 = $2(1 – b) = $2(1 – 0.30) = $1.40

33.

23$

06.0

12.0

40.1

$g

k DP

10

=−

=−

=

P/E = $23.33/$2 = 11.67

b.

PVG

O = P0 –

k E0= $23.33 –

66.6

$12

.0 00.2

$=

c.

g = RO

E × b = 0.20 × 0.20 = = 0.04 = 4.0%

D

1 = $2(1 – b) = $2(1 – 0.20) = $1.60

00

.20

$04

.012

.060

.1$

gk D

P1

0=

−=

−=

P/E = $20/$2 = 10.0

PV

GO

= P0 – k E

0= $20.00 –

12.0 00.2

$= $3.33

8. a.

k = rf + β (kM – rf ) = 6%

+ 1.25(14% – 6%

) = 16%

g = (2/3) × 9%

= 6%

D

1 = E0 × (1 + g) × (1 – b) = $3 × 1.06 × (1/3) = $1.06

60

.10

$06

.016

.006

.1$

gk D

P1

0=

−=

−=

b.

Leading P0 /E1 = $10.60/$3.18 = 3.33

Trailing P0 /E0 = $10.60/$3.00 = 3.53

c.

PVG

O = P0 –

k E0

= $10.60 – 15

.8$

16.0 3$

−=

The low

P/E ratios and negative PVG

O are due to a poor R

OE (9%

) that is less than the market

capitalization rate (16%).

d. Now

, you revise the following:

b = 1/3 g = 1/3 × 0.09 = 0.03 = 3.0% D

1 = E0 × 1.03 × (2/3) = $2.06

85.

15$

03.0

16.0

06.2

$g

k DV

10

=−

=−

=

V0 increases because the firm

pays out more earnings instead of reinvesting earnings at a

poor RO

E. This information is not yet know

n to the rest of the market.

18. a.

The value of a share of Rio N

ational equity using the Gordon grow

th model and the

capital asset pricing model is $22.40, as show

n below.

Calculate the required rate of return using the capital asset pricing m

odel:

k = rf + β (kM – rf ) = 4%

+ 1.8(9% – 4%

) = 13%

Calculate the share value using the G

ordon growth m

odel:

40

.22

$12

.013

.0)

12.0

1(20

.0$

gk

g)(1

DP

o0

=−

+×

=−

+×

=

b.

The sustainable growth rate of R

io National is 9.97%

, calculated as follows:

g = b × RO

E = Earnings Retention R

ate × ROE = (1 – Payout R

atio) × ROE =

%97.9

0997.0

35.

270$

16.

30$

16.

30$

20.3

$1

Equity

Beginning

Income

Net

Income

Net ividends

D1

==

×⎟⎠ ⎞⎜⎝ ⎛

−=

×⎟⎠ ⎞⎜⎝ ⎛

−

19. a. To obtain free cash flow

to equity (FCFE), the tw

o adjustments that Shaar should m

ake to cash flow

from operations (C

FO) are:

1. Subtract investment in fixed capital: C

FO does not take into account the investing

activities in long-term assets, particularly plant and equipm

ent. The cash flows

corresponding to those necessary expenditures are not available to equity holders and therefore should be subtracted from

CFO

to obtain FCFE.

2. Add net borrow

ing: CFO

does not take into account the amount of capital supplied to the

firm by lenders (e.g., bondholders). The new

borrowings, net of debt repaym

ent, are cash flow

s available to equity holders and should be added to CFO

to obtain FCFE.

b.

Note 1: R

io National had $75 m

illion in capital expenditures during the year. Adjustm

ent: negative $75 million

The cash flows required for those capital expenditures (–$75 m

illion) are no longer available to the equity holders and should be subtracted from

net income to obtain FC

FE.

Note 2: A

piece of equipment that w

as originally purchased for $10 million w

as sold for $7 m

illion at year-end, when it had a net book value of $3 m

illion. Equipment sales are unusual for

Rio N

ational. Adjustm

ent: positive $3 million

In calculating FCFE, only cash flow

investments in fixed capital should be considered. The $7

million sale price of equipm

ent is a cash inflow now

available to equity holders and should be added to net incom

e. How

ever, the gain over book value that was realized w

hen selling the equipm

ent ($4 million) is already included in net incom

e. Because the total sale is cash, not just

the gain, the $3 million net book value m

ust be added to net income. Therefore, the adjustm

ent calculation is: $7 m

illion in cash received – $4 million of gain recorded in net incom

e = $3 m

illion additional cash received that must be added to net incom

e to obtain FCFE.

Note 3: The decrease in long-term

debt represents an unscheduled principal repayment; there

was no new

borrowing during the year.

Adjustment: negative $5 m

illion The unscheduled debt repaym

ent cash flow (–$5 m

illion) is an amount no longer available

to equity holders and should be subtracted from net incom

e to determine FC

FE.

Note 4: O

n 1 January 2002, the company received cash from

issuing 400,000 shares of com

mon equity at a price of $25.00 per share.

No adjustm

ent Transactions betw

een the firm and its shareholders do not affect FC

FE. To calculate FC

FE, therefore, no adjustment to net incom

e is required with respect to the issuance of

new shares.

Note 5: A

new appraisal during the year increased the estim

ated market value of land

held for investment by $2 m

illion, which w

as not recognized in 2002 income.

No adjustm

ent The increased m

arket value of the land did not generate any cash flow and w

as not reflected in net incom

e. To calculate FCFE, therefore, no adjustm

ent to net income is required.

c.

Free cash flow to equity (FC

FE) is calculated as follows:

FCFE = N

I + NC

C – FC

INV

– WC

INV

+ Net B

orrowing

where

NC

C = non-cash charges

FCIN

V = investm

ent in fixed capital W

CIN

V = investm

ent in working capital

M

illion $ Explanation

NI =

$30.16 From

Exhibit 18B

NC

C =

+$67.17 $71.17 (depreciation and am

ortization from Exhibit 18B

)

– $4.00* (gain on sale from N

ote 2) FC

INV

= –$68.00

$75.00 (capital expenditures from N

ote 1) – $7.00* (cash on sale from

Note 2)

WC

INV

= –$24.00

–$3.00 (increase in accounts receivable from Exhibit 18A

+ –$20.00 (increase in inventory from

Exhibit 18A) +

–$1.00 (decrease in accounts payable from Exhibit 18A

) N

et Borrow

ing = +(–$5.00)

–$5.00 (decrease in long-term debt from

Exhibit 18A)

FCFE =

$0.33

*Supplemental N

ote 2 in Exhibit 18C affects both N

CC

and FCIN

V.

Ch. 17.

6. a.

Actual: (0.70 × 2.0%

) + (0.20 × 1.0%) + (0.10 × 0.5%

) = 1.65%

B

ogey: (0.60 × 2.5%) + (0.30 × 1.2%

) + (0.10 × 0.5%) = 1.91%

U

nderperformance = 1.91%

– 1.65% = 0.26%

b. Security Selection:

Market

Portfolio Perform

ance Index

Performance

Excess Perform

ance M

anager’s Portfolio W

eight C

ontribution

Equity 2.0%

2.5%

-0.5%

0.70

-0.35%

Bonds

1.0%

1.2%

-0.2%

0.20 -0.04%

C

ash 0.5%

0.5%

0.0%

0.10

0.00%

Contribution of security selection:

-0.39%

c.

Asset Allocation:

Market

Actual W

eight B

enchmark

Weight

Excess Weight

Index Return

Minus Bogey

Contribution

Equity 0.70

0.60 0.10

0.59%

0.059%

Bonds

0.20 0.30

-0.10 -0.71%

0.071%

C

ash 0.10

0.10 0.00

-1.41%

0.000%

Contribution of asset allocation:

0.130%

Summ

ary

Security selection -0.39%

A

sset allocation 0.13%

Excess perform

ance -0.26%

11.

a. The spreadsheet below

displays the monthly returns and excess returns for the V

anguard U

.S. Grow

th Fund, the Vanguard U

.S. Value Fund and the S&

P 500. Note that the

inception date for the Vanguard U

.S. Value Fund w

as 6/29/2000.

Monthly R

ates of Return: M

ay 2000 - April 2005

G

rowth

Value

Excess R

eturns M

onth Fund

Fund T-B

ills S&

P500 G

rowth Fund

Value Fund S&

P500 M

ay-00 -5.22

N/A

0.50-2.19

-5.72N

/A -2.69

Jun-00 9.09

N/A

0.492.39

8.61N

/A 1.91

Jul-00 -2.35

N/A

0.51-1.63

-2.86N

/A -2.15

Aug-00 10.01

8.77 0.52

6.079.48

8.24 5.55

Sep-00 -5.96

0.09 0.52

-5.35-6.48

-0.42 -5.86

Oct-00

-5.19 1.59

0.52-0.49

-5.721.07

-1.02N

ov-00 -18.67

-3.87 0.53

-8.01-19.20

-4.40 -8.54

Dec-00

-3.12 6.23

0.500.41

-3.615.74

-0.09

Jan-01 3.88

0.27 0.44

3.463.44

-0.17 3.02

Feb-01 -21.65

-0.45 0.42

-9.23-22.07

-0.87 -9.65

Mar-01

-15.24 -1.99

0.38-6.42

-15.61-2.37

-6.80Apr-01

15.16 4.34

0.337.68

14.834.00

7.35M

ay-01 -0.83

2.30 0.31

0.51-1.14

1.99 0.20

Jun-01 -5.94

-0.17 0.30

-2.50-6.24

-0.47 -2.80

Jul-01 -3.46

1.13 0.30

-1.07-3.75

0.83 -1.37

Aug-01 -9.00

-3.68 0.29

-6.41-9.28

-3.97 -6.70

Sep-01 -12.42

-8.36 0.22

-8.17-12.64

-8.58 -8.40

Oct-01

7.83 -0.68

0.181.81

7.64-0.86

1.63N

ov-01 10.95

8.40 0.16

7.5210.79

8.24 7.36

Dec-01

0.11 1.80

0.140.76

-0.041.66

0.61Jan-02

-5.25 1.68

0.14-1.56

-5.391.54

-1.70Feb-02

-7.18 -0.87

0.15-2.08

-7.33-1.02

-2.22M

ar-02 4.87

4.30 0.15

3.674.72

4.15 3.52

Apr-02 -10.22

-1.35 0.15

-6.14-10.37

-1.49 -6.29

May-02

-2.65 -0.43

0.15-0.91

-2.80-0.57

-1.05Jun-02

-9.30 -7.54

0.14-7.25

-9.45-7.68

-7.39Jul-02

-7.84 -9.18

0.14-7.90

-7.98-9.32

-8.04Aug-02

1.75 1.63

0.140.49

1.611.50

0.35Sep-02

-11.02 -

0.14-11.00

-11.15-10.98

-11.14O

ct-02 9.75

5.74 0.13

8.649.61

5.61 8.51

Nov-02

4.48 5.64

0.105.71

4.385.54

5.60D

ec-02 -8.19

-5.04 0.10

-6.03-8.29

-5.14 -6.13

Jan-03 -2.42

-2.87 0.10

-2.74-2.52

-2.97 -2.84

Feb-03 -0.94

-3.06 0.10

-1.70-1.04

-3.16 -1.80

Mar-03

3.19 -0.79

0.100.84

3.10-0.89

0.74Apr-03

6.77 8.41

0.108.10

6.688.31

8.01M

ay-03 2.90

7.34 0.09

5.092.81

7.25 5.00

Jun-03 0.68

2.05 0.08

1.130.61

1.97 1.05

Jul-03 3.63

0.57 0.08

1.623.55

0.50 1.55

Aug-03 1.53

2.19 0.08

1.791.45

2.11 1.71

Sep-03 -1.08

-0.74 0.08

-1.19-1.16

-0.82 -1.27

Oct-03

5.52 6.57

0.085.50

5.446.49

5.42N

ov-03 0.76

2.29 0.08

0.710.68

2.21 0.63

Dec-03

3.28 4.04

0.085.08

3.203.97

5.00Jan-04

3.04 2.89

0.081.73

2.972.82

1.65Feb-04

-0.06 1.53

0.081.22

-0.141.45

1.14M

ar-04 -1.03

-0.55 0.08

-1.64-1.11

-0.63 -1.72

Apr-04 -2.47

-2.63 0.08

-1.68-2.55

-2.71 -1.76

May-04

2.53 0.16

0.091.21

2.440.08

1.12Jun-04

1.43 2.53

0.111.80

1.322.42

1.69Jul-04

-7.49 -1.91

0.11-3.43

-7.60-2.02

-3.54Aug-04

-0.76 0.97

0.130.23

-0.890.85

0.10Sep-04

2.51 1.53

0.140.94

2.371.39

0.80O

ct-04 1.43

0.16 0.15

1.401.28

0.01 1.25

Nov-04

4.69 5.30

0.183.86

4.525.12

3.68D

ec-04 3.59

3.23 0.19

3.253.40

3.04 3.06

Jan-05 -3.46

-2.33 0.20

-2.53-3.66

-2.52 -2.73

Feb-05 0.77

3.65 0.22

1.890.55

3.43 1.68

Mar-05

-2.86 -2.37

0.23-1.91

-3.09-2.60

-2.15Apr-05

-2.42 -2.65

0.24-3.65

-2.65-2.88

-3.88S

td.Dev.

7.03 4.21

0.154.51

b. The standard deviations for the U

.S Grow

th Fund and the U.S. V

alue Fund are 7.03% and

4.21%, respectively, as show

n in the Excel spreadsheet above.

c. The betas for the U

.S. Grow

th Fund and the U.S. V

alue Fund are 1.432 and 0.799, respectively, as show

n in the Excel spreadsheets below:

GR

OW

TH FU

ND

SUM

MA

RY O

UTPU

T OF EXC

EL REG

RESSIO

N

Regression Statistics

Multiple R

0.919596

R S

quare 0.845657

Adj. R

Square

0.842996

Standard E

rror 2.799924

Observations

60.000000

AN

OV

A

df S

S

MS

F

Significance F

Regression

12491.311496

2491.31317.78651

3.31E-25

Residual

58454.695412

7.83958

Total 59

2946.006908

Coefficients

Std E

rror t S

tat P

-value

Intercept -0.632155

0.36381611-1.7376

0.0875945

S&

P 500

1.4319940.08032921

17.82663.308E

-25

VALU

E FUN

D

SUM

MA

RY O

UTPU

T OF EXC

EL REG

RESSIO

N

Regression Statistics

Multiple R

0.879497

R S

quare 0.773514

Adj. R

Square

0.769396

Standard E

rror 2.021425

Observations

57

AN

OV

A

df

SS

M

S

F S

ignificance F R

egression 1

767.547925767.548

187.84101 2.22E

-19R

esidual 55

224.7386534.08616

Total

56992.286579

Coefficients

Std E

rror t S

tat P

-value

Intercept 0.811917

0.26926013.01536

0.0038794

S&

P500

0.7992230.05831399

13.70552.223E

-19

d.

The formulas for the three m

easures are:

Sharpe:

p

fpσ

rr

−

Treynor:

p

fpβ

rr

−

Jensen:

{})r

r(β

rr

αf

Mp

fp

p−

+−

=

The values for the three m

easures are computed as follow

s:

U

.S. Grow

th Fund U

.S. Value Fund

Sharpe (─

1.16 ─ 0.21)/7.03 = ─

0.1945 (0.61 ─

0.19)/4.21 = 0.0999

Treynor (─

1.16 ─ 0.21)/1.432= ─

0.9551 (0.61 ─

0.19)/0.799 = 0.5184

Jensen ─

1.16 ─ {0.21 + 1.432 × (─

0.31 ─ 0.21)} = ─

0.6322

0.61 ─ {0.19 + 0.799 × (─

0.30 ─ 0.19)} =

0.8181

Note that, in the above calculations, rf and rM are calculated over different tim

e periods for the G

rowth Fund and the V

alue Fund because of the 6/29/2000 inception date of the Value Fund.

Also, there are som

e discrepancies due to rounding.

Hm

k 3 1. U

nderwriting is one of the services provided by w

hom?

2. Under firm

comm

itment underw

riting, who assum

es the full risk that the shares cannot be sold to the public at the stipulated offering price?

3. What is an EC

N?

4. The bid-ask spread exists because of the need for dealers to cover expenses and make a m

odest profit. True/False

5. Consider the follow

ing limit order book of a specialist. The last trade in the stock occurred at a price of

$40. If a market buy order for 100 shares com

es in, at what price w

ill it be filled?

Limit B

uy Orders

Limit Sell O

rdersPrice

SharesPrice

Shares$39.75

100$40.25

100$39.50

100$40.50

100

6. Assum

e you purchased 200 shares of XY

Z comm

on stock on margin at $80 per share from

your broker. If the initial m

argin is 60%, the am

ount you borrowed from

the broker is how m

uch? 7. Y

ou sold short 200 shares of comm

on stock at $50 per share. The initial margin is 60%

. Your initial

investment w

as how m

uch? 8. Y

ou short-sell 200 shares of Tuckerton Trading Co., now

selling for $50 per share. What is your

maxim

um possible loss? W

hat is your maxim

um possible gain?

9. You purchased 100 shares of A

BC

comm

on stock on margin at $50 per share. A

ssume the initial m

argin is 50%

and the maintenance m

argin is 30%. B

elow w

hat stock price would you get a m

argin call? A

ssume the stock pays no dividend and ignore interest on m

argin. 10. B

orrowing a security from

your broker in order to sell it, with the intention of repurchasing it later

when the price is low

er, is called what?

12. The margin requirem

ent on a stock purchase is 15%. Y

ou fully use the margin allow

ed to purchase 100 shares of M

SFT at $35. If the price drops to $32, what is your percentage loss?

13. What are the difference s betw

een active and passively managed m

utual funds? 15. U

nder SEC rules, the m

anagers of certain funds are allowed to deduct charges for advertising,

brokerage comm

issions, and other sales expenses, directly from the fund assets rather than billing

investors. These fees are known as…

..? 19. The Stone H

arbor Fund is a closed-end investment com

pany with a portfolio currently w

orth $300 m

illion. It has liabilities of $5 million and 9 m

illion shares outstanding. If the fund sells for $30 a share, w

hat is its premium

or discount as a percent of NA

V?

21. What is a R

EIT? 22. W

hat is market tim

ing? What funds are likely to be subject to it?

23. You purchased X

YZ stock at $50 per share. The stock is currently selling at $65. Y

our gains could be protected by placing a

A

) lim

it-buy order B) lim

it-sell order C) m

arket order D) stop-loss order

24. The market capitalization rate on the stock of A

berdeen Wholesale C

ompany is 10%

. Its expected RO

E is 12%

and its expected EPS is $5.00. If the firm's plow

-back ratio is 40%, its P/E ratio w

ill be __________.

25. Gagliardi W

ay Corporation has an expected R

OE of 15%

. Its dividend growth rate w

ill be __________ if it follow

s a policy of paying 30% of earning in the form

of dividends. 26. R

ose Hill Trading C

ompany is expected to have EPS in the upcom

ing year of $6.00. The expected RO

E is 18.0%

. An appropriate required return on the stock is 14%

. If the firm has a plow

back ratio of 60%

, its growth rate of dividends should be __________.

27. Rose H

ill Trading Com

pany is expected to have EPS in the upcoming year of $6.00. The expected R

OE

is 18.0%. A

n appropriate required return on the stock is 14%. If the firm

has a plowback ratio of

70%, its intrinsic value should be __________.

28. Cache C

reek Com

pany is expected to pay a dividend of $3.36 in the upcoming year. D

ividends are expected to grow

at 8% per year. The riskfree rate of return is 4%

and the expected return on the m

arket portfolio is 14%. Investors use the C

APM

to compute the m

arket capitalization rate, and the constant grow

th DD

M to determ

ine the value of the stock. The stock's current price is $84.00. U

sing the constant growth D

DM

, the market capitalization rate is __________.

29. Grott and Perrin, Inc. has expected earnings of $3 per share for next year. The firm

's RO

E is 20% and

its earnings retention ratio is 70%. If the firm

's market capitalization rate is 15%

, what is the

present value of its growth opportunities?

30. Annie's D

onut Shops, Inc. has expected earnings of $3.00 per share for next year. The firm's R

OE is

18% and its earnings retention ratio is 60%

. If the firm's m

arket capitalization rate is 12%, w

hat is the value of the firm

excluding any growth opportunities?

32. Cache C

reek Manufacturing C

ompany is expected to pay a dividend of $4.20 in the upcom

ing year. D

ividends are expected to grow at the rate of 8%

per year. The riskfree rate of return is 4%

and the expected return on the market portfolio is 14%

. Investors use the CA

PM to com

pute the m

arket capitalization rate on the stock, and the constant growth D

DM

to determine the

intrinsic value of the stock. The stock is trading in the market today at $84.00. U

sing the constant grow

th DD

M and the C

APM

, the beta of the stock is __________. 33. W

estsyde Tool Com

pany is expected to pay a dividend of $2.00 in the upcoming year. The risk-free

rate of return is 6% and the expected return on the m

arket portfolio is 12%. A

nalysts expect the price of W

estsyde Tool Com

pany shares to be $29 a year from now

. The beta of W

estsyde Tool Com

pany's stock is 1.20. Using a one-period valuation m

odel, the intrinsic value of W

estsyde Tool Com

pany stock today is 34. C

aribou Gold M

ining Corporation is expected to pay a dividend of $6 in the upcom

ing year. Dividends

are expected to decline at the rate of 3% per year. The risk-free rate of return is 5%

and the expected return on the m

arket portfolio is 13%. The stock of C

aribou Gold M

ining Corporation

has a beta of -0.50. Using the constant grow

th DD

M, the intrinsic value of the stock is

__________. 35. The Free cash flow

to the firm is $300m

in perpetuity, the cost of equity equals 14% and the W

AC

C is

10%. If the m

arket value of the debt is $1.0 billion, what is the value of the equity using the free

cash flow valuation approach?

36. The free cash flow to the firm

is reported as $405 million. The interest expense to the firm

is $76 m

illion. If the tax rate is 35% and the net debt of the firm

increased by $50, what is the free cash

flow to the equity holders of the firm

? 37. The free cash flow

to the firm is reported as $205 m

illion. The interest expense to the firm is $22

million. If the tax rate is 35%

and the net debt of the firm increased by $25, w

hat is the market

value of the firm if the FC

FE grows at 2%

and the cost of equity is 11%?

U

se the following to answ

er questions 38-40: In a particular year, Lost H

ope Mutual Fund earned a return of 2%

by making the follow

ing investments in

asset classes:

38. The total excess return on the m

anaged portfolio was __________.

39. The contribution of asset allocation across m

arkets to the total excess return was __________.

40. The contribution of security selection w

ithin asset classes to the total excess return was __________.

41. A portfolio generates an annual return of 13%

, a beta of 0.7 and a standard deviation of 17%. The

market index return is 14%

and a standard deviation of 21%. W

hat is the Treynor measure of the

portfolio if the risk free rate is 5%?

42. A

portfolio generates an annual return of 17%, a beta of 1.2 and a standard deviation of 19%

. The m

arket index return is 12% and a standard deviation of 16%

. What is the Treynor m

easure of the portfolio if the risk free rate is 4%

? 43. A


. The m


. What is the Sharpe m


? 44. A


. The m


. What is the Sharpe m


?





hat is the Jensen measure of the






hat is the Jensen measure of the


47. The portfolio that contains the benchm

ark asset allocation against which a m

anager will be m

easured is often called the _____________.

Solutions 1. ib

2. u 3. el com

net

4. t

5. 40.25 or less 6.

7.

400,6

)60

.1)(

80(

200B

orrowing

=−

=000

,6)

60)(.

50(

200Investm

ent=

=

8. Ans: There is no upper lim

it to the price of a share of stock, therefore no upper limit the price you w

ill have to pay to replace the 200 shares of Tuckerton. Tuckerton could go bankrupt w

ith a share price of $0. Y

ou could keep the entire proceeds from the short sale.

Maxim

um gain

= proceeds – m

inimum

possible replacement cost

=

200 ( $50 ) – 200 ( $0 )

=

$10,000

9. A

ns:71.

3530

.1

50.

150

=− −

=⎟⎠ ⎞

⎜⎝ ⎛P

or 2500=.7x100xX

10. sh s

12. Ans: M

argin = 35 x 100 x .15 = 525 Stock value at start = 100 x 35 = 3500 Stock value at end = 100 x 32 = 3200 D

ecrease in value = 300

Percentage loss = 300 / 525 = 57%

15. 12b-1 19. 8.5% d 23. B

24. 11.54 25. 10.5 26. 10.8%

27. $128.57 28. 12%

29. $70 30. $25 32. 0.9 33. $27.39 34. $150 35. 2bn 36. 406m

37. 2,397 38.

Excess R

eturn = .0200 – .0750 = –.0550 39.

(.10 – .50)(.1000) + (.90 – .50)(.0500) = –.0200

40.

(.1100 – .1000).10 + (.0100 – .05).90 = –.0350 41. 0.1143

42. 0.1083 43. 0.4706 44. Sh=0.6842 45. 0.017

46. 0.034 47. B

ogey

Ch 9 – Problem

s: 4, 9, 10, 11, 13, 14, 15, 18, 19, 24, 28, 31, 32, 33, 34, 39bcd L

egend: PVIF(n, r) = 1 / (1+r) n

PVA

F(n, r) = 1/r – 1/[r(1+r) n] FVA

F(n,r)= [(1+r) n-1]/r 4.

The bond price will be low

er. As tim

e passes, the bond price, which is now

above par value, w

ill approach par. 9.

a. The bond pays $50 every six m

onths. Current price:

[$50 × PV

AF(4%

, 6)] + [$1000 × PVIF(4%

, 6)] = $1,052.42

A

ssuming the m

arket interest rate remains 8%

semi=4%

per half year, price six m

onths from now

:

[$50 × PV

AF(4%

, 5)] + [$1000 × PVIF(4%

, 5)] = $1,044.52

b. R

ate of return =

months

six per

%00.4

0400.0

42.

052,1

$90.7

$50

$42.

052,1

$)

42.

052,1

$52.

044,1

($50

$=

=−

=−

+=

8% sem

i 10.

a. From

[$40 × PVA

F(x%, 40)] + [$1000 × PV

IF(x%, 40)] = $950 solve for x.

You w

ill find that the yield to maturity per 6 m

onths is x= 4.26%. This

implies a bond equivalent yield to m

aturity of: 4.26% × 2 = 8.52%

semi

Effective annual yield to m

aturity = (1.0426) 2 – 1 = 0.0870 = 8.70%

b.

Since the bond is selling at par, the yield per 6 months is the sam

e as the sem

i-annual coupon, 4%. The bond equivalent yield to m

aturity is 8%.


aturity = (1.04) 2 – 1 = 0.0816 = 8.16%

c.

Keeping other inputs unchanged except P = 1050, w

e find a bond equivalent yield to m

aturity of 7.52% sem

i, or 3.76% per 6 m

onths.


aturity = (1.0376) 2 – 1 = 0.0766 = 7.66%

11. Since the bond paym

ents are now m

ade annually instead of semi-annually, the

bond equivalent yield to maturity is the sam

e as the effective annual yield to m

aturity. The equation is [$80 × PVA

F(x%, 20)] + [$1000 × PV

IF(x%, 20)] = P

The resulting yields for the three bonds are:

Bond Price

Bond equivalent yield =

Effective annual yield $ 950

8.53%

$1000 8.00%

$1050

7.51%

9-1

The yields com

puted in this case are lower than the yields calculated w

ith semi-annual

coupon payments. A

ll else equal, bonds with annual paym

ents are less attractive to investors because m

ore time elapses before paym

ents are received. If the bond price is the sam

e with annual paym

ents, then the bond's yield to maturity is low

er. 13.

Rem

ember that the convention is to use sem

i-annual periods:

Price M

aturity (years)

Maturity

(half-years) Per 6 m

onths Y

TM

Bond equivalent sem

i YTM

$400.00

20.00 40.00

2.317%

4.634%

$500.00 20.00

40.00 1.748%

3.496%

$500.00

10.00 20.00

3.526%

7.052%

$376.89 10.00

20.00 5.000%

10.000%

$456.39

10.00 20.00

4.000%

8.000%

$400.00 11.68

23.36 4.000%

8.000%

14.

Zero 8%

coupon10%

coupon a.

Current prices

$463.19$1000

$1134.20 b.

Price one year from now

$500.25$1000

$1124.94

Price increase $ 37.06

$ 0.00 -$ 9.26

C

oupon income

$ 0.00$80.00

$ 100.00

Income

$ 37.06$80.00

$ 90.74

Rate of R

eturn 8.00%

8.00%

8.00%

15. The reported bond price is: 100 2/32 percent of par = $1,000.625 H

owever, 15 days have passed since the last sem

iannual coupon was paid, so

accrued interest equals: $35 × (15/182) = $2.885 The invoice price is the reported price plus accrued interest: $1003.51

18. The solution is obtained using Excel:

A

B

C

D

E 1

5.50%

coupon bond,

2

m

aturing March 15, 2014

3

Formula in C

olumn B

4

Settlem

ent date 2/22/2006

DA

TE(2006,2,22)

5

Maturity date

3/15/2014D

ATE

(2014,3,15)

6 A

nnual coupon rate 0.055

7

Yield to m

aturity 0.0534

8

Redem

ption value (% of face value)

100

9 C

oupon payments per year

2

10

11

12 Flat price (%

of par) 101.03327

PR

ICE

(B4,B

5,B6,B

7,B8,B

9) 13

Days since last coupon

160C

OU

PD

AY

BS

(B4,B

5,2,1)

9-2

14 D

ays in coupon period 181

CO

UP

DA

YS

(B4,B

5,2,1) 15

Accrued interest

2.43094(B

13/B14)*B

6*100/2

16 Invoice price

103.46393B

12+B15

19. The solution is obtained using Excel:

A

B

C

D

E

F G

1

Sem

iannual A

nnual 2

coupons

coupons 3

4 S

ettlement date

2/22/20062/22/2006

5 M

aturity date

3/15/2014

3/15/2014 6

Annual coupon rate

0.0550.055

7 B

ond price

102

102 8

Redem

ption value (% of face value)

100100

9 C

oupon payments per year

2

1 10

11

Yield to m

aturity (decimal)

0.051927

0.051889 12

13

14

Formula in cell E

11: Y

IELD

(E4,E

5,E6,E

7,E8,E

9) 24. a.

The bond sells for P=$1,124.72 based on the 3.5% yield to m

aturity:

[$40 × PV

AF(3.5%

, 60)] + [$1000 × PVIF(3.5%

, 60)] = $1,124.72

[n = 60; i = 3.5; FV = 1000; PM

T = 40]

Therefore, yield to call is x=3.368%

per 6 months, 6.736%

semi:

[$40 × PVA

F(x%, 10)] + [$1100 × PV

IF(x%, 10)] = $1,124.72

[n = 10; PV = 1124.72; FV

= 1100; PMT = 40]

b.

If the call price were $1050, w

e would set FV

= 1050 and redo part (a) to find that yield to call is x=2.976%


semi. W

ith a lower call

price, the yield to call is lower.

c.

Yield to call is x=3.031%


semi:

[$40 × PVA

F(x%, 4)] + [$1100 × PV

IF(x%, 4)] = $1,124.72

[n = 4; PV

= 1124.72 ; FV = 1100; PM

T = 40] 28. A

pril 15 is midw

ay through the semi-annual coupon period. Therefore, the invoice

price will be higher than the stated ask price by an am

ount equal to one-half of the sem

iannual coupon. The ask price is 101.125 percent of par, so the invoice price is:

$1,011.25 + (1/2 × $50) = $1,036.25

9-3

31. a. (1)

Current yield = C

oupon/Price = 70/960 = 0.0729 = 7.29%

(2)

YTM

= 3.993% per 6 m

onths or 7.986% sem

i bond equivalent yield

(3) R

ealized compound yield is 4.166%

(per 6 months), or 8.332%

semi

bond equivalent yield. To obtain this value, first calculate the future value of reinvested coupons. There w

ill be six payments of $35 each, reinvested

semiannually at a per period rate of 3%

: 35 x FV

AF(6, 3%

) = $226.39 [PV = 0; PM

T = $35; n = 6; i = 3%]

The bond w

ill be selling at par value of $1,000 in three years, since coupon is forecast to equal yield to m

aturity. Therefore, total proceeds in three years w

ill be $1,226.39. To find realized compound yield on a sem

iannual basis (i.e., for six half-year periods), w

e solve:

$960 × (1 + x

realized ) 6 = $1,226.39 ⇒ x

realized = 4.166% (per 6 m

onths)

b.

Shortcomings of each m

easure:

(1)

Current yield does not account for capital gains or losses on bonds

bought at prices other than par value. It also does not account for reinvestment

income on coupon paym

ents.

(2) Y

ield to maturity assum

es that the bond is held to maturity and that all

coupon income can be reinvested at a rate equal to the yield to m

aturity.

(3) R

ealized compound yield (horizon yield) is affected by the forecast of

reinvestment rates, holding period, and yield of the bond at the end of the investor's

holding period. 32. a.

The yield to maturity of the par bond equals its coupon rate, 8.75%

. All else

equal, the 4% coupon bond w

ould be more attractive because its coupon rate is far

below current m

arket yields, and its price is far below the call price. Therefore, if

yields fall, capital gains on the bond will not be lim

ited by the call price. In contrast, the 8.75%

coupon bond can increase in value to at most $1050, offering a m

aximum

possible gain of only 5%

. The disadvantage of the 8.75% coupon bond in term

s of vulnerability to a call show

s up in its higher promised yield to m

aturity.

b. If an investor expects rates to fall substantially, the 4%

bond offers a greater expected return.

c.

Implicit call protection is offered in the sense that any likely fall in yields

would not be nearly enough to m

ake the firm consider calling the bond. In

this sense, the call feature is almost irrelevant.

33. Market conversion value = value if converted into stock = 20.83 × $28 = $583.24

C

onversion premium

= Bond value – m

arket conversion value

= $775 – $583.24 = $191.76

9-4

34. a. The call provision requires the firm

to offer a higher coupon (or higher prom

ised yield to maturity) on the bond in order to com

pensate the investor for the firm

's option to call back the bond at a specified call price if interest rates fall sufficiently. Investors are w

illing to grant this valuable option to the issuer, but only for a price that reflects the possibility that the bond w

ill be called. That price is the higher prom

ised yield at which they are w

illing to buy the bond.

b.

The call option reduces the expected life of the bond. If interest rates fall substantially so that the likelihood of call increases, investors w

ill treat the bond as if it w

ill "mature" and be paid off at the call date, not at the stated m

aturity date. O

n the other hand if rates rise, the bond must be paid off at the m

aturity date, not later. This asym

metry m

eans that the expected life of the bond will be

less than the stated maturity.

c.

The advantage of a callable bond is the higher coupon (and higher promised

yield to maturity) w

hen the bond is issued. If the bond is never called, then an investor w

ill earn a higher realized compound yield on a callable bond issued at

par than on a non-callable bond issued at par on the same date. The

disadvantage of the callable bond is the risk of call. If rates fall and the bond is called, then the investor receives the call price and w

ill have to reinvest the proceeds at interest rates that are low

er than the yield to maturity at w

hich the bond w

as originally issued. In this event, the firm's savings in interest paym

ents is the investor's loss.

39.

b. (3) The yield on the callable bond m

ust compensate the investor for the risk

of call.

Choice (1) is w

rong because, although the owner of a callable bond receives

principal plus a premium

in the event of a call, the interest rate at which he

can subsequently reinvest will be low

. The low interest rate that m

akes it profitable for the issuer to call the bond m

akes it a bad deal for the bond’s holder.

C

hoice (2) is wrong because a bond is m

ore apt to be called when interest

rates are low. There w

ill be an interest saving for the issuer only if rates are low

.

c. (3)

d.

(2)

9-5

Ch 14 Problem

s: 1, 2, 4, 5, 6, 7, 8, 24 C

h 15 Problems 1, 3, 5, 6, 7, 8, 11, 14, 16, 17

CH

14 1.

c is false. This is the description of the payoff to a put, not a call. 2.

c is the only correct statement.

4.

Cost

Payoff Profit

Call option, X

= 85 3.82

5.00 1.18

Put option, X = 85

0.15 0.00

-0.15 C

all option, X = 90

0.40 0.00

-0.40 Put option, X

= 90 1.80

0.00 -1.80

Call option, X

= 95 0.05

0.00 -0.05

Put option, X = 95

6.30 5.00

-1.30 5.

In terms of dollar returns:

Price of Stock Six M

onths From N

ow

Stock price:$80

$100 $110

$120 A

ll stocks (100 shares) 8,000

10,000 11,000

12,000 A

ll options (1,000 shares)

0

0 10,000

20,000 B

ills + 100 options 9,360

9,360 10,360

11,360

In terms of rate of return, based on a $10,000 investm

ent:

Price of Stock Six Months From

Now

Stock price:

$80 $100

$110 $120

All stocks (100 shares)

-20%

0%

10%

20%

All options (1,000 shares)

-100%

-100%

0%

100%

Bills + 100 options

-6.4%

-6.4%

3.6%

13.6%

All options

All stocks

Bills plus options

ST

100

–100 0

– 6.4

Rate of return (%

)

100

110

6.

a. Purchase a straddle, i.e., both a put and a call on the stock. The total cost of the straddle w

ould be: $10 + $7 = $17

b. Since the straddle costs $17, this is the am

ount by which the stock w

ould have to m

ove in either direction for the profit on either the call or the put to cover the investm

ent cost (not including time value of m

oney considerations). 7.

a. Sell a straddle, i.e., sell a call and a put to realize prem

ium incom

e of:

$4 + $7 = $11

b.

If the stock ends up at $50, both of the options will be w

orthless and your profit w

ill be $11. This is your maxim

um possible profit since, at any other

stock price, you will have to pay off on either the call or the put. The stock

price can move by $11 (your initial revenue from

writing the tw

o at-the-m

oney options) in either direction before your profits become negative.

c.

Buy the call, sell (w

rite) the put, lend the present value of $50. The payoff is as follow

s:

Final Payoff

PositionInitial O

utlayS

T < XS

T > X

Long call C

= 7 0

ST – 50

Short put -P = -4

-(50 – ST )

0 Lending

50/(1 + r) (1/4)50

50 Total

7 – 4 + [50/(1 + r) (1/4)] S

TS

T

The initial outlay equals: (the present value of $50) + $3 In either scenario, you end up w

ith the same payoff as you w

ould if you bought the stock itself.

8. a.

By w

riting covered call options, Jones receives premium

income of $30,000.

If, in January, the price of the stock is less than or equal to $45, he will keep

the stock plus the premium

income. Since the stock w

ill be called away from

him

if its price exceeds $45 per share, the most he can have is:

$450,000 + $30,000 = $480,000

(We are ignoring interest earned on the prem

ium incom

e from w

riting the option over this short tim

e period.) The payoff structure is: Stock price

Portfolio value Less than $45

(10,000 times stock price) + $30,000

Greater than $45

$450,000 + $30,000 = $480,000

This strategy offers some prem

ium incom

e but leaves the investor with

substantial downside risk. A

t the extreme, if the stock price falls to zero,

Jones would be left w

ith only $30,000. This strategy also puts a cap on the final value at $480,000, but this is m

ore than sufficient to purchase the house. b.

By buying put options w

ith a $35 strike price, Jones will be paying $30,000 in

premium

s in order to insure a minim

um level for the final value of his

position. That minim

um value is: ($35 × 10,000) – $30,000 = $320,000

This strategy allows for upside gain, but exposes Jones to the possibility of a

moderate loss equal to the cost of the puts. The payoff structure is:

Stock price Portfolio value

Less than $35 $350,000 – $30,000 = $320,000

Greater than $35

(10,000 times stock price) – $30,000

c.

The net cost of the collar is zero. The value of the portfolio will be as

follows:


Less than $35 $350,000

Betw

een $35 and $4510,000 tim

es stock price G

reater than $45 $450,000

If the stock price is less than or equal to $35, then the collar preserves the $350,000 in principal. If the price exceeds $45, then Jones gains up to a cap of $450,000. In betw

een $35 and $45, his proceeds equal 10,000 times the

stock price.

The best strategy in this case is (c) since it satisfies the two requirem

ents of preserving the $350,000 in principal w

hile offering a chance of getting $450,000. Strategy (a) should be ruled out because it leaves Jones exposed to the risk of substantial loss of principal.

O

ur ranking is: (1) c

(2) b (3) a

24. a.

Joe’s strategy

Final Payoff

PositionInitial O

utlayS

T < 1200 S

T > 1200Stock index

1200

ST

ST

Long put (X = 1200)

60

1200 – ST

0 Total

1260

1200 S

TProfit = payoff – 1260

-60

ST – 1260

Sally’s Strategy

Final Payoff

PositionInitial O

utlayS

T < 1170 S

T > 1170Stock index

1200

ST

ST

Long put (X = 1170)

45

1170 – ST

0 Total

1260

1170 S

TProfit = payoff – 1245

-75

ST – 1245

Profit

JoeS

ally

-60-75

11701200

ST

b.

Sally does better when the stock price is high, but w

orse when the stock price is

low. (The break-even point occurs at S = $1185, w

hen both positions provide losses of $60.)

c. Sally’s strategy has greater system

atic risk. Profits are more sensitive to the

value of the stock index. 25.

This strategy is a bear spread. The initial proceeds are: $9 – $3 = $6 The payoff is either negative or zero:

Position S

T < 50 50 < S

T < 60S

T > 60

Long call (X = 60)

0 0

ST – 60

Short call (X = 50)

0 – (S

T – 50) – (S

T – 50)

Total 0

– (ST – 50)

–10

B

reakeven occurs when the payoff offsets the initial proceeds of $6, w

hich occurs at a stock price of S

T = $56.

0 S

T50

60

6-10

- 4P

rofit

Payoff

26.

Buy a share of stock, w

rite a call with X

= 50, write a call w

ith X = 60, and buy a call

with X

= 110.

Position S

T < 50 50 < S

T < 60 60 < S

T < 110 S

T > 110

Buy stock

ST

ST

ST

ST

Short call (X = 50)

0 – (S

T – 50) – (S

T – 50) – (S

T – 50) Short call (X

= 60) 0

0 – (S

T – 60) – (S

T – 60) Long call (X

= 110) 0

0 0

ST – 110

Total S

T 50

110 – ST

0

The investor is making a volatility bet. Profits w

ill be highest when volatility is low

so that the stock price ends up in the interval betw

een $50 and $60. C

H15

1. Put values also increase as the volatility of the underlying stock increases. W

e see this from

the parity relationship as follows:

C = P + S

0 – PV(X

) – PV(D

ividends)

Given a value of S and a risk-free interest rate, if C

increases because of an increase in volatility, so m

ust P in order to keep the parity equation in balance.

Num

erical example:

Suppose you have a put w

ith exercise price 100, and that the stock price can take on one of three values: 90, 100, 110. The payoff to the put for each stock price is:

Stock price 90

100110

Put value 10

00

N

ow suppose the stock price can take on one of three alternate values also centered

around 100, but with less volatility: 95, 100, 105. The payoff to the put for each

stock price is: Stock price

95 100

105Put value

5 0

0

The payoff to the put in the low

volatility example has one-half the expected value

of the payoff in the high volatility example.

3. N

ote that, as the option becomes progressively m

ore in the money, its hedge ratio

increases to a maxim

um of 1.0:

X

Hedge ratio

X

H

edge ratio 115

85/150 = 0.567

50

150/150 = 1.000 100

100/150 = 0.667

25 150/150 = 1.000

75 125/150 = 0.833

10

150/150 = 1.000 5.

a. W

hen S = 130, then P = 0.

When S = 80, then P = 30.

The hedge ratio is: [(P

u – Pd )/(uS

0 – dS0 ) = [(0 – 30)/(130 – 80)] = –3/5

b.

Riskless portfolio

S =80 S = 130

3 shares 240

390 5 puts

150 0

Total 390

390

Present value = $390/1.10 = $354.545 c.

Portfolio cost = 3S + 5P = $300 + 5P = $354.545

Therefore 5P = $54.545 ⇒ P = $54.545/5 = $10.91

6. The hedge ratio for the call is: [(C

u – Cd )/(uS

0 – dS0 )] = (20 – 0)/(130 – 80) = 2/5

Riskless portfolio

S =80 S = 130

2 shares 160

260 Short 5 calls

0 -100

Total 160

160

–5C

+ 200 = $160/1.10 = $145.455 ⇒ C

= $10.91 Put-call parity relationship: P = C

– S0 + PV

(X)

$10.91 = $10.91 + ($110/1.10) – $100 = $10.91

7. d1 = 0.3182

N(d1 ) = 0.6248

d2 = –0.0354

N(d2 ) = 0.4859

X

e –rT = $47.56

C = S

0 N(d

1 ) − Xe –rT N

(d2 ) = $8.13

8. P = $5.69

This value is from

our Black-Scholes spreadsheet, but note that w

e could have derived the value from

put-call parity:

P = C – S

0 + PV(X

) = $8.13 – $50 + $47.56 = $5.69 11.

The call price will decrease by less than $1. The change in the call price w

ould be $1 only if: (i) there w

ere a 100% probability that the call w

ould be exercised;

and (ii) the interest rate were zero.

14. The call option w

ith a high exercise price has a lower hedge ratio. The call option

is less in the money. B

oth d1 and N(d1 ) are low

er when X

is higher. 16.

The call option’s implied volatility has increased. If this w

ere not the case, then the call price w

ould have fallen. 17.

The put option’s implied volatility has increased. If this w

ere not the case, then the put price w

ould have fallen.


Week 7:

Equity Valuation (Chp. 13.2-13.4)

Slide 2Week 7


Intrinsic Value and Market Price

� Problem: determine the value of equity (i.e., of common shares)

� Market Price (P0,MP): consensus value of all potential traders

� Intrinsic Value (P0,IV):

─ Self assigned value

─ Variety of models are used for estimation

� Trading Signal

─ P0,IV > P0,MP: Buy

─ P0,IV < P0,MP: Sell or Short Sell

─ P0,IV = P0,MP: Hold or Fairly Priced

Slide 3Week 7


Realized vs. Expected Holding Period Return (1)

� Remember our definition of HPR (slide 10 of

week 2):

� Useful to evaluate an investment ex-post (i.e., when it is done and all cash-

flows and prices are observable and known).

� If we think about doing an investment and want to evaluate it ex-ante � we

need to look at the Expected HPR!

─ We know the current price P0.

─ We must estimate the expected

dividend payments and the expected

price at the end of the holding period.

( )( ) ( )

MP

t

i

iMPMPt

tP

DEPPE

rE,0

1

,0, ∑=

+−

=

MP

t

i

iMPMPt

tP

DPP

r,0

1

,0, ∑=

+−

=

Slide 4Week 7


Realized vs. Expected Holding Period Return (2)

� Again, two ways to come up with the expected HPR:

─ Market-Oriented: use the CAPM to estimated the return of a stock

required by the market (we refer to this rate as E(k))

─ Look at the firm, estimate future dividends and estimate the expected

HPR (we refer to this rate as E(r))

� Before, we called the difference between E(r) and E(k) the alpha of a stock.

� Trading Signal

─ E(r) > E(k): Buy (positive alpha case)

─ E(r) < E(k): Sell or Short Sell (negative alpha case)

─ E(r) = E(k): Hold or Fairly Priced (zero alpha case)

� Comparing P0,IV and P0,MP or E(r) and E(k) always gives the same signal!

Slide 5Week 7


For a 1-Year Holding Period (1)

� Consider the definition

of the Expected HPR:

� Assume that we know P0,MP, the required rate of return E(k) and have

estimates for E(P1) and E(D1).

� Goal: determine if we should buy or sell this stock

� Two ways to solve this problem:

1. Assume that E(k)=E(r) (i.e., the expected rate of return equals the

required rate of return) and check whether P0,IV=P0,MP (i.e., the

intrinsic value of the stock equals its market price)

2. Assume that P0,IV=P0,MP (i.e., the intrinsic value of the stock equals its

market price) and check whether E(k)=E(r) (i.e., the expected rate of

return equals the required rate of return)

( ) ( )

MP

MPMP

P

DEPPErE

,0

1,0,1

1)(+−

=

Slide 6Week 7


For a 1-Year Holding Period (2)Solution 1

� Solution 1: we assume that E(k)=E(r)

� Then, we calculate the IV out of the equation for the Expected HPR:

� Then we compare the quoted price P0,MP and the calculated intrinsic value

P0,IV:

─ P0,IV > P0,MP : buy

─ P0,IV < P0,MP : sell

─ P0,IV = P0,MP : fairly priced

( ) ( ))(1

,11

,0kE

PEDEP

MP

IV+

+=

( ) ( )

IV

IVMP

P

DEPPEkE

,0

1,0,1)(

+−=

Slide 7Week 7


For a 1-Year Holding Period (3)Solution 2

� Solution 1: we assume that P0,MP=P0,IV

� The, we calculate the expected rate of return using the equation for the

Expected HPR:

� Then we compare the expected rate of return E(r) and the required rate of

return E(k) (could come from the CAPM):

─ E(r) > E(k): Buy (positive alpha case)

─ E(r) < E(k): Sell or Short Sell (negative alpha case)

─ E(r) = E(k): Hold or Fairly Priced (zero alpha case)

( ) ( )

MP

MPMP

P

DEPPErE

,0

1,0,1)(

+−=

Slide 8Week 7


For a 1-Year Holding Period (4)Example

� You expect the price of IBX stock to be E(P1,MP)=$59.77 per share a year from

now. Its current market price is P0,MP=$50, and you expect it to pay a

dividend one year from now of E(D1)=$2.15 per share. The stock has a beta

of 1.15, the risk-free rate is 6% per year, and the expected rate of return on

the market portfolio is 14% per year. Determine if this stock is a good

investment!

� Solution 1:

─ Determine E(k): E(k) = 6%+1.15×(14%-6%) = 15.2%

─ Determine P0,IV = ($2.15+$59.77)/1.152 = $53.75

─ Compare P0,IV and P0,MP: $53.75 > $50 � BUY!

Slide 9Week 7


For a 1-Year Holding Period (5)Example

� You expect the price of IBX stock to be E(P1,MP)=$59.77 per share a year from

now. Its current market price is P0,MP=$50, and you expect it to pay a

dividend one year from now of E(D1)=$2.15 per share. The stock has a beta

of 1.15, the risk-free rate is 6% per year, and the expected rate of return on

the market portfolio is 14% per year. Determine if this stock is a good

investment! (same problem as on the previous slide)

� Solution 2:

─ Determine E(r): E(r) = ($59.77-$50+$2.15)/$50 = 0.2384

─ Determine E(k): E(k) = 6%+1.15×(14%-6%) = 15.2%

─ Compare E(r) and E(k): 23.84% > 15.2% � BUY!

Slide 10Week 7


For a 1-Year Holding Period (6)Exercise

� Consider again the problem discussed on the previous two slides. What does

the expected price of the stock in one year, i.e. E(P1,MP), has to be to make

the stock be priced fairly?

Slide 11Week 7


A More General Model: Dividend Discount Model (1)Motivation

� Consider the 1-year definition of the intrinsic value:

� What is E(P1,MP)? Assume that the stock sells for its intrinsic value in one year,

i.e. in one year P1,IV=P1,MP. We can estimate P1,IV and plug it into the equation

for P0,IV.

( ) ( ))(1.11

,0kE

PEDEP MP

IV+

+=

( ) ( ) ( ) ( ))(1.22

,1,1kE

PEDEPEPE MPIVMP

+

+==

( )( )

( ) ( )( )( )2

221,0

11 kE

PEDE

kE

DEP MP

IV+

++

+=

Slide 12Week 7


A More General Model: Dividend Discount Model (2)Equation

� If you continue to substitute for expected future market prices (i.e.,

for E(P2,MP), E(P3,MP), etc.) indefinitely you get:

� Description:

─ P0,IV…Intrinsic value of stock at t=0

─ E(k)...required return/discount rate

─ E(Dt)…expected dividend payment in period t

( )( )

( )( )( )

( )( )( )

...111 3

3

2

21,0 +

++

++

+=

kE

DE

kE

DE

kE

DEP IV

Slide 13Week 7


A More General Model: Dividend Discount Model (3)Equation

� How to use this model? � we need more assumptions on future dividends

� Assumption 1: constant dividend � E(D1)=E(D2)=…=E(D)

� Assumption 2: dividend that grows at the constant rate g

)(

)(,0

kE

DEP IV =

( )( )( )gkE

DEP IV

−= 1

,0

( ) ( )gDDE +×= 101

( ) ( )202 1 gDDE +×=

Slide 14Week 7


A More General Model: Dividend Discount Model (4)Example

� High Flyer Industries has just paid its annual dividend of D0=$3 per share.

The dividend is expected to grow at a constant rate of g=8% until infinity.

The beta of High Flyer stock is 1.0, the risk-free rate is 6%, and the market

risk premium is 8%.

� What is the intrinsic value of the stock?

─ E(k)=6%+1.0×8%=14%

─ P0,IV=$54

� What is the intrinsic value if the beta changes to 1.25?

─ E(k)=6%+1.25×8%=16%

─ P0,IV=$40.50

( )( )

54$08.014.0

08.013$,0 =

−

+×=IVP

Slide 15Week 7


A More General Model: Dividend Discount Model (5)Exercise

� IBX’s stock dividend at the end of this year is expected to be E(D1)=$2.15,

and it is expected to grow at g=11.2% per year forever. If the required rate

of return on IBX stock is E(k)=15.2% per year, what is its current intrinsic

value, P0,IV?

� What is IBX next year’s expected intrinsic value, P1,IV?

� If an investor were to buy IBX stock now and sell it after receiving the $2.15

dividend a year from now, what is the expected holding-period return?

Slide 16Week 7


A More General Model: Dividend Discount Model (6)Some More Remarks

� Constant Growth Dividend Discount Model:

─ Only valid if g is less than k

─ What happens if g is larger than k? Why can this not be the case?

� Implications of the constant growth DDM: a stock’s value will be greater

─ The larger the expected dividend per share

─ The lower the market capitalization rate, k

─ The higher the expected growth rate of dividends, g

Slide 17Week 7


Multistage Growth Dividend Discount Models (1)

� Consider a firm that quickly grows in its early years and then, once it matures,

yields a dividend with constant growth until infinity.

─ Dividends in the early years: $.32, $.41, $.50, $.60

─ Constant growth of dividends thereafter: g=9.375%

─ The market capitalization rate (i.e., E(k)) equals 13.1%.

� The current intrinsic value is

( )4

,4

432,0131.1131.1

60$.

131.1

50$.

131.1

41$.

131.1

32$. MP

IV

PEP ++++=

Slide 18Week 7


Multistage Growth Dividend Discount Models (2)

� Determine E(P4,MP): assume that the stock trades for its intrinsic value in four

years; thus we must calculate the intrinsic value of a constant growth

dividend discount model in four years

� Plug E(P4,MP) back into our equation for P0,IV:

( ) ( ) ( )( )

62.17$09375.0131.0

09375.0160$.,4,4 =

−

+×== IVMP PEPE

08.12$131.1

62.17$

131.1

60$.

131.1

50$.

131.1

41$.

131.1

32$.4432,0 =++++=IVP

Slide 19Week 7


Multistage Growth Dividend Discount Models (3)Exercise

� Consider the firm of the previous example. Now assume that its beta equals

1.0 (the risk-free rate is 5% and the market risk premium is 6%).

─ Calculate the new discount rate.

─ Determine the current intrinsic value of this company.

Slide 20Week 7


Determination of Dividend Growth Rate (1)

� Problem: how to come up with an estimate of dividend growth rate g

� Solution: look at the firm’s earnings (rather profits) and investment strategy;

basically a firm has two choices to spend its profits

1. Reinvest them into the firm

2. Pay them out as dividends

3. A combination of (1) and (2)

Slide 21Week 7


Determination of Dividend Growth Rate (2)Equation

� The dividend growth rate depends on the reinvestment rate and on the return

generated on these reinvestments.

� Description:

─ b…earnings reinvestment rate, earnings retention rate

─ ROE…return on equity; rate of return earned on reinvestments

─ g...dividend growth rate

bROEg ×=

Slide 22Week 7


Determination of Dividend Growth Rate (3)Example

� Consider a firm with $100 million of assets (all equity-financed) and 3 million

shares outstanding. The Return on Equity (ROE) equals 15%. Earnings

account for $15 million or $5 per share.

─ Management decides to reinvest b=60%. i.e., 0.6×15 = $9 million

─ The capital stock of the firm increases by $9 million, i.e., 9% (given the

initial capital of $100 million).

─ Endowed with 9% more capital the firm earns 9% more income per

year and dividends grow by 9% per year.

%960.0%15 =×=×= bROEg

Slide 23Week 7


Determination of Dividend Growth Rate (4)Different Retention Rates

� What is the influence of different retention rates on the firm value? If two

firms are identical but one reinvests 60% and the other one reinvests 0%

which one is going to have a higher firm value?

� Answer: this depends on the profitability of the reinvestment, i.e., compare

the ROE to the required rate of return as implied by the CAPM.

� Example: Consider a firm with $100 million of assets (all equity-financed)

and 3 million shares outstanding. The Return on Equity (ROE) equals 15%.

The risk-adjusted discount rate implied by the CAPM is k=12.5%. Earnings

account for $15 million or $5 per share. Management decides to reinvest

b=60%.

Slide 24Week 7


Determination of Dividend Growth Rate (5)Example

� From slide 22 we know that if b=60% then g=9%.

� The value of the firm in this case is:

� If the firm does not reinvest, there is no dividend growth and the value is:

( ) ( )14.57$

09.0125.0

2$1,0 =

−=

−=

gk

DP IV

40$125.0

5$10 ===

k

DP

( ) 2$6.015$1 =−×=D

Slide 25Week 7


Determination of Dividend Growth Rate (6)Exercise

� Consider the firm from before. Now assume that the ROE equals 12.5%. Does

an earnings retention ratio (i.e., b or reinvestment ratio) of 60% increase firm

value relative to the case of 0% earnings retention ratio? Explain your result.

� Consider the firm from before. Now assume that the ROE equals 10%. Does

an earnings retention ratio (i.e., b or reinvestment ratio) of 60% increase firm

value relative to the case of 0% earnings retention ratio? Explain your result.

Fin

ance 3

050

Hom

ework

#1

Sprin

g 2

007

Consid

er the fo

llow

ing 3

investm

ents:

Investm

ent

E(R

)F

A.1

9.2

158

Cov(a,b

) = .0

014562

B.1

0.0

632

Cov(a,c) =

-.0055285

C.1

2.1

404

Cov(b

,c) = .0

034125

1.

On g

raph p

aper, p

lot th

e three in

vestm

ents. U

se F o

n th

e x-ax

is, and E

(R) o

n th

e y-ax

is.

2.

Calcu

late the p

ortfo

lio E

(R) an

d F

for each

of th

e follo

win

g co

mbin

ations:

10%

A - 9

0%

B10%

A - 9

0%

C10%

B - 9

0%

C

50%

A - 5

0%

B50%

A - 5

0%

C50%

B - 5

0%

C

90%

A - 1

0%

B90%

A - 1

0%

C90%

B - 1

0%

C

Plo

t each p

oin

t on th

e grap

h

3.

Fin

d th

e min

imum

varian

ce portfo

lio p

roportio

ns fo

r each tw

o asset p

ortfo

lio,

(A,B

); (A,C

); (B,C

). The m

inim

um

varian

ce portfo

lio can

be fo

und b

y th

e form

ula

Where

W1

= p

roportio

n o

f the p

ortfo

lio in

investm

ent 1

, and

(1

- W1 )=

pro

portio

n o

f the p

ortfo

lio in

investm

ent 2

Calcu

late the E

(R) an

d F

for each

min

imum

varian

ce portfo

lio. P

lot each

varian

ce- min

imizin

g

portfo

lio o

n th

e grap

h.

Draw

a curv

e show

ing th

e portfo

lio p

ossib

ilities for each

of th

e three p

airs

(A,B

); (A,C

); (B,C

). Use a d

ifferent co

lor fo

r each cu

rve.

4.

Suppose in

vesto

rs could

only

invest in

A,B

,C o

r som

e com

bin

ation o

f any tw

o (b

ut n

ot all

three).

a)Id

entify

on y

our g

raph a seg

men

t where all in

vesto

rs would

wan

t to select a p

ortfo

lio.

b)

Mig

ht an

y in

vesto

r wan

t to in

vest in

A alo

ne? W

hat ab

out B

? What ab

out C

?

c)W

ould

investo

rs wan

t to in

vest in

any co

mbin

ation o

f A&

B?, A

&C

?, B&

C?

Fin

ance 3

050

Hom

ework

#2

S

prin

g 2

007

I.V

isit the w

ebsite h

ttp://fin

ance.y

ahoo.co

m an

d g

ather th

e follo

win

g in

form

ation fo

r Pfizer,

Inc, (tick

er sym

bol: P

FE

):

1.

The clo

sing p

rice for P

FE

’s com

mon sto

ck o

n T

uesd

ay. 4

/10/0

7

2.

PFE

’s curren

t annual d

ivid

end

3.

PFE

’s EP

S fo

r last yea

r

4.

The cu

rrent (as o

f 4/1

0/0

7) an

alysts’ co

nsen

sus estim

ate of P

FE

’s EP

S g

row

th fo

r the

nex

t five y

ears (per an

num

)

II.A

ssum

ing th

e follo

win

g in

form

ation:

E(r

m) =

12.5

.0%

Rf =

5.2

5%

PFE

’s beta =

0.7

0*

III.C

alculate th

e follo

win

g:

a.B

ased o

n p

art II, what rate o

f return

should

PFE

’s shareh

old

ers require o

n th

eir investm

ent?

b.

If PFE

retains 6

2%

of its earn

ings, an

d can

earn a retu

rn o

f 10%

on rein

vested

funds, at w

hat

rate can d

ivid

ends g

row

indefin

itely?

c.B

ased o

n y

our an

swer to

(a) and in

form

ation o

btain

ed in

part I, if P

FE

’s div

iden

d (P

art I,

item 2

) is expected

to g

row

indefin

itely at th

e rate found in

part (b

), what is a sh

are of P

FE

stock

worth

today

? (A

ssum

e that th

e div

iden

d y

ou fo

und in

part I is D

0 )

d.

By h

ow

much

is PFE

stock

overp

riced o

r underp

riced w

hen

com

parin

g y

our an

swer to

part

(c) with

the cu

rrent p

rice (part I, item

1)?

e.W

hat is P

FE

’s div

iden

d y

ield?

f.If P

FE

’s earnin

gs g

row

at the estim

ated g

row

th rate fo

r 5 y

ears (item I, p

art 5), w

hat w

ill EP

S

be at th

e end o

f 2011 (b

egin

with

the E

PS

for ‘0

6 [p

art 1, item

3])?

g.

Usin

g y

our an

swer to

(f), and assu

min

g th

at PFE

’s (trailing) P

/E ratio

is 11 at th

e end o

f

2011, w

hat sh

ould

a share o

f stock

sell for at th

at time?

h.

Assu

me th

at div

iden

ds will n

ot grow

for th

e nex

t 5 y

ears. If you b

uy a sh

are of P

FE

stock

at the cu

rrent p

rice (part I, item

1), receiv

e the cu

rrent d

ivid

end (p

art I, item 2

) for each

of

the n

ext 5

years, th

en sell th

e stock

for th

e price fo

und in

(g), w

hat w

ill be y

our d

ollar-

weig

hted

averag

e annual rate o

f return

(Calcu

late IRR

usin

g P

0 , D1 -D

5 and P

5 )?

*Ig

nore an

y actu

al beta fo

r PFE

that y

ou m

ay fin

d in

yah

oo o

r any o

ther so

urce.

Finance 3

050

Midterm

#1

Part I - P

roblem Solving

1.a.

Suppose y

ou sell sh

ort 3

00 sh

ares of JN

J at $55. Y

ou put in

$8,250 of y

our o

wn m

oney

for m

argin purposes. If th

e price o

f JNJ falls to

$47, h

ow m

uch do you gain

or lo

se as a

percen

tage of your original investm

ent?

b.

If the p

rice of JN

J falls to $45 an

d a d

ividend of $

1/sh

are is paid

, what is th

e % m

argin in

your acco

unt?

2.

Municip

al bond A pays in

terest at the rate o

f 3.5%, w

hile C

orporate b

ond B pays in

terest

at the rate o

f 5.50%. If L

ynette p

ays tax

at a marg

inal rate o

f 36%, w

hich

bond sh

ould

she ch

oose to

invest in

? Support y

our an

swer w

ith fig

ures.

3.

Suppose y

ou buy 400 sh

ares of C

SCO at $

25.00 usin

g $5,000 of y

our o

wn m

oney an

d

borro

wing th

e other $

5,000 on m

argin. If th

e main

tenance m

argin m

ust b

e 25%, h

ow lo

w

can th

e price o

f CSCO go befo

re a marg

in call is issu

ed? (Ig

nore in

terest)

Midterm

#2

1.

If inflatio

n in

a certain eco

nomy is ex

pected

to be 1

50%, w

hat n

ominal rate o

f interest

must b

e charg

ed to

earn a real rate o

f return of 8

%?

2Douglas earn

s the fo

llowing rates o

ver fo

ur y

ears of in

vestin

g: r

1 = .2

2, r

2 = .1

2, r

3 = -.0

8,

r4 =

.09. W

hat is th

e geometric av

erage o

f these retu

rns?

3.

Suppose y

ou m

anage a p

ortfo

lio with

E(r) =

16% an

d F = 30% T

he risk

-free rate is 2%.

a.If a clien

t of y

ours ch

ooses to

invest 6

5% of h

er wealth

in your p

ortfo

lio an

d th

e rest in t-

Bills, w

hat w

ould be th

e E(r) an

d F of h

er complete p

ortfo

lio?

b.

Ignorin

g part (a) ab

ove, if y

our clien

t wanted

to ach

ieve an

E(r) o

f 10% on her allo

cation

portfo

lio, w

hat p

roportio

n of h

er wealth

would sh

e need

to in

vest in

your p

ortfo

lio?

Final E

xam

1.

A sto

ck is ex

pected

to pay a d

ividend of $

1.48/sh

are at the en

d of each

of th

e next th

ree

years. A

t that tim

e, the sto

ck is ex

pected

to have y

ear-end earn

ings o

f $5.95 an

d a P

/E ratio

of

19. If in

vesto

rs require a retu

rn of 1

1% on th

is stock, h

ow m

uch sh

ould a sh

are be w

orth

today?

2.

Fairv

iew In

dustrial P

roducts In

c. Is expected

to have y

ear-end earn

ings (E

1 )of

$7.85/sh

are. Fairv

iew in

tends to

plowback

60% of earn

ings o

n which

a return (R

OE) o

f 12% is

expected

. What is th

e PVGO fo

r Fairv

iew if sh

areholders req

uire a 1

0% retu

rn on th

eir

investm

ent?

3.

BMC eq

uipment in

c. has ju

st paid

a dividend (D

0 ) of $

3.28. D

ividends are ex

pected

to

grow at a rate o

f 5.2% in

defin

itely. If B

MC is co

rrectly (fairly

) priced

at $31.95, w

hat is th

e

mark

et capitalizatio

n rate (k

) for th

is stock?

4.

Suppose y

ou buy one p

ut o

ptio

n co

ntract (1

00 sh

ares) on W

hitb

ury co

. with

an ex

ercise

price o

f $40 an

d six

months to

expiratio

n. T

he cu

rrent p

rice of th

e stock is $

44.35 an

d

the p

remium on th

e optio

n is $

.68. If th

e price o

f the sto

ck is $

38.75 on th

e expiratio

n

date o

f the o

ptio

n, w

hat is y

our p

rofit (o

r loss) in

dollars an

d as a p

ercent o

f your o

riginal

investm

ent (all p

rices given are p

er share)?

5.

Find th

e valu

e of a call o

ptio

n with

the fo

llowing ch

aracteristics:

Today’s p

rice of th

e underly

ing sto

ck (S

0 ):$50

Price o

f the sto

ck in

one y

ear (S1 ):

$65 or $

37

Exercise p

rice of th

e call optio

n (X

):$45

Risk

-free interest rate (r

f ):3.5%

1st H

omew

ork Assignm

ent

Due on

September 16

th before class

Important inform

ation (read carefully): a.

The first page of the homew

ork that you turn in must be the standardized C

over Page (i.e., the first page of this docum

ent). Clearly indicate on your cover sheet

which exercises you solved and are ready to present in class (these are the

exercises which are graded).

b. M

ake a copy of your solutions before you turn them in.

c. Show

your calculations. You m

ight get (partial) credit for wrong solutions if

your way of approaching the question m

akes sense. No credit w

ill be given to correct solutions that are not accom

panied by the appropriate calculations. Problem

1 A

T-bond expires in 4 years (assume that the bond w

as just issued). It has a coupon rate of 8%

, a face value of $1000 and pays coupons annually. The asked price is quoted as 102:10 w

hile the bid price is 101:20. a.

What is the asked price expressed in dollar term

s? b.

What is the bid price expressed in dollar term

s? c.

If you hold the bond until maturity, w

hat is going to be your holding period return? W

hat is going to be the annualized holding period return? d.

What is the internal rate of return if you invest into the bond?

Solution Problem 1

a. The asked price is P(asked)=(102+10/32)%

of $1000=$1023.125 b.

The bid price is (101+20/32)% of $1000 = $1016.25

c. Price at m

aturity = 1000; intermediate C

Fs are 4*80=320; price at which you

bought the bond is the ask price: HPR

=(1000+320)/1023.125 – 1=0.29; the annualized H

PR is (1+0.29)^0.25-1 = 0.0658

d. The bond pays 4 paym

ents of $80 every year in addition to the par of $1000 in the end of the fourth year. The price of the bond is your answ

er to part a., $1023.125. Y

ou need to find the interest rate that this cash flow stream

provides (N=4, PV

= -1023.125, FV

=1000, PMT

=80, P/YR

=1, I/YR

=?). The answ

er is 0.0731. A

lternatively, you can specify the CFs explicitly: C

Fj=-1023.125, CFj=80,

CFj=80, C

Fj=80, CFj=1080, IR

R/Y

R=???. T

he answer is 0.0731.

- 1 -

Problem 2

a. Y

ou own $2847. The interest rate is 6.43%

p.a. and compounding happens daily

(assume that there are 250 business days per year). W

hat is your future value after 3 years?

b. N

ow again you ow

n $2847 and you want to invest them

for 20 years. What m

ust be the interest rate such that you end up w

ith exactly $5000 after 20 years if com

pounding happens yearly? c.

Again you start out w

ith $2847. The interest rate is 4.65% p.a.. H

ow long do you

have to

invest your

money

in order

to get

$3500 if

there is

monthly

compounding?

Solution Problem 2

a. $2847*(1+6.43%

/250)^(250*3)=$3452.65 (N=250*3=750; PV

=-2847; I/Y

R=6.43 (enter like this); P/Y

R=250; FV

=?) b.

I/YR

=2.856% (N

=20; P/YR

=1; FV=5000; I/Y

R=?) or 2847*(1+r)^20=5000;

r=(5000/2847)^(1/20)-1=0.02856 c.

N=53.4 (I/Y

R=4.65; FV

=3500; P/YR

=12; N=?); 2847*(1+0.0465/12)^N

=3500; N

=ln(3500/2847)/ln(1+0.0465/12)=53.4 months

Problem 3 to 4: The follow

ing information is relevant for solving Problem

3 to 4. C

onsider the following price and dividend history for stock X

YZ:

Year

Beginning-of-Y

ear PriceD

ividend Paid at Year-End

2002 $100

$4 2003

$110 $4

2004 $90

$4 2005

$95 $4

An investor buys three shares of X

YZ at the beginning of 2002, buys another tw

o shares at the beginning of 2003, sells one share at the beginning of 2004, and sells all four rem

aining shares at the beginning of 2005. Problem

3: a.

What are the arithm

etic and geometric average 1-year holding period returns of

this investment?

b. W

hat is the 2-years holding period return in 2004 (i.e., the 2-years holding period return in the period 2002 to 2004)?

c. W

hat is the 3-years holding period return of this investment in 2005?

Solution Problem 3

- 2 -

(A) The tim

e-weighted average returns are based on year-by-year rates of return.

Year

Return = (C

apital Gain+D

ividend)/price 2002-2003

(110-100+4)/100=14.00%

2003-2004 (90-110+4)/110=-14.55%

2004-2005

(95-90+4)/90=10.00%

Arithm

etic Mean: (14%

-14.55%+10%

)/3=3.15%

Geom

etric Mean: [(1+0.14)*(1-0.1455)*(1+0.1)]^(1/3)-1=2.33%

(B

) (90-100+8)/100 = -0.02 (C

) (95-100+12)/100 = 0.07 Problem

4: W

hat is the investor’s internal rate of return? (Hint: C

arefully prepare a table with all the

cash flows for the four dates.)

Solution Problem 4

Prepare a table of Cash-Flow

s Tim

e C

ash-Flow

Explanation 0

-300 B

uy three shares at 100 1

-208 B

uy another two shares at 110 plus dividend incom

e on three shares held

2 90+5*4=110

Dividends on 5 shares plus sale of one share at $90

3 4*95+4*4=396

Dividend on four shares, plus sale of four shares at 95

Calculate the internal rate of return: -300+-208/(1+r)^1+110/(1+r)^2+396/(1+r)^3=0

(P/YR

=1; CFj=-300; C

Fj=-208; CFj=110; C

Fj=396; IRR

/YR

=?) IR

R=-0.166%

Problem

5 to 8: The following inform

ation is relevant for solving Problem 5 to 8.

Assum

e that you manage a risky portfolio, called “Y

our Fund”, with an expected rate of

return of 17% and a standard deviation of 27%

. The T-Bill rate (i.e., the risk-free rate) is

7%. The risky portfolio includes the follow

ing investments in the given proportions:

Stock A

27%Stock B

33%Stock C

40% Problem

5:

- 3 -

a. Y

our client chooses to invest 70% of a portfolio in “Y

our Fund” and 30% in the

T-Bill. W

hat is the expected return and standard deviation of your client’s portfolio?

b. W

hat are the investment proportions of your client’s overall portfolio, i.e. your

client’s position in stock A, stock B

, stock C and the T-B

ill? c.

What is the rew

ard-to-variability ratio (S) of “Your Fund” and your client’s

overall portfolio? d.

Draw

the CA

L of “Your Fund” on an expected return/standard deviation diagram

. W

hat is the slope of the CA

L? Show the position of your client on your fund’s

CA

L. Solution Problem

5 a.

E(rp)=0.3*7%+0.7*17%

=14%; SD

p=0.7*27%=18.9%

b.

T-Bills = 30%

; Stock A = 0.7*27%

=18.9%; Stock B

= 0.7*33%=23.1%

; Stock C

= 0.7*40%=28.0%

c.

Your S = (17-7)/27=0.3704; your client’s S = (14-7)/18.0 = 0.3704

d.

E(r)

σ

7

27

14 17P

CA

L ( slope=.3704)%

%

18.9

client

Problem

6: Suppose the sam

e client from the previous problem

decides to invest in “Your Fund” a

proportion y of his total investment budget so that his overall portfolio w

ill have an expected rate of return of 15%

. a.

What is the proportion y?

b. W

hat are your client’s investment proportions in your three stocks and the T-B

ill? c.

What is the standard deviation of the rate of return on your client’s portfolio?

- 4 -

Solution Problem 6

a. E(rp)=(1-y)*7%

+y*17%=15%

; 15%-7%

=y*(-7%+17%

); y=(15-7)/10=0.8 b.

T-Bills = 20%

; Stock A = 0.8*27%

=21.6%; Stock B

= 0.8*33%=26.4%

; Stock C

= 0.8*40%=32.0%

c.

SDp=0.8*27%

=21.6%

Problem 7:

Suppose the same client from

the previous problem decides this tim

e to invest in “Your

Fund” a proportion y of his total investment budget so that the rate of return of his overall

portfolio will have a standard deviation of 20%

. a.

What is the investm

ent proportion y? b.

What is the expected rate of return on your client’s portfolio?

Solution Problem 7

a. 20%

=y*27%; y=20%

/27%; y=0.7407

b. E(rp)=(1-0.7407)*7%

+0.7407*17%=14.407%

Problem

8: Y

ou estimate that a passive portfolio invested to m

imic the S&

P 500 stock index yields an expected rate of return of 13%

with a standard deviation of 25%

. Draw

the CM

L and “Y

our Fund’s” CA

L on an expected return/standard deviation diagram.

a. W

hat is the slope of the CM

L? b.

Characterize in one short paragraph the advantage of “Y

our Fund” over the passive fund.

Solution Problem 8

a. Slope of the C

ML: (13-7)/25=0.24 (scan the graph from

the solution manual)

b. M

y fund allows an investor to achieve a higher expected rate of return for any

given standard deviation than would a passive strategy.

Problem 9 to 10: The follow

ing information is relevant for solving Problem

9 to 10. Im

agine that the returns of the two stocks, A

mgen and C

oca Cola, w

hich caught your attention w

hen you last flipped through the WSJ depend on the overall m

arket conditions in the follow

ing way:

Am

gen would m

ake a return of -20% (i.e. a loss) if m

arkets are bearish, 18%, if they are

normal, and 50%

, if markets are bullish. The corresponding num

bers are -15%, 20%

, and 10%

for Coca C

ola. The probability of Bear M

arkets is 20%, of N

ormal M

arkets is 50%,

and of Bull M

arkets is 30%. This inform

ation is summ

arized in the table below:

- 5 -

B

ear Market

Norm

al Market

Bull M

arket Probability

0.2 0.5

0.3 R

eturn of Am

gen -20%

18%

50%

R

eturn of Coca C

ola -15%

20%

10%

Problem

9 W

hat are the expected returns and standard deviations of the returns of Am

gen and Coca

Cola?

Solution Problem 9: expected return

Expected Return of A

mgen=E

[rA] = (Probability of B

ear Market)*(R

eturn of Am

gen in B

ear Market) + (Probability of N

ormal M

arket)*(Return of A

mgen in N

ormal M

arket) + (Probability of B

ull Market)*(R

eturn of Am

gen in Bull M

arket) = = 0.2 *(-20%

)+0.5*(18%)+0.3*(50%

)=0.2 *(-0.2)+0.5*(0.18)+0.3*(0.50)= 20% = 0.2

Expected Return of C

oca Cola=E

[rC

C]=(Probability of Bear M

arket)*(Return of C

oca C

ola in Bear M

arket) + (Probability of Norm

al Market)*(R

eturn of Coca C

ola in Norm

al M

arket) + (Probability of Bull M

arket)*(Return of C

oca Cola in B

ull Market) =

= 0.2 *(-15%)+0.5*(20%

)+0.3*(10%) = 0.2 *(-0.15)+0.5*(0.20)+0.3*(0.10) = 10%

= 0.1 Solution Problem

9: standard deviations V

ariance[rA] = (Probability of B

ear Market) * ( R

eturn of Am

gen in Bear M

arket -E

[rA]) 2 + (Probability of N

ormal M

arket) * ( Return of A

mgen in N

ormal M

arket -E

[rA]) 2 + (Probability of B

ull Market) * ( R

eturn of Am

gen in Bull M

arket -E[rA

]) 2 = = (0.2) * (-0.2-0.2) 2 + (0.5)*(0.18-0.2) 2+(0.3)*(0.5-0.2) 2

= (0.2)*(-0.4) 2 + (0.5)*(-0.02) 2+(0.3)*(0.3) 2

= 0.0592 (in decimals) or 592 if you used percentages

SD is the square root of the variance: 0.0592^(1/2)= 0.2433 (or 24.33%

) V

ariance[rCC

] = (Probability of Bear M

arket)*( Return of C

oca Cola in B

ear Market -

E[rC

C] ) 2

+ (Probability of Norm

al Market)*( R

eturn of Coca C

ola in Norm

al Market -E

[rCC

]) 2

+ (Probability of Bull M

arket)*( Return of C

oca Cola in B

ull Market -E

[rCC

]) 2

= (0.2)*(-0.15-0.1) 2 + (0.5)*(0.2-0.1) 2+(0.3)*(0.1 – 0.1) 2

= (0.2)*(-0.25) 2 + (0.5)*(0.1) 2+(0.3)*(0) 2

= 0.0175 (in decimals) or 175 if you used percentages

SD is the square root of the variance: 0.0175^(1/2)= 0.1323 (or 13.23%

) Problem

10 A

ssume that of your $10,000 portfolio you invest $9000 in Stock A

mgen and $1000 in

Coca C

ola.

- 6 -

a. C

alculate the returns on your portfolio for every market condition, i.e., B

ear M

arket, Norm

al Market and B

ull Market.

b. W

hat is the expected return on your portfolio? c.

What is the standard deviation of the returns on your new

ly formed portfolio?

Solution Problem 10

The weight on A

mgen in this portfolio is w

(Am

gen) = 9/10 = 0.9. The weight on C

oca C

ola is w(C

oca Cola) = 1/10 = 0.1.

a. The returns in the different m

arket conditions are: i.

Bear M

arket: -19.5%

ii. N

ormal M

arket: 18.2%

iii. B

ull Market: 46%

b.

E[rP]= w(A

mgen)*E[rA

] + w(C

oca Cola)*E[rC

C] = 0.9*0.2+0.1*0.1 = 0.18+0.01

= 0.19 (in decimals); alternatively, you can calculate 0.2*(-19.5) + 0.5*18.2 +

0.3*46 = 19 c.

Variance[rP] = (Probability of B

ear Market)*(R

eturn of Portfolio in Bear M

arket -E

[rP]) 2 + (Probability of Norm

al Market)*( R

eturn of Portfolio in Norm

al M

arket-E[rP]) 2 + (Probability of B

ull Market)*(R

eturn of Portfolio in Bull

Market -E

[rP]) 2 = = (0.2)*(-0.195-0.19) 2 + (0.5)*(0.182-0.19) 2+(0.3)*(0.46 – 0.19) 2 = (0.2)*(-0.3850) 2 + (0.5)*(-0.008) 2+(0.3)*(0.27) 2

= 0.0515 SD

= 0.0515^(1/2)=0.2269

- 7 -

FINA

N 3050

Introduction to Investments

Cover Page

2nd H

omew

ork Assignm

ent

Student name:

Student ID

:

M

ark exercises solved:

Problem 1

Problem

2

Problem 3

Problem

4

Problem

5

Problem 6

Problem

7

Problem 8

Problem

9

Problem 10

C

omm

ents:

- 1 -

2nd H

omew

ork Assignm

ent

Important inform








b. M


c. Show







1-3:

Abigail G

race has a $900,000 fully diversified portfolio. She subsequently inherits

comm

on stock of company “M

yStock” worth $100,000. H

er financial advisor provided

her with the follow

ing forecasted information:

E

xpected Monthly

Returns

Standard Deviation

of Monthly R

eturns O

riginal Portfolio 0.67%

2.37%

MyStock

1.25%2.95%

The correlation coefficient of “M

yStock” stock returns with the original portfolio returns

is 0.40.

Problem 1:

Assum

ing Grace keeps the “M

yStock” stock, calculate the:

a. Expected m

onthly return of the new portfolio w

hich includes the “Mystock” stock

b. C

ovariance of “MyStock” stock returns w

ith the original portfolio returns.

c. Standard deviation of her new

portfolio which includes the “M

yStock” stock.

Solution Problem 1:

- 2 -

a. (0.9 ×

0.67) + (0.1 × 1.25) = 0.728%

b. C

OV

= 0.4 × 2.37 ×

2.95 = 2.7966

c. SD

= [(0.9^2 × 2.37^2) + (0.1^2 ×

2.95^2) + (2 × 0.9 ×

0.1 × 2.7966)]^0.5 =

2.2673%

Problem 2:

Assum

ing Grace sells the “M

yStock” stock and invests the proceeds in risk-free

government securities yielding 0.42%

per month, calculate the:

a. Expected m

onthly return of the new portfolio w

hich includes the government

securities

b. C

ovariance of the government security returns w

ith the original portfolio returns.

[Hint: think about how

much covariation there can be betw

een a risky and a risk-

free asset?]

c. Standard deviation of her new

portfolio which includes the governm

ent securities.

Solution Problem 2:

a. (0.9 ×

0.67) + (0.1 × 0.42) = 0.645%

b. C

OV

= 0 × 2.37 ×

0 = 0

c. SD

= [(0.9^2 × 2.37^2) + (0.1^2 ×

0) + (2 × 0.9 ×

0.1 × 0)]^0.5 = 2.133%

Problem 3:

Based on conversations w

ith her husband, Grace is considering selling the $100,000 of

“MyStock” stock and acquiring $100,000 of “O

therStock” comm

on stock instead.

“OtherStock” stock has the sam

e expected return and standard deviation as “MyStock”

stock. Her husband com

ments, “It does not m

atter whether you keep all of the “M

yStock”

stock or replace it with $100,000 of “O

therStock” stock.” State whether her husband’s

comm

ent is correct or incorrect. Justify your response.

Solution Problem 3:

The comm

ent is not correct. It also depends on the covariances between each security and

the original portfolio.

- 3 -

Problem 4:

An investor can design a risky portfolio based on tw

o stocks, A and B

. Stock A has an

expected return of 18% and a standard deviation of return of 20%

. Stock B has an

expected return of 14% and a standard deviation of return of 5%

. The correlation

coefficient between the returns of A

and B is 0.5. The risk-free rate of return is 10%

.

The proportion of the optimal risky portfolio that should be invested in stock A

is ?

A. 40%

B. 100%

C. 0%

D. 60%

[Hint: think about the condition that defines the optim

al risky portfolio – slide 4 of week 4

– and check this condition for the four choices.]

Solution Problem 4:

A. 0.4*18 + 0.6*14 = 15.6; V

ariance = 97 SD

= 9.8; S = (15.6-10)/9.8 = 0.57

B. 1.0*18 + 0.0*14 = 18.0; SD

= 20; S = (18-10)/20 = 0.4

C. 0.0*18 + 1.0*14 = 14.0; SD

= 5; S = (14-10)/5 = 0.8

D. 0.6*18 + 0.4*14 = 16.4; V

ariance = 172 SD

= 13.1; S = (16.4-10)/13.1 = 0.5

Problem 5:

Karen K

ay, a portfolio manager at C

ollins Asset M

anagement, is using the C

apital Asset

Pricing Model for m

aking recomm

endations to her clients. Her research departm

ent has

developed the information show

n in the following table:

R

eturn Forecast by R

esearch Dep.

Standard Deviation

Beta

Stock X

14.0%36%

0.8

Stock Y

17.0%25%

1.5

Market Index

14.0%15%

1.0

Risk-Free A

sset 5.0%

- 4 -

Calculate the expected return and alpha for each stock.

Solution Problem 5:

For stock X:

E(r) = 5%

+ 0.8×

(14%-5%

) = 12.2%

Alpha(x) =

14% - 12.2%

= 1.8%

For stock Y:

E(r) = 5%

+ 1.5×

(14%-5%

) = 18.5%

Alpha(y) =

17% - 18.5%

= -1.5%

Problem 6:

Imagine that this tim

e you have two stocks nam

ed AB

C and X

YZ w

hose returns depend

on the state of the world econom

y. The market portfolio constructed of the shares of

10000 firms (including the ones of A

BC

and XY

Z) has a return that also depends on the

state of the world econom

y.

R

ecessionR

egular B

oom

Probability 0.25

0.5 0.25

Return of A

BC

10%

20%

-10%

R

eturn of XY

Z

-20%

25%

10%

Return of M

arket Portfolio 0%

5%

10%

R

eturn of Riskless A

sset 5%

5%

5%

a. W

hat is asset AB

C’s m

arket Beta?

b. W

hat is asset XY

Z’s market B

eta?

c. Im

agine that you invest half of your funds in AB

C, and the other half in X

YZ,

what is the m

arket Beta of this portfolio?

Solution Problem 6:

(a) For asset AB

C:

Expected R

eturn of AB

C: 0.25×

10 + 0.5×

20 + 0.25×

(-10) = 10%

- 5 -

Expected R

eturn of Market: 0.25×

0 + 0.5×

5 + 0.25×

10 = 5%

(b) For asset X

YZ:

Expected R

eturn of XY

Z: 0.25×(-20) +

0.5×25 +

0.25×10 =

10%

Expected R

eturn of Market: 0.25×

0 + 0.5×

5 + 0.25×

10 = 5%

(c) For a portfolio: Beta(p) =

(-2)×0.5 +

3×0.5 =

0.5

Problem 7:

Biopharm

a is a pharmaceutical com

pany. Biopharm

a’s annual stock returns have a CA

PM beta

of 1.25 (i.e. β=1.25). The market portfolio’s return is 13%

, and the risk-free rate is 5%.

a. W

hat is the required expected return for Biopharm

a according to the CA

PM?

b. The firm

has the opportunity to develop a new drug. This project requires an initial

outlay of $400,000 and it will bring expected revenues of $100,000 in each of the next

6 years. The riskiness of this project is the same as the overall riskiness of B

iopharma.

Should the managem

ent team of B

iopharma do the project? O

r should it rather not do

it? Explain your answer.

Solution Problem 7:

(a) Required expected return:

- 6 -

(b) Investment decisions:

The PV

equals 378,448.27 and the Net Present V

alue (NPV

) is, as a consequence,

negative (-400,000 + 378,448.27).

Alternatively you can calculate the project’s IR

R (C

Fj -400,000; then six times the C

Fj

100,000) IR

R = 12.98%

IR

R < 15%

do not undertake the project.

Problem 8:

If the simple C

APM

is valid, is the following situation possible? Explain your answ

er.

PortfolioE

xpected Return

Beta

A

20%1.4

B

25%1.2

Solution Problem 8:

Not possible. Portfolio A

has a higher beta than Portfolio B, but the expected return for

portfolio A is low

er.

Problem 9:

If the simple C

APM

is valid, is the following situation possible? Y

ou also know that the

Market R

isk Premium

(E(rM )-rf ) equals 20%. Explain your answ

er. [Hint: plug everything

you know into the C

APM equation – slide 14 of w

eek 4 – and verify that the implied risk-

free rate makes sense.]

PortfolioE

xpected Return

Beta

A

25%1.5

B

5%0.5

Solution Problem 8:

Not possible. Plug the values into the C

APM

equation and solve for the risk-free rate:

25 = rf + 1.5*20 and 5 = rf + 0.5*20 the risk-free rate w

ould have to be minus 5%

to

make these relationships hold – this does not m

ake sense!

- 7 -

Problem 10:

If the simple C

APM

is valid, is the following situation possible? Explain your answ

er.

[Hint: Think about the “optim

ality” of the Market portfolio (refer to slide 4 and slide 7 of

week 4). Verify that the M

arket portfolio is optimal in that sense.]

Portfolio E

xpected Return

Standard Deviation

Risk-Free

10%0%

M

arket 18%

24%

A

16%12%

Solution Problem

10:

Not possible. The rew

ard-to-variability ratio for Portfolio A is better than that of the

market, w

hich is not possible according to the CA

PM, since the C

APM

predicts that the

market portfolio is the m

ost efficient portfolio.

S(A) = (16-10)/12 = 0.5

S(M) = (18-10)/12 = 0.33

- 8 -

FINA

N 3050


Cover Page

3rd H

omew

ork Assignm

ent

Student name:

Student ID

:

M


Problem 1

Problem

2

Problem 3

Problem

4

Problem

5

Problem 6

Problem

7

Problem 8

Problem

9

Problem 10

Problem

11

Problem 12

C

omm

ents:

- 1 -

3rd H

omew

ork Assignm

ent Problem

1:

Find the price of a coupon bond (par value of $1000) which pays 12%

coupon semi-

annually, matures in 5 years, and has a yield to m

aturity of 10%.

Solution: FV=1000, PM

T=60, i=5% (per 6-m

onth period), n=10→ PV

=-1077.22

Problem 2:

A coupon bond w

hich pays coupon of $100 annually has a par value of $1000, matures in

5 years, and is selling today at a $72 discount from par value (i.e., at $928). Find the yield

to maturity of this bond.

Solution: FV=1000, PM

T=100, n=5, PV= -928 →

i=12%

Problem 3:

Rank the follow

ing bonds in order of descending duration (note that you should not

actually calculate the duration).

Bond

Coupon

Tim

e to Maturity

Yield to M

aturity

A

15%

20 years 10%

B

15%

15 years 10%

C

0%

20 years 10%

D

8%

20 years 10%

E 15%

15 years

15%

Solution:

C:

Highest m

aturity, zero coupon

D:

Highest m

aturity, next-lowest coupon

A:

Highest m

aturity, same coupon as rem

aining bonds

B:

Lower yield to m

aturity than bond E

E: H

ighest coupon, shortest maturity, highest yield of all bonds.

- 2 -

Problem 4:

Consider a zero-coupon bond (face value of $1000) that m

atures in 3 years. The YTM

is

10%. Find the price, the duration and the convexity of this bond.

Solution:

1. Price: 1000/1.1

3 = 751.3148

2. D

uration = 3

3. C

onvexity = 1/1.12×3×(3+1)×1 = 9.92

Problem 5:

Consider a 15%

-coupon bond that matures in 3 years (face value of $1000). The Y

TM is

10%. Find the price, the duration and the convexity of this bond.

Solution:

C

onvexity: 1/1.12×(1×2×0.1213 + 2×3×0.1103 + 3×4×0.7685) = 8.369

Problem 6:

You purchased a 5-year annual interest coupon bond one year ago. Its coupon interest

rate was 6%

and its par value was $1,000. A

t the time you purchased the bond, the yield

to maturity w

as 4% (and the tim

e to maturity w

as 5 years). If you sold the bond after

receiving the first interest payment and the bond's yield to m

aturity had changed to 3%

(and the time to m

aturity is now 4 years), your annual total rate of return on holding the

bond for that year would have been __________.

- 3 -

Solution: 7.57%

1. B

uy price (one year ago; 5 years to maturity; Y

TM 4%

): 1089.04

2. Sell price (4 years to m

aturity; YTM

3%): 1111.51

3. C

apital Gain: 22.47

4. Interest Incom

e: 60

5. H

olding Period Return = (60+22.47)/1089.04 = 0.0757

Problem 7:

Consider the follow

ing $1,000 par value zero-coupon bonds:

C

alculate the expected one-year interest rates one year (from year 1 to 2), tw

o years

(from year 2 to 3), three years (from

year 3 to 4) and four years (from year 4 to 5) from

now.

Solution:

One Y

ear: (1+0.06) 1×(1+f1) = (1+0.07) 2 r = 0.0801

Two Y

ears: (1+0.07) 2×(1+f2) = (1+0.0799) 3 r = 0.10

Three Years: (1+0.0799) 3×(1+f3) = (1+0.0941) 4

r = 0.138

Four Years: (1+0.0941) 4×(1+r) = (1+0.1070) 5

r = 0.1601

Problem 8:

You w

ill be paying $10,000 a year in tuition expenses at the end of the next two years.

Bonds currently yield 8%

.

1. W

hat is the present value and duration of your obligation?

2. If you w

ant to invest only in one zero-coupon bond to imm

unize your obligation,

what does the m

aturity of this zero-coupon bond have to be?

- 4 -

3. V

erify that your answer to (2.) really im

munizes your obligation. For that

purpose, suppose the rates imm

ediately increase to 9%. W

hat happens to your net

position, that is, to the difference between the value of the bond and that of your

tuition obligation? What happens if rates fall to 7%

?

Solution:

1. The present value of the obligation is $17,832.65 and the duration is 1.4808 years.

2. To im

munize the obligation, invest in a zero-coupon bond m

aturing in 1.4808

years. Since the present value of the zero-coupon bond must be $17.832.65, the

face value must be: 17328.65 × 1.08

1.4804 = 19985.26

3. If the interest rate increases to 9%

, the zero-coupon bond would fall in value to

$17590.92. The PV of the tuition obligation w

ould fall to $17591.11, so that the

net position changes by $0.19.

4. If the interest rate decreases to 7%

, the zero-coupon bond would rise in value to

$18079.99. The PV of the tuition obligation w

ould rise to $18080.18, so that the

net position changes by $0.19.

Problem 9:

The risk-free rate of return is 10%, the required rate of return on the m

arket is 15%, and

High-Flyer stock has a beta coefficient of 1.5.

a. If the dividend per share expected during the com

ing year, D1 , is $2.50 and g=5%

,

at what price should a share sell?

b. If the share sells at $25, w

hat must be the m

arket’s expectation of the growth rate

of dividends (keeping all other information the sam

e)?

Solution:

(a) K=10%

+ 1.5×(15%-10%

)=17.5%

Therefore, P0 = $2.50/(0.175-0.05) = $20

(b) $25 = $2.50/(0.175-g) g = 7.5%

- 5 -

Problem 10:

Investment firm

SmartInvest w

ants to value firm SkiU

tah. The following inform

ation is

available for SkiUtah: risk-free rate = 5.0%

, expected market return = 12%

, beta = 1.09.

SmartInvest expects that SkiU

tah’s dividends are going to grow by 15%

in the first 3

years and by 8% thereafter. The current dividend (that has just been paid out) is $2.10.

Find the intrinsic value of SkiUtah using the m

ultistage DD

M and the C

APM

.

Solution:

Market capitalization rate: 0.05 + 1.09*(0.12-0.05) = 0.1263

Next 3-years dividends are: 2.10*1.15 = 2.415; 2.415*1.15 = 2.777; 2.777*1.15 = 3.194

Thereafter there is a constant growth rate of 8%

: 3.194*1.08/(0.1263-0.08) = 74.5

The intrinsic

value is:

2.415/1.1263^1 +

2.777/1.1263^2 +

3.194/1.1263^3 +

74.5/1.1263^3 = 58.71

Problem 11:

MF C

orp. has an RO

E of 16% and a retention rate of 50%

. The market capitalization rate

k equals 12%. If the com

ing year’s earnings are expected to be $2 per share, at what price

will the stock sell? W

hat price do you expect MF shares to sell for in three years?

Solution:

g=RO

E×b=0.16×0.5=0.08

D1 =$2×(1-0.50)=$1.00

P0 =$1.00/(0.12-0.08)=$25

P3 =P

0 ×(1+g) 3=$31.49

Problem 12:

Your prelim

inary analysis of two stocks has yielded the inform

ation set forth below. The

market capitalization rate k for both stock A

and stock B is 10%

per year.

Stock A

Stock B

- 6 -

Expected return on equity, RO

E14%

12%

Estimated earnings per share

$2.00 $1.65

Estimated dividends per share

$1.00 $1.00

Current m

arket price per share $27.00

$25.00

Determ

ine the intrinsic value of each stock and decide in which, if either, of the tw

o

stocks would you choose to invest.

Solution:

Stock A: b=50%

; g = 0.14×0.5 = 7.0%; V

0 = 1/(0.10-0.07) = $33.33

Stock B: b=1-$1/$1.65=0.394; g = 0.12×0.394 = 4.728%

; V0 = 1/(0.10-0.04728)=$18.97

You w

ould choose to invest in Stock A since its intrinsic value exceeds its price. Y

ou

might choose to sell short stock B

.

- 7 -

FINA

N 3050


Cover Page

4th H

omew

ork Assignm

ent

Student name:

Student ID

:

M


Problem 1

Problem

2

Problem 3

Problem

4

Problem

5

Problem 6

Problem

7

Problem 8

Problem

9

Problem 10

C

omm

ents:

- 1 -

4th H

omew

ork Assignm

ent

Important inform








b. M


c. Show







1:

Suppose you think Wal-M

art stock is going to appreciate substantially in value in the

next year. Say the stock’s current price, S0 , is $100, and the call option expiring in one

year has an exercise price, X, of $100 and is selling at a price, C

, of $10. With $10,000 to

invest, you are considering three alternatives:

a. Invest all $10,000 in the stock, buying 100 shares.

b. Invest all $10,000 in options, buying 1000 options.

c. B

uy 100 options for $1,000 and invest the remaining $9,000 in a m

oney market

fund paying 4% interest annually.

What is your rate of return for each alternative for four stock prices one year from

now?

Price of stock 1 year from

now

$80

$100 $110

$120

a. All Stocks

b. All O

ptions

c. Money m

arket fund + options

Solution:


now

$80

$100 $110

$120

- 2 -

a. All Stocks

800010000

1100012000

b. All O

ptions 0

0 10000

20000

c. Money m

arket fund + options9360

9360 10360

11360


now

$80

$100 $110

$120

a. All Stocks

-20%

0%

10%

20%

b. All O

ptions -100%

-100%0%

100%

c. Money m

arket fund + options-6.4%

-6.4%

3.6%

13.6%

Problem 2:

The comm

on stock of the P.U.T.T. C

orporation has been trading in a narrow price range

for the past month, and you are convinced it is going to break far out of that range in the

next three months. Y

ou do not know w

hether it will go up or dow

n, however. The current

price of the stock is $100 per share, the price of a three-month call option w

ith an

exercise price of $100 is $10, and a put with the sam

e expiration date and exercise price

costs $7.

Consider buying a straddle. D

raw a diagram

showing the payoff and profit of this

strategy at maturity. H

ow far w

ould the price have to move in either direction for you to

make a profit on your initial investm

ent?

Solution:

The straddle costs $17 stock price has to m

ove by at least $17 in either direction.

Problem 3:

A vertical com

bination is the purchase of a call with exercise price X

2 and a put with

exercise price X1 , w

ith X2 greater than X

1 . Graph the payoff to this strategy (do not graph

the profit, i.e., ignore the costs to establish the positions).

- 3 -

Solution:

See solution graph in the manual on page 14-4

Problem 4:

You w

rite a call option with X

= $50 and buy a call with X

= $60. The options are on the

same stock and have the sam

e maturity date. O

ne of the calls sells for $3; the other sells

for $9. Draw

a graph including the payoff and profit of this strategy at the option maturity

date. What is the break-even point for this strategy?

Solution:

Solution graph is in the manual on page 14-14.

Breakeven occurs w

hen the payoff offsets the initial proceeds of $6, which occurs at a

stock price of $56.

Problem 5:

Joseph Jones, a manager at C

omputer Science, Inc. (C

SI), received 10000 shares of

company stock as part of his com

pensation package. The stock currently sells at $40 a

share. Joseph would like to defer selling the stock until the next tax year. In January,

however, he w

ill need to sell all his holdings to provide for a down paym

ent on his new

house. Joseph is worried about the price risk involved in keeping his shares. A

t current

prices, he would receive $400,000 for the stock. If the value of his stock holdings falls

below $350,000, his ability to com

e up with the necessary dow

n payment w

ould be

jeopardized. On the other hand, if the stock value rises to $450,000 he w

ould be able to

maintain a sm

all cash reserve even after making the dow

n payment. Joseph considers

three investment strategies:

1. Strategy A

is to write January call options on the C

SI shares with a strike price of

$45. These calls are currently selling for $3 each.

2. Strategy B

is to buy January put options on CSI w

ith strike price $35. These

options also sell for $3 each.

3. Strategy C

is to write the January call options and buy the January put options

(this strategy is called a zero-cost collar).

- 4 -

Evaluate each of these strategies with respect to Joseph’s investm

ent goals. What are the

advantages and disadvantages of each? Which w

ould you recomm

end?

Solution:

By w

riting covered call options, Jones receives premium

income of $30,000. If, in

January, the price of the stock is less than or equal to $45, he will keep the stock plus the

premium

income. Since the stock w

ill be called away from

him if its price exceeds $45

per share, the most he can have is:

$450,000 + $30,000 = $480,000

(We are ignoring interest earned on the prem

ium incom

e from w

riting the option over

this short time period.) The payoff structure is:


Less than $45 (10,000 tim

es stock price) + $30,000

Greater than $45

$450,000 + $30,000 = $480,000

This strategy offers some prem

ium incom

e but leaves the investor with substantial

downside risk. A

t the extreme, if the stock price falls to zero, Jones w

ould be left with

only $30,000. This strategy also puts a cap on the final value at $480,000, but this is more

than sufficient to purchase the house.

By buying put options w

ith a $35 strike price, Jones will be paying $30,000 in prem

iums

in order to insure a minim

um level for the final value of his position. That m

inimum

value is: ($35 × 10,000) – $30,000 = $320,000

This strategy allows for upside gain, but exposes Jones to the possibility of a m

oderate

loss equal to the cost of the puts. The payoff structure is:


Less than $35 $350,000 – $30,000 = $320,000

Greater than $35

(10,000 times stock price) – $30,000

The net cost of the collar is zero. The value of the portfolio will be as follow

s:


Less than $35 $350,000

- 5 -

Betw

een $35 and $4510,000 tim

es stock price

Greater than $45

$450,000

If the stock price is less than or equal to $35, then the collar preserves the $350,000 in

principal. If the price exceeds $45, then Jones gains up to a cap of $450,000. In between

$35 and $45, his proceeds equal 10,000 times the stock price.

The best strategy in this case is (c) since it satisfies the tw

o requirements of

preserving the $350,000 in principal while offering a chance of getting $450,000.

Strategy (a) should be ruled out because it leaves Jones exposed to the risk of substantial

loss of principal.

O

ur ranking is: (1) c

(2) b (3) a

Problem 6:

Draw

the payoff diagram for the follow

ing investment strategy using A

BC

share and call

options on AB

C shares:

1. B

uy a share of AB

C

2. W

rite a call on AB

C w

ith X=50

3. W

rite a call on AB

C w

ith X=60

4. B

uy a call on AB

C w

ith X=110

Solution:

Devise a portfolio using only call options and shares of stock w

ith the following

Buy a share of stock, w

rite a call with X

= 50, write a call w

ith X = 60, and buy a call w

ith X

= 110.

Position ST < 50

50 < ST < 60 60 < ST < 110

ST > 110 B

uy stock ST

ST ST

ST Short call (X

= 50) 0

– (ST – 50) – (ST – 50)

– (ST – 50) Short call (X

= 60) 0

0 – (ST – 60)

– (ST – 60) Long call (X

= 110) 0

0 0

ST – 110 Total

ST 50

110 – ST 0

The investor is making a volatility bet. Profits w

ill be highest when volatility is low

so that the stock price ends up in the interval betw

een $50 and Look at the picture in the book at page 513.

- 6 -

Problem 7:

A put option w

ith strike price $60 currently sells for $2. To your amazem

ent, a put on the

same firm

with sam

e expiration date but with a strike price of $62 also sells for $2. H

ow

can you exploit this obvious mispricing using a sim

ple zero-net-investment strategy (i.e.,

a strategy that has zero net investment costs) that gives you a non-negative payoff at

expiration of the options with certainty? A

lso draw the payoff diagram

at expiration in

order to verify that your strategy yields a non-negative payoff for every stock price ST .

Solution:

Buy the X

= 62 put (which should cost m

ore than it does) and write the X

= 60 put. Since the options have the sam

e price, the net outlay is zero. Your proceeds at

maturity m

ay be positive, but cannot be negative.

Position ST < 60

60 < ST < 62ST > 62

Long put (X = 62)

62 – ST 62 – ST

0 Short put (X

= 60) – (60 – ST)

0 0

Total 2

62 – ST 0

0S

T

2

6062

Payoff = P

rofit (because net investment = 0)

Problem

8:

Assum

e the current stock price, S0 , equals $100. In one year the stock price can either go

up to $200 or down to $50. The risk-free interest rate equals 8%

. Determ

ine the price of a

call with strike price, X

, being equal to $75.

- 7 -

Solution:

H=0.833

buy 5 stocks and write 6 options

Risk-free payoff equals: 5×200-6×125 = 250 and 5×50 = 250

5×100-6×C=250/1.08

C=44.75

Problem 9:

Assum

e the current stock price, S0 , equals $100. In one year the stock price can either go

up to $300 or down to $25. The risk-free interest rate equals 8%

. Determ

ine the price of a

call with strike price, X

, being equal to $75.

Solution:

H=0.8181=

buy 9 stocks and write 11 options

Risk-free payoff equals: 9×300-11×225 = 225 and 9×25 = 225

9×100-11×C=225/1.08

C=62.88

Problem 10:

Consider the call options given in Problem

8 and Problem 9. D

etermine the appropriate

prices of put options with a strike price of X

=$75 using the put-call parity relationship.

Solution:

P = C-S

0 +PV(X

)

Given the data in Problem

9: P = 44.75 – 100 + 75/1.08 = 14.19

Given the data in Problem

10: P = 62.88 – 100 + 75/1.08 = 32.32

- 8 -


Week 3: Asset allocation: risky vs. risk-free assets (Chp. 5.5,5.6)

Asset allocation: optimal risky portfolios (Chp. 6.1,6.2)

Slide 2Week 3


Allocating Capital between Risky and Risk-Free AssetsOverview

� Goal: split investment between safe and risky assets

� Safe asset (risk-free asset): T-bills

� Risky asset: portfolio of stocks

� Issues:

─ Examine the risk-return tradeoff─ Influence of investor’s risk aversion on the allocation

Slide 3Week 3


Allocating Capital between Risky and Risk-Free AssetsEquation: Expected Returns and Standard Deviation

� Notation – Input:

─ rf…risk-free rate─ E(rp)…expected return of the risky asset

─ y...share invested in risky asset (in %); (1-y)…share invested in risk-free asset (in %)

─σrf/p…standard deviation of the risk-free asset (rf) or the risky asset (p)� Notation – Output:

─ E(rc)…expected return of the combined portfolio

─σc…standard deviation of the combined portfolio (c)

( ) ( ) ( ) fpc ryrEyrE ×−+×= 1pc y σσ ×=

Slide 4Week 3


Allocating Capital between Risky and Risk-Free AssetsExample: Expected Returns and Standard Deviation

� Consider the following data:─Risk-free asset:

• Rate of return: rf = 7%

• Standard deviation of return (SD): σrf= 0%─Risky asset:

• Expected rate of return: E(rp) = 15%

• Standard deviation of return (SD): σp = 22%─ y=50%

( ) ( ) 11750.011550.0 =×−+×=crE 112250.0 =×=cσ

Slide 5Week 3


Allocating Capital between Risky and Risk-Free AssetsInvestment Opportunity Set

Slide 6Week 3


Allocating Capital between Risky and Risk-Free AssetsEquation: Slope of the Capital Allocation Line

� The slope S can be calculated in the following way:

� The slope has a nice interpretation: ─ It represents the ratio of the risk premium to the standard deviation

─ It measures the additional return for taking on one more unit of risk─ It is called the “reward-to-variability ratio” or “Sharpe Ratio”

( )

c

fc rrES

σ

−=

Slide 7Week 3


Allocating Capital between Risky and Risk-Free AssetsExercise

� For each of the following combinations determine ─ the expected rate of return,

─ the risk premium, ─ the standard deviation,

─ the reward-to-variability ratio and ─ the point in the graphic on slide 5.

� Case 1: y=1.0

� Case 2: y=0.0

� Case 3: y=0.75

Slide 8Week 3


Allocating Capital between Risky and Risk-Free AssetsExercise

� Assume that you manage a risky portfolio with an expected rate of return of

14% and a standard deviation of 22%. The risk-free rate is 5%. The risky portfolio includes the following investments in the given proportions: 60% in a

U.S. Stock Portfolio and 40% in an Emerging Market Stock Portfolio.

� Suppose a client would like to invest in your risky portfolio a proportion y of

his total investment budget so that his overall portfolio will have an expected rate of return of 10%.

─ What is the proportion y?─ What are your client’s investment proportions in the U.S. Stock

Portfolio, the Emerging Market Stock Portfolio and the T-Bill?

Slide 9Week 3


Allocating Capital between Risky and Risk-Free AssetsInvesting more than 100% into the risky asset (1)

� If we invest 100% into the risky asset, we get to point P on the CAL.

� How can we invest even more in the risky asset?� Answer: by borrowing money at the risk-free rate (leverage) and invest it into

the risky asset

Slide 10Week 3



� Example: assume we want to invest 125% into the risky asset

─ Portfolio weights: y = 125% and (1-y) = -25% (i.e., borrowing)

─ Expected return of the combined portfolio:

─ Standard deviation of the combined portfolio:

─Reward-to-variability ratio:

( ) ( ) 17725.111525.1 =×−+×=crE

5.272225.1 =×=cσ

36.05.27

717=

−=S

Slide 11Week 3



� Usually investors cannot borrow at the risk-free rate.

� Consider again the example where you borrow 25%. But now you can only borrow at a rate of 9%: rB = 9%

─ Portfolio weights: y = 125% and (1-y) = -25%

─ Expected return of the

combined portfolio:

─ Standard deviation of the combined portfolio:

─Reward-to-variability ratio:

( ) ( ) 5.16925.111525.1 =×−+×=crE

5.272225.1 =×=cσ

27.05.27

95.16=

−=S

Slide 12Week 3



Slide 13Week 3


Allocating Capital between Risky and Risk-Free AssetsOptimal allocation between risky asset and risk-free asset

� How to determine the optimal allocation between risky asset and risk-free asset?

─ The investor’s risk aversion determines the “optimal” point on the CAL.

─Greater levels of risk aversion lead to larger proportions of the risk-free

rate.

─ Lower levels of risk aversion lead to larger proportions of the risky asset.

Slide 14Week 3


Allocating Capital between Risky and Risk-Free AssetsCapital Market Line

� Capital Market Line: is the Capital Allocation Line using the market index

portfolio as the risky asset.

� Historical evidence:

Slide 15Week 3


Optimal Risky Portfolios (Chapter 6)Equation: Expected Portfolio Return

� Consider a portfolio of a stock (S) and a risky corporate bond (B)

─ wS…proportion of funds invested in stock─ wB…proportion of funds invested in bond

─ Because we only have two risky assets: wS+wB= 1

─ E(rS)…expected return of stock S─ E(rB)…expected return of bond B

─ E(rP)…expected return of portfolio P

� Expected return of a portfolio P of stock S and bond B:

( ) ( ) ( )BBsSp rEwrEwrE ×+×=

Slide 16Week 3


Optimal Risky PortfoliosExample: Portfolio of Two Risky Assets (1)

� Consider the following scenario analysis:

� What are the expected rates of return?

─ Stock fund:

─ Bond fund:

-4%27%30%Boom

6%13%40%Normal

16%-11%30%Recession

Rate of Return for Bond FundRate of Return for Stock FundProbabilityScenario

( ) ( ) 10273.134.113. =×+×+−×=SrE

( ) ( ) 643.64.163. =−×+×+×=BrE

Slide 17Week 3



� What are the variances and standard deviations?

─ Stock fund:

─ Bond fund:

( ) ( ) ( )

75.760

60643.664.6163.2222

==

=−−×+−×+−×=

B

B

σ

σ

( ) ( ) ( )

92.146.222

6.22210273.10134.10113.2222

==

=−×+−×+−−×=

S

S

σ

σ

Slide 18Week 3



� What about the risk and return characteristics of a portfolio with 60% invested

in the stock fund and 40% invested in the bond fund?� First, calculate the portfolio return in each scenario:

─Recession:

─Normal:

─ Boom:

� Second, calculate the expected portfolio return:

( ) 2.0164.0116.0 −=×+−×

2.1064.0136.0 =×+×

( ) 6.1444.0276.0 =−×+×

( ) ( ) 4.86.143.2.104.2.03. =×+×+−×=PrE

Slide 19Week 3



� Third, calculate the portfolio variance and portfolio standard deviation:

( ) ( ) ( )

92.5

016.354.86.143.04.82.104.04.82.03.02222

=

=−×+−×+−−×=

p

p

σ

σ

Slide 20Week 3



� Notice that…─ the expected portfolio return equals the average of the expected returns of the two assets

─ the portfolio standard deviation DOES NOT equal the average of the standard deviations of the two assets

─ the portfolio standard deviation is actually less than that of either asset (5.92 < 14.92 & 5.92 < 7.75), i.e., the portfolio of the two assets is less risky than either asset individually (the POWER OF DIVERSIFICATION)

( ) 4.864.106. =×+×=PrE

92.5052.1275.74.92.146. ≠=×+×

Slide 21Week 3



� Why is this the case?─ Inverse relationship between the performance of the two funds.

─ In a recession: stocks perform poorly but bonds perform well

─ In a boom: stock perform very well but bonds fall

� How can we measure the tendency of the returns of two assets to vary either in tandem or in opposition to each other:

─Correlation (notation: ρS,B)─Covariance (notation Cov(rS,rB) or σS,B)

Slide 22Week 3


Optimal Risky PortfoliosEquation: Covariance Between Two Risky Assets

� Notation:

─ S…number of different states of the world to take into consideration─ pi…the probability of state i to take place

─ rS,i…return of stock S in state i─ rB,i…return of bond B in state I

─ E(rS)…average return of stock S (across all states of the world)─ E(rB)…average return of bond B (across all states of the world)

� Covariance:

( ) ( )( ) ( )( )BiB

S

i

SiSiBS rErrErprrCov −×−×=∑=

,

1

,,

Slide 23Week 3


Optimal Risky PortfoliosEquation: Correlation Between Two Risky Assets

� Covariance is, however, difficult to interpret � correlation coefficient

� Notation: σS…standard deviation of stock S, σB…standard deviation of bond B

� Correlation coefficient:

� Correlation ranges from values of -1 to +1─ -1 indicates perfect negative correlation: i.e., if the return of stock Sgoes up by +10%, the return of bond B goes down by -10%

─+1 indicates perfect positive correlation─ 0 indicates that the returns on the two assets are unrelated to each other

( )

BS

BSSB

rrCov

σσρ

×=

,

Slide 24Week 3


Optimal Risky PortfoliosEquation: Portfolio Variance

� Notation:

─ ws…weight invested in stock S─ wB…weight invested in bond B (has to be 1-ws)

─σS…standard deviation of stock S, ─σB…standard deviation of bond B

─ ρS,B…correlation between stock S and bond B─ Cov(rS,rB)…covariance between stock S and bond B

� Portfolio variance for two risky assets:

( ) ( )

( )BSBSBBSS

BSBBSSBBSSp

rrCovwwww

wwww

,2

2

2222

,

22222

×××+×+×=

=×××××+×+×=

σσ

ρσσσσσ

Slide 25Week 3



� Continued from slide 20; data on slide 16.� Calculate the covariance:

� Calculate the correlation:

� Calculate the portfolio standard deviation:

( ) ( ) ( ) ( ) ( ) ( ) ( )114

6410273.6610134.61610113.,

−=

=−−×−×+−×−×+−×−−×=BS rrCov

99.75.792.14

114−=

×

−=SBρ

( )

92.5

016.351144.6.2604.6.2226.222

=

=−×××+×+×=

P

P

σ

σ

Slide 26Week 3


Optimal Risky PortfoliosDiversification – Some Simple Rules

� If the correlation between two assets is equal to +1 (perfect positive correlation) there are no benefits from diversification.

� There are benefits to diversification whenever asset returns are less than perfectly correlated.

� If the correlation between two assets is equal to -1 (perfect negative

correlation) we can reduce the portfolio standard deviation all the way down to zero (maximum benefits from diversification).

Slide 27Week 3


Optimal Risky PortfoliosExercise: The Power of Diversification

� Consider the following input data: E(rB)=6%, E(rS)=10%, σB=12%, σS=25%,

ρBS=0

� Case 1: What is your expected return and standard deviation if you invest 100% into bonds.

� Case 2: What is your expected return and standard deviation if you invest 75% in bonds and only 25% in stocks?

� Case 3: What is your expected return and standard deviation if you invest 87.6% in bonds and only 12.4% in stocks?

� Compare the different portfolios and explain “the power of diversification”!� Compare Case 2 and Case 3: which portfolio would you prefer?

Slide 28Week 3


Optimal Risky PortfoliosThe Opportunity Set

Slide 29Week 3


Optimal Risky PortfoliosThe Mean-Variance Criterion

� Mean-Variance Criterion: how to compare two different portfolios?

─ If two portfolios have the same expected return, we prefer the one with the lower standard deviation.

─ If two portfolios have the same standard deviation, we prefer the one

with the higher expected return.

� Further: a portfolio A is said to dominate portfolio B ifand (portfolio A would lie to the northwest of portfolio B)

� Consider “Portfolio Z” and “Stocks” in the previous graph: which one

dominates?

( ) ( )BA rErE ≥

BA σσ ≤

Slide 30Week 3


Optimal Risky PortfoliosDiversification – Further Exercises

� Consider the following input data: E(rB)=6%, E(rS)=10%, σB=12%, σS=25%� Suppose that your are required to invest 50% of your portfolio in bonds and

50% in stocks.─ If the portfolio standard deviation of your portfolio is 15%, what must be the correlation coefficient between stock and bond returns?

─What is the expected rate of return on your portfolio?─Now suppose that in an alternative scenario the correlation between stock and bond returns is 0.22 but that you are free to choose whatever portfolio proportions you desire. Are you likely to be better or worse off than you were in part (a)?


Week 4:

Asset allocation: optimal risky portfolios (Chp. 6.3)

Capital Asset Pricing Model (Chp. 7.1-7.2)

Slide 2Week 4


Efficient DiversificationIncluding a Risk-Free Asset – Putting Things Together

� Problem: what is the optimal risky portfolio?

� Consider a risk-free asset (assume, for example, a risk-free rate of 5%) and think about how to draw a Capital Allocation Line.

� Remember that you prefer steep CALs (i.e., the steeper the better).

Slide 3Week 4


Efficient DiversificationExample: Including a Risk-Free Asset – Putting Things Together

� Consider the following information: E(rBOND)=6%, E(rSTOCK)=10%,

σBOND=12%, σSTOCK=25%, rf=5%, ρSTOCK,BOND = 0.2� Evaluate the following “potential” CALs (through portfolio A or B)

─ Portfolio A: wSTOCK=12.94% and wBOND=87.06%

─ Portfolio B: wSTOCK=20% and wBOND=80%

� Sharpe Ratio of CALA: ─ E(rA) = 0.1294×10 + 0.8706×6 = 6.52%

─ 0.12942×252 + 0.87062×122 + 2×0.1294×25×0.8706×12×0.2 = 133.13─ σA = 11.54─ SA = (6.52-5)/11.54 = 0.13

� Exercise:─ What is the Sharpe Ratio of CALB?─ Which CAL (i.e., which risky portfolio) do you prefer?

Slide 4Week 4


Efficient DiversificationFinding the Optimal Risky Portfolio

� Optimal Risky Portfolio � the portfolio that yields the steepest CAL!� Solution: draw a line starting at the risk-free rate that is a tangent to the

investment opportunity set

S = (8.68-5)/17.97 = 0.20

Slide 5Week 4


Efficient DiversificationComplete Portfolio

� Remember the two steps of the investment process:─ Finding the optimal risky portfolio

─Combining the optimal risky portfolio with the risk-free asset (i.e., find the optimal point on the CAL that goes through the optimal risky

portfolio)

� Complete portfolio: depends on the investor’s risk aversion

� Note that all investors hold the same optimal risky portfolio but investors differ in their allocation of funds into this risky portfolio.

Slide 6Week 4


Capital Asset Pricing Model (CAPM)Motivation

� Idea: if all investors follow the concepts we have just developed, what are the implications for financial markets and asset prices?

� To answer this question, we need an equilibrium theory like the CAPM!� The CAPM is a theory but it has important real-world implications for

investors and firms.� Key assumptions underlying the CAPM:

─ Single-period investment horizon; investments are limited to traded

financial assets; no taxes, and transaction costs─ Investors are rational mean-variance optimizers─Homogeneous expectations (everybody has the same information on

expectations and risk)

Slide 7Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Optimal Risky Portfolio

� All investors will hold the same portfolio of risky assets.─Why? Because all investors are identical: same mean-variance analysis,

same universe of securities, identical time horizon, etc. � all investors get the same efficient frontier and the same tangency portfolio

� This tangency portfolio (i.e., optimal risky portfolio) equals the market portfolio and contains all securities.

─Why? Because if the optimal risky portfolio does not include some stock

ABC � no one would buy stock ABC (i.e., no demand) � price of

stock ABC goes down � stock ABC becomes more attractive/cheaper � at some point becomes part of the optimal portfolio

Slide 8Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (1)

� What is the portfolio risk that investors are exposed to when holding the optimal risky portfolio?

� Excursus: decomposing portfolio risk─Component 1: Firm-specific risk (also called unique risk, diversifiable

risk, or nonsystematic risk, or idiosyncratic risk); e.g. firm wins a big deal, firm faces litigation

─Component 2: Market risk (also called systematic, or nondiversifiablerisk); e.g. inflation, employment rates, economic cycle

Slide 9Week 4



� Diversification (i.e., including many firms in the risky portfolio)

reduces firm-specific risk.

� Diversification does not reduce market risk because all firms are

exposed to market risk.

Total Risk

Slide 10Week 4



Slide 11Week 4



� Implications of the CAPM:

─ Investors hold the market portfolio, i.e., investors hold an optimally diversified portfolio.

─All firm-specific risk is diversified away.─ Investors are only exposed to market risk.

� Remember that we used the volatility of returns as our measure of risk

� the volatility of return, however, measures overall risk (firm-specific and market risk).

� In the CAPM, investors are only compensated for market risk � we need an appropriate measure for market risk!

Slide 12Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (5)Equation

� How to measure systematic (i.e., market) risk of a specific stock: look at the covariance of the stock’s return with the market’s return and normalize by the variance of the market’s return

� This measure of systematic risk is called Beta.� Notation:

─ Cov(ri,rM)…Covariance between the returns of stock i and the returns of the market

─ …variance of the returns of the market

( )2

,

M

Mii

rrCov

σβ =

2

Mσ

Slide 13Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (6)Example

� Problem: Given the following information, calculate the Beta of Stock A

� Solution:─Calculate the covariance between Stock A and the market portfolio

─Calculate the variance of the market portfolio and then the Beta

10%12%70%Boom

-7%-14%30%Recession

Market PortfolioStock AProbabilityScenario

( ) ( ) ( ) ( ) ( ) 82.929.4102.4127.09.472.4143.0, =−×−×+−−×−−×=MA rrCov

( ) ( ) 69.609.4107.09.473.0222 =−×+−−×=Mσ 53.1

69.60

82.92==iβ

Slide 14Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (1)Equation

� Our insight that investors are only compensated for the exposure to systematic

risk has important implications for the expected returns on individual assets.� The rate of return on any asset exceeds the risk-free rate by a risk

premium equal to the asset’s systematic risk measure (its beta) times the risk premium of the market portfolio.

� Notation:─ rf...risk-free rate of return

─ E(ri)…expected rate of return for some stock i

─ βi…Beta of stock i─ E(rM)…expected rate of

return of the market

( ) ( )( )fMifi rrErrE −×+= β

Slide 15Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (2)Example

� Consider the following information for a stock XYZ: E(rM) = 14%, rf = 6%,

βXYZ = 1.2

� What is the expected return of stock XYZ, i.e. E(rXYZ)?

� What happens if your read the Wall Street Journal and a well-known asset manager believes that the stock is going to yield an expected return of 17%?

─According to the CAPM, the stock should yield 15.6%.─ The difference between the 17% and the 15.6% (which is called alpha

or αααα) implies some mispricing of the stock.

( )( ) ( ) 6.156142.16 =−×+=−×+ fMif rrEr β

Slide 16Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (3)Equation: Alpha

� The Alpha of stock A is the difference between some “subjective” expected

rate of return (coming from an expert, a research department, personal beliefs, past evidence) and the expected rate of return according to the CAPM.

─ Positive alpha: stock is expected to earn more than expected according

to the CAPM � the stock is a good investment─Negative alpha: vice versa

� Notation:─ E(ri)…expected rate of return for some stock i according to the CAPM

─ SE(ri)…”subjective” expected rate of return for some stock i

( ) ( ) ( ) ( )( )[ ]fMifiii rrErrSErErSE −×+−=−= βα

Slide 17Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (4)

� Security Market Line (SML):

Graphical representation of the relationship between an asset’s

expected return and Beta

� Slope of the SML: risk premium of the market portfolio

� CML or CAL (in contrast): graphs the relationship between

an asset’s expected return and standard deviation

Slide 18Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (5)Exercise

� Consider the following information:─ Stock XYZ: expected return of 12% by some expert; Beta of 1.0

─ Stock ABC: expected return of 13% by some expert; Beta of 1.5

─ Market: expected return of 11%─ Risk-free rate equals 5%

� What is the expected return according to the CAPM and the alpha of each

stock?� According to the CAPM, which stock is a better buy?

Slide 19Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (6)Exercise

� If the CAPM holds, is the following situation possible?

� Does the given data imply anything about the Betas of Portfolio A and B?

25%40%B

35%30%A

Standard DeviationExpected ReturnPortfolio

Slide 20Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Portfolios (1)Equation

� Same argument as for individual securities: only the systematic risk matters �therefore, we need the portfolio beta!

� Equation: the beta of a portfolio is simply the weighted average of the betas of the stocks in the portfolio where the weights are the appropriate portfolio

weights.

� What is the Beta of the Market Portfolio?

Slide 21Week 4


Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Portfolios (2)Example

� Example: market risk premium = 7.5%; rf=5%; calculate the portfolio beta,

the portfolio expected return and the portfolio risk premium?

� Portfolio Beta = 1.2×0.5 + 0.8×0.3 + 0×0.2 = 0.84� Portfolio Expected Return = 5+0.84×7.5 = 11.3%� Portfolio Risk Premium = 11.3 – 5 = 6.3%

0.20.0Gold

0.30.8Con Edison

0.51.2Microsoft

Portfolio WeightBetaAsset

Slide 22Week 4


Capital Asset Pricing Model (CAPM)Implications for Firms: Capital Budgeting (1)

� Consider the following problem: as a manager of a firm you have to decide on

the execution of a risky project

� How should you make this decision?

� Answer 1: you should do the project if it is “profitable”.─ But what does “profitable” mean?─Does an expected internal rate of return of 1% justify execution of the

project? What do you think?

Slide 23Week 4


Capital Asset Pricing Model (CAPM)Implications for Firms: Capital Budgeting (2)

� Answer 2: The project has to yield a return that is acceptable for the investors given the risk of the project.

� The CAPM can be used to obtain this cutoff IRR or “hurdle rate” for the project.

� Example: suppose Silverado Springs Inc. is considering a new spring-water

bottling plant; the expected IRR is 14%; research shows the Beta of similar products is 1.3; the risk-free rate is 4%; market risk premium is estimated to be

8%

─ The hurdle rate of the project should be: 4 + 1.3×8 = 14.4%─ Because the IRR is less than the hurdle rate � the project has a negative

net present value � should not be done.

Slide 24Week 4


Capital Asset Pricing Model (CAPM)How to take it?

� Logical inconsistency:─ If a passive strategy is costless and efficient, why would anyone follow

and active strategy?─ But if no one does any security analysis, what brings about the efficiency

of the market portfolio?

� The CAPM depends on important assumptions but its popularity and use is some indication that its predictions are reasonable.

Slide 25Week 4


Capital Asset Pricing Model (CAPM)Betas of AT&T Over Time (1966-1996)

� Betas are not observed but estimated! Therefore, a “true” Beta does not exist.

0

0.2

0.4

0.6

0.8

1

1.2

Slide 26Week 4


Capital Asset Pricing Model (CAPM) Betas by Industries

1.05Steel

0.60Energy Utilities1.20Shipping

0.75Telephone Utilities1.25Chemicals

0.85Banks1.30Producer Goods

0.90Liquor1.45Consumer Durables

1.00Food1.60Electronics

1.00Agriculture1.80Airlines

BetaIndustryBetaIndustry


Week 10:

Options Valuation

(15.1, 15.2, Put-Call Parity p. 519 to 521)

Slide 2Week 10


Option Values

� Intrinsic value - profit that could be made if the option was immediately exercised

─Call: stock price minus exercise price

─ Put: exercise price minus stock price

� Time value - the difference between the option price and the

intrinsic value

Slide 3Week 10


Factors Influencing Option Values: Calls

increases

increases

decreases

increases

Effect on value

Asymmetric payoff of options: Longer maturity implies more chances for S to go up

Asymmetric payoff of options: zero payoff if not exercised, ST-X if exercised

payoff of call equals ST-X if call is exercised

payoff of call equals ST-X if call is exercised

Explanation

Time to expiration

Volatility of stock price

Exercise price (X)

Stock price (S)

Factor

� Exercise: prepare a similar table for the determinants of put option values.

Slide 4Week 10


Factors Influencing Option Values: Exercise

� Consider the following information:

─Which put option is written on the stock with the lower price?� Consider the following information:

─Which call option is written on the stock with higher volatility?

50

50

X

0.5

0.5

T

100.25B

100.20A

Price of OptionVolatility of StockPut

55

50

X

0.5

0.5

T

755B

1055A

Price of OptionS0Call

Slide 5Week 10


Binomial Option Pricing (1)

100

200

50

Stock Price

C

75

0

Call Option

Value X = 125

� Two-state (binomial) option pricing

� Assume a stock price can only take two possible values at option expiration:

─ Increase by a factor u=2─Decrease by a factor d=0.5

� Assume a call option with X=$125 and

T=1. Assume further the risk-free rate of interest is 8%.

Slide 6Week 10



� Consider the following portfolio: one share of stock and 2 calls written (X =

125)

� What are the payoffs to this portfolio in each state?

100-2××××C

50=200-2××××75

50=50-0

Portfolio

Payoff

Slide 7Week 10



� We created a risk-free portfolio with a payout pf $50. What is the value of a

risk-free profit of $50 in one period?

� Note: we can use the risk-free rate to discount ONLY because the payoffs are risk-free, i.e. do not depend on the price of the stock any more)

� $46.30 also has to be the value of the portfolio (the stock plus the two written

calls) today! Products with the same payoffs must have the same value (replication/arbitrage argument). Therefore, the price of the call is:

30.46$08.01

50$=

+

85.26$

30.46$2100$

=

=×−

C

C

Slide 8Week 10


Binomial Option Pricing: Exercise

� Consider the same stock price dynamics as before (i.e., the stock price can go

up to $200 or go down to $50; the current stock price equals $100). The riskfree rate is again 8%.

� You want to determine the price of a call option with X=$125.

� Consider the following portfolio – buy 0.5 shares and write one option:─Determine the payoffs of this portfolio at the end of the period in each of

the two states.─Determine the present value of this payoff.

─Determine the price of the call option.

Slide 9Week 10



� How to come up with a portfolio that creates a

risk-free payoff?� Portfolio weights depend on the hedge ratio:

� The hedge ratio equals the ratio of the weight in the

stock and the weight in the option.� For the previous example: twice as much weight in

the option as in the stock

100

200

50

Stock Price

C

75

0

Call Option

Value X = 125

du

du

SS

CCH

−

−=

5.050200

075=

−

−=H

Slide 10Week 10


Binomial Option Pricing: Exercise

� Reconsider the example from before. Now we

change the exercise price of the option to X=$100

� What is the new hedge ratio?

� What does the portfolio with risk-free payoffs look

like, for example?

� Determine the risk-free payoffs?

� Determine the call option price.

100

200

50

Stock Price

C

??

??

Call Option

Value X = 100

Slide 11Week 10


Put-Call Parity Relationship (1)

� Prices of European Puts and European Calls are linked together in an important

relationship – the Put-Call Parity Relationship.� Consider the following portfolio: buy a call option and write a put

option, each with the same exercise price X and the same expiration date T

ST-XST-XTotal

0-(X-ST)Minus payoff of put written

ST-X0Payoff of call held

ST>XST≤X

Slide 12Week 10



Slide 13Week 10



� Why is the combined payoff called “Leveraged Equity”?� Consider the following portfolio: buy the stock and take out a loan with face

value X at the risk-free rate

ST-XST-XTotal

-X-XMinus payoff of paying back the loan

STSTPayoff of stock

ST>XST≤X

Slide 14Week 10



� By the argument of arbitrage/replication the two portfolios – (A) long a call plus short a put, and (B) long the stock plus loan with face value of X – must

have the same price/value today

C - P = S0 - X/(1+rf)T

� Assume S0=$110, call price (T=0.5 years, X=$105) equals $14, put price

(T=0.5 years, X=$105) equals $5, risk-free rate is 5%─Verify that put-call parity holds:

• $14-$5 = $9

• $110-$105/(1+0.05)0.5 = $7.53

• Violation of put-call parity

Slide 15Week 10



� Example continued: what does the violation of put-call parity imply � a risk-free profit (i.e., arbitrage)

─ Buy the “cheap” portfolio, i.e. the stock plus the loan

─ Sell the “expensive” portfolio, i.e. the call and the short put

Cash-Flow (CF) in 0.5 years

1.47

-5

+14

+102.47

-110

CF at t=0

-105-105Borrow X/(1+rf)T

-(ST-105)0Sell call

00Total

0105- STBuy put

+ ST+STBuy stock

ST≥105ST<105Position

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