Question

It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.6. The company would like to use the CME December futures contract on the S&P 500 to change the beta of the portfolio to 0.3 during the period July 16 to November 16. The December S&P 500 index futures price is currently 1,000, and each contract is on $250 times the index.

(a) What position should the company take? How many December S&P 500 futures contracts should the company buy or sell now?

(b) Suppose that the company changes its mind and decides to increase the beta of the portfolio from 1.6 to 1.8. What position in futures contracts should it take? How many December S&P 500 futures contracts should the company buy or sell now?

Answer 1

We know, stock index futures are used by the speculators or the portfolio managers tp hedge against their potential future losses intheir equity stock holdings /equity portfolio.
A stock index measures the stock market performance by comparing the past share price movements with the current share prices. They are the benchmarks for evaluating the market performance by considering a hypothetical portfolio of stocks. For example, S&P 500 is a popular stock market Index for the American stock exchange which includes the 75% of all the stocks traded in the US markets or rather, we can say, the stocks of the biggest 500 US companies.

The underlying asset for a stock index future is the particular stock index. For example, with S&P 500 Index Future, the investor can buy or sell the future contracts based on the S&P index to speculate or hedge against their position depending upon the appreciation or depreciation of the S&P index value. The value of the S&P 500 future contract is always $250 times of the then prevailing S&P index value.

The formula for the no. of contracts to be shorted by using the stock index future can be expressed mathematically as :-  

N = E / A , { this is the equation no. 1}

where, N= no. of contracts to be bought or sold
E = value of the equity portfolio
A = the value of the stock underlying one future contract

As per the given problem, the current S&P 500 index future price is 1000 & each cotract is $250 times the index.

Therefore, A = 250 * 1000 = 250,000

In the above given problem, we have to chage the of the portfolio, say from, ₁ to _2, where (₁ > ₂).

Hence to reduce the of the portfolio from ₁ to  ₂ , a short position is needed to be taken.

The no. of contracts required to take a short postion= (₁ - ₂ ) * E/A {derived from the equation no. 1}

In other case, if we need to increase the of the portfolio from ₁ to ₂, such that, (₁ < ₂), along postion is needed to be taken.

The no. of contracts required to take a long position = (₂ - ₁) * E/A {derived from the equation no. 1}

(a) In the given problem, ₁ = 1.6 &  ₂ = 0.3, i.e. ( _{1 > 2})

The value of the portfolio or E = $100 million = $100,000,000

A = 1000 * 250 = 250,000

Since the co. needs to decrease the beta of the portfolio from ₁ to ₂,

Therefore, the company needs to take a short position.

The no. of contracts required to take a short postion as per the aforesaid formula :- (₁ - ₂ ) * E/A

{(1.6 - 0.3) * 100,000,000} / 250,000

= (1.3 * 100,000,000) / 250,000

= 130,000,000 / 250,000

= 520

Hence, 520 no. of contracts required to take a short position by the company.

(b)

In the given problem, ₁ = 1.6 &  ₂ = 1.8, i.e. ( _{1 < 2})

The value of the portfolio or E = $100 million = $100,000,000

A = 1000 * 250 = 250,000

Since the co. needs to increase the beta of the portfolio from ₁ to ₂,

Therefore, the company needs to take a long position.

The no. of contracts required to take a long postion as per the aforesaid formula :- (₂ - ₁ ) * E/A

{(1.8 - 1.6) * 100,000,000} / 250,000

= (0.2 * 100,000,000) / 250,000

= 20,000,000 / 250,000

= 80

Hence, 80 no. of contracts required to take a long position by the company.