Question

Risk aversion is embedded in what part of CAPM. Show the process.

Answer 1

The Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk and expected return for assets, particularly stocks. CAPM is widely used throughout finance for pricing risky securities and generating expected returns for assets given the risk of those assets and cost of capital.

ERi​=Rf​+βi​(ERm​−Rf​)

where:

ERi​=expected return of investment

Rf​=risk-free rate [risk aversion]

βi​=beta of the investment

(ERm​−Rf​)=market risk premium​

Investors expect to be compensated for risk and the time value of money. The risk-free rate in the CAPM formula accounts for the time value of money. The other components of the CAPM formula account for the investor taking on additional risk.

Process: Let (σM, rM) denote the point corresponding to the market portfolio M. All portfolios chosen by a rational investor will have a point (σ, r) that lies on the so-called capital market line r = rf + rM − rf σM σ,--(1)

the efficient frontier for investments. It tells us the expected return of any efficient portfolio, in terms of its standard deviation, and does so by use of the so-called price of risk rM − rf σM , (2) the slope of the line, which represents the change in expected return r per one-unit change in standard deviation σ. If an individual asset i (or portfolio) is chosen that is not efficient, then we learn nothing about that asset from (1). It would seem useful to know, for example, how ri −rf , the expected excess rate of return is related to M. The following formula involves just that, where σM,i denotes the covariance of the market portfolio with individual asset i: Theorem 1.1 (CAPM Formula) For any asset i ri − rf = βi(rM − rf ), where βi = σM,i σ 2 M , is called the beta of asset i. This beta value serves as an important measure of risk for individual assets (portfolios) that is different from σ 2 i ; it measures the nondiversifiable part of risk. More generally, for any portfolio p = (α1, . . . , αn) of risky assets, its beta can be computed as a weighted average of individual asset betas: rp − rf = βp(rM − rf ), where βp = σM,p σ 2 M = Xn i=1 αiβi . Before proving the above theorem, we point out a couple of its important consequences and explain the meaning of the beta. For a given asset i, σ 2 i tells us the risk associated with its own fluctuations about its mean rate of return, but not with respect to the market portfolio. For example if asset i is uncorrelated with M, then βi = 0 (even presumably if σ 2 i is huge), and this tells us that there is no risk associated with this asset (and hence no high expected return) in the sense that the variance σ 2 i can be diversified away (recall our example in Lecture Notes 4 of diversification of n uncorrelated assets). The idea would be to collect a large number of uncorrelated (and uncorrelated with M) assets and form a portfolio with equal proportions thus dwindling the variance to 0 so it becomes like the risk-free asset with deterministic rate of return rf . So, in effect, in the world of the market you are not rewarded (via a high expected rate of return) for taking on risk that can be diversified away. So we can view βi as a measure of nondiversifiable risk, the correlated-with-the-market part of risk that we can’t reduce by diversifying. This kind of risk is sometimes called market or systematic risk. It is not true in general that higher beta value βi implies higher variance σ 2 i , but of course a higher beta value does imply a higher expected rate of return: you are rewarded (via a high expected rate of return) for taking on risk that can’t be diversified away; everyone must face this kind of risk.