Question

(20 marks) Question 4 Pension funds pay lifetime annuities to recipients. If a firm remains in business indefinitely. the pension obligation will resemble a perpetuity. Suppose that you are managing a pension fund with obligations to make perpetual payments of $3 million per year to beneficiaries. The yield to maturity on all bonds is 10%. To fully fund and immunize the obligations, you want to set up an immunization strategy using the following two high-grade corporate bonds. Delta Bond A 4-year maturity bond has a 7% coupon rate, paid annually. : A 30-year maturity bond has a 3% coupon rate, paid semi-annually. It sells today for $337.47. The Macaulay duration of this bond is 12.17 Gamma Bond : years. Required: Determine the number of each of these corporate bonds you will want to hold to both fully fund and immunize the obligations (20 marks)

Answer 1

Step 1 - Current price of Delta bond: FV = 1,000; N = 4; PMT = coupon rate*par value = 7%*1,000 = 70; rate (YTM) = 10%, solve for PV

PV = 904.90

Step 2- Macaulay duration calculation for Delta bond: If interest rate (YTM) drops by 100 bp then

change in price/price = - (duration * change in (1+y))/(1+y)

Price when YTM = 9%: FV = 1,000; N = 4; PMT = 70; rate = 9%, solve for PV. PV = 935.21

Change in price = 935.21 - 904.90 = 30.30

Change in yield = -0.01

1+y = 1+YTM = 1+10% = 1.10

Applying Macaulay duration equation,

(30.30/904.90) = -Duration*(-0.01/1.10)

Duration D = 3.68 years

Step 3- Current price of Gamma bond = 337.47

Macaulay duration of Gamma bond = 12.17 years

Step 4 - In order to immunize, the duration of the pension obligation has to match the duration of the portfolio made up of Delta and Gamma bonds. The portfolio duration is the weighted average duration of both bonds. If w is the fraction to be invested in Delta bonds then the fraction to be invested in Gamma bonds will be (1-w).

Duration for the pension obligation will be (1+y)/y where y = yield on all bonds

= (1+10%)/10% = 11 years

For immunization,

11 = 3.68w + 12.17(1-w)

11 - 12.17 = 3.68w - 12.17w

w = 0.14

Present Value of the pension obligation = pension obligation/YTM = 3/10% = 30 million

0.14*30 million = 4.14 million invested in Delta bonds

(1-0.14)*30 million = 25.86 million invested in Gamma bonds

Market value = par value*number of bonds

4.14 million = 1,000*n

n = 4.14 million/1,000 = 4,135.96 or 4,136 (number of Delta bonds to be bought)

n = 25.86 million/1,000 = 25,864.04 or 25,864 (number of Gamma bonds to be bought)