CHAPTER 9
Interest Rate Risk II
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All Rights Reserved.
Overview
 This chapter discusses a market valuebased model for assessing and
managing interest rate risk:
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Duration
Computation of duration
Economic interpretation
Immunization using duration
*Problems in applying duration
9-2
Price Sensitivity and Maturity
 In general, the longer the term to
maturity, the greater the sensitivity to
interest rate changes
 Example: Suppose the zero coupon
yield curve is flat at 12%. Bond A pays
$1790.85 in five years. Bond B pays
$3207.14 in ten years, and both are
currently priced at $1000.
9-3
Example continued...
– Bond A: P = $1000 = $1790.85/(1.06)10
– Bond B: P = $1000 = $3207.14/(1.06)20
 Now suppose the annual interest rate
increases by 1% (0.5 % semiannually).
– Bond A: P = $1762.34/(1.065)10 = $954.03
– Bond B: P = $3105.84/(1.065)20 = $910.18
 The longer maturity bond has the
greater drop in price because the
payment is discounted a greater
number of times.
9-4
Coupon Effect
 Bonds with identical maturities will
respond differently to interest rate
changes when the coupons differ. This
is easily understood by recognizing
that coupon bonds consist of a bundle
of “zero-coupon” bonds. With higher
coupons, more of the bond’s value is
generated by cash flows which take
place sooner in time. Consequently, it
is less sensitive to changes in R.
9-5
Remarks on Preceding Slides
 In general, longer maturity bonds
experience greater price changes in
response to any change in the
discount rate
 The range of prices is greater when the
coupon is lower
– A 6% bond will have a larger change in
price in response to a 2% change than an
8% bond
– The 6% bond has greater interest rate risk
9-6
Extreme Examples With Equal Maturities

Consider two ten-year maturity instruments:
– A ten-year zero coupon bond
– A two-cash flow “bond” that pays $999.99
almost immediately and one penny ten years
hence


Small changes in yield will have a large
effect on the value of the zero but almost
no impact on the hypothetical bond
Most bonds are between these extremes
– The higher the coupon rate, the more similar the
bond is to our hypothetical bond with higher
value of cash flows arriving sooner
9-7
Duration
 Duration
– Weighted average time to maturity using
the relative present values of the cash
flows as weights
– Combines the effects of differences in
coupon rates and differences in maturity
– Based on elasticity of bond price with
respect to interest rate
– The units of duration are years
9-8
Macaulay Duration
 Macaulay Duration
Where
D = Macaulay duration (in years)
t = number of periods in the future
CFt = cash flow to be delivered in t periods
N= time-to-maturity
DFt = discount factor
9-9
Duration
 Since the price (P) of the bond equals
the sum of the present values of all its
cash flows, we can state the duration
formula another way:
 Notice the weights correspond to
the relative present values of the
cash flows
9-10
Semiannual Cash Flows
 It is important to see that we must
express t in years, and the present
values are computed using the
appropriate periodic interest rate. For
semiannual cash flows, Macaulay
duration, D is equal to:
9-11
Duration of Zero-coupon Bond
 For a zero-coupon bond, Macaulay
duration equals maturity since 100% of
its present value is generated by the
payment of the face value, at maturity
 For all other bonds, duration < maturity
9-12
Computing duration
 Consider a 2-year, 8% coupon bond,
with a face value of $1,000 and yieldto-maturity of 12%
 Coupons are paid semi-annually
 Therefore, each coupon payment is
$40 and the per period YTM is (1/2) ×
12% = 6%
 Present value of each cash flow
equals CFt ÷ (1+ 0.06)t where t is the
period number
9-13
Duration of 2-year, 8% bond:
Face value = $1,000, YTM = 12%
t
years CFt
PV(CFt)
1
0.5
40
37.736
Weight W × years
(W)
0.041
0.020
2
1.0
40
35.600
0.038
0.038
3
1.5
40
33.585
0.036
0.054
4
2.0
1,040 823.777
0.885
1.770
P = 930.698
1.000
D=1.883
(years)
9-14
Special Case
 Maturity of a consol: M = .
 Duration of a consol: D = 1 + 1/R
9-15
Features of Duration
 Duration and maturity
– D increases with M, but at a decreasing
rate
 Duration and yield-to-maturity
– D decreases as yield increases
 Duration and coupon interest
– D decreases as coupon increases
9-16
Economic Interpretation
 Duration is a measure of interest rate
sensitivity or elasticity of a liability or
asset:
[ΔP/P]  [ΔR/(1+R)] = -D
Or equivalently,
ΔP/P = -D[ΔR/(1+R)] = -MD × ΔR
where MD is modified duration
9-17
Economic Interpretation
 To estimate the change in price, we
can rewrite this as:
ΔP = -D[ΔR/(1+R)]P = -(MD) × (ΔR) × (P)
 Note the direct linear relationship between
ΔP and -D
9-18
Dollar Duration
 Dollar duration equals modified
duration times price
 Dollar duration = MD × Price
 Using dollar duration, we can compute
the change in price as
ΔP = -Dollar duration × ΔR
9-19
Semi-annual Coupon Payments
 With semi-annual coupon payments
the percentage change in price is
ΔP/P = -D[ΔR/(1+(R/2)]
9-20
Immunization
 Matching the maturity of an asset
investment with a future payout
responsibility does not necessarily
eliminate interest rate risk
 Matching durations will immunize
against changes in interest rates
9-21
An Example
 Consider three loan plans, all of which
have maturities of 2 years. The loan
amount is $1,000 and the current
interest rate is 3%.
 Loan #1 is a two-payment loan with
two equal payments of $522.61 each.
 Loan #2 is structured as a 3% annual
coupon bond.
 Loan #3 is a discount loan, which has
a single payment of $1,060.90.
9-22
Duration as Index of Interest Rate Risk
Yield
Loan Value
2%
3%
ΔP N D
Equal
$1014.68 $1000 $14.68 2 1.493
Payment
3% Coupon $1019.42 $1000 $19.42 2 1.971
Discount
$1019.70 $1000 $19.70 2 2.000
9-23
Balance Sheet Immunization
 Duration is a measure of the interest
rate risk exposure for an FI
 If the durations of liabilities and assets
are not matched, then there is a risk
that adverse changes in the interest
rate will increase the present value of
the liabilities more than the present
value of assets is increased
9-24
Duration Gap
 Suppose that a 2-year coupon bond is
the only loan asset (A) of an FI. A 2-year
certificate of deposit is the only liability
(L). If the duration of the coupon bond
is 1.8 years, then:
Maturity gap: MA - ML = 2 -2 = 0, but
Duration Gap: DA - DL = 1.8 - 2.0 = -0.2
– Deposit has greater interest rate sensitivity
than the bond, so DGAP is negative
– FI exposed to rising interest rates
9-25
Immunizing the Balance Sheet of an FI
 Duration Gap:
– From the balance sheet, E=A-L. Therefore,
DE=DA-DL. In the same manner used to
determine the change in bond prices, we
can find the change in value of equity
using duration.
– DE = [-DAA + DLL] DR/(1+R) or
– DE = -[DA - DLk]A(DR/(1+R))
9-26
Duration and Immunizing
 The formula shows 3 effects:
– Leverage adjusted D-Gap
– The size of the FI
– The size of the interest rate shock
9-27
An Example
 Suppose DA = 5 years, DL = 3 years and
rates are expected to rise from 10% to
11%. (Rates change by 1%). Also, A =
100, L = 90 and E = 10. Find change in
E.
 DE = -[DA - DLk]A[DR/(1+R)]
= -[5 - 3(90/100)]100[.01/1.1] = - $2.09.
 Methods of immunizing balance sheet.
– Adjust DA , DL or k.
9-28
Immunization and Regulatory Concerns
 Regulators set target ratios for an FI’s
capital (net worth):
– Capital (Net worth) ratio = E/A
 If target is to set D(E/A) = 0:
– DA = DL
 But, to set DE = 0:
– DA = kDL
9-29
Limitations of Duration
– Immunizing the entire balance sheet need
not be costly
– Duration can be employed in combination
with hedge positions to immunize
– Immunization is a dynamic process since
duration depends on instantaneous R
– Large interest rate change effects not
accurately captured
 Convexity
– More complex if nonparallel shift in yield
curve
9-30
Convexity
 The duration measure is a linear
approximation of a non-linear
function. If there are large changes in
R, the approximation is much less
accurate. All fixed-income securities
are convex. Convexity is desirable, but
greater convexity causes larger errors
in the duration-based estimate of price
changes.
9-31
*Convexity
 Those who are familiar with calculus
may recognize that duration involves
only the first derivative of the price
function. We can improve on the
estimate using a Taylor expansion. In
practice, the expansion rarely goes
beyond second order (using the
second derivative). This second order
expansion is the convexity adjustment.
9-32
*Modified Duration &
Convexity
– DP/P = -D[DR/(1+R)] + (1/2) CX (DR)2 or
DP/P = -MD DR + (1/2) CX (DR)2
– Where MD implies modified duration and
CX is a measure of the curvature effect.
– CX = Scaling factor × [capital loss from 1bp
rise in yield + capital gain from 1bp fall in
yield]
– Commonly used scaling factor is 108
9-33
*Calculation of CX
 Example: convexity of 8% coupon, 8%
yield, six-year maturity Eurobond
priced at $1,000
CX = 108[DP-/P + DP+/P]
= 108[(999.53785-1,000)/1,000 +
(1,000.46243-1,000)/1,000)]
= 28
9-34
*Duration Measure: Other Issues
 Default risk
 Floating-rate loans and bonds
 Duration of demand deposits and
passbook savings
 Mortgage-backed securities and
mortgages
– Duration relationship affected by call or
prepayment provisions
9-35
*Contingent Claims
 Interest rate changes also affect value
of off-balance sheet claims
– Duration gap hedging strategy must
include the effects on off-balance sheet
items such as futures, options, swaps,
caps, and other contingent claims
9-36
Pertinent Websites
Bank for International
Settlements
Securities Exchange
Commission
The Wall Street Journal
www.bis.org
www.sec.gov
www.wsj.com
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