Question

Providence Health Care is obligated to make a payment of $300,000 in exactly three years. In order to provide for this obligation, their financial officer decides to purchae a combination of one-year zero-coupon bonds and four-year zero-coupon bonds. Each of these is sold to yield an annual effective yield of 4%. How much of each type of bond should be purchased so that the present value and duration conditions of Redington immunization are satisfied? Is the convexity condition also satisfied at i = 4%?

The answer in the back of the book says: $88,899.64 of one-year bond, $177,799.27 of four-year bond; yes.

But whats the process of getting this answer using the appropriate formulas?

Answer 1

Let's Calculate the duration of the liability:

Period(X)	PV of Cashflow(W)[Discount factor@4%]	XW
1	0	0
2	0	0
3	$3,00,000*1/(1.04)3 = $ 266698.91	$800096.72
Total	$266698.91	$800096.72

Duration of liability = XW/W =  $800096.72/ $266698.91= 3 years.

Duration of 1 year Maturity Zero Coupon Bond = 1

Duration of 4 year Maturity Zero Coupon Bond =4

Total Investment to be made today = $266698.91

Investments should be made in two bonds in such a ratio that the duration of bond portfolio is 3.

Lets invest W1 in 1 year maturity bonds and the balance of investment amount in 4 years maturity bonds

W1(1) + (1-W1)(4)= 3

W1 =1/3

W2 =2/3

Investment in 1 year Maturity = 1/3 * $266698.91 = $ 88,899.64

Investment in 4 year Maturity = 2/3 * $266698.91 = $177,799.27