Question $3.$

Consider a pension plan that will pay $$10,000$ once a year for a $5-$year period $(5$ annual payments$).$

The first payment will come in exactly $5$ years $($at the end of year $5)$ and the last payment in $9$ years

$($at the end of year $9).$

a$.$ What is the duration of the pension obligation? The current interest rate is $10\%$ per year

for all maturities.

$(3$ marks$)$

b$.$ To generate the scheduled pension payments, the pension fund wants to invest the

present value of the future payouts in bonds and match the duration of its obligation in

part a$).$ If the fund uses $5-$year and $10-$year zero$-$coupon bonds to construct its

investment position, how much money $($dollar amount$)$ ought to be placed in each bond

now? What should be the total face value $($not current market value$)$ of each zero$-$

coupon bond held?

$(3$ marks$)$

c$.$ Right after the fund made its investment outlined in part b$),$ market interest rates for all

maturities dropped from $10\%$

p$.$a$.$to $9\%$ p$.$a$.$ Show that the investment position

constructed in part b$)$ can still fund $($approximately$)$ the future payments by showing

that the fund's net investment is close to $0$ at the end of year $9$ after making all the

scheduled payments. Assume that interest rates will remain at $9\%$ p$.$a$.$ Any excess cash

from the $5-$year investment will be reinvested at $9\%$ and any fraction of the $10-$year

bonds held can be sold at the going market price at any time to fund the annual

payments.

$(6$ marks$)$

$($Total for Question: $12$ marks$)$