692 Part Eight Risk Management
bre44380_ch26_673-706.indd 692 09/30/15 12:09 PM
To hedge interest-rate risk, the CFO has to design the debt issue so that any change in
interest rates has the same (and thus offsetting) impact on the PV of the lease payments and
the PV of the debt. There are two ways to accomplish this:- Zero-maintenance. Issue debt requiring interest and principal payments of exactly
$2 million per year for 20 years. This debt would be similar to a real-estate mortgage
with level payments. In this case, lease payments would exactly cover debt service
in each year. The PVs of the lease payments and the offsetting debt would always be
identical, regardless of the level of future interest rates. - Match duration. Issue debt with the same duration as the lease payments. Here debt
service does not have to match the lease payments in each (or any) year. If durations
are matched, then small changes in interest rates, say from 10% down to 9.5% or up
to 10.5%, will have the same impact on the PVs of the lease payments and the debt.
The duration-matching strategy is usually more convenient, but it is not zero-maintenance
because durations will drift out of line as interest rates change and time passes. Thus the CFO
will have to revisit and reset the hedge. She will have to execute a dynamic strategy to make
duration-matching work.
Let’s work out the duration-matching strategy. The duration of the lease payments is
7.5 years:
Duration = ___^1
PV
{[ PV(C 1 ) × 1] + [PV(C 2 ) × 2] + [PV(C 3 ) × 3] + ⋯}= ____^1
17.0
{
[____^2
1.10× 1
]+
[_____^2
1.10^2× 2
]+ ⋯ +
[______^2
1.10^20× 20
]
}
= 7.5 yearsThe duration of a “zero-maintenance” debt-issue, with debt service of exactly $2 million per
year for 20 years, would of course also be 7.5 years.
Many other debt instruments have a duration of 7.5 years. For example, you can check
that a 12-year bond with a 10% coupon has a 7.5-year duration. But suppose the CFO finds it
more convenient to raise $17 million by issuing a zero-coupon note with a maturity of exactly
7.5 years.^30 The note has only one cash payment at 7.5 years and therefore a duration of
7.5 years. Is Potterton now hedged against interest rate risk?
Figure 26.4 plots the PVs of the lease payments (on the left) and the 7.5-year note (on the
right) as a function of the interest rate. On the right we have also plotted the PV curve for the
“zero-maintenance” debt package with debt service exactly matching the lease payments. All
the PV curves are downward-sloping but convex; note how each curve comes down steeply at
low interest rates but flattens out at higher interest rates.
Now compare the slope of the PV curve for the lease payments (and also the “zero-
maintenance” debt package) to the slope of the 7.5-year zero. The slopes are identical at
the current 10% interest rate because the duration is identical at this rate. As we pointed
out in Chapter 4, (modified) duration measures the percentage change in bond price for a
1 percentage-point change in the interest rate.^31 If the interest rate falls to 9.5% or increases
to 10.5%, the PVs of the lease cash flows and the zero-coupon debt change by the same
amount. Potterton is therefore hedged, so long as the interest rate does not stray too far from
the current level of 10%.(^30) The principal of the zero-coupon note is $34.75 million. The PV of this promised payment is 34.75/(1.10)7.5 = $17 million.
(^31) The slope equals (minus) modified duration, defined as −D/(1 + y), where D = duration and y = the current interest rate. If durations
are equal, modified durations must also be equal.