Capital-Asset Pricing

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 65 in Conjugate Duality in Economic Analysis, 2026, pp 501-508 from Springer

Abstract: Abstract The analysis here focuses on the allocation of risk and the implication of risk for asset pricing. Working with a simple mean/variance utility, we calculate portfolio demand. We obtain the capital-asset pricing model (Sharpe, W. F. (1964, September). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, XIX (3), 425–442). By setting the aggregate portfolio demand equal to the aggregate supply (the market portfolio of all assets), we solve to obtain the equilibrium price of assets. The risk premium on an asset is proportional to the covariance of its payoff with the payoff on the market portfolio. Via primal and dual efficiency problems, we also study efficient risk sharing. How can the market portfolio be divided among investors so that the risk is shared efficiently? We find a simple necessary and sufficient condition for efficient risk sharing. The aggregate risk of the market portfolio is divided among the investors according to risk aversion. Each investor holds a fraction of the market portfolio plus some positive or negative quantity of the risk-free asset (each lends or borrows via the risk-free asset). The payoff for one investor is perfectly correlated with the payoff for another. The standard deviation of the payoff is inversely proportional to absolute risk aversion. An investor having high risk aversion holds mostly the risk-free asset, whereas an investor having low risk aversion makes a leveraged investment in the market portfolio. We obtain first and second theorems of efficient risk sharing, a variant of the first and second theorems of welfare economics. Asset-market equilibrium shares risk efficiently. Conversely, an efficient allocation of risk can be obtained as a market equilibrium, for some asset price.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_65

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