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Chapter FI 11
Managing Bond Portfolios
Section 11.1: Interest Rate Risk
The sensitivity of bond prices to changes in market interest rate is the interest rate risk.
Example
If a bond is issued with an 8% coupon when competitive yields are 8%, then it will
sell at par value
If the market rate rises to 9%, who would purchase an 8% coupon bond at par value?
o Therefore bond prices must fall until its expected return increases to the
competitive level of 9%.
If the market rate falls to 7%, the 8% coupon on the bond is attractive compared to
alternative investment opportunities in the market. Therefore bond price will go up.
Figure 11.1: Change in bond price as a fraction of change in YTM
Properties:
Bond prices decrease when yields rise.
The relationship is convex. That is decrease in yields have bigger impacts on price than
increase in yields of equal magnitude.
The propositions below summarize the observations:
1. Bond prices and yields are inversely related: as yields increase bond prices fall; as
yields fall bond prices rise.
2. An increase in a bonds yield-to-maturity results in a smaller price change than a
decrease in yield of equal magnitude.
3. Prices of long-term bonds tend to be more sensitive to interest rate changes than
prices of short term bonds. Compare A and B.
4. The sensitivity of bond prices to changes in yields increases at a decreasing rate as
maturity increases. In other words, interest rate risk is less than proportional to bond
maturity.
Maturity of B is six times more than maturity of A. But the interest rate
sensitivity of B does not increase 6 times more than that of A.
5. Interest rate risk is inversely related to the bonds coupon rate. Prices of high coupon
bonds are less sensitive to changes in interest rates than prices of low coupon bonds.
Compare bonds Band C.
6. Bonds with higher initial yield-to-maturity are less sensitive to changes in yields.
These properties imply that maturity is a major determinant of interest rate risk. But
maturity alone is mot sufficient to measure interest rate sensitivity.
Example
Table 11.1: Prices of 8% annual coupon bonds:
YTM
8%
9%
%change in price
T=1 yr
1000
990.83
-0.92%
T=10 yrs
1000
935.82
-6.42
T=20 yrs
1000
908.71
-9.13%
T=10 yrs
463.19
422.41
-8.80%
T=20 yrs
214.55
178.43
-16.84%
Table 11.2: Prices of zero coupon bonds:
YTM
8%
9%
%change in price
T=1 yr
925.93
917.43
-0.92%
Why bond characteristics such as coupon rate or YTM affect interest rate sensitivity
Coupon Bonds
Table 11.1 gives bond prices for 8% annual coupon bonds at different yield to
maturity and times to maturity
The shortest term bond falls in value by less than 1% when interest rate increases
from 8% to 9%
The 10-year bond falls by 6.4%
The 20-year bond by more than 9%.
Zero Coupon Bonds
Table 11.2 gives bond prices for zero coupon bonds.
For both maturities beyond one year, the price of the zero coupon bond falls by a
greater proportional amount than the price of the 8% coupon bond.
In some sense, a zero coupon bond represents a longer term investment than an equal
time to maturity coupon bonds.
Therefore, time to maturity of bonds is not the perfect measures of the long term or short
term nature of the bonds.
Each payment of coupon bond can be considered to have its own maturity date
It is useful to view a coupon bond as a portfolio of coupon payments.
The effective maturity of the bond should be measured as average of the
maturities of all the cash flows paid out by the bond.
High coupon rate bond has a higher fraction of its value tied to coupons rather
than payments of par value, so the portfolio is more heavily weighted toward the
earlier payments, which give lower effective maturity.
Therefore price sensitivity falls as coupon rate increases.
Duration
Macaulays Duration
A measure of the effective maturity of a bond, defined as the weighted average of the
times until each payment, with weights proportional to the present value of the payment.
That is, weight for each payment time is the proportion of the total value of the bond
accounted by that payment. This proportion is just the present value of the payment
divided by the bond price.
If wt is the weight associated with each cash flow at time t, and CFt is the cash flow at
time t, then
CFt
wt =
(1 + y ) t
Bond Pr ice
where, y is the yield to maturity of the bond
The duration is captured as follows:
T
D = t * wt
t =1
This represents weighted average of the times until the receipt of each of the bonds
payments.
When rates interest change, the percentage change in a bonds price is proportional to
its duration.
The proportional change in a bonds price can be related to the change in its yield to
maturity, y according to the following rule:
(1 + y )
P
= D
P
(1 + y )
(BKM 11.2)
The above equation implies that bond price volatility is proportional to the bonds
duration.
o Duration is a measure of interest rate exposure.
o This relationship is key to interest rate risk management.
Practitioners use slightly different form of equation (11.2)
D
They define modified duration as D =
and rewrite equation 11.2 as follows:
(1 + y )
P
= D y
P
(BKM 11.3)
Modified duration measures bonds exposure to interest rate volatility
Example:
A bond with maturity of 30 years has a coupon rate of 8% (paid annually) and a yield to
maturity of 9%. Its price is $897.26, and its duration is 11.37 years. What will happen to
the bond price if the bonds yield to maturity increases to 9.1%
Excel Function
= DURATION (settlement date, maturity date, coupon rate, yield to maturity, payments
periods per year)
Settlement date, maturity date: format is DATE(year, month, day)
If we dont have a settlement date, use arbitrary (Jan 1, 2000) as the settlement date
and use a maturity date accordingly (depending upon number of years to maturity)
Coupon rate and yield to maturity are entered in decimals
Convexity: Section 11.3
The duration rule is a good approximation for small changes in bond yield, but it is less
accurate for longer changes.
Equation 11.3 is as follows:
P
= D y
P
This states that the percentage change in the value of a bond approximately equals the
product of modified duration times the change in the bonds yield.
This predicts that the relationship is linear. The graph of percentage change in bond
price as a function of the change in its yield would plot as a straight line with slope
( D ) .
However, we know from the bond pricing relationships that the relationship between
bond prices and yields is not linear.
Figure 11.6: Bond Pricing Convexity
Duration approximation (the straight line) always understates the value of the bond.
It underestimates the increase in bond price when the yield falls.
It overestimates the decline in price when the yield rises.
The true price-yield relationship is convex shaped.
The curvature of the price-yield curve is called the convexity of the bond.
Convexity allows us to improve the duration approximation for bond price changes.
Equation (11.3) can be modified as follows:
P
1
= D y + Convexity (y ) 2
P
2
(BKM 11.5)
The second term is always positive for a bond with positive convexity, regardless of
whether the yield rises or falls.
Equation (11.5) will always predicts a higher bond price than equation (11.3)
2
If y is small, then (y ) be close to zero, and therefore it will add little to the
approximation. In that case linear approximation by duration will be sufficiently
accurate.
Example: Convexity
Let a bond has a 30-year maturity, an 8% coupon and sells at an yield to maturity of 8%.
Because the coupon rate equals YTM, the bond sells at a par-value or $1000.
The modified duration of the bond at its initial yield is 11.26 years and its convexity is
212.4
If the bonds yield increases from 8% to 10%, the bond price will fall to $811.46, a
decline of 18.85%.
The duration rule (without convexity) would predict a price decline of
P
= D y = 11.26*0.02 = -0.2252 = -22.52%
P
Which is higher than the bond price actually falls.
The duration with convexity rule predicts
P
1
= D y + Convexity (y ) 2
P
2
1
= 11.26 * 0.02 + * 212.4 * (0.02) 2 = 0.1827 = 18.27%
2
This is closer to the exact change in bond price.
If the change in yield is smaller, say 0.1%, convexity would have smaller effect. The
price of the bond actually would fall to $988.85 a decline of 1.115%
The duration would predict
P
= D y = -11.26 * 0.001 = -0.01126 = -1.126%
P
Duration with convexity would predict
P
1
= D y + Convexity (y ) 2
P
2
1
= 11.26 * 0.001 + * 212.4 * (0.001) 2 = 0.01115 = 1.115%
2
Hence, in the case of smaller change in yield, the duration is quite accurate, even without
accounting for convexity.
Question: Why do investors like convexity?
Convexity is generally a desirable trait.
Bonds with more convexity gain more in price when yields fall than they lose when
yields rise.
Higher convexity bonds have greater price increase and smaller price decrease when
interest rates fluctuate by larger amounts.