Question

A portfolio manager has a $33m position in an equity portfolio which tracks the YAL100 index. The manager is concerned about the possibility of a short term fall in the index and consequent decrease in the value of his portfolio. The fund manager decides to hedge using futures written on the YAL100 index. The current value of the index is 7,500 points with a continuously compounded dividend yield of 3.8%. The portfolio has a beta of 1.1 with respect to the index. The relevant futures contract has 6 months to maturity and has a contract multiple of $25 per full index point. The risk-free rate of interest is 4.5%.

(a) Calculate the futures position required to hedge the portfolio using a beta hedge.

(b) After 3 months the spot price of the index falls to 7,200 points and the futures position is closed out. What will be the new quoted futures price and what will be the gain or loss on the futures and spot positions and the return on the hedged portfolio?

Answer 1

Solution:

we have, Current futures price (S) = 7500, Dividend yield(y) = 3.8% continuosly, Risk free rate = 4.5% continuosly(assumed), Contract period = 6 months

First, we will calculate 6 months futures price

F = S * e(r-y)*t

= 7500 * e(0.045 - 0.038)*0.5

= 7500 * 2.7180.0035 (Value of e=2.718 approximately)

= 7500 * 1.0035

= 7526.25

a) No. of Futures contract position to hedge the portfolio

= Vp (Bt - Bp) / F * m * Bf , Where

Vp = value of portfolio, Bt= target beta, Bp = beta pf portfolio, F= Futures value, m= contract multiple, Bf = beta of futures

= 330,00,000 (0 - 1.1) / 7526.25 * 25 * 1

= - 363,00,000 / 188,156.25

= - 192.92 i.e we will sell 193 Contracts @ 7526.25

b) After 3 months futures position is closed out, Spot price = 7200 points

So, the new quoted futures price for remaining 3 months maturity will be

F = S * e(r-y)*t

= 7200 * e(0.045 - 0.038)* 0.25

= 7200 * 2.7180.0004

= 7200 * 1.0004

= 7202.88

So,

Gain from futures position = ( 7526.25 - 7202.88) * 193 * $25

= $ 15,60,260.25

% fall in index points = 300 / 7500 *100 = 4%

Therefore Portfolio value will fall by (4 * 1.1) = 4.4%

i.e Loss on Portfolio = 330,00,000 * 4.4% = $ 14,52,000

Net gain from Hedgeing = $15,60,260.25 - $ 14,52,000

= $ 1,08,260.25