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in a competitive securities market. To understand how market equilibrium is formed we must
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 share”often called the intrinsic value of the share”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 year™s dividends
will conform to Sigma™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 Sigma™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 Fund™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.

Sigma™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, Sigma™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.
Bodie’Kane’Marcus: II. Portfolio Theory 7. Capital Asset Pricing © The McGraw’Hill
Essentials of Investments, and Arbitrage Pricing Companies, 2003
Fifth Edition Theory




217
7 Capital Asset Pricing and Arbitrage Pricing Theory



TA B L E 7.2
Capital market expectations of Sigma™s portfolio manager and optimal portfolio weights

BU TD

Expected annual dividend ($/share) 6.40 3.80
Discount rate Required return* (%) 16 10
Expected end-of-year price† ($/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 weight‡ .8070 .1930

*Based on assessment of risk.

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

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
Sigma™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 Sigma™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
Bodie’Kane’Marcus: II. Portfolio Theory 7. Capital Asset Pricing © The McGraw’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 Sigma™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, Sigma™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.
Bodie’Kane’Marcus: II. Portfolio Theory 7. Capital Asset Pricing © The McGraw’Hill
Essentials of Investments, and Arbitrage Pricing Companies, 2003
Fifth Edition Theory




219
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



Sigma™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 BU™s stock price falls, Sigma will desire more shares for two reasons: (1) an income
effect”at a lower price Sigma can purchase more shares with the same budget”and (2) a
substitution effect”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 Funds™ Demands for Stock
We will see shortly that index funds play an important role in portfolio selection, so let™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 fund”named Index”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%)
Bodie’Kane’Marcus: II. Portfolio Theory 7. Capital Asset Pricing © The McGraw’Hill
Essentials of Investments, and Arbitrage Pricing Companies, 2003
Fifth Edition Theory

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