Please explain what this problem wants us to do? And how is the duration of 4 years from?

Problem: Pension funds pay lifetime annuities to recipients. If a firm remains in business indefinitely, the pension obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make perpetual payments of $2 million per annum to beneficiaries. The yield to maturity on all bonds is 16%.

a. If the duration of five-year maturity bonds with coupon rates of 10% (paid annually) is five years and the duration of 20-year maturity bonds with coupon rates of 8% (paid annually) is 12 years, how much of each of these coupon bonds (in market value) will you want to hold to both fully fund and immunise your obligation?

b. What will be the par value of your holdings in the 20-year coupon bond?

Answer:

a. PV of obligation = $2 million/0.16 = $12.5 million

Duration of obligation = 1.16/0.16 = 7.25 years

Call w the weight on the five-year maturity bond (with duration of 4 years). Then:

(w 4) + [(1 - w) 11] = 7.25 w = 0.6786

Therefore:

0.6786 $12.5 = $8.48 million in the five-year bond and

0.3214 $12.5 = $4.02 million in the 20-year bond.

b. The price of the 20-year bond is:

[80 annuity factor(16%,20)] + [1000 PV factor(16%, 20)] = $525.69

Therefore, the bond sells for 0.5257 times its par value, so that:

Market value = par value 0.5257

$4.02 million = par value 0.5257 par value = $7.64 million

Another way to see this is to note that each bond with par value $1000 sells for $525.69.

If total market value is $4.02 million, then you need to buy:

$4 020 000/407.11 = 7 643 bonds

Therefore, total par value is $7 643 000.