Lecture 5- Principles of Financial Economics

7/31/2019 Lecture 5- Principles of Financial Economics

1/25

Lecture 5

Capital Allocation between theRisky and the Risk-free Asset


2/25

Slide 6-2

w1 = proportion of funds in Security 1w2 = proportion of funds in Security 2

r1 = expected return on Security 1r2 = expected return on Security 2

1wn

1i

i

Two-Security Portfolio: Return

2211P rwrwr


3/25

Slide 6-3

1

2 = variance of Security 1

2

2 = variance of Security 2

Cov(r1,r2) = covariance of returns forSecurity 1 and Security 2

Two-Security Portfolio: Risk

)r,r(Covww2ww 21212

2

2

2

2

1

2

1

2

p


4/25

Slide 6-4

Range of values for1,2

+ 1.0 > > -1.0

If = 1.0, the securities would be

perfectly positively correlated

If= - 1.0, the securities would be

perfectly negatively correlated

Correlation Coefficients:

Possible Values


5/25

Slide 6-5

1,2 = Correlation coefficient of returns

1 = Standard deviation of returns for

Security 12 = Standard deviation of returns for

Security 2

Covariance

212,121 )r,r(Cov


6/25

Slide 6-6

Possible to split investment fundsbetween safe and risky assets

Risk free asset: proxy; T-bills

Risky asset: stock (or a portfolio)

Allocating Capital Between

Risky & Risk Free Assets


7/25Slide 6-7

Examine risk/return tradeoff

Demonstrate how different degrees of riskaversion will affect allocations betweenrisky and risk free assets

Allocating Capital Between

Risky & Risk Free Assets


8/25Slide 6-8

The Risk-Free Asset

Perfectly price-indexed bond the only

risk free asset in real terms; T-bills are commonly viewed as the

risk-free asset;

Money market funds - the mostaccessible risk-free asset for mostinvestors.


9/25Slide 6-9

Portfolios of One Risky Asset

and One Risk-Free Asset Assume a risky portfolio P defined by :

E(rp) = 15% and p = 22%

The available risk-free asset has:

rf= 7% and

rf= 0%

And the proportions invested:

y% in P and (1-y)% in rf


10/25Slide 6-10

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

rc = complete or combined portfolio

If, for example, y = .75E(rc) = .75(.15) + .25(.07)

= .13 or 13%

Expected Returns for

Combinations


11/25Slide 6-11

rf

pc=

Since

y

= 0, then

* Rule 4 in Chapter 5

*

Variance on the Possible

Combined Portfolios


12/25Slide 6-12

Possible Combinations

E(r)

E(rp) = 15%

rf= 7%

22%0

P

F

c

E(rc) = 13% C


13/25Slide 6-13

CAL (Capital Allocation Line)

E(r)

E(rp) = 15%

rf= 7%

p= 22%0

P

F

) S = 8/22

E(rp) - rf= 8%


14/25Slide 6-14

Greater levels of risk aversion lead tolarger proportions of the risk free rate

Lower levels of risk aversion lead tolarger proportions of the portfolio of riskyassets

Willingness to accept high levels of riskfor high levels of returns would result inleveraged combinations

Risk Aversion and Allocation


15/25Slide 6-15

c = .75(.22) = .165 or 16.5%

If y = .75, then

c= 1(.22) = .22 or 22%

If y = 1

c

=0(.22) = .00 or 0%

If y = 0

Combinations Without Leverage


16/25Slide 6-16

Borrow at the Risk-Free Rate and invest in

stock Using 50% Leverage

rc

= (-.5) (.07) + (1.5) (.15) = .19

c = (1.5) (.22) = .33

Using Leverage with

Capital Allocation Line


17/25Slide 6-17

Borrowing

If investors can borrow at the risk-free rateof rf= 7%, they can construct portfolios that

may be plotted on the CAL to the right ofP.

The leveraged portfolio has a higher

standard deviation than the unleveragedposition in the risky asset.


18/25Slide 6-18

Leveraged Position

Suppose the investment budget is $300,000 andthe investor borrows an additional $120,000investing the total available funds in the risky

asset. This is a leveraged position in the risky asset

which is financed in part by borrowing. This reflects a short position in the risk-free

asset. Rather than lending at 7%, the investor borrows

at the rate of 7%.


19/25

Slide 6-19

Effect of Leverage on the Reward-

to-variability ratio The distribution of the portfolio rate of

return still exhibits the same reward-to-

variability ratio. However the borrowing rate is likely to be

higher.


20/25

Slide 6-20

Higher borrowing rate

Assume that the borrowing rate is 9 percent. Calculate the reward-to-variability

ratio if the expected return on the portfolioof risky assets is 15 per cent and itsstandard deviation is 22 per cent.


21/25

Slide 6-21

CAL with Higher Borrowing

RateE(r)

9%

7%) S = .36

) S = .27P

p = 22%


22/25

Slide 6-22

CAL is kinked

With a higher borrowing rate (than the lendingrate) the CAL is kinked at point P.

To the left of P the investor is lending at 7 percent and the slope of the CAL is 0.36

To the right of P, y>1, the investor is borrowingat 9 per cent to finance extra investments in the

risky asset and the slope is 0.27


23/25

Slide 6-23

Indifference Curves and Risk

AversionCertainty equivalent ofportfolio Ps expected return

for two different investors

P

E(r)

rf=7%

A = 4

A = 2

p = 22%

E(rp)=15%


24/25

Slide 6-24

Certainty Equivalent Rate of Return

More risk averse investors have a steeper ICs

Less risk averse investors have flatter ICs

Portfolios utility value is its certainty equivalentrate of return to the investor

The certainty equivalent rate of return of theportfolio is the rate that risk-free investments

would need to offer with certainty to beconsidered equally attractive as the riskyportfolio.


25/25

Slid 6 25

CAL with Risk Preferences

P

E(r)

7%Lender

Borrower

p = 22%

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Lecture 5- Principles of Financial Economics