McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 7Chapter 7Chapter 7Chapter 7

Capital Allocation Between The Capital Allocation Between The Risky And The Risk-Free AssetRisky And The Risk-Free Asset

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It’s possible to split investment funds between safe and risky assets.

Risk free asset: proxy; T-billsRisky asset: stock (or a portfolio).

Capital Allocation Decision: A choice among broad investment classes rather than among the specific securities within each asset class.It is considered as the most important portfolio construction.

1. Allocating Capital: Risky & Risk Free Assets

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The Most Basic Asset Allocation Choice

The choice of how much of the portfolio to place in rf money market instruments vs other risky assetsSuppose

P; is the investor’s portfolio of risky assets, F; is the risk-free asset.

P consists of two mutual funds: invested in stocks, invested in L-T Bonds

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Assume that Initial Portfolio $300,000 $90,000 is invested in the MM Funds

$210,000 in risky securities$113,400 in equity (E)

$56,600 L-T Bonds (B)

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The weigt of Equity and Bonds in the Risky Portfolio, P.

54.0000,210

400,113: 1 WE

46.0000,210

600,96: 2 WB

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The weight of the risky portfolio, P, in the complete portfolio;

Risky assets;

Risk-free assets;

70.0000,300

000,210y

30.0000,300

000,901 y

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The weights of each risky assets in the complete portfolio;

378.0000,300

400,113E

70.0327.0378.0

322.0000,300

600,96

B

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Example 7.1

With the same assets suppose you want to decrease risk by reducing the allocation to the risky portfolio from y=0.70 to y=0.56.

Risky portfolio: 300,000 x 0.56 = $168,000

You sold 210,000-168,000= $42,000 of your risky holdings and purchased new shares of risk-free asset

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Holding in the risk-free asset increase to 90,000+ 42,000= $132,000 or,

300,000 x (1-.56)= $132,000We leave the proportions of each assets in the risky portfolio unchanged. We sold;.54 x 42,000= $22,680 of E and.46 x 42,000= $19,320 of B.

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Question 1

What will be the dollar value of your position in E, and its proportion in your whole portfolio, if you decide to hold 50% of your investment budget in risk-free instrument?

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Solution

Risky asset proportion is decreased from 70% to 50%.

50% in risky asset.

Proportion of E in the overall portfolio

.50 x 54% = 27%

300,000 x .27 = $81,000

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2. The Risk-free Asset

We commonly view the T-Bills as the “risk-free asset”. Their short-term nature makes their values insensitive to interest rate fluctuations.

There are other money market instruments that may be used by the investors as risk-free assets such as certificate of deposits and commercial papers.

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Issues

Examine risk/return tradeoff.

Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets.

3. Portfolios of One Risky Asset and One Risk-Free Asset

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Suppose investor has already decided on the combination of the risky portfolio.

y: proportion of the investment allocated to the risky portfolio, P,

1-y: proportion of the investment allocated to the risk-free asset, F.

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rf = 7% rf = 0%

E(rp) = 15% p = 22%

y = % in p (1-y) = % in rf

Example Using Chapter 7.3 Numbers

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

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

= yE(rp)+rf-yrf

E(rc) = rf + y [E(rp)- rf]

Expected Returns for Combinations

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E(rc) = .07+y(.15-.07)

E(rc) = .07 + .08y

For example, y = .75

= .07 + .75(.15-.07)

= .13 or 13%

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Possible Combinations

E(r)

E(rp) = 15%

rf = 7%

22%0

P

F

c

E(rc) = 13%C

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pc =

Since rf

y

= 0, then

* Rule 4 in Chapter 6

*

Variance For Possible Combined Portfolios

σc=22y

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

If y = .75, then

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

If y = 1

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

If y = 0

Combinations Without Leverage

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The base rate of return for any portfolio is the risk-free rate. In addition, the portfolio is expected to earn a risk premium that depends of the risk premium of the risky portfolio and the investor’s position in the risky asset (y).

Investors are assumed to be risk averse and unwilling to take on a risky position without a positive risk premium.

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E(rc) = rf + y [E(rp)- rf]

y =σc /σp

Slope;

cpp

cc rfrErfrE

22

87])([)(

36.022

8)(

p

p rfrES

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CAL (Capital Allocation Line)

E(r)

E(rp) = 15%

rf = 7%

p = 22%0

P

F

) S = 8/22E(rp) - rf = 8%

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S; equals to an increase in the expected return of the complete portfolio for unit of additional standard deviation. (Incremental return per incremental risk). For this reason; reward-to-variability ratio.y=.5 equally divided between P and FE(rc)= 7+ .5 x (15-7) =11%

Risk premium 4%δc= .5 x 22= 11%S= 4 / 11 = .36 the same as that of the portfolio P,

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Suppose the investment budget is 300,000 and we borrow an additional $150,000, investing into risky assets.

Borrow at the Risk-Free Rate and invest in stock.

Using 50% Leverage,

Capital Allocation Line with Leverage

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y = 450,000/300,000 = 1.5

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

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

S = (.19-.07)/.33 = .36

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In the real world, of course, nongovernment investors can not borrow at the risk free rate. Our borrowing cost will exceed the lending rate which is 7%. Suppose borrowing rate is rfB= 9%. Then the new slope is;

S = (.15-.09)/.22 = .27

CAL with Higher Borrowing Rate

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CAL with Higher Borrowing Rate

E(r)

9%

7%) S = .36

) S = .27

P

p = 22%

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The CAL will therefore “kinked” at P.

To the left at P, the investor is lending at 7% with slope 0.36

To the right of P, the investor is borrowing at 9% with slope 0.27

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In practice, you can borrow and make investment in risky portfolio. => margin purchasingMargin purchase may not exceed 50% of the purchase value. Therefore if you have 300,000, you can lend up to 300,000 to purchase additional stock.You would have $600,000 on the asset side and 300,000 on the liability side of your account, y=2.

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Question

Suppose that there is a shift upward in the expected rate of return on the risky asset, from 15% to 17%. If all other parameters remain unchanged, what will be the slope of the CAL for y that is smaller than and equal to 1 and that is greater than 1.

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Solution

S (Lending) = (.17-.07)/.22 = 0.45

S (Borrowing) = (.17-.09)/22 = 0.36

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4. Risk Tolerance and Allocation

We have shown how to develop CAL, the graph of all feasible risk-return combinations available from different asset allocation choices.Now we must choose one optimal portfolio, C, from the set of feasible choices.Individual investors differences in risk aversion imply that, given an identical opportunity set ( a rf and slope), different investors will choose different positions in the risky asset.

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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 portfolio of risky assets.Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations.

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Utility Function

The utility of investors with a given expected return and std. deviation;

U = E ( r ) - .005 A 2

Where

U = utility

E ( r ) = expected return on the asset or portfolio

A = coefficient of risk aversion (degree of risk aversion)

= variance of returns

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U ↑, E(r)↑ U↓ ≠ σ2↑Risk-neutral investors, A=0Higher level of risk aversion, larger values for A.Investor attempts to maximize utility, by choosing the best allocation to the risky portfolio, y.

E(rc) = rf + y[E(rp)- rf]Optimal position for risk-averse investors in the risky asset.

222 005.)(005.)( pypccy ArfrEyrfrEUMax

2*

01.

)(

p

p

A

rfrEy

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Example 7.3

Rf= 7% E(rp)= 15% σp=22%, A=4

41% in the risky asset 1-y=59% in risk-free assetE(rc)= 7+[.41 x (15-7)]=10.28%σc=.41 x 22 = 9.02%Risk premium of the complete portfolio;E(rc)- rf= 10.28- 7= 3.28%S= 3.28/ 9.02= .36 (reward to variability ratio)

41.22401.

7152

*

y

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Indifference Curves

A graphical way of presenting this decision problem is to use indifference curve analysis.

Indifference curve is a graph in the expected return- standard deviation plane of all points that result in a given (same) level of utility. The curve displays the investor’s required trade-off between expected return and standard deviation.

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How to build an Indifference Curve

Consider an investor with A=4 invests all her money into risk-free portfolio rf=5%Because variance is 0. U=5 => We try to hold the utility value at 5.Expected return when holding a risky portfolioSay δ=1%U= E(r)- .005 x A x δ2

5= E(r)- .005 x 4 x 12

E(r)=5.02%We can repeat this calculation for many levels of δ, finding the value of E(r) necessary to maintain U=5. Plotting these combinations gives us the indifference curve.

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Choosing Optimal Complete Portfolio

The more risk-averse investor (with greater A) has steeper indifference curves than the less risk averse investor (with lower A).Higher indifference curves correpond to higher level of utility.The investor attempts to find the complete portfolio on the highest possible indifference curve.The optimal complete portfolio is the point at which the highest possible indifference curve touches the CAL.

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CAL with Risk Preferences

E(r)

7%

P

Lender

Borrower

p = 22%

The lender has a larger A when

compared to the borrower

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Question

If an investor’s A = 3, how does the optimal asset mix change?. What are the new E(rc) and σc?

Suppose that the borrowing rate is 9%, is greater than the lending rate, 7%. Show graphically how the optimal portfolio choice of some investors will be affected by the higher borrowing rate. Which investors will not be affected by the borrowing rate.

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Solution

Before y=.41

E(rc)= 7+ (15-7) x .55 =11.4 (before 10.28)

δc =.55 x 22=12.1 (before 9.02)

55.22301.

715

01.

)(22

p

p

A

rfrEy

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Solution

Which investors will not be affected by the borrowing rate y<1=> They are lending rather than borrowing. So are not affected by the borrowing rate.

The least risk-averse investors hold 100% in the risky portfolio so y=1.

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Solution

Any investor who is more risk tolerant (A<1.65) would borrow if the borrowing rate 7%

AAA

rfrEy

p

p

84.4

8

2201.

8

01.

)(1

22

65.184.4

8A

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Solution

For borrowers

Suppose A=1.1

Investor will borrow an amount equal to 50% of her own investment.

201.

)(

p

Bp

A

rfrEy

50.184.41.101.

715

y

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Solution

Raise the borrowing rate rfB=9%

Only 13% of her investment will be borrowed.

13.184.41.101.

915

y

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Question

You manage a risky portfolio with expected rate of return of 18% and standard deviation of 28%. The T-bill rate is 8%.

1. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected value and standard deviation of the rate of return on his portfolio?

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Expected return = (0.7 x 18%) + (0.3 x 8%) = 15%

Standard deviation = 0.7x28%= 19.6%

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2. Suppose that your risky portfolio includes the following investments in the given proportions:

Stock A: 25%

Stock B: 32%

Stock C: 43%

What are the investments proportions of your client’s overall portfolio, including the position in T-bills?

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Investment proportions:

30.0% in T-bills0.7 25% =17.5% in Stock A

0.7 32% = 22.4% in Stock B

0.7 43% = 30.1% in Stock C

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3. What is the reward-to variability ratio (S) of your risky portfolio? Your client’s?

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Your reward-to-variability ratio:

Client's reward-to-variability ratio:

3571.028

818S

3571.028

818S

3571.06.19

815S

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4. Draw the CAL of your own portfolio. What is the slope of the CAL? Show the position of your client on your fund’s CAL.

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ClientP

0

5

10

15

20

25

30

0 10 20 30 40

E(r)%

CAL (Slope = 0.3571)

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a. E(rC) = rf + y[E(rP) – rf] = 8 + y(18 8)If the expected return for the portfolio is 16%, then:

16 = 8 + 10 y

Therefore, in order to have a portfolio with expected rate of return equal to 16%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills.

b. Client’s investment proportions: 20.0% in T-bills

0.8 25% =20.0% in Stock A0.8 32% = 25.6% in Stock B0.8 43% = 34.4% in Stock C

c. sdC = 0.8 x sdP = 0.8 x 28% = 22.4%

8.010

816y

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a. sdC = y x 28%

If your client prefers a standard deviation of at most 18%, then:

y = 18/28 = 0.6429 = 64.29% invested in the risky portfolio

b. E(rC) = 8 + 10y = 8 + (10 x 0.6429)

= 8 + 6.429 = 14.429%

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a.

Therefore, the client’s optimal proportions are: 36.44% invested in the risky portfolio and 63.56% invested in T-bills.

b. E(rC) = 8 + 10y* = 8 + (10 x 0.3644) = 11.644%

C = 0.3644 x 28 = 10.203%

3644.044.27

10

285.301.0

818

A01.0

r)r(Ey

22P

fP*

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a. Slope of the CML;

The diagram follows.My fund allows an investor to achieve a higher mean for any given standard deviation than would a passive strategy, i.e., a higher expected return for any given level of risk.

20.025

813

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CML and CAL

0

2

4

6

8

10

12

14

16

18

0 10 20 30

Standard Deviation

Ex

pe

cte

d

Re

trun

CAL: Slope = 0.3571

CML: Slope = 0.20

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