BUFN 722. ch9 Interest Rate Risk – part 2 Duration, convexity, etc. Overview. This chapter discusses a market valuebased model for assessing and managing interest rate risk: Duration Computation of duration Economic interpretation Immunization using duration
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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= termtomaturity & r = yield to maturity (per period basis).
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D = Snt=1[t (Present Value of Ct/Price)]
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[dP/P] [dR/(1+R)] = D
Or equivalently,
dP/P = D[dR/(1+R)] = MD × dR
where MD is modified duration.
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dP = D[dR/(1+R)]P = (MD) × (dR) × (P)
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(dP/P)/(dR/R) = D[dR/(1+(R/2)]
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= [5  3(90/100)]100[.01/1.1] =  $2.09.
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CX = Scaling factor × [capital loss from 1bp rise in yield + capital gain from 1bp fall in yield]
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CX = 108[DP/P + DP+/P]
= 108[(999.537851,000)/1,000 + (1,000.462431,000)/1,000)]
= 28.
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%ΔP  DUR x Δi/(1+i)
i 5%, from 10% to 15%
ΔAsset Value = %ΔP x Assets
= 2.7 x .05/(1+.10) x $100m
= $12.3m
ΔLiability Value = %ΔP x Liabilities
= 1.03 x .05/(1+.10) x $95m
= $4.5m
ΔNW = $12.3m  ($4.5m) = $7.8m
DURgap = DURa  [L/A x DURl]
= 2.7  [(95/100) x 1.03]
= 1.72
%ΔNW =  DURgap x Δi/(1+i)
=  1.72 x .05/(1+.10)
= .078 = 7.8%
ΔNW = .078 x $100m
= $7.8m
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Friendly Finance Company
Assets Liabilities

Cash and Deposits $ 3 m  Commercial Paper $ 40 m

Securities  Bank Loans
less than 1 year $ 5 m  less than 1 year $ 3 m
1 to 2 year $ 1 m  1 to 2 year $ 2 m
greater than 2 year $ 1 m  greater than 2 year $ 5 m

Consumer Loans  LongTerm Bonds
less than 1 year $ 50 m  and other longterm
1 to 2 year $ 20 m  debt $ 40 m
greater than 2 year $ 15 m 
 Capital $ 10 m
Physical capital $ 5 m 
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1. Reduce DURa = 0.98 DURgap = 0.98  [(95/100) x 1.03] = 0
2. Raise DURl = 2.80 DURgap = 2.7  [(95/100) x 2.80] = 0
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The weightedaverage time to maturity on an
Investment, using the relative values of the cash flows as weights
N N
CFt tPVt t
t = 1(1 + R)tt = 1
D = N = N
CFt PVt
t = 1 (1 + R)t t = 1
This measure is known as Macaulay’s Duration
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Table 3
1 CFt CFt X t Percent of Initial
t CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Investment Recovered
48.08
46.23
44.45
42.74
41.10
39.52
38.00
767.22
1,067.34
.5
1
1.5
2
2.5
3
3.5
4
50
50
50
50
50
50
50
1,050
0.9615
0.9246
0.8890
0.8548
0.8219
0.7903
0.7599
0.7307
24.04/1,067.34 = 0.02
46.23/1,067.34 = 0.04
66.67/1,067.34 = 0.06
85.48/1,067.34 = 0.08
102.75/1,067.34 = 0.10
118.56/1,067.34 = 0.11
133.00/1,067.34 = 0.13
3,068.88/1,067.34 = 2.88
24.04
46.23
66.67
85.48
102.75
118.56
133.00
3,068.88
3,645.61
3,645.61
1,067.34
D =
= 3.42 years
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%ΔP  DUR x ΔR/(1+R)
Modified Duration = Duration / (1 + R)
R 10% to 11% (.10 to .11) => D = + .01:
Table 4 10% coupon bond
%ΔP = 6.76 x .01/(1+.10)
= .0615 = 6.15%.
Actual decline = 6.23% (need to add correction for convexity)
The duration measure is a less accurate measure of price sensitivity the larger the change in interest rates
20% required return on 10% coupon bond, DUR = 5.72 years
%ΔP =  5.72 x .01/(1+.10)
= .0520 = 5.20%
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Securities that have cash flows which are sensitive (in amount or timing) to interest rate levels should use effective duration (ED) and effective dollar duration (EDD).
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Exhibit 7Modified duration versus coupon (8.5% yield to maturity)
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Exhibit.12 Gains from convexity versus modified duration
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General Formulas Incorporating Both Duration and Convexity
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Exhibit 11B DGFC for two different changes in interest rates
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example  estimate both the duration and convexity of a 6year bond paying 5 percent coupon annually and the annual yield to maturity is 6 % 6year Coupon Bond Par value =$1,000 Coupon =0.05 YTM =0.06 Maturity =6
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= 4.36 years
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