Chapter 5 Risk and Return: Past and Prologue Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Chapter 5

Risk and Return: Past and Prologue

Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

5.1 Rates of Return

5-2

Measuring Ex-Post (Past) Returns

One period investment: regardless of the length of the period.

Holding period return (HPR):

HPR = where

PS = Sale price (or P1)

PB = Buy price ($ you put up) (or P0)

CF = Cash flow during holding period

• Q:

• Q:

[PS - PB + CF] / PB

Why use % returns at all?

What are we assuming about the cash flows in the HPR calculation?

5-3

Annualizing HPRs

Q: Why would you want to annualize returns?

1. Annualizing HPRs for holding periods of greater than one year:– Without compounding (Simple or APR):

HPRann =

– With compounding: EAR

– HPRann =

where n = number of years held

HPR/n

[(1+HPR)1/n]-1

5-4


•An example: Suppose you buy one share of a stock today for $45 and you hold it for two years and sell it for $52. You also received $8 in dividends at the end of the two years.

•(PB = , PS = , CF = ):

•HPR =

•HPRann =

•The annualized HPR assuming annual compounding is (n = ):

•HPRann =

$45 $52 $8

(52 - 45 + 8) / 45 = 33.33%

0.3333/2 = 16.66%

2(1+0.3333)1/2 - 1 = 15.47%

Annualized w/out compounding

5-5


Annualizing HPRs for holding periods of less than one year:

– Without compounding (Simple): HPRann =

– With compounding: HPRann =

where n = number of compounding periods per year

HPR x n

[(1+HPR)n]-1

5-6

Measuring Ex-Post (Past) Returns•An example when the HP is < 1 year: •Suppose you have a 5% HPR on a 3 month investment. What is the annual rate of return with and without compounding?

•Without:

•With:

•Q: Why is the compound return greater than the simple return?

n = 12/3 = 4 so HPRann = HPR*n = 0.05*4 = 20%

HPRann = (1.054) - 1 = 21.55%

5-7

Arithmetic Average

Finding the average HPR for a time series of returns:• i. Without compounding (AAR or Arithmetic Average

Return):

• n = number of time periods

n

1T

Tavg n

HPRHPR

5-8

Arithmetic Average

AAR =

An example: You have the following rates of return on a stock: 2000 -21.56% 2001 44.63% 2002 23.35% 2003 20.98% 2004 3.11% 2005 34.46% 2006 17.62%

n

1T

Tavg n

HPRHPR

7

.1762).3446.0311.2098.2335.4463(-.2156HPRavg

17.51%

17.51%

5-9

Geometric Average

•With compounding (geometric average or GAR: Geometric Average Return):

GAR =

An example: You have the following rates of return on a stock: 2000 -21.56% 2001 44.63% 2002 23.35% 2003 20.98% 2004 3.11% 2005 34.46% 2006 17.62%

1 )HPR(1HPR

/1n

1TTavg

n

11.1762)1.34461.03111.20981.23351.4463(0.7844HPR 1/7avg 15.61%

15.61%

5-10


•Finding the average HPR for a portfolio of assets for a given time period:

•where VI = amount invested in asset I,

•J = Total # of securities•and TV = total amount invested;

•thus VI/TV = percentage of total investment invested in asset I

J

1IIavg HPRHPR

TV

VI

5-11

Measuring Ex-Post (Past) Returns•For example: Suppose you have $1000 invested in a stock portfolio in September. You have $200 invested in Stock A, $300 in Stock B and $500 in Stock C. The HPR for the month of September for Stock A was 2%, for Stock B the HPR was 4% and for Stock C the HPR was - 5%.

•The average HPR for the month of September for this portfolio is:

J

1IIavg HPRHPR

TV

VI

)(500/1000)(-.05 )(300/1000)(.04 )(200/1000)(.02HPRavg -0.9%

5-12

Measuring Ex-Post (Past) Returns• Measuring returns when there are investment

changes (buying or selling) or other cash flows within the period.

• An example: Today you buy one share of stock costing ___. The stock pays a __ dividend one year from now.

– Also one year from now you purchase a second share of stock for ____.

– Two years from now you collect a ___ per share dividend and sell both shares of stock for ___ a share. Q: What was your average (annual) return?A: It depends. There are different ways to measure

this.

$50 $2

$53

$2$54

5-13

Dollar-Weighted Return

i. Dollar-weighted return procedure (DWR):Find the internal rate of return for the cash flows (i.e. find the discount rate that makes the NPV of the net cash flows equal zero.)

5-14

Tips on Calculating Dollar Weighted Returns

This measure of return considers both changes in investment and security performance

Initial Investment is an _______

Ending value is considered as an ______

Additional investment is an _______

Security sales are an ______

outflow

inflow

outflow

inflow

5-15

Measuring Ex-Post (Past) Returnsi. Dollar-weighted return procedure (DWR):Find the internal rate of return for the cash flows (i.e. find the discount rate that makes the NPV of the net cash flows equal zero.)

•NPV =

•Solve for IRR:

•IRR =

Total Cash Flows Each Year Year 0 1 2 -$50 $ 2 $ 4 -$53 $108 Net -$50 -$51 $112

$0 = -$50/(1+IRR)0 - $51/(1+IRR)1 + $112/(1+IRR)2

7.117% average annual dollar weighted return

The DWR gives you an average return based on the stock’s performance and

the dollar amount invested (number ofshares bought and sold) each period.

5-16


Q: You are paying somebody to advise you which assets to buy, but you are deciding when to buy and sell shares. If you want to evaluate the quality of the investment advice you are getting, should you use dollar weighted returns to evaluate the quality of the investment advice?

Total Cash Flows Each Year Year 0 1 2 -$50 $ 2 $ 4 -$53 $108 Net -$50 -$51 $112

5-17

Time-Weighted Returns

ii. Time-weighted returns (TWR):

TWRs assume you buy ___ share of the stock at the beginning of each interim period and sell ___ share at the end of each interim period. TWRs are thus ___________ of the amount invested in a given period.

To calculate TWRs:

Calculate the return for each time period, typically a year.

Then calculate either an arithmetic (AAR) or a geometric average (GAR) of the returns.

one

one

independent

5-18

Time-Weighted Returns

Same example as before, initially buy one share at $50, in one year collect a $2 dividend, and you buy another share at $53. In two years you sell the stock for $54, after collecting another $2 dividend per share.

+$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

+$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

TWR Cash Flows

TWRs assume you buy one share of the stock at the beginning of each period and sell it at the end of each period after collecting any cash flow.

5-19


Same example as before, initially buy one share at $50, in one year collect a $2 dividend, and you buy another share at $53. In two years you sell the stock for $54, after collecting another $2 dividend per share.

Year 0-1 Year 1-2

0 1 1 2

-$50 $ 2 -$53 $ 2

+$53 +$54

Year 0-1 Year 1-2

0 1 1 2

-$50 $ 2 -$53 $ 2

+$53 $54

$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

TWR Cash Flows

Year 0-1Year 1-2

0 1 1 2

-$50 $ 2 -$53 $ 2

+$53 +$54 +$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

+$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

TWR Cash Flows

5-20


HPR for year 1:

HPR for year 2:

a) Calculating the arithmetic average TW return:

Arithmetic Average Return (AAR): Calculate the

arithmetic average

[$54 - $53 +$2] / $53 = 5.66%

[$53 + $2 - $50] / $50 = 10%

AAR = [0.10 + 0.0566] / 2 = 7.83%

+$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

+$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

TWR Cash Flows

5-21


b) Calculating the geometric average TW return (GAR):

GAR =

11.0566)(1.10HPR 1/2avg

HPR1 = 10%

HPR2 = 5.66%

7.81%

7.81%

1 )HPR(1HPR

/1n

1TTavg

n

+$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

+$54+$53

$ 2-$53$ 2-$50

2110

Year 1-2Year 0-1

TWR Cash Flows

5-22

Measuring Ex-Post (Past) ReturnsQ: When should you use the GAR and when should you use the

AAR?

A1: When you are evaluating PAST RESULTS (ex-post):

A2: When you are trying to estimate an expected return (ex-ante return):

Use the AAR (average without compounding) if you ARE NOT reinvesting any cash flows received before the end of the period.

Use the GAR (average with compounding) if you ARE reinvesting any cash flows received before the end of the period.

Use the AAR

5-23

5.2 Risk and Risk Premiums

5-24

Subjective expected returns

E(r) = Expected Returnp(s) = probability of a stater(s) = return if a state occurs1 to s states

E(r) = Expected Returnp(s) = probability of a stater(s) = return if a state occurs1 to s states

Measuring Mean: Scenario or Subjective Returnsa. Subjective or Scenario

E(r) = p(s) r(s)s

5-25

= [= [22]]1/21/2

E(r) = Expected Returnp(s) = probability of a staters = return in state “s”

Measuring Variance or Dispersion of Returns

a. Subjective or Scenario

Variance

s

2s

2 E(r)][rp(s)σ

5-26

Numerical Example: Subjective or Scenario Distributions

State Prob. of State Return

1 .2 - .05

2 .5 .05

3 .3 .15

E(r) = (.2)(-0.05) + (.5)(0.05) + (.3)(0.15) = 6%

2 = [(.2)(-0.05-0.06)2 + (.5)(0.05- 0.06)2 + (.3)(0.15-0.06)2]2 = 0.0049%2

= [ 0.0049]1/2 = .07 or 7%

s

2s

2 E(r)][rp(s)σ

5-27

Expost Expected Return &

Annualizing the statistics:

n

ii rr

n1

2)(1

1 : VarianceExpost 2

periods # periodannual

periods # rr periodannual

2σσ :Deviation Standard Expost

n

1T

Tn

HPRr HPR averager

nsobservatio #n

5-28

Average 0.011624 0.219762458

Variance 0.003725

Stdev 0.061031 n 60

n-1 59

Annualized

Average 0.139486

Variance 0.044697

Stdev 0.211418

(r - ravg)2 =

Annualizing the statistics:Annualizing the statistics:

n

ii rr

n1

2)(1

1 : VarianceExpost 2

n

1T

T

n

HPRr HPR averager

12 monthlyannual

12rr monthlyannual

2σσ :Deviation Standard Expost

nsobservatio #n

31 0.027334 0.000246811 3/1/200532 -0.088065 0.009937839 4/1/200533 0.037904 0.000690654 5/2/200534 -0.089915 0.010310121 6/1/200535 0.0179 3.93874E-05 7/1/200536 -0.017814 0.000866572 8/1/200537 -0.043956 0.003089121 9/1/200538 0.010042 2.50266E-06 10/3/200539 0.022495 0.00011818 11/1/200540 -0.029474 0.001689005 12/1/200541 0.05303 0.001714497 1/3/200642 0.09589 0.007100858 2/1/200643 -0.003618 0.000232311 3/1/200644 0.002526 8.27674E-05 4/3/200645 0.083361 0.005146208 5/1/200646 -0.016818 0.000808939 6/1/200647 -0.010537 0.000491104 7/3/200648 -0.001361 0.000168618 8/1/200649 0.04081 0.000851813 9/1/200650 0.01764 3.61885E-05 10/2/200651 0.047939 0.001318787 11/1/200652 0.044354 0.001071242 12/1/200653 0.02559 0.000195054 1/3/200754 -0.026861 0.001481106 2/1/200755 0.005228 4.09065E-05 3/1/200756 0.015723 1.68055E-05 4/2/200757 0.01298 1.83836E-06 5/1/200758 -0.038079 0.002470321 6/1/200759 -0.034545 0.002131602 7/2/200760 0.017857 0.000038854 8/1/2007

Monthly Source Yahoo financeHPRs

Obs DIS (r - ravg)2

1 -0.035417 0.002212808 9/3/20022 0.093199 0.006654508 10/1/20023 0.15756 0.021297275 11/1/20024 -0.200637 0.045054632 12/2/20025 0.068249 0.00320644 1/2/20036 -0.026188 0.001429702 2/3/20037 -0.00183 0.000181016 3/3/20038 0.087924 0.005821766 4/1/20039 0.050211 0.001489002 5/1/200310 0.004734 4.74648E-05 6/2/200311 0.099052 0.00764371 7/1/200312 -0.068896 0.006483384 8/1/200313 -0.016478 0.000789704 9/2/200314 0.109174 0.009516098 10/1/200315 0.019343 5.95893E-05 11/3/200316 0.019409 6.06076E-05 12/1/200317 0.02829 0.000277753 1/2/200418 0.095035 0.00695741 2/2/200419 -0.061342 0.005324028 3/1/200420 -0.085344 0.00940277 4/1/200421 0.018851 5.22376E-05 5/3/200422 0.079128 0.004556811 6/1/200423 -0.103832 0.013330149 7/1/200424 -0.028414 0.001603051 8/2/200425 0.004562 4.98687E-05 9/1/200426 0.105671 0.008844901 10/1/200427 0.061998 0.002537528 11/1/200428 0.041453 0.000889761 12/1/200429 0.028856 0.000296963 1/3/200530 -0.024453 0.001301505 2/1/2005

Monthly Source Yahoo financeHPRs

Obs DIS (r - ravg)2

1 -0.035417 0.002212808 9/3/20022 0.093199 0.006654508 10/1/20023 0.15756 0.021297275 11/1/20024 -0.200637 0.045054632 12/2/20025 0.068249 0.00320644 1/2/20036 -0.026188 0.001429702 2/3/20037 -0.00183 0.000181016 3/3/20038 0.087924 0.005821766 4/1/20039 0.050211 0.001489002 5/1/2003

10 0.004734 4.74648E-05 6/2/200311 0.099052 0.00764371 7/1/200312 -0.068896 0.006483384 8/1/200313 -0.016478 0.000789704 9/2/200314 0.109174 0.009516098 10/1/200315 0.019343 5.95893E-05 11/3/200316 0.019409 6.06076E-05 12/1/200317 0.02829 0.000277753 1/2/200418 0.095035 0.00695741 2/2/200419 -0.061342 0.005324028 3/1/200420 -0.085344 0.00940277 4/1/200421 0.018851 5.22376E-05 5/3/200422 0.079128 0.004556811 6/1/200423 -0.103832 0.013330149 7/1/200424 -0.028414 0.001603051 8/2/200425 0.004562 4.98687E-05 9/1/200426 0.105671 0.008844901 10/1/200427 0.061998 0.002537528 11/1/200428 0.041453 0.000889761 12/1/200429 0.028856 0.000296963 1/3/200530 -0.024453 0.001301505 2/1/2005

5-29

Using Ex-Post Returns to estimate Expected HPR

Estimating Expected HPR (E[r]) from ex-post data.

Use the arithmetic average of past returns as a forecast of expected future returns as we did and,

Perhaps apply some (usually ad-hoc) adjustment to past returns

Problems?

• Which historical time period?

• Have to adjust for current economic situation

• Unstable averages

• Stable risk

5-30

Characteristics of Probability Distributions

1. Mean: __________________________________ _

2. Median: _________________

3. Variance or standard deviation:

4. Skewness:_______________________________

5. Leptokurtosis: ______________________________

If a distribution is approximately normal, the distribution is fully described by _____________________

Arithmetic average & usually most likelyArithmetic average & usually most likely

Dispersion of returns about the meanDispersion of returns about the mean

Long tailed distribution, either sideLong tailed distribution, either side

Too many observations in the tailsToo many observations in the tails

Characteristics 1 and 3Characteristics 1 and 3

Middle observationMiddle observation

5-31

Normal Distribution

E[r] = 10%

= 20%Average = Median

Risk is the Risk is the possibility of getting possibility of getting returns different returns different from expected.from expected.

measures deviations measures deviations above the mean as well as above the mean as well as below the mean. below the mean.

Returns > E[r] may not be Returns > E[r] may not be considered as risk, but with considered as risk, but with symmetric distribution, it is symmetric distribution, it is ok to use ok to use to measure risk. to measure risk.

I.E., ranking securities by I.E., ranking securities by will give same results as will give same results as ranking by asymmetric ranking by asymmetric measures such as lower measures such as lower partial standard deviation.partial standard deviation.

5-32

rrNegative Positive

Skewed Distribution: Large Negative Returns Possible (Left Skewed)

Median

rr = averageImplication?

is an incomplete risk measure

5-33

Negative Positive

Skewed Distribution: Large Positive Returns Possible (Right Skewed)

Median

rr = average

rr

5-34

LeptokurtosisImplication?Implication?

is an incomplete is an incomplete risk measurerisk measure

5-35

Value at Risk (VaR)

Value at Risk attempts to answer the following question:

• How many dollars can I expect to lose on my portfolio in a given time period at a given level of probability?

• The typical probability used is 5%.

• We need to know what HPR corresponds to a 5% probability.

• If returns are normally distributed then we can use a standard normal table or Excel to determine how many standard deviations below the mean represents a 5% probability:

– From Excel: =Norminv (0.05,0,1) = -1.64485 standard deviations

5-36

Value at Risk (VaR)From the standard deviation we can find the corresponding

level of the portfolio return:

VaR = E[r] + -1.64485

For Example:

A $500,000 stock portfolio has an annual expected return of

12% and a standard deviation of 35%. What is the portfolio

VaR at a 5% probability level?

VaR = 0.12 + (-1.64485 * 0.35)

VaR = -45.57% (rounded slightly)

VaR$ = $500,000 x -.4557 = -$227,850

What does this number mean?

5-37

Value at Risk (VaR)

VaR versus standard deviation:• For normally distributed returns VaR is equivalent to

standard deviation (although VaR is typically reported in dollars rather than in % returns)

• VaR adds value as a risk measure when return distributions are not normally distributed. – Actual 5% probability level will differ from 1.68445

standard deviations from the mean due to kurtosis and skewness.

5-38

Risk Premium & Risk Aversion• The risk free rate is the rate of return that can be

earned with certainty.• The risk premium is the difference between the

expected return of a risky asset and the risk-free rate.

Excess Return or Risk Premiumasset =

Risk aversion is an investor’s reluctance to accept risk.

How is the aversion to accept risk overcome?

By offering investors a higher risk premium.

E[rasset] – rf

5-39

5.3 The Historical Record

5-40

Frequency distributions of annual HPRs, 1926-2008

5-41

Rates of return on stocks, bonds and bills, 1926-2008

5-42

Annual Holding Period Returns Statistics 1926-2008

From Table 5.3

• Geometric mean:

Best measure of compound historical return

• Arithmetic Mean:

Expected return

• Deviations from normality?

Geom. Arith. Excess

Series Mean% Mean% Return% Kurt. Skew.

World Stk 9.20 11.00 7.25 1.03 -0.16

US Lg. Stk 9.34 11.43 7.68 -0.10 -0.26

Sm. Stk 11.43 17.26 13.51 1.60 0.81

World Bnd 5.56 5.92 2.17 1.10 0.77

LT Bond 5.31 5.60 1.85 0.80 0.51

5-43

Deviations from Normality: Another Measure Portfolio

World Stock US Small Stock US Large Stock

Arithmetic Average .1100 .1726 .1143

Geometric Average .0920 .1143 .0934

Difference .0180 .0483 .0209

½ Historical Variance .0186 .0694 .0214

If returns are normally distributed then the following relationship among geometric and arithmetic averages holds:

Arithmetic Average – Geometric Average = ½ 2

•The comparisons above indicate that US Small Stocks may have deviations from normality and therefore VaR may be an important risk measure for this class.

5-44

Actual vs. Theoretical VaR 1926-2008

ActualVaR%

VaR% if NormalSeries

World Stk -21.89 -21.07US Lg. Stk -29.79 -22.92US Sm. Stk -46.25 -44.93World Bnd -6.54 -8.69US LT Bond -7.61 -7.25

These comparisons indicate that the U.S. Large Stock portfolio, the US small stock portfolio and the World Bond portfolio may exhibit differences from normality.

5-45

Annual Holding Period Excess Returns 1926-2008 From Table 5.3 of Text

Arith. RequiredSeries Avg% Return%

World Stk 7.25

US Lg Stk 7.68

US Sm Stk 13.51

World Bonds 2.17

US LT Bonds 1.85

If the risk free rate is currently 3%, then what return should an investor require for each asset class?Problems with this approach?

10.25

10.68

16.51

5.17

4.85

• Historical data

• Assumes all securities in the category are equally risky 5-46

5.4 Inflation and Real Rates of Return

5-47

Inflation, Taxes and ReturnsThe average inflation rate from 1966 to 2005 was _____.

This relatively small inflation rate reduces the terminal value of $1 invested in T-bills in 1966 from a nominal value of ______ in 2005 to a real value of _____.

Taxes are paid on _______ investment income. This reduces _____ investment income even further.

You earn a ____ nominal, pre-tax rate of return and you are in a ____ tax bracket and face a _____ inflation rate. What is your real after tax rate of return?

rreal [6% x (1 - 0.15)] – 4.29% 0.81%; taxed on nominal

4.29%

$10.08 $1.63

nominalreal

6%15% 4.29%

5-48

Real vs. Nominal RatesFisher effect: Approximation

real rate nominal rate - inflation rate

rreal rnom - i

Example rnom = 9%, i = 6%

rreal 3%

Fisher effect: Exact

rreal = or

rreal =

rreal =

The exact real rate is less than the approximate real rate.

[(1 + rnom) / (1 + i)] – 1

(rnom - i) / (1 + i)

(9% - 6%) / (1.06) = 2.83%

rreal = real interest rate

rnom = nominal interest rate

i = expected inflation rate

5-49

Exact Fisher Effect Explained

1) I want to be able to buy more Quantity or Qnew = Qold x (1 + rreal) BUT

2) The Price, P, is also rising Pnew = Pold x (1 + i) i = inflation

Total $ spent = Pnew x Qnew

Pnewx Qnew = Pold x Qold x [(1 + rreal) x (1 + i)]

or (1 + rnom)= (1 + rreal) x (1 + i)

5-50

Nominal and Real interest rates and Inflation

5-51

Historical Real Returns & Sharpe Ratios

Real Returns% Sharpe RatioSeriesWorld Stk 6.00 0.37US Lg. Stk 6.13 0.37Sm. Stk 8.17 0.36

World Bnd 2.46 0.24LT Bond 2.22 0.24

• Real returns have been much higher for stocks than for bonds• Sharpe ratios measure the excess return to standard deviation.

• The higher the Sharpe ratio the better.• Stocks have had much higher Sharpe ratios than bonds.

5-52

5.5 Asset Allocation Across Risky and Risk

Free Portfolios

5-53

Allocating Capital Between Risky & Risk-Free Assets

Possible to split investment funds between safe and risky assets

Risk free asset rf : proxy; ________________________ Risky asset or portfolio rp: _______________________

Example. Your total wealth is $10,000. You put $2,500 in risk free T-Bills and $7,500 in a stock portfolio invested as follows:– Stock A you put ______– Stock B you put ______– Stock C you put ______

$2,500

$3,000

$2,000

T-bills or money market fund

risky portfolio

$7,500

5-54

Weights in rp

– WA =

– WB =

– WC =

The complete portfolio includes the riskless

investment and rp.

$2,500 / $7,500 = 33.33%

$3,000 / $7,500 = 40.00%

$2,000 / $7,500 = 26.67%

100.00%

Your total wealth is $10,000. You put $2,500 in risk free T-Bills and $7,500 in a stock portfolio invested as follows

Wrf = ; Wrp =

In the complete portfolio

WA = 0.75 x 33.33% = 25%; WB = 0.75 x 40.00% = 30%

WC = 0.75 x 26.67% = 20%;

25% 75%

Stock A $2,500Stock A $2,500

Stock B $3,000Stock B $3,000

Stock C $2,000Stock C $2,000

Wrf = 25%


5-55

• Issues in setting weights– Examine ___________________

– Demonstrate how different degrees of risk aversion will affect __________ between risky and risk free assets

risk & return tradeoff

allocations


5-56

rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%

y = % in rpy = % in rp (1-y) = % in rf(1-y) = % in rf

Example

5-57

E(rC) = E(rC) =

Expected Returns for Combinations

E(rC) =

For example, let y = ____

E(rC) =

E(rC) = .1175 or 11.75%C = yrp + (1-y)rf

C = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5%

c =

rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


yE(rp) + (1 - y)rf

yrp + (1-y)rf

Return for complete or combined portfolio

0.75

(.75 x .14) + (.25 x .05)

5-58

Complete portfolio

Varying y results in E[rC] and C that are ______ ___________ of E[rp] and rf and rp and rf

respectively.

E(rc) = yE(rp) + (1 - y)rf

c = yrp + (1-y)rf

linearcombinations

This is NOT generally the case for the of combinations of two or more risky assets.

5-59

E(r)

E(rp) = 14%

rf = 5%

22%0

P

F

Possible Combinations

E(rp) = 11.75%

16.5%

y =.75

y = 1

y = 0

5-60

E(r)

E(rp) = 14%

rf = 5%

22%0

P

F

Possible Combinations

E(rp) = 11.75%

16.5%

y =.75

y = 1

y = 0

5-61

Combinations Without Leverage

Since σrf = 0

σc= y σp

If y = .75, thenσc=

If y = 1σc=

If y = 0σc=

rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


75(.22) = 16.5%

1(.22) = 22%

0(.22) = 0%

E(rc) = yE(rp) + (1 - y)rf

y = .75E(rc) =

y = 1E(rc) =

y = 0E(rc) =

(.75)(.14) + (.25)(.05) = 11.75%

(1)(.14) + (0)(.05) = 14.00%

(0)(.14) + (1)(.05) = 5.00%

5-62

Using Leverage with Capital Allocation Line

Borrow at the Risk-Free Rate and invest in stock

Using 50% Leverage

E(rc) =

c =

rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


(1.5) (.14) + (-.5) (.05) = 0.185 = 18.5%

(1.5) (.22) = 0.33 or 33%

E(r)E(r)

E(rE(rpp) = 14%) = 14%

rrff = 5%= 5%

22%22%00

PP

FF

Possible CombinationsPossible Combinations

E(rE(rpp) = 11.75%) = 11.75%

16.5%16.5%

E(rE(rpp) = 11.75%) = 11.75%

16.5%16.5%

y =.75y =.75

y = 1y = 1

E(rE(rCC) =18.5%) =18.5%

33%33%

y = 1.5

y = 1.5

y = 0y = 0

5-63

Risk Aversion and Allocation Greater levels of risk aversion lead investors to

choose larger proportions of the risk free rate

Lower levels of risk aversion lead investors to choose larger proportions of the portfolio of risky assets

Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

E(r)E(r)

E(rE(rpp) = 14%) = 14%

rrff = 5%= 5%

22%22%00

PP

FF

Possible CombinationsPossible Combinations

E(rE(rpp) = 11.75%) = 11.75%

16.5%16.5%

E(rE(rpp) = 11.75%) = 11.75%

16.5%16.5%

y =.75y =.75

y = 1y = 1

E(rE(rCC) =18.5%) =18.5%

33%33%

E(rE(rCC) =18.5%) =18.5%

33%33%

y = 0y = 0

y = 1.5

5-64

E(r)E(r)E(r)E(r)

E(rE(rpp) = 14%) = 14%

rrff = 5% = 5%

= 22%= 22%00

PP

FFFF

rprprprp

) Slope = 9/22) Slope = 9/22

E(rE(rpp) - ) - rrff = 9% = 9%

CALCAL(Capital(CapitalAllocationAllocationLine)Line)

P or combinations of P or combinations of P & Rf offer a return P & Rf offer a return per unit of risk of per unit of risk of 9/22.9/22.

5-65

Quantifying Risk Aversion

25.0 pfp ArrE E(rp) = Expected return on portfolio p

rf = the risk free rate

0.5 = Scale factor

A x p2 = Proportional risk premium

The larger A is, the larger will be the _________________________________________ investor’s added return required to bear risk

5-66

Quantifying Risk AversionRearranging the equation and solving for A

Many studies have concluded that investors’ average risk aversion is between _______

σ

rrEA

p

fp

2.50

)(

2 and 4

5-67

Using A

What is the maximum A that an investor could have and still choose to invest in the risky portfolio P?

Maximum A =

E(r)E(r)

E(rE(rpp) = 14%) = 14%

rrff = 5%= 5%

= 22%= 22%00

PP

FF

rprp

) Slope = 9/22) Slope = 9/22

E(rE(rpp) ) -- rrff = 9%= 9%


σ

rrEA

p

fp

2.50

)(

0.220.5

0.050.14A

2

3.719

3.719

5-68

“A” and Indifference Curves

The A term can used to create indifference curves. Indifference curves describe different combinations of

return and risk that provide equal utility (U) or

satisfaction. U = E[r] - 1/2Ap

2

Indifference curves are curvilinear because they exhibit

diminishing marginal utility of wealth.• The greater the A the steeper the indifference curve and all else equal, such investors will invest less in risky assets.

• The smaller the A the flatter the indifference curve and all else equal, such investors will invest more in risky assets.

5-69

Indifference Curves• Investors want

the most return for the least risk.

• Hence indifference curves higher and to the left are preferred.

II22

II11

II33

U = E[r] - 1/2Ap2

123 I I I

5-70

E(r)

rf = 5%

00

PP

F

CAL(CapitalAllocationLine)

A=3A=3A=3A=3

QQSS

5-71

E(r)

rf = 5%

00

PP

F


A=2A=2

A=3A=3

SST

5-72

5.6 Passive Strategies and the Capital Market Line

5-73

A Passive Strategy

• Investing in a broad stock index and a risk free investment is an example of a passive strategy.

– The investor makes no attempt to actively find undervalued strategies nor actively switch their asset allocations.

– The CAL that employs the market (or an index that mimics overall market performance) is called the Capital Market Line or CML.

5-74

Excess Returns and Sharpe Ratios implied by the CML

Excess Return or Risk Premium

Time Period Average

Sharpe Ratio

1926-2008 7.86 20.88 0.371926-1955 11.67 25.40 0.461956-1984 5.01 17.58 0.281985-2008 5.95 18.23 0.33

The average risk premium implied by the CML for large common stocks over the entire time period is 7.86%.

• How much confidence do we have that this historical data can be used to predict the risk premium now?

5-75

Active versus Passive Strategies• Active strategies entail more trading costs than

passive strategies.• Passive investor “free-rides” in a competitive

investment environment.• Passive involves investment in two passive

portfolios– Short-term T-bills– Fund of common stocks that mimics a broad

market index– Vary combinations according to investor’s risk

aversion.

5-76

Selected Problems

5-77

Problem 1

• V(12/31/2004) = V (1/1/1998) x (1 + GAR)7

• = $100,000 x (1.05)7 =

$140,710.04

→→5-78

Problem 2

a. The holding period returns for the three scenarios are:Boom:

Normal:

Recession:

E(HPR) =

2(HPR)

(50 – 40 + 2)/40 = 0.30 = 30.00%(43 – 40 + 1)/40 = 0.10 = 10.00%

(34 – 40 + 0.50)/40 = –0.1375 = –13.75%[(1/3) x 30%] + [(1/3) x 10%] + [(1/3) x (–13.75%)] = 8.75%

]8.75%) – (–13.75% x [(1/3) ]8.75%) – (10% x [(1/3) ]8.75%) – (30% x [(1/3)σ 222(HPR)

2 0.031979

17.88%σ(HPR) →5-79

Problem 2 Cont.

b. E(r) =

=

(0.5 x 8.75%) + (0.5 x 4%) = 6.375%

0.5 x 17.88% = 8.94%

→

Risky E[rp] = 8.75%Risky p = 17.88%

5-80

Problems 3 & 4

3. For each portfolio: Utility = E(r) – (0.5 4 2 )

We choose the portfolio with the highest utility value, which is Investment 3.

Investment E(r) U

1 0.12 0.30 -0.0600

2 0.15 0.50 -0.3500

3 0.21 0.16 0.1588

4 0.24 0.21 0.1518

→→5-81

Problems 3 & 4 Cont.

4. When an investor is risk neutral, A = _ so that the portfolio with the highest utility is the portfolio with the _______________________.

So choose ____________.

highest expected return0

Investment 4

→→5-82

Problem 5

(95 – 90 + 4)/90 = 10.00%2009-2010

(90 – 110 + 4)/110 = –14.55%2008-2009

(110 – 100 + 4)/100 = 14.00%2007-2008

Return = [(capital gains + dividend) / price]

a. TWRYear

3

10.00%14.55%14.00%AAR

3.15%

110]0.1455)x1.[1.14x(1GAR 1/3 2.33%

Dividends on four shares,plus sale of four shares at $95 per share

396

Dividends on five shares,plus sale of one share at $90

110

Purchase of two shares at $110,plus dividend income on three shares held

-208

Purchase of three shares at $100 per share

-300

3

2

1

0

ExplanationCash flow

Time

IRR)(1

$396

IRR)(1

$110

IRR)(1

$208

IRR)(1

$300$0

3210

-0.1661%

b. DWR

a. T

WR

→→ 5-83

Documents

Chapter 5 Risk and Return: Past and Prologue Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin