Question

Suppose a hedge fund follows the following strategy: Each month it holds \(100 million of an S&P 500 Index fund and writes out-of-the-money put options on \)100 million of the index with exercise price 5% lower than the current value of the index. Suppose the premium it receives for writing each put is $.25 million, roughly in line with the actual value of the puts.

a. Calculate the Sharpe ratio the fund would have realized in the period October 1982–September 1987. Compare its Sharpe ratio to that of the S&P 500. Use the data from the previous problem available at the Online Learning Center, and assume the monthly risk-free interest rate over this period was .7%.

b. Now calculate the Sharpe ratio the fund would have realized if we extend the sample period by one month to include October 1987. What do you conclude about performance evaluation and tail risk for funds pursuing option like strategies?

Short answer

a.0 .275

b. 0.080

Step-by-step solution

Calculation of Sharpe ratio ‘a’

From the data available at the link, let’s calculate the rate of return for each month between October 1982 to September 1987.

Since the end of the month value for S&P 500 in September 1982 was $120.42, hence, exercise price for the October Put:

$\begin{array}{rcl}\text{Exercise price}& =& \text{Exercise price in \%×Current value of index}\\ & =& 0.95\times \$120.42\\ & =& \$114.3990\\ & & \end{array}$

The rate of return earned by hedge fund on index:

$\begin{array}{rcl}\text{Rate of return on hedge funds}& =& \left(\frac{\text{Price at the end of month}}{\text{Price in the beginning of month}}\right)-1\\ & =& \left(\frac{\$133.72}{\$120.42}\right)-1\\ & =& 0.11045\text{or 11.045\%}\\ & & \end{array}$

(Because October end of month value was 133.72)

Assuming the hedge fund invests 0.25 million along with $100 million beginning of month value, the end of month value of the fund:

$$\begin{gathered} \text{Ending value = Investment}\times \left(1+\text{Rate of return}\right) \\ =\$100.25\text{Million}\times \text{1.11045} \\ \text{= \$ 111.32 Million} \end{gathered}$$

The rate of return for the month:

$\begin{array}{rcl}\text{Rate of return on hedge funds}& =& \left(\frac{\text{Price at the end of month}}{\text{Price in the beginning of month}}\right)-1\\ & =& \left(\frac{\$111.322}{\$100}\right)-1\\ & =& 0.11322\text{or 11.322\%}\end{array}$

The May end of month value for the index was $150.55, and therefore the payout for the writer of a put option on one unit of the index is:

$\begin{array}{rcl}\text{Payout for put option}& =& \text{Exercise price of may put}-\text{Price at the end of month}\\ & =& \$152.0475-\$150.55\\ & =& \$1.4975\\ & & \end{array}$

The rate of return the hedge fund earns on the index:

$\begin{array}{rcl}\text{Rate of return on hedge funds}& =& \left(\frac{\text{Price at the end of month}}{\text{Price in the beginning of month}}\right)-1\\ & =& \left(\frac{150.55}{160.05}\right)-1\\ & =& -0.05936\text{or}-\text{5.936\%}\\ & & \end{array}$

The payout of 1.4975 per unit of the index reduces the hedge fund’s rate of return by:

$$\begin{gathered} \text{Payout =}\frac{\text{Payout per unit of index}}{\text{Price in the beginning of month}} \\ =\frac{1.4975}{160.05} \\ =0.00936\text{or 0.936\%} \end{gathered}$$

The rate of return the hedge fund earns :

$$\begin{gathered} \text{Rate of return = Rate of return on hedge - Payout rate} \\ =-5.936\%-0.936\% \\ =-6.872\% \end{gathered}$$

The end of month value of the fund is:

$$\begin{gathered} \text{Ending value = Current value×exercise price} \\ =\$100.25\text{Million}\times \text{0.93128} \\ \text{= \$ 93.361 Million} \end{gathered}$$

The rate of return for the month is:

$\begin{array}{rcl}\text{Rate of return on hedge funds}& =& \left(\frac{\text{Price at the end of month}}{\text{Price in the beginning of month}}\right)-1\\ & =& \left(\frac{\$93.361}{\$100}\right)-1\\ & =& -0.06639\text{or}-6.639\%\\ & & \end{array}$

For the period October 1982-September 1987:

Mean monthly return = 1.898%

Standard deviation = 4.353%

$\begin{array}{rcl}\text{Sharpe ratio}& =& \frac{\text{E}\left({\text{r}}_{\text{p}}-{\text{r}}_{\text{f}}\right)}{\text{P}}\\ & =& \left(\frac{0.1898-0.007}{0.4353}\right)\\ & =& 0.0275\\ & & \end{array}$

Calculation of Sharpe ratio ‘b’

Given values for October 1982 – October 1987

Mean monthly return = 1.238%

Standard deviation = 6.724%

$\begin{array}{rcl}\text{Sharpe ratio}& =& \frac{\text{E}\left({\text{r}}_{\text{p}}-{\text{r}}_{\text{f}}\right)}{\text{P}}\\ & =& \left(\frac{0.1238-0.007}{0.06724}\right)\\ & =& 0.080\\ & & \end{array}$

On extending the sample period, the mean monthly return decreases but the standard deviation jumps by more than 2% points. This implies that there is considerable trail risk for the fund.