## Sigmas Demand for Shares

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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.

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 shares is not difficult. First, we revise Table 9.2 to recompute the expected return on BU at different 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 9.3. The first four columns in Table 9.3 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 is changed.

Sigma's demand curve for BU stock is given by the Desired Shares column in Table 9.3 and is plotted in Figure 9.2. 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

CHAPTER 9 The Capital Asset Pricing Model 261

 (%) BU Optimal Proportion Desired BU 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

Figure 9.2 Supply and demand for BU shares.

Figure 9.2 Supply and demand for BU shares. 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 9.2 shows that when the price exceeds \$44, Sigma wants a short position in BU.

The demand curve for BU shares assumes that the price and therefore expected return of TD remain 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 9.2 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 9.3. 