Pg 101, #18.17 Hull, Chapter 19: The Greek Letters - PQ set

[Dr. Jayanthi Sankaran]

Hi David,

As referenced above and cited below:

Problem 18.17

A fund manager has a well-diversified portfolio that mirrors the performance of the S&P 500 and is worth $360 million. The value of the S&P 500 is 1,200, and the portfolio manager would like to buy insurance against a reduction of more than 5% in the value of the portfolio over the next six months. The risk-free interest rate is 6% per annum. The dividend yield on the portfolio is 4% per annum, the dividend yield on the S&P 500 is 3% per annum, and the volatility of the index is 30% per annum. The portfolio has a beta of 1.5.

Hull's approach

When the value of the portfolio goes down 5% in six months, the total return from the portfolio, including dividends, in the six months is -5 + 2 = -3% i.e. -6% per annum. This is 12% per annum less than the risk-free interest rate. Since the portfolio has a beta of 1.5 we would expect the market to provide a return of 8% per annum less than the risk-free interest rate i.e. we would expect the market to provide a return of -2% per annum. Since the dividends on the market index are 3% per annum, we would expect the market index to have dropped at the rate of 5% per annum or 2.5% per six months i.e. we would expect the market to have dropped to 1170. A total of 450,000 = (1.5 x 300,000) put options on the S&P 500 with exercise price 1170 and exercise date in six months are therefore required.

My question is: When the value of the portfolio goes down 5% in six months, the total return from the portfolio, including dividends, in the six months is -5 + 2 = -3% i.e. -6% per annum. However, we would expect the market to provide a return of -2% per annum, and since the dividends on the market index are 3% per annum, we would expect the market index to have dropped at the rate of 5% per annum. Why is it that in the case of the portfolio, the dividend yield of 2% for six months stems the decline of 5% in portfolio value. Whereas, in the case of the S&P 500 the dividend yield of 3% per annum causes a further decline in the return? Is it because in the case of the portfolio, dividends are reinvested whereas in the case of the S&P 500 they are not?

Thanks!
Jayanthi

 

[David Harper CFA FRM]

Hi @Jayanthi Sankaran

It does look that way, doesn't it!? I really had to write these assumptions down to figure this out. For me, the problem is Hull's muddy explanation and the fact we are switching between (i) annual and semi-annual and (ii) between price return and total return. Arggh. But ultimately I think he's consistent because here is how I interpret the "final" answer:

Riskfree rate, Rf = 6.0% (for both of course). Please note "price" versus "total" where total return = price return + dividend yield; all returns are per annum

• Portfolio:
  • Threshold price return = -10%
  • Threshold total return = -6% = -10% + 4% div
  • Excess total return = -12% = -6% threshold total return - 6% Rf
• Market
  • Threshold price return = -5%;
  • Threshold total return = -2% = -5% + 3% div
  • Excess total return = -8% = -2% threshold total return - 6% Rf
    • such that per CAPM: Portfolio's excess total return = beta*Market's excess total return --> -12% = -8%*1.5

I hope that shows that, in the case of the S&P 500 div yield of 3%, Hull does include it: the market's implied total return of -2% is reduced to a price return of -5%. Thanks,

 

[Dr. Jayanthi Sankaran]

Thanks David for the clear and lucid explanation. And for taking the time to work through this problem in detail. Much appreciate it!

Jayanthi

 

[David Harper CFA FRM]

sure thing @Jayanthi Sankaran I really do like to work through other people's problems, it helps keep me sharp and open to other approaches!

 

[Dr. Jayanthi Sankaran]

Thanks for saying that David - much appreciate it!

Jayanthi