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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