## Demand For Stocks And Equilibrium Prices

So far we have been concerned with efficient diversification, the optimal risky portfolio and its risk-return profile. We haven't had much to say about how expected returns are determined in a competitive securities market. To understand how market equilibrium is formed we need to connect the determination of optimal portfolios with security analysis and the actual buy/sell transactions of investors. We will show in this section how the quest for efficient diversification leads to a demand schedule for shares. In turn, the supply and demand for shares determine 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 9.1. Investors can also invest in a money market fund (MMF) which 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, that 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. Therefore, 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 9.2.

The expected returns in Table 9.2 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 estimated 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 9.1 and computes the optimal portfolio proportions corresponding to the tangency portfolio. These proportions, combined with the total investment budget, yield the fund's

260 PART III Equilibrium in Capital Markets

Figure 9.1 Sigma's efficient frontier and optimal portfolio.

Figure 9.1 Sigma's efficient frontier and optimal portfolio.

buy orders. With a budget of $220 million, Sigma wants a position in BU of $220,000,000 X .8070 = $177,540,000, or $177,540,000/39 = 4,552,308 shares, and a position in TD of $220,000,000 X .1930 = $42,460,000, which corresponds to 1,088,718 shares.

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