Question

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?

Short answer

#tag_title#Question#tag_content#Derive the formula for the difference between the expected gross rate of return and risk-free rate of return using Equation 17.37, then, use the Cauchy-Schwarz inequality to show that the absolute value of the difference between expected returns has an upper limit. Finally, define and derive the Sharpe ratio bound in terms of the stochastic discount rate and approximate the coefficient of variation of the given stochastic discount factor. Discuss the equity premium paradox using historical data and how it implies high values for relative risk aversion. #tag_title#Answer#tag_content#The formula for the difference between the expected gross rate of return and risk-free rate of return is given by \(E(R_i)-R_f = -R_f \operatorname{Cov}(m, R_i)\), where \(R_i\) is the rate of return on a risky asset, \(R_f\) is the risk-free rate, and \(m\) is the stochastic discount factor. Using the Cauchy-Schwarz inequality, we can establish an upper limit for the absolute value of the difference between expected returns as \(|E(R_i) - R_j| \leq R_j \sigma_{m} \sigma_{k}\). The upper bound for the Sharpe ratio can be derived as \(\frac{\sigma_m}{E(m)}\). The approximation for the coefficient of variation of the given stochastic discount factor is \(CV(m) \approx \gamma \sigma_{\ln \Delta c}\). The equity premium paradox refers to the puzzlingly high difference between the returns on risky assets and risk-free assets, which implies a high level of risk aversion among investors. This can be discussed using historical data on the average Sharpe ratio of a broad stock market index and the implications of a high relative risk aversion parameter \(\gamma\).

Step-by-step solution

Write down Equation 17.37

Equation 17.37 states that \(R_f = E(R_i m)\), where \(R_f\) is the risk-free rate, \(R_i\) is the rate of return on risky asset, and \(m\) is the stochastic discount factor.

Apply the expectation operator on both sides

Take expectations on both sides: \(E(R_f) = E(E(R_i m)) = E(R_i m)\)

Rearrange the equation to express \(E(R_i)\) in terms of other variables

Now, express \(E(R_i)\) in terms of other variables: \(E(R_i) = \frac{E(R_i m)}{E(m)}\)

Subtract \(R_f\) from both sides of the equation and simplify

Subtract \(R_f\) from both sides and simplify: \(E(R_i)-R_f = -R_f \operatorname{Cov}(m, R_i)\) Part b: Using the Cauchy-Schwarz inequality

Write down the Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality states that for any two random variables \(x\) and \(y\), \(|\operatorname{Cov}(x, y)| \leq \sigma_{x} \sigma_{y}\).

Apply the inequality to the given variables

Apply the inequality to \(x = m\) and \(y = R_i - R_j\): \(|\operatorname{Cov}(m, R_i - R_j)| \leq \sigma_{m} \sigma_{k}\)

Simplify the inequality

By rearranging the terms: \(|E(R_i) - R_j| \leq R_j \sigma_{m} \sigma_{k}\) Part c: Sharpe ratio bound derivation

Write down the definition of the Sharpe ratio

Sharpe ratio \(=\frac{E(R_i)-R_f}{\sigma_{R_i}}\)

Apply the result from part (b) to the Sharpe ratio

Recall that the upper bound for the difference in returns is given by \(R_j \sigma_{m} \sigma_{k}\). As a result, the upper bound on the Sharpe ratio is \(\frac{R_j \sigma_m \sigma_k}{\sigma_{R_i}}\).

Simplify the expression

After canceling out the \(\sigma_{R_i}\) terms, the upper bound for the Sharpe ratio is given by \(\frac{\sigma_m}{E(m)}\). Part d: Approximating the coefficient of variation

Write down the given equation for the coefficient of variation

Coefficient of variation (CV) of \(m\) is given by \(\sqrt{e^{\gamma^2} \tan \alpha}-1\).

Apply the approximation

Apply the given approximation: \(CV(m) \approx \gamma \sigma_{\ln \Delta c}\) Part e: Equity premium paradox Since part (e) requires searching the internet for historical data on the average Sharpe ratio and discussing your own thoughts about the equity premium paradox, this part cannot be provided in a step-by-step manner. However, instructions are given below on how to proceed: 1. Search for historical data on the average Sharpe ratio for a broad stock market index over the past 50 years. 2. Use the value of \(\sigma_{\ln \Delta c} \approx .01\) and the result from part (d) to discuss the implication of high relative risk aversion parameter \(\gamma\). 3. Explain the equity premium paradox and provide your own thoughts on the matter.