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

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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 right” discount bonds for a fixed future

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


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

bonds (paying annual coupons):





Years to Maturity











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

1. Suppose that the yield curve will remain unchanged for the following five years and you

have decided to use bond A to fund the liability. That is, you want to invest in bond

A and invest the coupons at the prevailing interest rates to produce a future value at

the end of year five of 1 million. How much should you invest in bond A?

2. Now suppose that right after you invested in bond A, the yield curve makes a parallel

move down by 1% to 9%. What is the future value five years from now of your

investment? What is the future value if the yield curve moves up by 1% to 11%?

Please explain why the future value changes differently depending on the direction of

the change in the yield curve.


Part (b) shows that the future value of your investment is sensitive to interest rate

fluctuations and you face the risk that your future liabilities may not be met. You

should try to immunize” this interest rate risk. But how? Do the following:

(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).) This technique is called duration matching”: if you adjust your portfolio

over time so that its duration always matches the years to the payment date of

your fixed liabilities, you will approximately immunize the risk of parallel shifts

in the yield curve.

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