20 - Bond Duration and Convexity

Notes

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Economics of Capital Markets

Page 1Version 1.0 Outline

Bond Duration and ConvexityBond Duration and Convexity

Major Topics:



IntroductionIntroduction

Notes

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Bond Duration and Convexity IntroductionBond Duration and Convexity Introduction

We already know the relationship between bond prices and yields (or required rates of return)– Some others may be new



1. The value of a bond is inversely related to changes in the investor’s required rate of return.

– The higher the required rate of return, the lower the bond value.

Bond Duration and Convexity Introduction (Continued)


Notes

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$0

$50

$100

$150

$200

0% 2% 4% 6% 8% 10% 12% 14% 16%

Yield

Pric

e

( ) ( )

++

+−= nn

B0 y1

Fy1

11yCP

1. Prices and returns on bonds are inversely related





2 The market value of a bond will be less than the par value if the investor’s required rate is above the coupon interest rate



Notes

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Premium bonds

Par bonds

Discount bonds





3. Long-term bonds have greater interest rate risk than short term bonds.

$0

$50

$100

$150

$200

$250

0% 2% 4% 6% 8% 10% 12% 14% 16%Yield

Pric

e

10 Year20 Year5 Year

20

10

5

51020



Notes

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3. Corollary: Low coupon bonds have greater interest rate sensitivity than high coupon bonds.

$0$50

$100$150$200

1% 3% 5% 7% 9% 11%

13%

15%

Yield

Pric

e

CouponZero

Coupon

Zero

Zero

Coupon





The sensitivity of a bond’s value to changing interest rates depends on both the length of time to maturity and on the pattern of cashflows provided by the bond



Notes

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DurationDuration



We want to know how the price of a bond changes as the yield changes

Bond Duration and Convexity DurationBond Duration and Convexity Duration

Notes

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To derive the elasticity, use

Bond Duration and Convexity Duration (Continued)


∑= +

++

=n

1tnt

B0 y)(1

F y)(1

C P



Take the first derivative with respect to y to get



+

++

+++

+++

=

++

+++

++

+= ++

nn21

1n1n32

B0

y)(1nF

y)(1nC ...

y)(12C

y)(11C

y11-

y)(1(-n)F

y)(1(-n)C ...

y)(1(-2)C

y)(1(-1)C

dydP

Notes

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Multiplying through by

where



Dy1

1-

P1

y)(1nF

y)(1nC ...

y)(12C

y)(11C

y11-

P1

dydP

B0

nn21B0

B0

+=

∗

+

++

+++

+++

=

B0P1

0 P1

y)(1nF

y)(1nC ...

y)(12C

y)(11C D B

0nn21 >∗

+

++

+++

++

=



Now we can get



0 D- P

y1 dy

dP η B0

B0 <=

+=

Notes

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Definition– Duration measures the responsiveness of a

bond’s price to interest rate changes:

B0

n

B0

n

1tt

Py)(1

nF

P

y)(1Ct

D Duration +++⋅

==∑=





Despite the name, the duration is not the same as the time to maturity



Notes

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Rewrite D as

B

n

nnt

B

nn

1tB

0

tt

Py)(1

n

γ,F γ tw

Py)(1

nF

P

y)(1Ct

tD

+=+=

++

+⋅

=

∑

∑=





The duration shows the “actual” weighted length of time needed to recover the current cost of the bond,



B0P

Notes

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For a zero coupon bond, duration equals the maturity



+

=

++

+⋅

= ∑=

B0

n

B

nn

1tB

0

tt

P1

y)(1Fn

Py)(1

nF

P

y)(1Ct

tD

0



But

soD = n



nB

0 y)(1FP+

=

Notes

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For non-zero bonds, D < n– Generally, the higher C and the higher y, the

shorter is durationAlso, the longer the maturity, the greater the duration so the greater is the price change for a small yield change





ConvexityConvexity

Notes

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We noticed earlier that the price-yield curve is convex to the origin

Bond Duration and Convexity ConvexityBond Duration and Convexity Convexity



Consider this situation

Bond Duration and Convexity Convexity (Continued)


y

PB

PB*

y*

Tangent Line

Notes

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The tangent line shows the arte of change of PB to a change in y at the tangency point





Notice that, starting at y*, if we move away from y*, the price change will always be underestimated by the tangent



y

PB

PB*

y*

Underestimate

Underestimate

Notes

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The accuracy of using the tangent depends on the convexity of the price-yield curve





Expand the price equation using a Taylor Series Expansion– Write the first two terms plus the error as



( ) error dydy

Pd21 dy

dydp dP 2

2

B0

2B0 ++=

Notes

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Dividing by , we get

– The first term is the duration for a small yield change



( )B0

B0

2

2

B0

2

B0

B0

B0

B0

Perror

Pdy

dyPd

21

Pdy

dydP

PdP

++=

B0P

Duration



The second term corrects for the convexity– The second derivative is called the dollar

convexity of a bond



2

B0

2

dyPd $Convexity =

Notes

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It is easy to show that the second derivative of the price-yield relationship is



( )( )

( )( )∑

=++ +

++

++

=n

1t2n2t2

B0

2

y1F1nn

y1C1tt

dyPd



The duration and the convexity adjustment term can be summed to get the estimated price change due to duration and convexity



Notes

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Example: 25-year, 6% coupon selling to yield 9%– Duration = 10.62– Convexity = 182.92– Assume the yield rises 200 basis points to 11%





Example (Continued)

Percentage change in price due to duration:= -duration*dy= -(10.62)*(0.02)= -21.24%

Percentage change in price due to convexity:= 0.5*convexity*(dy)2

= 0.5*182.92*0.022

= 3.66%

Percentage change in price = -21.24% + 3.66% = -17.58%



Notes

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Consider two coupon bonds, A and B– B has greater convexity

Bond Duration and Convexity Application of Duration and ConvexityBond Duration and Convexity Application of Duration and Convexity

y

PB

PB*

y*

B

A

Same Duration



Consider two coupon bonds (Continued)

– Bond B will have a greater price no matter what happens to the yield

– Also,

Bond Duration and Convexity Application of Duration and Convexity (Continued)


More is Bon Gain Falls YieldLess is Bon Loss Rises Yield

⇒⇒

Notes

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The market takes B’s greater convexity into account when pricing B– The market prices the convexity since the

convexity adds a risk element





Pricing depends on expectations– If investors expect very small yield changes, the

advantages of holding B are small – both A and Boffer the same approximate price (i.e., return)

» There is little paid to the convexity

– If a big change is expected, B will be priced higher than A

» B has more risk



Notes

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There are three properties of bonds due to convexity

1. As the required yield increases (decreases), the convexity of a bond decreases (increases).» This is referred to as positive convexity





There are three properties (Continued)

1. As the required yield increases: Implication– As the yield rises, the price declines but is “slowed”

by the decline in the duration– As the yield falls, the price rises but is “enhanced”

by the increase in the duration



Notes

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1. As the required yield increases: Graphically



y

PB

↑↓

↓↑

Duration ,y AsDuration ,y As




2. For a given yield and maturity, the lower the coupon, the greater the convexity of a bond• A zero coupon bond has the highest convexity

3. For a given yield and duration, the lower the coupon, the smaller the convexity• A zero coupon bond has the lowest convexity



Notes

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583.780%/25-year25.180%/5-year182.926%/25-year20.856%/5-year160.729%/25-year19.459%/5-year

ConvexityBond(Coupon/Maturity)

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20 - Bond Duration and Convexity