Question

Wang Life Insurance Company issues a three year annuity that pays 40,000 at the end of each year. Wang uses the following three bonds to absolutely match the cash flow under this annuity:

1. A zero coupon bond which matures in one year for 1,000.
2. A two year bond which matures for 1,200 and pays an annual coupon of 100. This bond is priced using an annual yield of 7%.
3. A three year bond which matures for 2,000 and pays annual coupons of 75. This bond has a price of 1,750.

It costs Wang 104,000 to purchase all three bonds to absolutely match this annuity.

Calculate the one year spot interest rate.

Answer 1

The One year Spot interest rate is 5.2335%. Details are as below:

Computation of One year Spot interst rate for Price and Cost of Bonds
Particulars	Amount
Required Cash Flow of Year 3	40000
Cash Flow of 3 year Bond at Year 3 ( Maturity 2000 + Coupen 75)	2075
No. of 3 years bonds Purchased for 40000 cash flow at year 3(40000/2075)	19.27711
Price per bond	1750
Cost of three year bonds	33734.94
Required Cash Flow of Year 2	40000
Less:- Coupen interest of three year Bonds ( 19.277 bonds * $ 75 coupen)	1445.783
Amount required from two year bond	38554.22
Cash Flow 2 year Bond at Year 2 ( Maturity 1200 + Coupen 100)	1300
No. of 2 years bonds Purchased for 38554.22 cash flow at year 2 (38554.22/1300)	29.65709
Price per Bond using effective annual yield of 7% (As per Working Note 1)	1228.928
Cost of two year bonds (29.65709 bonds @ 1228.928)	36446.44
Required Cash Flow of Year 1	40000
Less:- Coupen interest of three year Bonds ( 19.277 bonds * $ 75 coupen)	1445.783
Less:- Coupen interest of two year Bonds ( 29.65709 bonds * $ 100 coupen)	2965.709
Amount required from zero coupen bond	35588.51
Less:- Cost of Zero Coupen Bonds (working Note 2)	33818.62
Implied interest in Zero Coupen Bonds	1769.884
	0.052335
One year spot interest rate = interest/ Cost = 1769.884/33818.62 = 0.052335 or say 5.2335%
Working Note 1: Price of two year bond (at 7% effective yield)
Year	Cash Flow	Discount Factor @ 7% [1/(1+0.07)Year]	Present Value
1	100	0.93458	93.458
2	1300	0.87344	1135.470
Total			1228.928
Working Note 2 : Cost of Zero Coupen Bonds
Total Cost of Pruchasing all the three bonds			104000
Less:- Cost of three year Bond			-33734.9
Less:- Cost of two year Bond			-36446.4
Cost of Zero Coupen bonds			33818.62