Bond Price Elasticity and Duration

8/2/2019 Bond Price Elasticity and Duration

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Bond Price Elasticity

Business 4179


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Bond Price Elasticity

The sensitivity of bond prices (BP) to changes in the

required rate of return (I) is commonly measured by bond

price elasticity (BPe), which is estimated as

iinchangepercentBPinchangepercentBPe


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Example of Elasticity

If the required rate of return changes from 10 percent to 8

percent, the bond price of a zero coupon bond will rise

from $386 to $463. Thus the bond price elasticity is

997.%20

%9.19

%10

%10%8386$

386$463$

e

BP


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997.%20

%9.19

%10

%10%8

386$

386$463$

e

BP

Example of Elasticity

This implies that for each 1 percent change in interest

rates, bond prices change by 0.997 percent in the oppositedirection.


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Bond Price Elasticity and Bond Price

Theorums

The following table demonstrates how bond price elasticity

measures the effects of a given change in interest rates on

bonds with different coupon rates. Zero coupon or stripped bonds have the longest durations

because there are no intermediate cash flows, hence they

exhibit the greatest elasticity.

The higher the coupon rate, the lower the elasticity allother things being equal.


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Sensitivity of 10-year bonds with different

coupon rates to interest rate changes

Effects of a Decline in the Required Rate of Return

(1)Bonds withA CouponRate of:

(2)Initial PriceOf Bonds

(When i=10%)

(3)Price of

Bonds wheni=8%

(4)=[(3)-(2)]/(2)Percentage

Changein Bond Price

(5)Percentage

Changein i

(6)=(4)/(5)Bond PriceElasticity

(BPe)

0% $386 $463 +19.9% -20% -.9975 693 799 +15.3 -20% -.765

10 1,000 1,134 +13.4 -20% -.670

15 1,307 1,470 +12.5 -20% -.624

Effects of an Increase in the Required Rate of Return:

(1)Bonds with

a CouponRate of:

(2)Initial Price

Of Bonds(When i=10%)

(3)Price of

Bonds wheni=12%

(4)=[(3)-(2)]/(2)Percentage

Change in BondPrice

(5)Percentage

Changein i

(6)=(4)/(5)Bond Price

Elasticity(BP

e)

0% $386 $322 -16.6% +20% -.830

5 693 605 -12.7 +20% -.635

10 1,000 887 -11.3 +20% -.565

15 1,307 1,170 -10.5 +20% -.525


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Bond Price Sensitivity and Term to

Maturity

The following chart explores the impact of the term to

maturity on bond price sensitivity

clearly, the longer the term to maturity, the greater the

bond price elasticity.

When interest rates rise, the bond price will rise by a

greater percentage, than the fall in bond price in response

to an equal but opposite increase in interest rates.


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Sensitivity of 10-year bonds with different

coupon rates to interest rate changes

Effects of a Decline in the Required Rate of Return on a 10% Coupon Rate Bond

(1)Bonds witha Term to

Maturity of:


(When i=10%)

(3)Price of

Bonds wheni=8%

(4)=[(3)-(2)]/(2)Percentage

Changein Bond Price

(5)Percentage

Changein i


(BPe)

1 $1,000 $1,019.40 +1.9% -20% -.095

5 1,000 1,079.87 +8.0 -20% -.4

10 1,000 1,134.21 +13.4 -20% -.67

30 1,000 1,225.20 +22.5 -20% -1.126

Effects of an Increase in the Required Rate of Return on a 10% Coupon Rate Bond

(1)Bonds witha Term to

Maturity of:


(When i=10%)

(3)Price of

Bonds when

i=12%

(4)=[(3)-(2)]/(2)Percentage

Change in Bond

Price

(5)Percentage

Change

in i


(BPe

)1 $1,000 $982.19 -1.8% +20% -.09

5 1,000 927.88 -7.2 +20% -.36

10 1,000 887.02 -11.3 +20% -.565

30 1,000 838.92 -16.0 +20% -.80


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Bond Prices and Term to Maturity

R e q u ire d re t 1 0 .0 0 % 1 0 .0 0 % 1 0 .0 0 % 1 0 .0 0 % 1 0 .0 0 %

C o u p o n R a t 10 .00 % 8 .00% 12 .00 % 6 .00% 14 .00 %

Term to M at B o n d P r ice B o n d P rice B o n d P rice B o n d P r ic e B o n d P r ice

1 $ 1 ,0 0 0 .0 0 $ 9 8 1 .8 2 $ 1 ,0 1 8 .1 8 $ 9 6 3 .6 4 $ 1 ,0 3 6 .3 6

2 $ 1 ,0 0 0 .0 0 $ 9 6 5 .2 9 $ 1 ,0 3 4 .7 1 $ 9 3 0 .5 8 $ 1 ,0 6 9 .4 2

3 $ 1 ,0 0 0 .0 0 $ 9 5 0 .2 6 $ 1 ,0 4 9 .7 4 $ 9 0 0 .5 3 $ 1 ,0 9 9 .4 7

4 $ 1 ,0 0 0 .0 0 $ 9 3 6 .6 0 $ 1 ,0 6 3 .4 0 $ 8 7 3 .2 1 $ 1 ,1 2 6 .7 9

5 $ 1 ,0 0 0 .0 0 $ 9 2 4 .1 8 $ 1 ,0 7 5 .8 2 $ 8 4 8 .3 7 $ 1 ,1 5 1 .6 3

6 $ 1 ,0 0 0 .0 0 $ 9 1 2 .8 9 $ 1 ,0 8 7 .1 1 $ 8 2 5 .7 9 $ 1 ,1 7 4 .2 1

7 $ 1 ,0 0 0 .0 0 $ 9 0 2 .6 3 $ 1 ,0 9 7 .3 7 $ 8 0 5 .2 6 $ 1 ,1 9 4 .7 4

8 $ 1 ,0 0 0 .0 0 $ 8 9 3 .3 0 $ 1 ,1 0 6 .7 0 $ 7 8 6 .6 0 $ 1 ,2 1 3 .4 0

9 $ 1 ,0 0 0 .0 0 $ 8 8 4 .8 2 $ 1 ,1 1 5 .1 8 $ 7 6 9 .6 4 $ 1 ,2 3 0 .3 6

1 0 $ 1 ,0 0 0 .0 0 $ 8 7 7 .1 1 $ 1 ,1 2 2 .8 9 $ 7 5 4 .2 2 $ 1 ,2 4 5 .7 8

1 1 $ 1 ,0 0 0 .0 0 $ 8 7 0 .1 0 $ 1 ,1 2 9 .9 0 $ 7 4 0 .2 0 $ 1 ,2 5 9 .8 0

1 2 $ 1 ,0 0 0 .0 0 $ 8 6 3 .7 3 $ 1 ,1 3 6 .2 7 $ 7 2 7 .4 5 $ 1 ,2 7 2 .5 5

1 3 $ 1 ,0 0 0 .0 0 $ 8 5 7 .9 3 $ 1 ,1 4 2 .0 7 $ 7 1 5 .8 7 $ 1 ,2 8 4 .1 3

1 4 $ 1 ,0 0 0 .0 0 $ 8 5 2 .6 7 $ 1 ,1 4 7 .3 3 $ 7 0 5 .3 3 $ 1 ,2 9 4 .6 7

1 5 $ 1 ,0 0 0 .0 0 $ 8 4 7 .8 8 $ 1 ,1 5 2 .1 2 $ 6 9 5 .7 6 $ 1 ,3 0 4 .2 4

1 6 $ 1 ,0 0 0 .0 0 $ 8 4 3 .5 3 $ 1 ,1 5 6 .4 7 $ 6 8 7 .0 5 $ 1 ,3 1 2 .9 5

1 7 $ 1 ,0 0 0 .0 0 $ 8 3 9 .5 7 $ 1 ,1 6 0 .4 3 $ 6 7 9 .1 4 $ 1 ,3 2 0 .8 6

1 8 $ 1 ,0 0 0 .0 0 $ 8 3 5 .9 7 $ 1 ,1 6 4 .0 3 $ 6 7 1 .9 4 $ 1 ,3 2 8 .0 6

1 9 $ 1 ,0 0 0 .0 0 $ 8 3 2 .7 0 $ 1 ,1 6 7 .3 0 $ 6 6 5 .4 0 $ 1 ,3 3 4 .6 0

2 0 $ 1 ,0 0 0 .0 0 $ 8 2 9 .7 3 $ 1 ,1 7 0 .2 7 $ 6 5 9 .4 6 $ 1 ,3 4 0 .5 4

2 1 $ 1 ,0 0 0 .0 0 $ 8 2 7 .0 3 $ 1 ,1 7 2 .9 7 $ 6 5 4 .0 5 $ 1 ,3 4 5 .9 5

2 2 $ 1 ,0 0 0 .0 0 $ 8 2 4 .5 7 $ 1 ,1 7 5 .4 3 $ 6 4 9 .1 4 $ 1 ,3 5 0 .8 6

2 3 $ 1 ,0 0 0 .0 0 $ 8 2 2 .3 4 $ 1 ,1 7 7 .6 6 $ 6 4 4 .6 7 $ 1 ,3 5 5 .3 3

2 4 $ 1 ,0 0 0 .0 0 $ 8 2 0 .3 1 $ 1 ,1 7 9 .6 9 $ 6 4 0 .6 1 $ 1 ,3 5 9 .3 9

2 5 $ 1 ,0 0 0 .0 0 $ 8 1 8 .4 6 $ 1 ,1 8 1 .5 4 $ 6 3 6 .9 2 $ 1 ,3 6 3 .0 8

2 6 $ 1 ,0 0 0 .0 0 $ 8 1 6 .7 8 $ 1 ,1 8 3 .2 2 $ 6 3 3 .5 6 $ 1 ,3 6 6 .4 4

2 7 $ 1 ,0 0 0 .0 0 $ 8 1 5 .2 6 $ 1 ,1 8 4 .7 4 $ 6 3 0 .5 1 $ 1 ,3 6 9 .4 9

2 8 $ 1 ,0 0 0 .0 0 $ 8 1 3 .8 7 $ 1 ,1 8 6 .1 3 $ 6 2 7 .7 4 $ 1 ,3 7 2 .2 6

2 9 $ 1 ,0 0 0 .0 0 $ 8 1 2 .6 1 $ 1 ,1 8 7 .3 9 $ 6 2 5 .2 2 $ 1 ,3 7 4 .7 8

3 0 $ 1 ,0 0 0 .0 0 $ 8 1 1 .4 6 $ 1 ,1 8 8 .5 4 $ 6 2 2 .9 2 $ 1 ,3 7 7 .0 8

T e r m t o M a t u r i ty a n d B o n d P r i c e

0

20 0

40 0

60 0

80 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 15 16 1 7 1 8 1 9 2 0 21 22 2 3 2 4 2 5 2 6 27 28

Y e a r s L e f t t o M a t u r i t y

BondP

rice(

$)


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Duration

An alternative measure of bond price sensitivity is the

bonds duration.

Duration measures the life of the bond on a present value

basis.

Duration can also be thought of as the average time to

receipt of the bonds cashflows.

The longer the bonds duration, the greater is its sensitivity

to interest rate changes.


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Duration and Coupon Rates

A bonds duration is affected by the size of the coupon rate

offered by the bond.

The duration of a zero coupon bond is equal to the bonds

term to maturity. Therefore, the longest durations are

found in stripped bonds or zero coupon bonds. These are

bonds with the greatest interest rate elasticity.

The higher the coupon rate, the shorter the bonds duration.

Hence the greater the coupon rate, the shorter the duration,and the lower the interest rate elasticity of the bond price.


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Duration

The numerator of the duration formula represents the present value of

future payments, weighted by the time interval until the payments

occur. The longer the intervals until payments are made, the larger will

be the numerator, and the larger will be the duration. The denominatorrepresents the discounted future cash flows resulting from the bond,

which is the bonds present value.

maturitytoyieldsbondthei

providedarepaymentsthewhichattimethet

bondthebygeneratedpaymentprincipalorcoupontheCwhere

i

C

i

tC

DUR

t

n

tt

t

n

tt

t

'

:

)1(

)1(

)(

1

1


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Duration Example

As an example, the duration of a bond with $1,000 par value and a 7

percent coupon rate, three years remaining to maturity, and a 9 percent

yield to maturity is:

years

DUR

80.2)09.1(

1070$

)09.1(

70$

)09.1(

70$

)09.1(

)3(1070$

)09.1(

)2(70$

)09.1(

70$

321

321


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Duration Example ...

As an example, the duration of a zero-coupon bond with $1,000 par

value and three years remaining to maturity, and a 9 percent yield to

maturity is:

years

DUR

0.3

)09.1(

1000$

)09.1(

)3(1000$

3

3


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23

Duration is a handy tool because it can encapsule interest rate exposure in a

single number.

rather than focus on the formula...think of the duration calculation as a

process... semi-annual duration calculations simply call for halving the annual

coupon payments and discounting every 6 months.


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Duration Rules-of-Thumb duration of zero-coupon bond (strip bond) = the term left until

maturity.

duration of a consol bond (ie. a perpetual bond) = 1 + (1/R)

where: R = required yield to maturity duration of an FRN (floating rate note) = 1/2 year


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25

Other Duration Rules-of-ThumbDuration and Maturity

duration increases with maturity of a fixed-income asset, but at a

decreasing rate.

Duration and Yield

duration decreases as yield increases.

Duration and Coupon Interest

the higher the coupon or promised interest payment on the security, the

lower its duration.


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26

Economic Meaning of Duration duration is a direct measure of the interest rate sensitivity or elasticity

of an asset or liability. (ie. what impact will a change in YTM have on

the price of the particular fixed-income security?)

interest rate sensitivity is equal to:

dP = - D [ dR/(1+R)]

P

Where: P = Price of bond

C = Coupon (annual)

R = YTM

N = Number of periods

F = Face value of bond


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27

Interest Rate Elasticity the percent change in the bonds price

caused by a given change in interest rates

(change in YTM)


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28

Economic Meaning of Duration interest rate sensitivity is equal to:

dP = - D [ dR/(1+R)]

P

dP/P = change in bond price

[ dR/(1+R)] = change in interest rate

Obviously, the relationship is an inverse function of Duration (D)


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29

Example of Calculation of Interest

Rate Sensitivity given:

n = 6 years (Eurobond ... annual coupon payments)

8 percent coupon

8 YTM

if yields are expected to rise by 10%, what impact will that have on the price of

the bond?

the first step is to calculate the duration of the bond.

If there were no coupon payments the duration would be = 6.

since there are coupon payments the duration must be less than 6 years.

D = 4.993 years the second step is to calculate the % change in price for the bond.

= -(4.993)(.1/1.08) = - 0.4623 = - 46.23%


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30

Immunization fully protecting or hedging an FIs equity holders against

interest rate risk.

elimination of interest rate risk by matching the duration of

both assets and liabilities. (not their average lives or final

maturities).

when immunized:

the gains or losses on reinvestment income that result from an

interest rate change are exactly offset by losses or gains from thebond proceeds on sale of the bond.


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459K. Hartviksen

Example of Bond Price

The Canada 10.25 1 Feb 04 is quoted at 123.95 yielding 5.27%. Thismeans that for a $1,000 par value bond, these bonds are trading apremium price of $1,239.50

The figure represents bond prices as of June 17, 1998.

This bond has 5 years and 8 months (approximately) until maturity =5+(8/12) = 5.7 years

Bond Price = $102.50(PVIFAn=5.7 ,r=5.27%) + $1,000 / (1.0527)5.7

= $102.50(PVIFAr=5.27%%, n= 5.7) + $746.21= $102.50(4.8156653) + $746.21

= $493.61 + $743.42 = $1,237.03

Can you explain why the quoted price might differ from your answer?


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Example of a Duration Calculation

Example

Assume a 10% coupon bond with three years left to maturity and a required return of 8%.

Coupon Rate = 10.00%

Required Return = 8.00%

Time Cashflow PVIF Present Value Weight

Time

Weighted

CFs

0

0.5 50 0.96225 $48.11 4.55% 0.022767679

1 50 0.925926 $46.30 4.38% 0.043816419

1.5 50 0.890973 $44.55 4.22% 0.063243554

2 50 0.857339 $42.87 4.06% 0.0811415172.5 50 0.824975 $41.25 3.90% 0.097598077

3 1050 0.793832 $833.52 78.89% 2.36662759

Bond Price = $1,056.60 100.00% 2.675194837 =Duration


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459K. Hartviksen

Sensitivity Analysis of Bonds

B o n d

C o u p o n

R a t e

T i m e to

m a t u r i ty

Y i e l d to

M a tu ri ty P V IF A P V IF B o n d P ric e

B o n d X 7 .0 % 1 3 6 .0 % 8 .8 5 2 6 8 3 0 .4 6 8 8 3 9 $ 1 ,0 8 8 .5 3

7 .0 % 1 2 6 .0 % 8 .3 8 3 8 4 4 0 .4 9 6 9 6 9 $ 1 ,0 8 3 .8 4

7 .0 % 1 0 6 .0 % 7 .3 6 0 0 8 7 0 .5 5 8 3 9 5 $ 1 ,0 7 3 .6 0

7 .0 % 5 6 .0 % 4 .2 1 2 3 6 4 0 .7 4 7 2 5 8 $ 1 ,0 4 2 .1 2

7 .0 % 1 6 .0 % 0 .9 4 3 3 9 6 0 .9 4 3 3 9 6 $ 1 ,0 0 9 .4 3

B o n d Y 5 .0 % 1 3 8 .0 % 7 .9 0 3 7 7 6 0 .3 6 7 6 9 8 $ 7 6 2 .8 9

5 .0 % 1 2 8 .0 % 7 .5 3 6 0 7 8 0 .3 9 7 1 1 4 $ 7 7 3 .9 2

5 .0 % 1 0 8 .0 % 6 .7 1 0 0 8 1 0 .4 6 3 1 9 3 $ 7 9 8 .7 0

5 .0 % 5 8 .0 % 3 .9 9 2 7 1 0 .6 8 0 5 8 3 $ 8 8 0 .2 2

5 .0 % 1 8 .0 % 0 .9 2 5 9 2 6 0 .9 2 5 9 2 6 $ 9 7 2 .2 2


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459K. Hartviksen

Prices over time

Time To M Bond X Bond Y

13 $1,088.53 $762.89

12 $1,083.84 $773.92

10 $1,073.60 $798.70

5 $1,042.12 $880.221 $1,009.43 $972.22

0 1000 1000

Bond Prices over Time

0

500

1000

1500

13 12 10 5 1 0

Years Left Until Maturity

Bond

Pric

e


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Duration of a Portfolio

Bond portfolio mangers commonly attempt to immunize

their portfolio, or insulate their portfolio from the effects of

interest rate movements.

For example, a life insurance company knows that they need $100million 30 years from now cover actuarially-determined claims

against a group of life insurance policies just no sold to a group of 30

year olds.

The insurance company has invested the premiums into 30-year

government bonds. Therefore there is no default risk to worry about.The company expects that if the realized rate of return on this bond

portfolio equals the yield-to-maturity of the bond portfolio, there

wont be a problem growing that portfolio to $100 million. The

problem is, that the coupon interest payments must be reinvested and

there is a chance that rates will fall over the life of the portfolio.


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Duration of a Portfolio ...

The life insurance company example illustrates a keep risk

in fixed-income portfolio management - interest rate risk.

The portfolio manager, before-the-fact calculates the bond

portfolios yield-to-maturity. This is an ex ante

calculation. As such, a nave assumption assumption is

made that the coupon interest received each year is

reinvested at the yield-to-maturity for the remaining years

until the bond matures. Over time, however, interest rates will vary and

reinvestment opportunities will vary from that which was

forecast.


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Duration of a Portfolio ...

The insurance company will want to IMMUNIZE their

portfolio from this reinvestment risk.

The simplest way to do this is to convert the entire bond

portfolio to zero-coupon/stripped bonds. Then the ex ante

yield-to-maturity will equal ex post(realized) rate of

return. (ie. the ex ante YTM is locked in since there are

no intermediate cashflows the require reinvestment).

If the bond portfolio manager matches the duration of thebond portfolio with the expected time when they will

require the $100 m, then interest rate risk will be

eliminated.

Bond Price Elasticity and Duration

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Transcript of Bond Price Elasticity and Duration