Question

Tree row bank has assets of $150 million, liabilities of $135 million, and equity of $15 million. The asset duration is six years and the duration of the liabilities is four years. Market interest rates are 10 percent. Tree row bank wishes to hedge the balance sheet with treasury bond futures contracts, which currently have a prim quote of $95 per $100 face value for the benchmark 20-year, 8 percent coupon underlying the contract, a market yield of 8.5295 percent, and a duration of 10.3725 years. A. Should the bank go short or long on the futures contracts to establish the correct macrohedge? B. How many contracts are necessary to fully hedge the bank? Need help with B - i don't know how figure out Df = 10.3725 years

Answer 1

Duration of a bond = sum of [time of cash flow*weight of cash flow] where

time of cash flow = time of annual payments made; weight of cash flow = present value (PV) of cash flow/total PV of cash flows = PV of cash flow/bond price

Duration calculation:

Annual cash flow (from T = 1 to T = 19) = annual coupon rate*par value = 8%*100 = 8

Annual cash flow (at T = 20) = annual coupon + par value = 8 + 100 = 108

Formula		CF/(1+annual yield)^n	PV of CF/Bond price	w*n
Time until payment (n)	Cash flow (CF)	PV of CF	Weight (w)	Weighted Time
1	8	7.37	0.08	0.08
2	8	6.79	0.07	0.14
3	8	6.26	0.07	0.20
4	8	5.77	0.06	0.24
5	8	5.31	0.06	0.28
6	8	4.90	0.05	0.31
7	8	4.51	0.05	0.33
8	8	4.16	0.04	0.35
9	8	3.83	0.04	0.36
10	8	3.53	0.04	0.37
11	8	3.25	0.03	0.38
12	8	3.00	0.03	0.38
13	8	2.76	0.03	0.38
14	8	2.54	0.03	0.37
15	8	2.34	0.02	0.37
16	8	2.16	0.02	0.36
17	8	1.99	0.02	0.36
18	8	1.83	0.02	0.35
19	8	1.69	0.02	0.34
20	108	21.01	0.22	4.42
	Bond price (BP)	95.00	Duration (in years)	10.3725

Bond duration = sum of weighted time column = 10.3725 years

a). The bank needs to sell futures contracts for hedging the balance sheet. If interest rates increase then equity value will decrease but since futures contracts value will also decrease, the bank can buy them to offset the decrease in equity value. Hence, equity will be hedged.

b). Number of futures contracts required = - (duration of assets - (liabilities/assets)*duration of liabilities)*asset value]/(duration of futures contract*price of futures contract)

One treasury bond futures contract is sold in minimum units of $100,000 so price of one treasury bond contract will be (100,000*95/100) = 95,000

Number of futures contracts = - [(6 - (135/150)*4)*150]/(10.3725*95,000) = -365 contracts (sell 365 contracts)