# Optimal number of contracts

• The optimal number of contracts (N*) to hedge a portfolio consisting of NA number of units and where Qf is the total number of futures being used for hedging

• In the case of a stock index the similar logic follows. If P is the value of the portfolio of stocks held by an investor and A is the current value of the stocks lying under one futures contract then the optimal hedge ratio, N*, should be equal to P / A
• In practical cases investors don’t typically have portfolios that trace the index. Hence the concept of β comes into play. Beta (β) is a measure of a stock's volatility in relation to the market. Then
• In order to change the beta (β) of the portfolio to (β*), we need to long or short the (N*) number of contracts depending on the sign of (N*)

Example

If you have a portfolio of $500,000 which mirrors S&P500. Each S&P500 contract is$250 times the index when the index is at 500. Calculate the number of contracts to be hedged?

Solution

A = 250 * 500 = 125,000. Then N* = 500,000/125,000 = 4 contracts should be shorted for
the hedge.

Example
The current value of the S&P 500 index is 1,457, and each S&P futures contract is for delivery of US$250 times the index. A long‐only equity portfolio with market value of US$300,100,000 has beta of 1.1. To reduce the portfolio beta to 0.75, how many S&P futures contract should you sell? (FRM Sample Paper 2009).

Solution
[(0.75 – 1.1)/ 1] * [300,100,000 / {250 * 1,457}] = ‐288.36 -> sell 288 contracts

Rolling forward a hedge:
• We can use a series of futures contracts to increase the life of a hedge
• Each time we switch from 1 futures contract to another we incur a type of basis risk
• Strip Hedge
• Stack and Roll Hedge

Expect 2 questions based on Stack and Roll hedge or MG case.
Please refer original case study for Stack and Roll Hedge.