a) Using FRAs FRA rate 4.82% (3 – 7), since the investment will take place in three months‟ time for a period of 4 months. If interest rates increase by 0.9% to 4.99%

Investment return = 4.79% × 4/12 × $24,000,000 = $383,200 Payment to BK Bank = (4.99% – 4.82%) × $24,000,000 × 4/12 = $(13,600 Net receipt = $369,600 Effective annual interest rate = 369,600/24,000,000 × 12/4 = 4.62%

If interest decrease by 0.9&% to 3.19%

Investment return = 2.99% × 4/12 × $24,000,000 = $239,200 Receipt from BK Bank = (4.82% – 3.19%) × 24,000,000 × 4/12 = $130,400 Net receipt = $369,600 Effective annual interest rate (as above) 4.62%

Using futures Need to hedge against a fall in interest rates, therefore go long in the futures market (i.e. buy futures).

TL needs March contracts as the investment will be made on 1 February. No. of contracts needed = $24,000,000/$2,000,000 × 4 months/ 3 months = 16 contracts.

Basis Current price (on 1/11) – futures price = total basis (100 – 4.09) – 94.76 = 1.15 Unexpired basis = 2/5 × 1.15 = 0.46 If interest rates increase by 0.9% to 4.99%

Investment return (from above) = $383,200 Expected futures price = 100 – 4.99 – 0.46 = 94.55 Loss on the futures market = (0.9455 – 0.9476) × $2,000,000 × 3/12 × 16 = $(16,800) Net return = $366,400 Effective annual interest 366,400/24,000,000 × 12/4 = 4.58%

If interest rates decrease by 0.9% to 3.19% Investment return (from above) = $239,200 Expected futures price = 100 – 3.19 – 0.46 = 96.35 Gain on the futures market = (0.9635 – 0.9476) × $2,000,000 × 3/12 × 16 = $127,200 Net return $366,400 Effective annual interest rate (as above) 4.58%

Using options on futures Need to hedge against to fall in interest rates, therefore buy call options. As before, TL needs 16 March call option contracts ($24,000,000/$2,000,000) × 4 months/3months). If interest rates increase by 0.9% to 4.99%

Exercise price 94.50 95.00 Futures price 94.55 94.55 Exercise? Yes No Gain in basis points 5 0 Underlying investment return (from above) $383,200 $383,200 Gain on options (0.0005 × 2,000,000 × 3/12 × 16) $4,000 $0 Premium

0.00432 × $2,000,000 × 3/12 × 16 $(34,560 0.00121 × $2,000,000 × 3/12 × 16 $(9,680) Net return $352,640 $337,520 Effective interest rate 4.41% 4.22%

Alternative presentation of calculation

Forward rate agreement FRA rate 4.82% (3 – 7), since the investment will take place in three months‟ time for a period of 4 months.

Note: Ref. rate = reference rate, which in this question is the inter-bank rate. Futures: March contracts to buy at 94.76 or 5.24% (100 – 94.76). No of contracts needed =$24,000,000/$2,000,000 × 4 months/3 months =16 contracts.

Basis Current basis (on 1/11) = (100 – spot interest rate) – current futures price = total basis = (100 – 4.09) – 94.76 = 1.15 Unexpired basis = 2/5 × 1.15 = 0.46

• So if interest rates increase by 0.9% to 4.99% the estimated futures price is 100 – (4.99 + 0.46) = 94.55% (or 5.45%). • If interest rates decrease by 0.9% to 3.19% the estimated futures price is 100 – (3.19 + 0.46) = 96.35 (or 3.65%).

Options on futures March call options at 5% (95.00) or 5.5% (94.50) can be chosen. This solution illustrates the outcome if 5% is chosen, which is the rate closest to the current inter-bank rate. Again 16 contracts will be needed.

Discussion The FRA offer from BK Bank gives a slightly higher return compared to the futures market; however, TL faces a credit risk with over-the-counter products like the FRA, where BK Bank may default on any money owing to TL if interest rates should fall. The March call option at the exercise price of 94.50 seems to fix the rate of return at 4.41%, which is lower than the return on the futures market and should therefore be rejected. The March call option at the exercise price of 95.00 gives a higher return compared to the FRA and the futures if interest rates increase, but does not perform as well if the interest rates fall. If TL takes the view that it is more important to be protected against a likely fall in interest rates, then that option should also be rejected. The choice between the FRA and the futures depends on TL‟s attitude to risk and return; the FRA gives a small, higher return, but carries a credit risk. If the view is that the credit risk is small and it is unlikely that BK Bank will default on its obligation, then the FRA should be chosen as the hedge instrument.

b) The delta value measures the extent to which the value of a derivative instrument, such as an option, changes as the value of its underlying asset changes. For example, a delta of 0.8 would mean that a company would need to purchase 1.25 option contracts (1/0.8) to hedge against a rise in price of an underlying asset of that contract size, known as the hedge ratio. This is because the delta indicates that when the underlying asset increases in value by $1, the value of the equivalent option contract will increase by only $0.80.

The option delta is equal to N(d1) from the Black-Scholes option pricing formula. This means that the delta is constantly changing when the volatility or time to expiry change. Therefore, even when the delta and hedge ratio are used to determine the number of option contracts needed, this number needs to be updated periodically to reflect the new delta.