# Risk Compensation

I try to understand the different ways to compensate for risk.

In the CAPM, when we plot the excess return against the risk, we find that portfolios of interest lie on the efficient frontier (i.e. the Markowitz-bullet). Among these, the tangency (i.e. market) portfolio has the highest Sharpe ratio, and is therefore to be preferred over the other ones.

The Sharpe ratio measures the compensation required for additional risk. It makes sense to me that if two portfolios have the same risk, but one has a higher excess return, it has a higher Sharpe ratio, and we should favor it over the other one. But, in my understanding, it also implies that portfolios with the same Sharpe ratio should be treated somewhat the same (e.g., if portfolio A is twice as risky as portfolio B, then we simply require twice the return, and we are compensated for the additional risk).

How is this linear relationship between risk and return justified? I can see that this could be explained by saying that identical Sharpe ratio is equivalent to identical Value at Risk (assuming normal distributions). Is this justification correct?

Another notion I encountered is the one of Value Added (VA), which, given a risk-aversion parameter $\lambda$, is $VA = r-\lambda \sigma^2$. Here, $r$ is the return, and $\sigma$ is the variance. In some contexts, it is stated that $VA$ tells us how much compensation we require for additional risk, and managers should seek to maximize VA (see Grinold & Kahn, Active Portfolio Management). The situation is similar to before: If two portfolios have the same VA, a manager with fixed risk preference would not prefer one portfolio over the other - even though one is riskier, it gives an adequate amount of higher return. (I understand that the concept of VA is quite heuristic, but I don't intend to question it here.) But obviously, VA is a nonlinear relation between risk and return, and in general, two portfolios with the same Sharpe ratio will not have the same VA.

In the CAPM, why don't we try to maximize VA instead of the Sharpe ratio? (and therefore obtain a different tangency/market portfolio).

• BTW, your objective function of $\max_{\mathbf{w}} \mathbf{w}' \boldsymbol{\mu} - \lambda \mathbf{w}'\Sigma\mathbf{w}$ is just the objective of an investor who only cares about mean and variance. You then obtain the classic result that everyone holds leveraged positions on the same portfolio, the tangency portfolio. (Leverage depends on investor's $\lambda$). Since everyone holds the tangency portfolio, market clearing then implies the tangency portfolio is the market portfolio and with some algebra, you obtain the CAPM. – Matthew Gunn Dec 16 '17 at 16:02

• A linear relationship between expected returns and covariance with a risk factor is a necessary consequence of a linear asset pricing function
• In theory, a CAPM relationship can be derived when a pricing kernel $S$ is affine in the return of the market portfolio. Different sets of assumptions lead to this affine relationship. Be aware that the CAPM is an empirical failure; do not use the CAPM for empirical asset pricing.

### A consequence of a linear asset pricing function

Let $p(X)$ be an asset pricing function that gives today's price of a random payoff $X$. A straightforward assumption is that $p$ should be a linear functional. Then by the Riesz representation theorem, there exists a pricing kernel (aka stochastic discount factor) $S$ such that $p$ can be written as an inner product of $S$ and $X$.

$$p(X) = \operatorname{E}[S X]$$

The basic intuition is that with linearity, you can go back and forth between (1) security prices and (2) state prices (prices for outcomes from the sample space $\omega \in \Omega$). With complete markets, state price density $S$ is uniquely determined; with incomplete markets, $S$ is not uniquely determined. An academic dream has been to use macroeconomic theory to derive a state price density that is empirically consistent with security prices.

### Rearranging above to write as a classic, regression beta model

If $X$ is a return, then $p(X) = 1$. Some algebraic manipulation of the above equation (eg. see John Cochrane's book Asset Pricing (revised)) allows you to represent this as a regression beta model:

\begin{align*} \operatorname{E}[R_i] - R^f &= \frac{\operatorname{Cov}(R_i, S)}{\operatorname{Var}(S)} \lambda_S \\ &= \beta_{R_i, S} \lambda_S\end{align*}

A linear asset pricing function $p$ implies that expected excess returns $\operatorname{E}[R_i] - R^f$ are linear with the covariance of $R_i$ and stochastic discount factor $S$.

The whole academic asset pricing game is about figuring out what $S$ is.

(Note that some papers have taken potshots at linearity, eg. Lamont and Thaler (2003), "Can the Market Add and Subtract?").

If you're a math guy, something to have in the back of your head is that all this mean-variance frontier stuff is just classic linear algebra in disguise. It's not super intuitive, but it's not that complicated either. For a technical exposition, perhaps see Hansen and Richard (1987) or for more intuition, Cochrane's exposition in chapter 5.3.

### An economic interpretation of $S$

Simple economic arguments say that pricing kernel $S$ should be given by the ratio of marginal utility of consumption:

$$S_{t, t+1} = \frac{u'(C_{t+1})}{u'(C_t)}$$.

If $S$ is affine in the market return, then you get the CAPM relationship between expected returns and covariance with market returns. A lot of different sets of assumptions can get you there.

### References

Cochrane, John, 2005 Asset Pricing (revised)