Question

Alpha Insurance Company is obligated to make payments of $2 million, $3 million, and $4 million at the end of the next three years, respectively. The market interest rate is 8% per annum.

i. Determine the duration of the company’s payment obligations.

ii. Suppose the company’s payment obligations are fully funded and immunized using both 6-month zero coupon bonds and perpetuities. Determine how much of each of these bonds the company will hold in the portfolio. (7 marks )

Answer 1

1) The duration of a recurring payment (or cash received) is nothing but the weighted average of the time of payment, which can be calculated as follows:

	1	2	3	4
	YEAR	Payment Obligation	PV:Nominal Value/(1+annual rate)^year	Share of cash flow in PV (col-3/sum(col-3))	Weighted time of payment col-4*col-1
	1	2000000	1851851.852	0.243690456	0.243690456
	2	3000000	2572016.461	0.338458967	0.676917934
	3	4000000	3175328.964	0.417850577	1.25355173
Sum			7599197.277		2.17416012

So the duration would be 2.174 years

2)

a) six months zero coupon bonds:

The primary point to understand here is that the immunization simply means that the bonds in which the company would invest to meet its future obligations can be safely assumed to have a non-fluctuating 8% interest rate. With that in mind, we can draw a cash flow structure which pays a semi annual interest rate of 4% (8%/2). The cash flow table would look like this:

1	2	3	4
	Payment Obligation	Nominal Value of Bond with Reinvested Int	Less the Payment Obligation (col-3-col-4)
0.5	-	XXX*(1.04)	XXX*(1.04)-(col-2)
1	20,00,000	(XXX*(1.04)-(col-2))*1.04
1.5	-
2	30,00,000
2.5	-
3	40,00,000
				Initial bond portfolio Value	XXX

Populating the rest of the values, and solving for XXX we get the initial bond portfolio value.

1	2	3	4
	Payment Obligation	Nominal Value of Bond with Reinvested Int	Less the Payment Obligation (col-3-col-4)
0.5	-	78,77,774	78,77,774
1	20,00,000	81,92,885	61,92,885
1.5	-	64,40,601	64,40,601
2	30,00,000	66,98,225	36,98,225
2.5	-	38,46,154	38,46,154
3	40,00,000	40,00,000	-
		Initial bond portfolio Value	75,74,783.10

Please note that this result can be calculated very easily using the goal seek function in EXCEL.

Ans. 75,74,783.10

b) Perpetuity:

The same obligation can be fulfilled by a perpetuity paying cash flows, the table for it can be drawn in the similar way.

	Payment Obligation	Nominal Value of perpetuity payments+ Reinvested Int	Less the Payment Obligation (col-3-col-4)
0.5	-	14,44,980	14,44,980
1	20,00,000	29,47,759	9,47,759
1.5	-	24,30,650	24,30,650
2	30,00,000	39,72,856	9,72,856
2.5	-	24,56,750	24,56,750
3	40,00,000	40,00,000	-
		Corpus for perpetuity	3,61,24,500.89

Ans: 3,61,24,500.89