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 approximately

$1$

$$million$,$ given the new flat 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.