Many results from the theory of finance are framed in terms of the expected
gross rate of return \(E\left(R_{i}\right)=E\left(x_{i}\right) / p_{i}\) on a
risky asset. In this problem you are asked to derive a few of these results.
a. Use Equation 17.37 to show that \(E\left(R_{i}\right)-R_{f}=\)
\\[
-R_{f} \operatorname{Cov}\left(m, R_{i}\right)
\\]
b. In mathematical statistics the Cauchy-Schwarz inequality states that for
any two random variables
\(x\) and \(y,|\operatorname{Cov}(x, y)| \leq \sigma_{x} \sigma_{y}\) Use this
result to show that
\\[
\left|E\left(R_{i}\right)-R_{j}\right| \leq R_{j} \sigma_{m} \sigma_{k}
\\]
c. Sharpe ratio bound. In finance, the "Sharpe ratio" is defined as the excess
expected return of a risky asset over the risk-free rate divided by the
standard deviation of the return on that risky asset. That is, Sharpe ratio
\(=\left[E\left(R_{4}\right)-R_{f}\right] / \sigma_{R_{i}} .\) Use the results
of part
(b) to show that the upper bound for the Sharpe ratio is \(\sigma_{m} / E(m) .\)
(Note: The ratio of the standard deviation of a random variable to its mean is
termed the "coefficient of variation," or \(C V\). This part shows that the
upper bound of the Sharpe ratio is given by the \(C V\) of the stochastic
discount rate.
d. Approximating the \(C V\) of \(m\). The stochastic discount factor, \(m,\) is
random because consumption growth is random. Sometimes it is convenient to
assume that consumption growth follows a "lognormal" distribution-that is, the
logarithm of consumption growth follows a Normal distribution. Let the
standard deviation of the logarithm consumption growth be given by
\(\sigma_{\ln \Delta c}\) Given these assumptions, it can be shown that \(C
V(m)=\sqrt{e^{r^{2}} \tan \alpha}-1 .\) Use this result to show that
an approximation to the value of this radical can be expressed as \(C V(m)
\cong \gamma \sigma_{\ln \Delta c}\)
e. Equity premium paradox. Search the Internet for historical data on the
average Sharpe ratio for a broad stock market index over the past 50 years.
Use this result together with the rough estimate that \(\sigma_{\ln 3 e}
\approx .01\) to show that parts
(c) and (d) of this problem imply a very high value for individual's relative
risk aversion parameter \(\gamma\). That is, the relatively high historical
Sharpe ratio for stocks can only be justified by our theory if people are much
more risk averse than is usually assumed. This is termed the "equity premium
paradox." What do you make of it?
