Assume you own the following portfolio:

It is $\frac{1}{15}$ and you want to hedge till $\frac{3}{31}.$ You have the following contracts on the S&P$500$ index

available to you. The correlation between your portfolio and the E$-$mini.S&P$500$ index is $-82$

a$.$ Which is the appropriate maturity of the contract?

b$.$ Should you go long or short on the futures contract?

c$.$ What are the optimal number of contracts?

d$.$ How effective can you expect your hedge to be$?$

Answer:

a$.$ Since contract has to expire after the hedge date the appropriate contract is lune

b$.$ I have a long position and the correlation is negative so I would take a long position in the futures

contract

c$.$ Number of futures contract is given by

$N_f=(\frac{{}_s}{{}_f})(\frac{S}{f})$

Rounding weights to $2$ decimal places:

${}_5=\mathrm{.}25**1\mathrm{.}2+\mathrm{.}25**1\mathrm{.}4+\mathrm{.}28**1\mathrm{.}3+\mathrm{.}23**1\mathrm{.}6=1\mathrm{.}4(roundingto1$ decimal place$)$

Market value of portfolio $(S)=13,000,000$

$f=2840**50=142,000$

$N_t={1\mathrm{.}4}^{**}(13,000,\frac{000}{142},000)=128\mathrm{.}2=128$ contracts

d$.$ Hedging effectiveness $=-{0\mathrm{.}82}^2=\mathrm{.}67=67\%$