This document discusses interest rate risk and bond portfolio risk management. It defines key concepts like duration, DV01, convexity, and gamma that measure a bond or portfolio's sensitivity to changes in interest rates. It describes strategies for constructing bond portfolios to match a target duration and convexity in order to minimize exposure to small interest rate movements. Larger rate changes require rebalancing as sensitivity estimates become less accurate.

1. 1
Risk Management: Duration,
DV01 and Convexity
Suggested readings: MPP chapters 5 and 6
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2. 2
Types of risk
 Interest rate risk
 Call/prepayment risk
 Reinvestment risk
 Credit risk
◦ Default risk
◦ Downgrade risk
 Liquidity risk
 etc…
 In this chapter we deal with interest rate risk
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3. 3
Highlights
 Measuring sensitivity of bond prices to
interest rate risk: duration, DV01
 Managing the interest rate risk of a portfolio
of assets and liabilities
 Measuring the sensitivity of duration to
changes in interest rates: convexity
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4. 4
Price sensitivity of bonds
120
1 yr. zero (par = 100)
100
80
Price
60
40
10 yr. zero (par = 100)
20
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
r (continuously compounded)
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5. 5
Linear Approximations
 How to measure sensitivity of bond prices
to interest rates?
 Calculus provides answer: use derivative
 More specifically, if f is a function of x, the
change in x is approximately (Taylor
expansion) f(x) - f(x0) ≈ f’(x0)(x-x0)
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6. 6
Sensitivity of zero prices
 Price of a zero paying KT at time T
KT KT 
rT T
B KT e
1 rT
T
e rT T
 Derivative (using continuous compounding):
B 
rT T
TK T e TB

rT
 Taylor approximation:
change in B ≈ -TB x (change in rT )

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7. 7
Dollar Duration
 Dollar duration ( $), or delta, is minus the
change in price for 1 percentage point (e.g.,
Price change for a 100 basis point
movement in interest rates). For a zero
maturing at T:
TB
$
100
 $ is minus the slope of the price-interest
rate curve: change in price ≈ - $ x (change
in (continuously compounded) int. rate)
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8. 8
Coupon Bonds
 Coupon bond = sum of zeros => change in
price = sum of changes in zero prices => delta
of coupon bond = sum of deltas of zeros
 For a bond of maturity T, paying Kt in period t
1 PV K 1 2 PV K 2 ... T PV KT
$
100
where: 
rt t Kt
PV K t Kte t
1 rt
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9. 9
Parallel shifts of the term structure
 Use of delta does not assume flat term
structure (interest rates are the same for all
different maturities)
– only that term structure makes a parallel
(or uniform shift), i.e. that all (continuously
compounded) interest rates change by the
same amount
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10. 10
DV01
 $ is a price change per percentage point.
Price change often expressed per basis
point (i.e. 0.01%), known as DV01 (“dollar
value of an ’01”).
DV01 = - $ / 100.
 Also called PVBP (“price value of a basis
point”) or PV01
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11. 11
Small vs. large changes
in interest rates
 Consider a 20 yr. bd with annual cp of 10%,
face=$100. Term structure is flat at 10%
(annual compounding) or 9.53%
(continuous compounding; 0.0953=ln(1.1))
 Assume term structure shifts up by: (1) 1
b.p. to 9.54%. (2) 200 b.p. to 11.53%
 Compare actual price change vs. delta-
based prediction. When is approximation
good?
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12. 12
Yield duration
The Duration of a bond is a
measure of how long on
average the holder of the bond
has to wait before receiving
cash payments.
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13. 13
Yield duration
 Yield duration, or Macaulay duration
1 K1 2 K2 T KT
2
... T
1 y 1 y 1 y
Dy
B
where y is the bond YTM (Cont. Comp.)
 Dy is a weighted average of cash-flow times
(where weights = present value of cash-
flows); in particular, for a zero, Dy = maturity
 Modified yield duration: Dymod = Dy/(1+y)
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14. 14
Yield duration
 Dy provides approximation of percentage change
in bond price (vs. price change for $) when yield
changes:
(change in price / price)
≈ - Dymod x (change in yield)
≈ - (Dy/(1+y)) x (change in yield)
 Yields on different bonds change differently when
non-flat terms structure changes (even in case of
parallel shifts) => can’t combine yield durations of
different bonds to find portfolio duration => can’t
use it in bond portfolio risk management (unless
we assume flat term structure and all bond yields
are the same)
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15. 15
Duration of a portfolio
 For a portfolio of n assets:
$ = N1 $1 + N2 $2 + … + Nn $n
 Where Ni is the number of units of security i
 For liabilities (e.g., short sales, bonds
issued by a bank, or customers’ deposits),
Ni < 0
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16. 16
Risk Management 101
 Value of portfolio (net of liabilities) = net equity:
Vne = N1B1 + N2B2 + … + NnBn
 Delta of net equity: $ne = N1 $1 + N2 $2 + … +
Nn $n
 Simplest interest rate risk management strategy
(“delta-matching”) consists in picking desired
delta ( $ne). Special case where $ne = 0 known
as immunization
 Need to use two bonds which you are free to
buy and sell (to set $ne to desired level without
changing Vne, i.e. without investing extra
money).
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17. 17
Example of Delta Matching
 Liability is 7 year zero coupon bond F=$100K
 Delta of this liability can be matched using a 3
year zero coupon bond with F=$100, and a 20
year coupon bond with F=$100 and annual
coupon payments of $5.
 Term structure continuous compounding

r1 
.1, r2 
.11, r3 
.115 , r4 
.12 , r5 
.125 , r6 
.13, r7 
.135 , r8 
.14 , r9 
.145 , r10 .15,

r11 
.1525 , r12 
.1550 , r13 
.1575 , r14 
.16 , r15 
.16 , r16 
.155 , r17 
.15, r18 
.145 , r19 
.14 , r20 .13
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18. 18
Problems with delta-matching
 Works well for small interest rate changes
 Requires rebalancing: deltas change over
time
 Assumes uniform term structure shifts
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19. 19
Forwards
 Forward <=> portfolio with two bonds, one
long, one short
 $forward = $ of long bond – $ of short
bond
 Risk management: given the existing $ of
your position ( $before) and your desired $
( $after), pick Nforward such that:
$after = $before + (Nforward x $forward)
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20. 20
Convexity
 Convexity = curvature of the price-int. rate
relationship = change in duration as int.
rates change (=> error in delta-based
approximation)
 Measure of convexity, gamma:
12 PV K 1 22 PV K 2 ... T 2 PV KT
$
10,000
 Kt
where PV K t Kte rt t
t
1 rt
 For zero with maturity T, $ =T2B/10,000
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21. 21
Mathematical justification
 The Taylor approximation can be improved
by taking second derivative into account:
f(x)≈f(x0)+f’(x0)(x-x0) +(1/2)f’’(x0)(x-x0)2
 More accurate estimate for bond price
change: change in price ≈ - $ (change in
int. rate) + (1/2) $ (change in int. rate)2
(using continuously compounded rates)
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22. 22
Example of Gamma Calculation
 3 year bond, semi-annual coupons, coupon yield
14%. Face value 1,000
t CFt DFt PV(Kt) T2xPV(kt)
0.5 70 0.951 66.59 16.65
1.0 70 0.905 63.34 63.34
1.5 70 0.861 60.25 135.56
2.0 70 0.819 57.31 229.24
2.5 70 0.779 54.52 340.73
3.0 1070 0.741 792.68 7,134.08
Sum 1094.68 7,919.58
• Value = 1094.68. Delta ( $)=28.1594,
Gamma ( $)=0.79196
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23. 23
Risk Management 102: matching
duration and convexity
 Gamma of portfolio = weighted sum of
gammas of components
 To achieve portfolio of desired gamma and
delta (e.g. $ = 0 to improve immunization)
Vne = N1B1 + N2B2 + … + NnBn
$ne = N1 $1 + N2 $2 + … + Nn $n
$ne = N1 $1 + N2 $2 + … + Nn $n
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24. 24
Barbells vs. bullets
 Barbelling = using a portfolio of short and
long maturity bonds (barbell) rather than
bonds of intermediate maturity (bullet)
 Barbell has greater convexity than bullet of
same duration
 Suggests that, whatever change in interest
rates, barbell always does better: is there
an arbitrage (exploited by doing butterfly
trade)?
 No, because of the effect of the passage of
time
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25. 25
Theta
 Theta, $, measures the effect of a change
in time (= minus the derivative of price with
respect to maturity)
$

r1 PV K1 
r2 PV K 2 
... rT PV KT
 Improved approximation of change in bond
price over small period:
change in price ≈ - $ x (change in int. rate)
+ (1/2) $ x (change in int. rate)2 + $ x
(change in time)
25