FIS Ch 4: Refinements in Interest Rate Risk Management

Question 1

Define Convexity

Answer 1

Convexity C
1
P
d2P
dr 2

Convexity measures the percentage change in the price of the security due to the
curvature of the price with respect to the interest rate r

Question 2

Using duration and convexity to approximate change in portfolio value from rate
changes

Answer 2

Using both duration and convexity, we obtain a more accurate approximation of the
impact of changes in interest rates on bond prices
dP D dr P 􀀀
1
2 C dr 2
P

Question 3

Convexity of Zero Coupon Bonds

Answer 3

The price of a zero coupon bond is given by:
Pzpr , t; Tq 100 er pTtq
The convexity of the zero-coupon bond is:
Cz
1
Pz

d2Pz
dr 2

1
Pz
rpT tq
2
Pz s
pT tq
2 X

Question 4

Convexity of a portfolio of securities

Answer 4

The convexity of a portfolio of securities is equal to the weighted average of the
convexities of the individual securities
CW
¸n
i1
wiCi

The weights wi are given by;
wi
Ni Pi
W

Ni = the units of securities 1,..,n in a portfolio

Pi = the price of the ith security

W
°ni
1 NiPi = the value of the portfolio

Ci = the convexity of security i

Question 5

Positive convexity of a bond

Answer 5

E

dP
P

D Erdr s 􀀀
1
2 C Erdr 2
s

We can assume Erdr s 0, and Erdr 2s is likely to be positive

It then follows that the expected return of the bond from convexity, E

dP
P

, will be
positive

The positive expected return from the bond price convexity will be
counterbalanced by a lower yield to maturity of the bond

Question 6

Change in value of hedged portfolio from duration hedging

Answer 6

Suppose we are long a coupon bond with price P, and we duration hedge the position
with k-units of a zero-coupon bond with price Pz . The change in value of the hedged
portfolio is:
dV dP 􀀀 k dPz

dP = change in value of the firm’s coupon bond

dPz = change in value of the zero-coupon bond

Question 7

Change in value of hedged portfolio from duration-convexity hedging

Answer 7

Use two zero-coupon bonds (with different maturities) to hedge a security with price P

Let P1 and P2 be the prices of these two bonds

Let D1, D2, C1, and C2 be their durations and convexities

Let k1 and k2 be the positions in these two bonds

The value of the hedged portfolio is given by:
V P 􀀀 k1 P1 􀀀 k2 P2

The change in the hedged portfolio value is:
dV dP 􀀀 k1 dP1 􀀀 k2 dP2

Question 8

Number of units of zero-coupon bonds to purchase to apply duration-convexity hedging

Answer 8

k1
P
P1

D C2 C D2
D1 C2 C1 D2

k2
P
P2

D C1 C D1
D2 C1 C2 D1

Question 9

Describe factor models

Answer 9

A factor model takes into account changes in the slope and curvature of the interest
rate term structure

Let T1,T2,..,Tn be n points on the interest rate curve

Let ri r pt, Ti q denote the corresponding spot rate on the interest rate curve

The factor model assumes that dri is driven by a set of common factors
1,2, …,m
dr1 11d1 􀀀 12d2 􀀀 … 􀀀 1mdm
dr2 21d1 􀀀 22d2 􀀀 … 􀀀 2mdm
…………………………………….
drn n1d1 􀀀 n2d2 􀀀 … 􀀀 nmdm
Ÿ The ij ’s determine the sensitivity of the dri ’s to changes in the factors dj
(j 1, ..,m)

Question 10

Define factor duration

Answer 10

The factor duration of an asset with price P with respect to factor j is:
Dj
1
P
dP
dj

For example, for the slope factor j 2, D2 measures the percentage impact of a
change in slope of the yield curve on the price P

Question 11

Factor duration of zero-coupon bond

Answer 11

The sensitivity of a zero coupon bond price to the factor j can be calculated from the
chain-rule:
dPzpt, Ti q
dj

dPzpt, Ti q
dri

dri
dj

dPzpt, Ti q
dri
ij
The factor duration of the zero coupon bond with respect to factor j is:
Dj,z
1
Pzpt, Ti q
dPzpt, Ti q
dj

1
Pzpt, Ti q

pTi tq Pzpt, Ti q ij

pTi tq ij X

Question 12

Using factor duration to approximate change in the bond price

Answer 12

dP
P D1 d1 D2 d2 D3 d3