In this chapter, we calculate the ‘Duration’ for financial security such as a Bond.
Identify what is the Economic meaning of this Duration? What is its usefulness for investors and institutions? [Read chapter 3 LG 3-8 first]
Suggested length: Two long paragraphs; One external citation!
80 Part 1 Introduction and Overview of Financial Markets
Duration and Coupon Interest. A comparison of Tables 3–7 and 3–8 indicates that the
higher the coupon or promised interest payment on the bond, the shorter its duration. This
is due to the fact that the larger the coupon or promised interest payment, the more quickly
investors receive cash flows on a bond and the higher are the present value weights of
those cash flows in the duration calculation. On a time value of money basis, the investor
recoups his or her initial investment faster when coupon payments are higher.
Duration and Rate of Return. A comparison of Tables 3–7 and 3–9 also indicates that
duration decreases as the rate of return on the bond increases. This makes intuitive sense
since the higher the rate of return on the bond, the lower the present value cost of waiting
to receive the later cash flows on the bond. Higher rates of return discount later cash flows
more heavily, and the relative importance, or weights, of those later cash flows decline
when compared to cash flows received earlier.
Duration and Maturity. A comparison of Tables 3–7 , 3–10 , and 3–12 indicates
that duration increases with the maturity of a bond, but at a decreasing rate. As matu-
rity of a 10 percent coupon bond decreases from four years to three years ( Tables 3–7
and 3–10 ), duration decreases by 0.75 years, from 3.42 years to 2.67 years. Decreas-
ing maturity for an additional year, from three years to two years ( Tables 3–10 and 3–12 ),
decreases duration by 0.81 years, from 2.67 years to 1.86 years. Notice too that for a cou-
pon bond, the longer the maturity on the bond the larger the discrepancy between matu-
rity and duration. Specifically, the two-year maturity bond has a duration of 1.86 years
(0.14 years less than its maturity), while the three-year maturity bond has a duration of 2.67
years (0.33 years less than its maturity), and the four-year maturity bond has a duration of 3.42
years (0.58 years less than its maturity). Figure 3–6 illustrates this relation between duration
and maturity for our 10 percent coupon (paid semiannually), 8 percent rate of return bond.
Economic Meaning of Duration
So far we have calculated duration for a number of different bonds. In addition to being a
measure of the average life of a bond, duration is also a direct measure of its price sensitiv-
ity to changes in interest rates, or elasticity.
13
In other words, the larger the numerical value
LG 3-8
1. The higher the coupon or promised interest payment on a security, the shorter is its duration.
2. The higher the rate of return on a security, the shorter is its duration.
3. Duration increases with maturity at a decreasing rate.
TABLE 3–11 Features of Duration
t CF t
1 __________
(1 + 4%)2t
CFt __________
(1 + 4%)2t
CFt × t __________
(1 + 4%)2t
½ 50 0.9615 48.08 24.04
1 50 0.9246 46.23 46.23
1½ 50 0.8890 44.45 66.67
2 1,050 0.8548 897.54 1,795.08
1,036.30 1,932.02
D =
1,932.02
________
1,036.30
= 1.86 years
TABLE 3–12 Duration of a Two-Year Bond with 10 Percent Coupon Paid Semiannually
and 8 Percent Rate of Return
13. In Chapter 22, we also make the direct link between duration and the price sensitivity of an asset or liability or of
an FI’s entire portfolio (i.e., its duration gap). We show how duration can be used to immunize a security or portfolio of
securities against interest rate risk.
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Chapter 3 Interest Rates and Security Valuation 81
of duration ( D ), the more sensitive the price of that bond (Δ P / P ) to (small) changes or
shocks in interest rates Δ r b /(1 + r b ). The specific relationship between these factors for
securities with annual compounding of interest is represented as:
14
ΔP/P ___________
Δrb/(1 + rb)
= -D
For securities with semiannual receipt (compounding) of interest, it is represented as:
ΔP/P ____________
Δrb/(1 + rb/2)
= -D
The economic interpretation of this equation is that the number D is the elasticity, or
sensitivity, of the bond’s price to small interest rate (either required rate of return or yield
to maturity) changes. The negative sign in front of the D indicates the inverse relationship
between interest rate changes and price changes. That is, – D describes the percentage
value decrease —capital loss—on the security (Δ P / P ) for any given (discounted) small
increase in interest rates [Δ r b /(1 + r b )], where Δ r b is the change in interest rates and
1 + r b is 1 plus the current (or beginning) level of interest rates.
The definition of duration can be rearranged in another useful way for interpretation
regarding price sensitivity:
ΔP ____
P
= -D
[
Δrb ______
1 + rb
]
or
ΔP ____
P
= -D
[
Δrb _______
1 + rb/2
]
for annual and semiannual compounding of interest, respectively. This equation shows that
for small changes in interest rates, bond prices move in an inversely proportional manner
14. In what follows, we use the Δ (change) notation instead of d (derivative notation) to recognize that interest rate
changes tend to be discrete rather then infinitesimally small. For example, in real-world financial markets the smallest
observed rate change is usually one basis point, or 1/100 of 1 percent.
Figure 3–6 Discrepancy between Maturity and Duration on a Coupon Bond
Maturity
(years)
Years
1 2 3 4 5
5
4
3
2
1
0
Duration
Maturity
Gap 5 Maturity 2 Duration
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82 Part 1 Introduction and Overview of Financial Markets
according to the size of D. Clearly, for any given change in interest rates, long duration
securities suffer a larger capital loss (or receive a higher capital gain) should interest rates
rise (fall) than do short duration securities.
15
The duration equation can be rearranged, combining D and (1 + r b ) into a single vari-
able D /(1 + r b ), to produce what practitioners call modified duration ( MD ). For annual
compounding of interest:
ΔP ____
P
= -MD × Δrb
where
MD = D ______
1 + rb
For semiannual compounding of interest:
ΔP ____
P
= -MD × Δrb
where
MD = D _______
1 + rb/2
This form is more intuitive than the Macaulay’s duration because we multiply MD by the
simple change in interest rates rather than the discounted change in interest rates as in the
general duration equation. Thus, the modified duration is a more direct measure of bond
price elasticity. Next, we use duration to measure the price sensitivity of different bonds to
small changes in interest rates.
15. By implication, gains and losses under the duration model are symmetric. That is, if we repeated the above exam-
ples but allowed interest rates to decrease by one basis point annually (or ½ basis point semiannually), the percentage
increase in the price of the bond (Δ P / P ) would be proportionate with D. Further, the capital gains would be a mirror
image of the capital losses for an equal (small) decrease in interest rates.
modified duration
Duration divided by 1 plus the
initial interest rate.
EXAMPLE 3–14 Four-Year Bond
Consider a four-year bond with a 10 percent coupon paid semiannually (or 5 percent paid
every 6 months) and an 8 percent rate of return ( r b ). According to calculations in Table
3–7 , the bond’s duration is D = 3.42 years. Suppose that the rate of return increases by 10
basis points (1/10 of 1 percent) from 8 to 8.10 percent. Then, using the semiannual com-
pounding version of the duration model shown above, the percentage change in the bond’s
price is:
ΔP ____
P
= -(3.42)
[
0.001
_____
1.04
]
= -0.00329
or
= -0.329%
The bond price had been $1,067.34, which was the present value of a four-year bond with
a 10 percent coupon and an 8 percent rate of return. However, the duration model predicts
that the price of this bond will fall by 0.329 percent, or by $3.51, to $1,063.83 after the
increase in the rate of return on the bond of 10 basis points.
16
16. That is, a price fall of 0.329 percent in this case translates into a dollar fall of $3.51. To calculate the dollar change
in value, we can rewrite the equation as Δ P = ( P )(- D )((Δ r b )/(1 + r b /2)) = ($1,067.34)(-3.42)(0.001/1.04) = $3.51.
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Chapter 3 Interest Rates and Security Valuation 83
D O Y O U U N D E R S T A N D :
14. When the duration of an asset is
equal to its maturity?
15. What the denominator of the
duration equation measures?
16. What the numerator of the duration
equation measures?
17. What the duration of a zero-coupon
bond is?
18. Which has the longest duration: a
30-year, 8 percent yield to maturity,
zero-coupon bond, or a 30-year, 8
percent yield to maturity, 5 percent
coupon bond?
19. What the relationship is between
the duration of a bond and its price
elasticity?
Large Interest Rate Changes and Duration
It needs to be stressed here that duration accurately measures the price sensitivity
of financial securities only for small changes in interest rates of the order of one or
a few basis points (a basis point is equal to one-hundredth of 1 percent). Suppose,
however, that interest rate shocks are much larger, of the order of 2 percent or 200
basis points or more. While such large changes in interest rates are not common,
this might happen in a financial crisis or if the central bank (see Chapter 4) sud-
denly changes its monetary policy strategy. In this case, duration becomes a less
accurate predictor of how much the prices of bonds will change, and therefore,
a less accurate measure of the price sensitivity of a bond to changes in interest
rates. Figure 3–7 is a graphic representation of the reason for this. Note the dif-
ference in the change in a bond’s price due to interest rate changes according to
the proportional duration measure ( D ), and the “true relationship,” using the time
value of money equations of Chapter 2 (and discussed earlier in this chapter) to
calculate the exact present value change of a bond’s price in response to interest
rate changes.
Specifically, duration predicts that the relationship between an interest rate
change and a security’s price change will be proportional to the security’s D
With a lower coupon rate of 6 percent, as shown in Table 3–8 , the bond’s duration, D,
is 3.60 and the bond price changes by:
ΔP ____
P
= -(3.60)
[
0.001
_____
1.04
]
= -0.00346
or
= -0.346%
for a 10-basis-point increase in the rate of return. The bond’s price drops by 0.346 percent,
or by $3.23, from $932.68 (reported in Table 3–8 ) to $929.45. Notice again that, all else
held constant, the higher the coupon rate on the bond, the shorter the duration of the
bond and the s m aller the percentage decrease in the bond’s price for a given increase in
interest rates.
Figure 3–7 Duration Estimated versus True Bond Price
True Relationship
Duration Model
Error
Error
P
P
2D
=
rb
(1 1 rb)
=
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84 Part 1 Introduction and Overview of Financial Markets
(duration). By precisely calculating the exact or true change in the security’s price using
time value of money calculations, however, we would find that for large interest rate
increases, duration overpredicts the fall in the security’s price, and for large interest rate
decreases, it underpredicts the increase in the security’s price. Thus, duration misestimates
the change in the value of a security following a large change (either positive or negative)
in interest rates. Further, the duration model predicts symmetric effects for rate increases
and decreases on a bond’s price. As Figure 3–7 shows, in actuality, the capital loss effect of
large rate increases tends to be smaller than the capital gain effect of large rate decreases.
This is the result of a bond’s price–interest rate relationship exhibiting a property called
convexity rather than linearity, as assumed by the simple duration model. Intuitively, this
is because the sensitivity of the bond’s price to a change in interest rates depends on the
level from which interest rates change (i.e., 6 percent, 8 percent, 10 percent, 12 percent). In
particular, the higher the level of interest rates, the smaller a bond’s price sensitivity to
interest rate changes.
convexity
The degree of curvature of
the price–interest rate curve
around some interest rate level.
EXAMPLE 3–15 Calculation of the Change in a Security’s Price
Using the Duration versus the Time Value of Money
Formula
To see the importance of accounting for the effects of convexity in assessing the impact
of large interest rate changes, consider the four-year, $1,000 face value bond with a
10 percent coupon paid semiannually and an 8 percent rate of return. In Table 3–7 we
found this bond has a duration of 3.42 years, and its current price is $1,067.34. We repre-
sent this as point A in Figure 3–8 . If rates rise from 8 percent to 10 percent, the duration
model predicts that the bond price will fall by 6.577 percent; that is:
ΔP ____
P
= -3.42(0.02/1.04) = -6.577%
or from a price of $1,067.34 to $997.14 (see point B in Figure 3–8 ). However, using time
value of money formulas to calculate the exact change in the bond’s price after a rise in
rates to 10 percent, we find its true value is:
Vb = 50 [
1 –
1
______________
[1 + (0.10/2)]
2(4)
__________________
0.10/2
] + 1,000/[1 + (0.10/2)]
2(4)
= $1,000
This is point C in Figure 3–8 . As you can see, the true or actual fall in price is less than the
duration predicted fall by $2.86. The reason for this is the natural convexity to the price–
interest rate curve as interest rates rise.
Reversing the experiment reveals that the duration model would predict the bond’s
price to rise by 6.577 percent if yields were to fall from 8 percent to 6 percent, resulting
in a predicted price of $1,137.54 (see point D in Figure 3–8 ). By comparison, the true
or actual change in price can be computed, using time value of money formulas and a
6 percent rate of return, as $1,140.39 (see point E in Figure 3–8 ). The duration model has
underpredicted the true bond price increase by $2.85 ($1,140.39 – $1,137.54).
An important question for managers of financial institutions and individual savers
is whether the error in the duration equation is big enough to be concerned about. This
depends on the size of the interest rate change and the size of the portfolio under manage-
ment. Clearly, for a large portfolio the error will also be large.
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Chapter 3 Interest Rates and Security Valuation 85
Figure 3–8 Price–Interest Rate Curve for the Four-Year 10 Percent Coupon Bond
Rate of Return (%)
Price (Vb)
6 8
E
D
A
B
C
10
$1,140.39
$1,137.54
$1,067.34
$1,000.00
$997.14
0
Note that convexity is a desirable feature for an investor or FI manager to capture in
a portfolio of assets. Buying a bond or a portfolio of assets that exhibits a lot of convexity
or curvature in the price–interest rate relationship is similar to buying partial interest rate
risk insurance. Specifically, high convexity means that for equally large changes of interest
rates up and down (e.g., plus or minus 2 percent), the capital gain effect of a rate decrease
more than offsets the capital loss effect of a rate increase.
So far, we have established the following three characteristics of convexity:
1. Convexity is desirable. The greater the convexity of a security or portfolio of securities,
the more insurance or interest rate protection an investor or FI manager has against rate
increases and the greater the potential gains after interest rate falls.
2. Convexity diminishes the error in duration as an investment criterion. The larger the
interest rate changes and the more convex a fixed-income security or portfolio, the
greater the error the investor or FI manager faces in using just duration (and duration
matching) to immunize exposure to interest rate shocks.
3. All fixed-income securities are convex. That is, as interest rates change, bond prices
change at a nonconstant rate.
To illustrate the third characteristic, we can take the four-year, 10 percent coupon,
8 percent rate of return bond and look at two extreme price–interest rate scenarios. What
is the price on the bond if rates fall to zero, and what is its price if rates rise to some very
large number such as infinity? Where r b = 0:
Vb =
50
_______
(1 + 0)
1
+
50
_______
(1 + 0)
2
+ . . . +
1,050
_______
(1 + 0)
8
= $1,400
The price is just the simple undiscounted sum of the coupon values and the face value of
the bond. Since interest rates can never go below zero, $1,400 is the maximum possible
price for the bond. Where r b = ∞:
Vb =
50
________
(1 + ∞)
1
+
50
________
(1 + ∞)
2
+ . . . +
1,050
________
(1 + ∞)
8
= $0
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86 Part 1 Introduction and Overview of Financial Markets
Figure 3–9 The Natural Convexity of Bonds
`0
Price
$1,400
Price–Interest Rate Curve
Convexity
Rate of Return (r)
SUMMARY
This chapter applied the time value of money formulas presented in Chapter 2 to the valu-
ation of financial securities such as bonds and equities. With respect to bonds, we included
a detailed examination of how changes in interest rates, coupon rates, and time to maturity
affect their price and price sensitivity. We also presented a measure of bond price sensi-
tivity to interest rate changes, called duration. We showed how the value of duration is
affected by various bond characteristics, such as coupon rates, interest rates, and time to
maturity.
CHAPTER NOTATION
r = required rate of return
CF t = cash flow received on a security at end of period t
n = number of periods in the investment horizon
PV = present value of a security
E ( r ) = expected rate of return
P or
__
P = current market price for a security
RCF t = realized cash flow in period t
_
r = realized rate of return
V b = the price on a bond
M = par or face value of a bond
INT = annual interest payment on a bond
T = number of years until a bond matures
r b = annual interest rate used to discount cash flows on a bond
r s = interest rate used to discount cash flows on equity
Div t = dividend paid at the end of year t
g = constant growth rate in dividends each year
D = duration on a security measured in years
N = last period in which the cash flow is received or number of periods to maturity
MD = modified duration = D /(1 + r )
As interest rates go to infinity, the bond price falls asymptotically toward zero, but by defi-
nition a bond’s price can never be negative. Thus, zero must be the minimum bond price
(see Figure 3–9 ). In Appendix 3B to this chapter (available through Connect or your course
instructor) we look at how to measure convexity and how this measure of convexity can
be incorporated into the duration model to adjust for or offset the error in the prediction of
security price changes for a given change in interest rates.
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