A manager has a $100 million portfolio that consists of 50% stock and 50% bonds.

Question

A manager has a $100 million portfolio that consists of 50% stock and 50% bonds. The beta of the stock position is 1.3. The modified duration of the bond position is 2. The manager wishes to achieve an effective mix of 70% stock and 30% bonds. The price and beta of the stock index futures contracts are $412,562 and 1 respectively. (The futures price includes the effect of the index multiplier.) The price, modified duration, and yield beta of the bond futures contracts are $99,580, 4, and 1 respectively. What is the appropriate strategy?

Short 88 bond futures contracts and go long 63 stock index futures contracts.

Go long 176 bond futures contracts and short 51 stock index futures contracts.

Short 176 bond futures contracts and go long 63 stock index futures contracts.

Go long 88 bond futures contracts and short 15 stock index futures contracts.

Short 88 bond futures contracts and go long 63 stock index futures contracts.

Go long 176 bond futures contracts and short 51 stock index futures contracts.

Short 176 bond futures contracts and go long 63 stock index futures contracts.

Go long 88 bond futures contracts and short 15 stock index futures contracts.

Explanation / Answer

Soln : As per the given condition, at present the portfolio contains $50 mn of stocks and $50mn of bonds

Now, the protfolio is to be made of 70% stocks and 30% bond or we can say that 70 mn of stocks and 30mn of bond is required

Keeping the beta and duration of the stock and bond same as of original portfolio i.e. beta = 1.3 and modified duration = 2

Now, beta can be additive and weighted amount is to be taken to calculate the portfolio beta.

So, Let X be the amount of stock future to be invested

Portfolio stock beta = w1*betastocks + w2* betafutures = 1.3

(50/70)*1.3 + (X/70)* 1 = 1.3

On calculating we get the value of X = 26 million or we can say number of stock future index required = 26mn/412562 = 63

Now, in case of bonds, lets calculate the combined duration of the bond added and existing, Let the value of the future bonds be Y

In that case duration will be = (50/30)* 2 + (4/1.1)*Y/30 = 2

On solving we get the value of no. of bond future = Y/99580 = -88

Option a should be correct in this case.

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