Question

 

1. Suppose a bank has an asset duration of 4 years and a liability duration of 2.75 years. This bank has $1,050 million in assets and $780 million in liabilities. They are planning on trading in a Treasury bond future which has a duration of 7.5 years and which is selling right now for $98,500 for a $100,000 contract. How many futures contracts does this bank need to fully hedge itself against interest rate risk?

2. Suppose a Eurodollar time deposit futures contract has a duration of .6 years and has a current market price of $965,000. Market interest rates are 7.5 percent and are expected to fall to 6.5 percent. What is the change in this futures contract's market price from this change in interest rates?

3. An investor purchases one September T-bond futures contract at 116-210. The settlement price for the contract on next day is 118-120. What is the marked-to-market gain/loss for the investor?

4. Suppose a $100,000 T-Bond futures contract whose underlying bond’s duration is 6 years and has a current market price of $98,765. Market interest rates are 3 percent today but are expected to rise to 4 percent. What is the expected change in this futures contract's market price as a result of this change in interest rates?

5. ABC Bank’s asset portfolio has an average duration of 4 years and its liability portfolio has an average duration of 3 years. The bank has $500 million in total assets and $400 million in liabilities. ABC Bank is thinking about hedging its risk by using a Treasury bond futures contract whose underlying’s duration is 5 years and has a price of $97,650. How many futures contracts will it need to hedge its risk?

Answer 1

(1).

Asset Duration = 4 years = D(a) and Liability Duration = 2.75 years = D(l)

Asset Size = $ 1050 million and Liability Size = $ 780 million

k = Ratio of Liabilities to Assets = (780 / 1050) = 0.7428 ~ 0.743

As D(a) > k x D(l) = 4 > 0.743 x 2.75, the duration gap is positive which implies that an increase in the interest rate erodes asset value by a greater amount than it erodes liability values, thereby leading to a reduction in equity. The opposite happens in case of interest rate decrease. Assets gain more in value than liabilities, thereby elevating equity. A positive duration gap is hedged by taking a position in the T-bond futures that benefits from a rise in interest rates (as positive duration gaps face the risk of shrinking equity (assets - liabilities) when interest rates rise).Such a position would entail selling short the T-bonds so that a negative (decrease) in equity owing to interest rate rise is compensated by the difference between the selling and buying price of the T-Bond Future contracts.

Price of each contract = $ 98500, Face Value of Contract = $ 100000 and Duration of Futures = 7.5 years

Number of Futures Required = [D(a) - k x D(l)] x Asset Value / Duration of Futures x Price of Futures

= [4 - (780/1050) x 2.75] x 1050 / [7.5 x (98500/100000)] = 278.17 ~ 278 contracts as contract number can't be a decimal number.

(2).

Bond maturity value = $965,000 * (1+7.5%)^0.5

= $1,000,533

Market value at 6.5% yield = $1,000,533 / (1+6.5%)^0.5

= $969,520

Therefore, change in value = $969,520 - $9,65,000

= $4,520

(3).

Marking-to-market refers to the process of reconciling an assets daily loss/gain using either the last traded price or an average of the last few traded prices. This price which is used to reconcile an asset's daily loss/gain is known as the settlement price.

Settlement Price = $ 118-120 where 120 is actually 12 points with each point being worth $31.25

Settlement Price (in $) = 118 x 1000 + 12 x 31.25 = $118375

Purchase Price = $ 116-210 where again 210 is 21 points with each point being worth $31.25

Purchase Price (in $) = 116 x 1000 + 21 x 31.25 = $ 116656.25

Marked to Market Gain = 118375 - 116656.25 = $ 1718.75 (as settlement price is greater than the purchase price the investor has a marked to market gain instead of a loss).

(4).

(5).