Question

Suppose you are holding a stock portfolio worth $20 million at current market prices. The stock portfolio has a beta of 1.3. You are given the following data: • Today’s date: September 2020 • December NZSE10 futures expiration: Dec. 2020 (in exactly 3 months’ time from today) • Current NZSE10 index: 1,950.00 • December NZSE10 futures price (actual trade price): 1,989.00 • Current 3-month risk-free interest rate: 6 percent per annum (continuously compounded) • Dividend yield on NZSE10 index: 0.00 percent per annum (continuously compounded) • Futures contract size: One futures contact is equal to $25 times the index value.

a. What is the theoretical price today for the December NZSE10 futures contract?

b. If the December NZSE10 futures is not at fair value as at today’s date, what arbitrage trade should you do? Also calculate the arbitrage profit at maturity for one futures contract that you may trade.

c. You are worried that market may crash between now and December, when the futures contract on the NZSE10 index will expire. How many December NZSE10 futures contracts should you trade (indicate long or short) to reduce the overall portfolio beta of your current portfolio to 0.50. (Ignore basis risk between the NZSE10 futures and the cash index). In your calculations use the actual traded December NZSE10 futures price, rather than the theoretical futures price that you calculated in part a. of this question. That is, for this part of the question only you can assume the actual December NZSE10 futures price equals the theoretical futures price.

Answer 1

Information provided in the question

Portfolio value = $ 20 million

Portfolio

Spot price of Index = 1950

Futures:

Duration = 3 months (September - December)

Spot price = 1989

Interest = 6% p.a. (continuously compounded)

Dividend = 0% p.a. (continuously compounded)

1 Future = $25 times index value

Solution

a)

Theoretical futures price or fair futures price = Spot price + Cost to carry - Convenience yield

Cost to carry for the index futures = Interest

Convienience yield = Dividend = Nil (since dividend yield is 0% p.a.)

Theoretical futures price = Spot price of index * (where e represents continuous compounding function, t=time, r=rate) = 1950 * = 1950 * = 1950 * 1.0151 = 1979.445

b)

If the December NZSE10 futures is not at fair value as at today's date, arbitrage opportunity exists. In the current example, actual futures price is more than theoretical futures price, i.e. futures contract is overpriced. December futures are trading at $1989.

The arbitrage strategy would be to sell the index futures @ $1989 and buy index fund at $1950

• Borrow $1950 at 6% p.a. continuously compounded for 3 months, Buy 1 index share at spot for $1950, Sell stock futures at $1989
• Repay borrowing with interest after 3 months = 1950 * = 1950 * = 1950 * 1.0151 = 1979.445
• Sell the index share in futures market on maturity date at $1989 and use the proceeds to repay borrowing
• Arbitrage profit per share = $1989 - $1979.445 = $9.555
• This profit is riskless and represents profit on one futures contract.

c) Cash market value of the portfolio = $20 million

Position to be takes in the futures market to hege against fall in portfolio value: Sell index futures

Value to be hedged = [Target - Current ] * Cash market value = [1.3 - 0.5]*$20 million = $16 million

Number of futures contracts to be sold = $16 million / (1989 * 25) = 322 contracts (round off, as decimal number of contracts cannot be purchased)

Proof of hedge:

Suppose the market crashes and index value falls by 10%, then portfolio value falls by 10 * 1.3 = 13%

i.e. Loss due to fall in portfolio value = $20 million * 13% = $2.6 million

Futures market position: Sell index @ $1989, buy index at 1989*90% = $1790.1, Gain per index = 1989 - 1790.1 = $198.9

Profit credited in margin account = $198.9 * 25 index * 322 contracts = $1.6 million

Hence, due to futures hedging fall in portfolio value of $2.6 million is now restricted to (2.6 - 1.6 = $1 million). In other words, the 13% change is now restricted to 5%. When the market index changes by 10%, the portfolio value changes by 5%, and the portfolio beta is reduced to 0.5.