Interest Rate Risk II
Chapter 9
Financial Institutions Management, 3/e
By Anthony Saunders
Irwin/McGraw-Hill
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Price Sensitivity and Maturity
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 $1762.34
in five years. Bond B pays $3105.85 in ten
years, and both are currently priced at
$1000.

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Example continued...
• Bond A: P = $1000 = $1762.34/(1.12)5
• Bond B: P = $1000 = $3105.84/(1.12)10


Now suppose the interest rate increases by 1%.
• Bond A: P = $1762.34/(1.13)5 = $956.53
• Bond B: P = $3105.84/(1.13)10 = $914.94
The longer maturity bond has the greater drop in
price.
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Coupon Effect

Bonds with identical maturities will respond
differently to interest rate changes when the
coupons differ. This is more readily
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.
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Price Sensitivity of 6% Coupon Bond
r
8%
6%
4%
Range
n
40
$802
$1,000
$1,273
$471
20
$864
$1,000
$1,163
$299
10
$919
$1,000
$1,089
$170
2
$981
$1,000
$1,019
$37
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Price Sensitivity of 8% Coupon Bond
r
10%
8%
6%
Range
n
40
$828
$1,000
$1,231
$403
20
$875
$1,000
$1,149
$274
10
$923
$1,000
$1,085
$162
2
$981
$1,000
$1,019
$38
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Remarks on Preceding Slides
The 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.

• The 6% bond shows greater changes in price in
response to a 2% change than the 8% bond. The
first bond is riskier.
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Duration

Duration
• Combines the effects of differences in coupon
rates and differences in maturity.
• Based on elasticity of bond price with respect
to interest rate.
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Duration
Duration
D = Snt=1[Ct• t/(1+r)t]/ Snt=1 [Ct/(1+r)t]
Where

D = duration
t = number of periods in the future
Ct = cash flow to be delivered in t periods
n= term-to-maturity & r = yield to maturity.
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Duration

Duration
• Weighted sum of the number of periods in the
future of each cash flow, (weighted by
respective fraction of the PV of the bond as a
whole).
• For a zero coupon bond, duration equals
maturity since 100% of its present value is
generated by the payment of the face value, at
maturity.
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Advantages to Duration Measure:
1. Simplicity
 2. Can be used to determine elasticity
between price and YTM:
(DP/P)/(Dr/r) = -D[r/(1+r)]
 We can rewrite this as:
DP = -D[P/(1+r)] Dr


Note the direct relationship between DP and -D.
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Duration as Index of Interest Rate Risk:

The greater the duration, the greater the
price sensitivity and the greater the risk.
Higher duration indicates that it takes a
longer time to recover the PV of the bond.
This agrees with intuition once we realize
that ONLY a zero-coupon bond has duration
equal to maturity. ALL other bonds will
have duration LESS than maturity.
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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 an installment loan with two
equal payments of $522.61. Loan #2 is a
discount loan, which has a single payment
of $1,060.90. Loan #3 is structured as a 3%
annual coupon bond.
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Duration as Index of Interest Rate Risk
Yield
Loan Value 2%
3%
Installment $1014.67 $1000
Discount
$1019.70 $1000
Coupon
$1019.41 $1000
DP
$28.98
$38.84
$38.27
n
2
2
2
D
1.493
2.000
1.97
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Limits to Duration Measure

Duration relationship may not hold if the
bond has a call or prepayment provision.
• Convexity.
• Negative Convexity.
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Special Case and an Adjustment
Maturity of a consol: M = .
 Duration of a consol: D= 1 + 1/R
 Adjusting for semi-annual payments
dP/P = -D[dR/(1+ (1/2)R]

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Immunizing 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))
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Duration and Immunizing

The formula shows 3 effects:
• Leverage adjusted D-Gap
• The size of the FI
• The size of the interest rate shock
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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.
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*Limitations of Duration
• Only works with parallel shifts in yield curve.
• 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.
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*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. Recall 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).
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*Modified duration
• 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.
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*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.
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