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connect the determination of optimal portfolios with security analysis and actual buy/sell

transactions of investors. We will show in this section how the quest for efficient diversifica-

tion leads to a demand schedule for shares. In turn, the supply and demand for shares deter-

mine equilibrium prices and expected rates of return.

Imagine a simple world with only two corporations: Bottom Up Inc. (BU) and Top Down

Inc. (TD). Stock prices and market values are shown in Table 7.1. Investors can also invest in

a money market fund (MMF) that yields a risk-free interest rate of 5%.

Sigma Fund is a new actively managed mutual fund that has raised $220 million to invest

in the stock market. The security analysis staff of Sigma believes that neither BU nor TD will

grow in the future and, therefore, each firm will pay level annual dividends for the foreseeable

future. This is a useful simplifying assumption because, if a stock is expected to pay a stream

of level dividends, the income derived from each share is a perpetuity. The present value of

each shareâ€”often called the intrinsic value of the shareâ€”equals the dividend divided by

the appropriate discount rate. A summary of the report of the security analysts appears in

Table 7.2.

The expected returns in Table 7.2 are based on the assumption that next yearâ€™s dividends

will conform to Sigmaâ€™s forecasts, and share prices will be equal to intrinsic values at year-

end. The standard deviations and the correlation coefficient between the two stocks were esti-

mated by Sigmaâ€™s security analysts from past returns and assumed to remain at these levels for

the coming year.

Using these data and assumptions Sigma easily generates the efficient frontier shown in

Figure 7.1 and computes the optimal portfolio proportions corresponding to the tangency port-

folio. These proportions, combined with the total investment budget, yield the Fundâ€™s buy

orders. With a budget of $220 million, Sigma wants a position in BU of $220,000,000 .8070

$177,540,000, or $177,540,000/39 4,552,308 shares, and a position in TD of

$220,000,000 .1930 $42,460,000, which corresponds to 1,088,718 shares.

Sigmaâ€™s Demand for Shares

The expected rates of return that Sigma used to derive its demand for shares of BU and TD

were computed from the forecast of year-end stock prices and the current prices. If, say, a

share of BU could be purchased at a lower price, Sigmaâ€™s forecast of the rate of return on BU

would be higher. Conversely, if BU shares were selling at a higher price, expected returns

would be lower. A new expected return would result in a different optimal portfolio and a

different demand for shares.

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217

7 Capital Asset Pricing and Arbitrage Pricing Theory

TA B L E 7.2

Capital market expectations of Sigmaâ€™s portfolio manager and optimal portfolio weights

BU TD

Expected annual dividend ($/share) 6.40 3.80

Discount rate Required return* (%) 16 10

Expected end-of-year priceâ€ ($/share) 40 38

Current price 39 39

Expected return (%): Dividend yield (div/price) 16.41 9.74

Capital gain (P1 P0)/P0 2.56 2.56

Total rate of return 18.97 7.18

Standard deviation of rate of return (%) 40 20

Correlation coefficient between BU and TD ( ) .20

Risk-free rate (%) 5

Optimal portfolio weightâ€¡ .8070 .1930

*Based on assessment of risk.

â€

Obtained by discounting the dividend perpetuity at the required rate of return.

â€¡

Using footnote 3 of Chapter 6, we obtain the weight in BU as

2

[E(rBU) rf ] [E(rTD) rf ]

TD BU TD

wBU

2 2

[E(rBU) rf ] [E(rTD) rf ] [E(rBU) rf E(rTD) rf ]

TD BU BU TD

The weight in TD equals 1.0 wBU.

F I G U R E 7.1

45

Sigmaâ€™s efficient

CAL

Optimal Portfolio frontier and optimal

40

wBU 80.70% portfolio

wTD 19.30%

35

Mean 16.69%

Expected return (%)

30 Standard deviation 33.27% Efficient frontier

of risky assets

25

20

BU

15

10 Optimal portfolio

TD

5

0

0 20 40 60 80 100

Standard deviation (%)

We can think of Sigmaâ€™s demand schedule for a stock as the number of shares Sigma would

want to hold at different share prices. In our simplified world, producing the demand for BU

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

218 Part TWO Portfolio Theory

TA B L E 7.3

Calculation of Sigmaâ€™s demand for BU shares

Current Price Capital Gain Dividend Yield Expected Return BU Optimal Desired BU

($) (%) (%) (%) Proportion Shares

45.0 11.11 14.22 3.11 .4113 2,010,582

42.5 5.88 15.06 9.18 .3192 1,652,482

40.0 0 16.00 16.00 .7011 3,856,053

37.5 6.67 17.07 23.73 .9358 5,490,247

35.0 14.29 18.29 32.57 1.0947 6,881,225

46 Supply 5 million shares

44

Sigma Equilibrium

Price per share ($)

42 demand price $40.85

Aggregate

40

Index fund

(total) demand

demand

38

36

34

32

3 1 0 1 3 5 7 9 11

Number of shares (millions)

F I G U R E 7.2

Supply and demand for BU shares

shares is not difficult. First, we revise Table 7.2 to recompute the expected return on BU at dif-

ferent current prices given the forecasted year-end price. Then, for each price and associated

expected return, we construct the optimal portfolio and find the implied position in BU. A few

samples of these calculations are shown in Table 7.3. The first four columns in Table 7.3 show

the expected returns on BU shares given their current price. The optimal proportion (column

5) is calculated using these expected returns. Finally, Sigmaâ€™s investment budget, the optimal

proportion in BU, and the current price of a BU share determine the desired number of shares.

Note that we compute the demand for BU shares given the price and expected return for TD.

This means that the entire demand schedule must be revised whenever the price and expected

return on TD are changed.

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

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7 Capital Asset Pricing and Arbitrage Pricing Theory

40

Supply 4 million shares

Aggregate

40

Price per share ($)

demand

Sigma

39 demand

Equilibrium

39

price $38.41

Index fund

demand

38

38

37

3 2 1 0 1 2 3 4 5 6

Number of shares (millions)

F I G U R E 7.3

Supply and demand for TD shares

Sigmaâ€™s demand curve for BU stock is given by the Desired Shares column in Table 7.3

and is plotted in Figure 7.2. Notice that the demand curve for the stock slopes downward.

When BUâ€™s stock price falls, Sigma will desire more shares for two reasons: (1) an income

effectâ€”at a lower price Sigma can purchase more shares with the same budgetâ€”and (2) a

substitution effectâ€”the increased expected return at the lower price will make BU shares

more attractive relative to TD shares. Notice that one can desire a negative number of shares,

that is, a short position. If the stock price is high enough, its expected return will be so low that

the desire to sell will overwhelm diversification motives and investors will want to take a short

position. Figure 7.2 shows that when the price exceeds $44, Sigma wants a short position

in BU.

The demand curve for BU shares assumes that the price of TD remains constant. A similar

demand curve can be constructed for TD shares given a price for BU shares. As before, we

would generate the demand for TD shares by revising Table 7.2 for various current prices of

TD, leaving the price of BU unchanged. We use the revised expected returns to calculate the

optimal portfolio for each possible price of TD, ultimately obtaining the demand curve shown

in Figure 7.3.

Index Fundsâ€™ Demands for Stock

We will see shortly that index funds play an important role in portfolio selection, so letâ€™s see

how an index fund would derive its demand for shares. Suppose that $130 million of investor

funds in our hypothesized economy are given to an index fundâ€”named Indexâ€”to manage.

What will it do?

Index is looking for a portfolio that will mimic the market. Suppose current prices and mar-

ket values are as in Table 7.1. Then the required proportions to mimic the market portfolio are:

wBU 195/(195 156) .5556 (55.56%); wTD 1 .5556 .4444 (44.44%)

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

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