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Security Pricing in an Imperfect Capital Market

Published online by Cambridge University Press:  19 October 2009

James C.T. Mao
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Extract

A perfect capital market is a key assumption in recent theories of security pricing. It is assumed that the costs of transactions, information-gathering, and portfolio management are all zero, and that no investor is so large as to exert an appreciable effect on either the risk-free interest rate or the yield on risky securities. If, in this perfect capital market, investors have identical decision horizons and homogeneous expectations, then there is a unique optimal portfolio of risky securities. Since this unique portfolio must include every security in proportion to its relative valuation in the capital market, it is referred to as the “market” portfolio. When the capital market reaches equilibrium, the expected return of every security will be a linear function of the expected return of the market portfolio. From this relationship Lintner and Mossin have separately derived valuation formulas that express the market price of a security as a function of the security[s end-of-period expected value, its risk as measured by the variance and covariances of this end-of-period value, the market price of risk within the portfolio, and the risk-free rate of interest.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 6 , Issue 4 , September 1971 , pp. 1105 - 1116
Copyright
Copyright © School of Business Administration, University of Washington 1971

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References

1 See, for example, Lintner, John, “Security Prices, Risk, and Maximal Gains from Diversification,” Journal of Finance 20 (December 1965), pp. 587–616Google Scholar; Sharpe, William F., “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk,” Journal of Finance 19 (September 1964), pp. 425–442Google Scholar; and Mossin, J., “Security Pricing and Investment Criteria in Competitive Markets,” American Economic Review 59 (December 1969), pp.749–756Google Scholar. A summary and discussion of the assumptions underlying Lintner[s and Sharpe[s models can also be found in Fama, Eugene F., “Risk, Return and Equilibrium: Some Clarifying Comments,” Journal of Finance, 23 (March 1968), pp. 29–40.Google Scholar

2 Lintner, “Security Prices, Risk, and Maximal Gains,” p. 598. See also his paper, “The Aggregation of Investor[s Diverse Judgments and Preferences in Purely Competitive Security Markets,” Journal of Financial and Quantitative Analysis 4 (December 1969), pp. 347–400.CrossRefGoogle Scholar

3 Mossin, “Security Pricing,” p. 752.

4 See the papers cited in footnote 2.

5 Jensen, Michael C., “Risk, the Pricing of Capital Assets, and the Evaluation of Investment Portfolios,” Journal of Business 42 (April 1969), pp. 186–192.Google Scholar

6 Since these two questions are discussed in my “Essentials of Diversification Strategy,” Journal of Finance, 25 (December 1970), pp. 1109–1121, only the main results are summarized here.CrossRefGoogle Scholar

7 Tobin, James, “Liquidity Preference as Behavior Towards Risk,” Review of Economic Studies 25 (February 1958), pp. 65–85.CrossRefGoogle Scholar

8 Unless otherwise stated, the return on a security is defined here as the excess over the risk-free rate of interest.

9 This formula is derived by solving the equation α where α is the relative gain in θ desired by the investor. See my paper mentioned in footnote 6.

10 In the diagonal model presented below, the coefficient of correlation, ρ, between any two securities equals , where all terms are as defined in the model. For a sample of 224 stocks, Latane and Young found that b and a. had average values of 1.00 and 0.30 respectively and that σπ had a value of .2111, all covering the period 1953–60.[See Latané, Henry A. and Young, William E., “Test of Portfolio Building Rules,” Journal of Finance 24 (September 1969), pp. 597–598, tables 1 and 2.] These figures imply an average value of 4/9 for ρ.CrossRefGoogle Scholar

11 This analysis agrees with recent empirical studies on the relationship between the number of securities in a portfolio and the reduction in portfolio variance. See, for example, Evans, John L. and Archer, Stephen H., “Diversification and the Reduction of Dispersion: An Empirical Analysis,” Journal of Finance 23 (December 1968), pp. 761–767Google Scholar, and Latané, Henry A. and Young, William E., “Test of Portfolio Building Rules,” Journal of Finance 24 (September 1969), pp. 595–612.Google Scholar

12 For a formal proof, see my forthcoming paper, “Essentials of Portfolio Diversification Strategy,” Journal of Finance

13 Computer algorithms, such as IBM's “1401 Portfolio Selection Program (1401–F1–04X),” are available for implementing this procedure.

14 Since the value of θ is constant, the value of φ must also be constant.

15 It should be noted that , but .

16 Since the random variables , underlying λk are measured in percentages, we must redefine the variables by multiplying them by Tk, the total dollar value of the portfolio, in order to calculate the corresponding price of risk in dollars. This transformation magnifies the portfolio return by a multiple of Tk, and the portfolio risk by a multiple of . Hence, if λk, is the price of risk in percentages,λk/ Tk is the corresponding measure in dollars.

17 Lintner, “Security Prices, Risk, and Maximal Gains.”