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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, plot1

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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3 Answers 3

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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

It is straight line because it is defined by y=mx+c equation. No other reason. It is not adequate and certainly something that you can to gain infinite returns. You want to make a model that accounts for convexity like you have , or use a multi factor model like Fama French or your own model which is evolving and is a N factor polynomial function ,please do so . Only reason why a linear approximation is used is because it is simple and sort of works. You can read about fama french model here .

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Fama French has more factors, but it's still a straight-line model. –  justin-- Nov 15 '12 at 17:50
    
My point , you want a curve add higher powers it you can justify. Everyone knows CAPM is not sufficient Risk vs Return gauge. –  ash Nov 15 '12 at 17:54

The CAPM is just a model, not the truth nobody believes it. So you shouldn't apply it blindly. As a model it is useful but it has larger defects than this.

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