# Why is the CAPM securities market line straight?

Let $\gamma$ be the expected return, in terms of its exponential growth rate, of the market asset. If we set $\gamma=\mu-\sigma^2/2$ as explained by the Doléans-Dade exponential, then the expected return of a balanced portfolio with fraction $\beta$ invested in the market asset, and the remainder lent or borrowed at risk-free rate, is $$R = r_f + \beta(\mu-r_f) - \beta^2\sigma^2/2.$$ I have plotted $R$ against $\beta$ in the following chart,

where for purposes of the chart $r_f=0.04$, $\gamma=0.13$, and $\sigma=0.2$. I know this effect is not my imagination, because Fernholz and others have quantified the "excess returns" of a balanced portfolio (where the green line lies above the red line) in their framework of "stochastic portfolio theory", and I myself have noticed this and alluded to it in my answer to How to calculate compound returns of leveraged ETFs?

Risk aversion notwithstanding, I find it absurd to think that unlimited expected gains are available simply by being highly leveraged in the market. So why does the CAPM use a straight line as if this were the case?

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The efficient frontier should be expressed in terms of arithmetic returns since only these returns can account for cross-sectional aggregation. Hence, if you assume the log returns of the risky portfolio are $X_{p} \sim N(\mu,\sigma^{2})$, then you first have to convert it to log-normal moments before combining it with the risk-free rate, $r_{f}$. However, it should be noted that while the median equals the mean for the normal distribution, that is not the case for the log normal distribution. Hence, the median arithmetic return will not show the same linear relationship against the standard deviation that the mean does. Similarly, if using CVaR as a measure of risk, then the risk will increase by even more as leverage increases due to the non-normality of the log normal distribution.

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Thank you! You've explained it very well for me. Sorry, not registered can't upvote for now. –  justin-- Nov 15 '12 at 20:09