Many institutions have fixed future liabilities to meet (such as pension payments) and they

fund these future liabilities using default-free fixed income securities. When discount bonds

of all maturities are available, these institutions can simply buy discount bonds to fund their

liabilities. For example, if there is a fixed liability equal to 1 million dollars five years from

now, an institution can buy a discount bond maturing in five years with a face value of 1

million dollars. Unfortunately, there may not be the ight" discount bonds for a fixed future

liability and coupon bonds must be used. Then an institution faces reinvestment risk on the

coupons.

For example, suppose that the yield curve is flat 10% and we have the following coupon

bonds (paying annual coupons):

Bond	Prices	Principal	Coupon	Years to Maturity
A	118.95	100	15	5
B	130.72	100	15	10

and we have a 1 million liability five years from now.

(a) Construct a portfolio of the two coupon bonds so that the future value of the

portfolio is 1 million and the duration of this portfolio is equal to five years,

assuming that the yield curve will remain at flat 10%.

(b) Show that if immediately after you purchased this portfolio the yield curve makes

a permanent parallel downward or upward move of 1%, the future value of this

portfolio at the end of year 5 will still be approximately 1 million. You have

immunized the portfolio of the risk associated with parallel movements of the

yield curve by buying a portfolio of coupon bonds so that the duration of the

portfolio matches the number of years to the payment of the fixed liability.

(c) Suppose now that you have held your portfolio for one year after a 1% decrease

in the yield curve to 9% which occurred immediately after you constructed your

initial portfolio with a duration of five. There are now four years to the payment

of the fixed liability. Use the money at your disposal (the market value of your

investment at the end of year one) to construct a portfolio of coupon bonds with

duration equal to four years and a future value four years hence equal to approxi-

mately 1 million, given the new at yield curve at 9%. Show that if the yield curve

then makes a parallel upward or downward move of 1%, the future value of your

portfolio four years from now will be unchanged. You have approximately funded

your liability of 1 million at the end of the fifth year. (Compare the difference

between the future value of your portfolio and your fixed liability here and in part

(b).)