Transcript Saunders Cornett Chapter 9

Chapter 9
Interest Rate Risk
II
McGraw-Hill/Irwin
© 2008 The McGraw-Hill Companies, Inc., All Rights Reserved.
Overview

This chapter discusses a market valuebased model for assessing and managing
interest rate risk:





Duration
Computation of duration
Economic interpretation
Immunization using duration
* Problems in applying duration
9-2
Price Sensitivity and Maturity
9-3
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 $1762.34
in five years. Bond B pays $3105.85 in ten
years, and both are currently priced at
$1000.

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
10 = $914.94
 Bond B: P = $3105.84/(1.13)
The longer maturity bond has the greater drop in
price because the payment is discounted a
greater number of times.


9-4
Coupon Effect

9-5
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.
Consequently, less sensitive to changes in
R.
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
9-6
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
9-7
Remarks on Preceding Slides
9-8
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.


The 6% bond shows greater changes in price in
response to a 2% change than the 8% bond.
The first bond has greater interest rate risk.
Extreme examples with equal maturities

Consider two ten-year maturity instruments:




9-9
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 essentially 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.
Duration

9-10
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.
Duration
Duration
D = SNt=1[CFt• t/(1+R)t]/ SNt=1 [CFt/(1+R)t]
Where

D = duration
t = number of periods in the future
CFt = cash flow to be delivered in t periods
N= time-to-maturity
R = yield to maturity.
9-11
Duration

9-12
Since the price (P) of the bond must equal
the present value of all its cash flows, we
can state the duration formula another way:
D = SNt=1[t  (Present Value of CFt/P)]
 Notice that the weights correspond to
the relative present values of the cash
flows.
Duration of Zero-coupon Bond
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.
 For all other bonds:

duration < maturity
9-13
Computing duration
9-14
Consider a 2-year, 8% coupon bond, with a
face value of $1,000 and yield-to-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.

9-15
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)
Special Case
Maturity of a consol: M = .
 Duration of a consol: D = 1 + 1/R

9-16
Duration Gap
9-17
Suppose the bond in the previous example
is the only loan asset (L) of an FI, funded by
a 2-year certificate of deposit (D).
 Maturity gap: ML - MD = 2 -2 = 0
 Duration Gap: DL - DD = 1.885 - 2.0 = -0.115



Deposit has greater interest rate sensitivity than
the loan, so DGAP is negative.
FI exposed to rising interest rates.
Features of Duration

Duration and maturity:


Duration and yield-to-maturity:


D increases with M, but at a decreasing rate.
D decreases as yield increases.
Duration and coupon interest:

D decreases as coupon increases
9-18
Economic Interpretation

9-19
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.
Economic Interpretation


9-20
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.
Semi-annual Coupon Payments

With semi-annual coupon payments:
(ΔP/P)/(ΔR/R) = -D[ΔR/(1+(R/2)]
9-21
An example:
9-22
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.

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
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))
9-24
Duration and Immunizing

The formula shows 3 effects:



Leverage adjusted D-Gap
The size of the FI
The size of the interest rate shock
9-25
An example:
9-26
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.
9-27
Immunization and Regulatory Concerns

Regulators set target ratios for an FI’s
capital (net worth):


If target is to set D(E/A) = 0:


Capital (Net worth) ratio = E/A
DA = DL
But, to set DE = 0:

DA = kDL
*Limitations of Duration
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
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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.


9-28
Convexity
More complex if nonparallel shift in yield curve.
*Convexity

9-29
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.
*Convexity

9-30
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).
*Modified duration & Convexity
9-31
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]
8
 Commonly used scaling factor is 10 .

*Calculation of CX

9-32
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.
*Duration Measure: Other Issues
9-33
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.
*Contingent Claims

9-34
Interest rate changes also affect value of offbalance 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.
Pertinent Websites
Bank for International Settlements
www.bis.org
Securities Exchange Commission
www.sec.gov
The Wall Street Journal
www.wsj.com
9-35