Sharpe Ratio for a Two-Stock Portfolio with a Risk-Free Asset
Summary
The document explains how to express the Sharpe ratio of a portfolio containing two risky stocks as a function of one stock’s weight. With the other stock’s weight set to the remainder, expected return is a weighted average, while portfolio variance depends on both assets’ variances and their covariance. Subtracting the risk-free return and dividing by portfolio volatility gives the Sharpe ratio for that risky allocation.
When a risk-free asset is included, its return shifts the portfolio’s expected return while risk comes from the risky allocation, assuming the risk-free claim has no variance. The response describes the Sharpe ratio as the slope of the risk-return line and notes that the mean-variance utility parameter is not needed for this initial expression. It does not calculate a numerical result because the referenced table of asset inputs is absent from the text, and one secondary answer appears to misstate the portfolio weights.
Key ideas
- Set the two risky-asset weights to sum to one and express return as a function of one weight.
- Compute portfolio variance using both individual variances and the covariance term.
- The Sharpe ratio is excess expected return divided by portfolio standard deviation.
- A risk-free allocation changes expected return but contributes no variance under the stated assumption.
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# Set up sharpe ratio with 2 risky portfolio
# Set up sharpe ratio with 2 risky portfolio
You are considering an investment in the stock. In the stock market, there are two risky stocks (A and B) and a risk free claim, C (you can think of it as the t-bill). The covariances and returns of these three stocks are described in the following table: Assume that you have a mean-variance utility function with risk aversion A=5. That is, your utility function is
Let P be the risky portfolio that consists of stocks A and B. Let Wpa be the weight of stock A in portfolio P and let Wpb = 1-Wpa be the weight of stock B in the portfolio P. Write down the Sharpe ratio of portfolio P as a function of Wpa...
> I am having trouble with how to set this equation up to start, any guidance would be helpful to start this off
## Answer by Dr.Raghnar (score 1)
https://quant.stackexchange.com/a/18785
Sharpe Ratio is defined as the slope of the line that is the return in function of the variance of a portfolio composed by a risky and a risky-less asset.
Hence, if you have a bunch of risky assets (A,B) and a risky-less (C) you simply calculate the efficient portfolio for your risky asset (A,B), and then calculate the Sharpe ratio for the tangential portfolio.
In other words the properties of the combination of A,B assets (given $x_A + x_B = 1$) are, $$\mu_e = \mu_A x_A + \mu_B x_B = \mu_A x_A + \mu_B (1-x_A)=x_A(\mu_A - \mu_B)+\mu_B, $$ $$\sigma^2_e = \sigma^2_A x^2_A + \sigma^2_B x^2_B + \sigma_{AB}x_A x_B = x^2_A (\sigma^2_A+\sigma^2_B-2\sigma_{AB}) +2 x_A(\sigma_{AB} - \sigma^2_B) + \sigma^2_B,$$ with $mu$ the expected returns, and $\sigma^2$ the risks.
This results that when you consider the risk-free asset C in the mix you have $$\mu_p = (1-x_e) r_f + x_e \mu_e = r_f + x_e(\mu_e-r_f),$$ with $r_f$ the risk-free return, in your case the 4% return of C. The variance is given only by the risky part of the portfolio, therefore $$\sigma^2_p = x^2_e \sigma^2_e,$$ from which one can conveniently define the weight to be given to the efficient portfolio as $$x_e = \sigma_p / \sigma_e,$$ determining that $$\mu_p = (1-x_e) r_f + x_e \mu_e = r_f + \frac{(\mu_e-r_f)}{\sigma_e}\sigma_p.$$
The angular coefficient of such line, telling us that one can tune the expected return on the volatility using the risk-free rate as buffer (and eventually shorting on it), is known as Sharpe Ratio, $$\frac{(\mu_e-r_f)}{\sigma_e}.$$
If you substitute $\sigma_e$ and $\mu_e$ with the above given equations, you have the sharpe ratio in function of $x_A$ (or Wpa as you call it).
For this first assignment the Utility function is not needed, but I guess it will come handy later when it will ask to use it to calculate the desired weights.
## Answer by arodrisa (score 0)
https://quant.stackexchange.com/a/15005
Well, you need to calculate the expected return of your portfolio, and the volatility. Taking into account that your weights are 100% long in A, 100% short in B: $$ E(r)_A-E(r)_B $$ Same for the volatility of the portfolio.
Then just calculate the Sharpe ratio: $$ \frac{E(r)_{portf}-E(r)_{rf}}{Volatility_{portf}} $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.