Fall-02

Bond Price Volatility

Zvi Wiener

Based on Chapter 4 in Fabozzi Bond Markets, Analysis and Strategies http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

EMBAF

You Open a Bank!

You have 1,000 customers.

Typical CD is for 1-3 months with $1,000.

You pay 5% on these CDs.

A local business needs a $1M loan for 1 yr.

The business is ready to pay 7% annually.

Zvi Wiener

What are your major sources of risk?

How you can measure and manage it?

Fabozzi Ch 4 slide 2

Money Manager

value

Zvi Wiener

0 Original plan t

Fabozzi Ch 4

New plan Market shock D

slide 3

8% Coupon Bond

Yield to Maturity

8% 9% Price Change

T=1 yr.

T=10 yr. T=20 yr.

1,000.00 1,000.00 1,000.00

990.64

934.96

907.99

0.94% 6.50% 9.20%

Zero Coupon Bond

Yield to Maturity

8% 9% Price Change

T=1 yr.

924.56

T=10 yr. T=20 yr.

456.39

208.29

915.73

0.96% 414.64

9.15% 171.93

17.46%

Zvi Wiener Fabozzi Ch 4 slide 4

Price-Yield for option-free bonds

price

Zvi Wiener

yield

Fabozzi Ch 4 slide 5

Taylor Expansion

To measure the price response to a small change in risk factor we use the Taylor expansion.

Initial value y 0 , new value y 1 , change  y:

y

1 

y

0  

y f

(

y

1 ) 

f

(

y

0 ) 

f

' (

y

0 ) 

y

 1 2

f

" (

y

0 ) 

y

2  

Zvi Wiener Fabozzi Ch 4 slide 6

F(x)

Derivatives

Zvi Wiener Fabozzi Ch 4

x

slide 7

Properties of derivatives

 

f f

(

x

) (

x

)  

g g

( (

x x

) ) ' '    

f

' (

x

) 

g

' (

x

)

f

' (

x

) 

g

(

x

) 

f

(

x

) 

g

' (

x

)

d dx

  

f g

( (

x

)

x

)    

f

' (

x

)

g

(

x

) 

f

(

x

)

g

' (

x

)

g

2 (

x

)

d dx f

(

g

(

x

)) 

f

' (

g

(

x

))

g

' (

x

)

slide 8 Zvi Wiener Fabozzi Ch 4

Zvi Wiener

Zero-coupon example

P

0  100 ( 1 

y

)

T P

1  ( 1  100

y

 

y

)

T P

1 

P

0

P

0    ( 1  100

y

 

y

)

T

 100 ( 1 

y

)

T

  ( 1 

y

100 )

T Fabozzi Ch 4 slide 9

T 1 2 10 y=10%,  y=0.5%

Example

P 0 90.90

82.64

38.55

P 1 90.09

81.16

35.22

 P -0.45% -1.79% -8.65%

Zvi Wiener Fabozzi Ch 4 slide 10

Property 1

Prices of option-free bonds move in

OPPOSITE

direction from the change in yield.

The price change (in %) is

NOT

the same for different bonds.

Zvi Wiener Fabozzi Ch 4 slide 11

Property 2

For a given bond a

small

increase or decrease in yield leads very similar (but opposite) changes in prices.

What does this means mathematically?

Zvi Wiener Fabozzi Ch 4 slide 12

Property 3

For a given bond a

large

increase or decrease in yield leads to different (and opposite) changes in prices.

What does this means mathematically?

Zvi Wiener Fabozzi Ch 4 slide 13

Property 4

For a given bond a

large

change in yield the percentage price increase is greater than the percentage decrease.

What does this means mathematically?

Zvi Wiener Fabozzi Ch 4 slide 14

Zvi Wiener

What affects price volatility?

Linkage Credit considerations Time to maturity Coupon rate

Fabozzi Ch 4 slide 15

Bond Price Volatility

Consider only IR as a risk factor Longer TTM means higher volatility Lower coupons means higher volatility Floaters have a very low price volatility Price is also affected by coupon payments Price value of a Basis Point (PVBP)= price change resulting from a change of 0.01% in the yield.

slide 16 Zvi Wiener Fabozzi Ch 4

Duration and IR sensitivity

Zvi Wiener Fabozzi Ch 4 slide 17

Duration

F. Macaulay (1938) Better measurement than time to maturity.

Weighted average of all coupons with the corresponding time to payment.

Bond Price = Sum[ CF t /(1+y) t ] suggested weight of each coupon: w t = CF t /(1+y) t /Bond Price

What is the sum of all w t ?

Zvi Wiener Fabozzi Ch 4 slide 18

Duration



P P

 

D MC

   ( 1  1 

y y

)   The bond price volatility is proportional to the bond’s duration.

Thus duration is a natural measure of interest rate risk exposure.

Zvi Wiener Fabozzi Ch 4 slide 19

Modified Duration

D

* 

D

1 

y



P

 

D

* 

y P

The percentage change in bond price is the product of modified duration and the change in the bond’s yield to maturity.

Zvi Wiener Fabozzi Ch 4 slide 20

Duration

P



C

( 1 

y

)  ( 1 

C y

) 2    ( 1 

C y

)

n



M

( 1 

y

)

n Macaulay Duration

 1

P

   1 ( 1 

C y

)  2

C

( 1 

y

) 2   

nC

( 1 

y

)

n



nM

( 1 

y

)

n

  

slide 21 Zvi Wiener Fabozzi Ch 4

Duration

P



C

( 1 

y

)  ( 1 

C y

) 2    ( 1 

C y

)

n



M

( 1 

y

)

n dP



dy



C

( 1 

y

) 2   2

C

( 1 

y

) 3    ( 1 

nC



y

)

n

 1  ( 1 

nM



y

)

n

 1

dP dy

  1  1

y

  

C

( 1 

y

)  2

C

( 1 

y

) 2   

nC

( 1 

y

)

n

 ( 1

nM



y

)

n

  

slide 22 Zvi Wiener Fabozzi Ch 4

Duration

Modified Duration



Macaulay

1 

y Duration dP

 

P



Modified dy Duration Zvi Wiener Fabozzi Ch 4 slide 23

Zvi Wiener

Measuring Price Change

dP



dP dy



dy

1 2

d

2

P

(

dy

) 2

dy

2 

error dP P

 

Ddy



Conv

(

dy

) 2 2 

error P Fabozzi Ch 4 slide 24

The Yield to Maturity

The

yield to maturity of a fixed coupon bond

y is given by

p

(

t

) 

i n

  1

c i e

 (

T i



t

)

y Zvi Wiener Fabozzi Ch 4 slide 25

Macaulay Duration

Definition of duration, assuming t=0.

D



i n

  1

T i c i e



T i y p Zvi Wiener Fabozzi Ch 4 slide 26

Macaulay Duration

D



t T

  1

t w t



Bond

1 Pr

ice t T

  1

t

( 1

CF t



y

)

t

A weighted sum of times to maturities of each coupon.

What is the duration of a zero coupon bond?

slide 27 Zvi Wiener Fabozzi Ch 4

Zvi Wiener

$

Meaning of Duration

dp dy



d dy i n

  1

c i e



T i y

 

Dp Fabozzi Ch 4

r

slide 28

r

Parallel shift

upward move Downward move Current TS T

slide 29 Zvi Wiener Fabozzi Ch 4

Comparison of two bonds

Coupon bond with duration 1.8853

Zero-coupon bond with equal duration must have 1.8853 years to maturity.

Price (at 5% for 6m.) is $964.5405

If IR increase by 1bp (to 5.01%), its price will fall to $964.1942, or 0.359% decline.

At 5% semiannual its price is ($1,000/1.05

3.7706

)=$831.9623

If IR increase to 5.01%, the price becomes: ($1,000/1.0501

3.7706

)=$831.66

0.359% decline.

Zvi Wiener Fabozzi Ch 4 slide 30

D

Duration

Zero coupon bond 0 3m 6m 1yr 15% coupon, YTM = 15% 3yr 5yr Maturity 10yr 30yr

slide 31 Zvi Wiener Fabozzi Ch 4

Example

A bond with 30-yr to maturity Coupon 8%; paid semiannually YTM = 9% P 0 = $897.26

D = 11.37 Yrs if YTM = 9.1%, what will be the price?

 P/P =  y D*  P = -(  y D*)P = -$9.36

P = $897.26 - $9.36 = $887.90

Zvi Wiener Fabozzi Ch 4 slide 32

What Determines Duration?

Duration of a zero-coupon bond equals maturity.

Holding ttm constant, duration is higher when coupons are lower.

Holding other factors constant, duration is higher when ytm is lower.

Duration of a perpetuity is (1+y)/y.

slide 33 Zvi Wiener Fabozzi Ch 4

What Determines Duration?

Holding the coupon rate constant, duration not always increases with ttm.

Zvi Wiener Fabozzi Ch 4 slide 34

Zvi Wiener

$

Convexity

C

  2

p



y

2

Fabozzi Ch 4

r

slide 35

Example

10 year zero coupon bond with a semiannual yield of 6%

P

 100 ( 1  0 .

03 ) 20  $ 55 .

368 The duration is 10 years, the modified duration is:

D

*  ( 1  10 0 .

03 )  9 .

71 The convexity is

C

 1

d

2

P dy

2   ( 1  100 0 .

5

y

) 20    98 .

97

Zvi Wiener Fabozzi Ch 4 slide 36

Example

If the yield changes to 7% the price change is 

P

  9 .

71  $ 55 .

37  0 .

01  98 .

97  $ 55 .

37  0 .

01 2  0 .

5   $ 5 .

375  $ 0 .

274   $ 5 .

101 

P

 100 ( 1  0 .

035 ) 20  100 ( 1  0 .

03 ) 20   $ 5 .

111

slide 37 Zvi Wiener Fabozzi Ch 4

FRM-98, Question 17

A bond is trading at a price of 100 with a yield of 8%. If the yield increases by 1 bp, the price of the bond will decrease to 99.95. If the yield decreases by 1 bp, the price will increase to 100.04. What is the modified duration of this bond?

A. 5.0

B. -5.0

C. 4.5

D. -4.5

slide 38 Zvi Wiener Fabozzi Ch 4

FRM-98, Question 17

D Modified



P

 

P

 2

P



y

 100 .

04  99 .

95 100  0 .

0002  4 .

5

Zvi Wiener Fabozzi Ch 4 slide 39

FRM-98, Question 22

What is the price of a 10 bp increase in yield on a 10-year par bond with a modified duration of 7 and convexity of 50?

A. -0.705

B. -0.700

C. -0.698

D. -0.690

Zvi Wiener Fabozzi Ch 4 slide 40

FRM-98, Question 22



P

 

D

 

y

 

y

2 2

C



P

   7  $ 100  0 .

001  0 .

001 2 2 50  $ 100  0 .

6975

Zvi Wiener Fabozzi Ch 4 slide 41

Portfolio Duration

Similar to a single bond but the cashflow is determined by all Fixed Income securities held in the portfolio.

Zvi Wiener Fabozzi Ch 4 slide 42

Bond Price Derivatives

D* - modified duration, dollar duration is the negative of the first derivative:

f

' (

y

0 ) 

dP dy

 

D

* 

P

0 Dollar convexity = the second derivative, C - convexity.

f

" (

y

0 ) 

d

2

P dy

2 

C



P

0

Zvi Wiener Fabozzi Ch 4 slide 43

Zvi Wiener

Duration of a portfolio

P

(

y

) 

i N

  1

x i P i

(

y

)

dP dy



i N

  1

x i dP i

(

y

)

dy dP dy

 

D

* 

P

 

i N

  1

x i D i

*

P i Fabozzi Ch 4 slide 44

Zvi Wiener

ALM Duration

D A

  1

A



A



r D L

  1

L D A



L

  1

A



L

 (

A

 

r L

) 

L



r

Does NOT work!

Wrong units of measurement Division by a small number

Fabozzi Ch 4 slide 45

Duration Gap

A - L = C, assets - liabilities = capital

D C



D A A



D L C L C D gap



D A



D L L A D gap A C



D C Zvi Wiener Fabozzi Ch 4 slide 46

Zvi Wiener

ALM Duration

1

VaR P



P



r

A similar problem with measuring yield

Fabozzi Ch 4 slide 47

Do not think of duration as a measure of time!

Zvi Wiener Fabozzi Ch 4 slide 48

Key rate duration Principal component duration Partial duration

Zvi Wiener Fabozzi Ch 4 slide 49

Very good question!

Cashflow: Libor in one year from now Libor in two years form now Libor in three years from now (no principal) What is the duration?

Zvi Wiener Fabozzi Ch 4 slide 50

Home Assignment

What is the duration of a floater?

What is the duration of an inverse floater?

How coupon payments affect duration?

Why modified duration is better than Macaulay duration?

How duration can be used for hedging?

Zvi Wiener Fabozzi Ch 4 slide 51

Home Assignment Chapter 4 Ch. 4: Questions 1, 2, 3, 4, 15.

Calculate duration of a consul (perpetual bond).

slide 52 Zvi Wiener Fabozzi Ch 4

End Ch. 4

Zvi Wiener Fabozzi Ch 4 slide 53

Understanding of Duration/Convexity

What happens with duration when a coupon is paid?

How does convexity of a callable bond depend on interest rate?

How does convexity of a puttable bond depend on interest rate?

Zvi Wiener Fabozzi Ch 4 slide 54

Callable bond

The buyer of a callable bond has written an option to the issuer to call the bond back.

Rationally this should be done when … Interest rate fall and the debt issuer can refinance at a lower rate.

Zvi Wiener Fabozzi Ch 4 slide 55

Puttable bond

The buyer of a such a bond can request the loan to be returned.

The rational strategy is to exercise this option when interest rates are high enough to provide an interesting alternative.

Zvi Wiener Fabozzi Ch 4 slide 56

Zvi Wiener

Embedded Call Option

regular bond strike callable bond r

Fabozzi Ch 4 slide 57

Zvi Wiener

Embedded Put Option

puttable bond regular bond r

Fabozzi Ch 4 slide 58

Convertible Bond

Stock Payoff Convertible Bond Straight Bond

Stock

Zvi Wiener Fabozzi Ch 4 slide 59

Zvi Wiener

Timing of exercise

European American Bermudian Lock up time

Fabozzi Ch 4 slide 60

Zvi Wiener

Macaulay Duration

P

(

y

) 

i T

 

t

( 1

C t



y

)

t D M

 1

P t T

  1 ( 1

tC t



y

)

t

Modified duration

D

*  1

D M



y Fabozzi Ch 4 slide 61

Bond Price Change



P

 

D

* 

P

 

y

 1 2

C



P

 

y

2  

Zvi Wiener Fabozzi Ch 4 slide 62

Duration of a coupon bond

P

(

y

) 

t T

  1 ( 1

C t



y

)

t dP

(

y

)

dy



t T

  1 ( 1  

tC t y

)

t

 1   1

D



y P D

 1

P t T

  1 ( 1

tC t



y

)

t Zvi Wiener Fabozzi Ch 4 slide 63

Exercise

Find the duration and convexity of a consol (perpetual bond).

Answer: (1+y)/y.

Zvi Wiener Fabozzi Ch 4 slide 64

FRM-98, Question 29

A and B are perpetual bonds. A has 4% coupon, and B has 8% coupon. Assume that both bonds are trading at the same yield, what can be said about duration of these bonds?

A. The duration of A is greater than of B B. The duration of A is less than of B C. They have the same duration D. None of the above

Zvi Wiener Fabozzi Ch 4 slide 65

FRM-97, Question 24

Which of the following is NOT a property of bond duration?

A. For zero-coupon bonds Macaulay duration of the bond equals to time to maturity.

B. Duration is usually inversely related to the coupon of a bond.

C. Duration is usually higher for higher yields to maturity.

D. Duration is higher as the number of years to maturity for a bond selling at par or above increases.

Zvi Wiener Fabozzi Ch 4 slide 66

FRM-99, Question 75

You have a large short position in two bonds with similar credit risk. Bond A is priced at par yielding 6% with 20 years to maturity. Bond B has 20 years to maturity, coupon 6.5% and yield of 6%. Which bond contributes more to the risk of the portfolio?

A. Bond A B. Bond B C. A and B have similar risk D. None of the above

slide 67 Zvi Wiener Fabozzi Ch 4

Portfolio Duration and Convexity

D

*

p P p



i N

  1

D i

*

x i P i

Portfolio weights

w i



x i P i P p D

*

p



i N

  1

w i D i

*

C

*

p



i N

  1

w i C i

*

Zvi Wiener Fabozzi Ch 4 slide 68

Example

Construct a portfolio of two bonds: A and B to match the value and duration of a 10-years, 6% coupon bond with value $100 and modified duration of 7.44 years.

A. 1 year zero bond - price $94.26

B. 30 year zero - price $16.97

slide 69 Zvi Wiener Fabozzi Ch 4

 100   7 .

44  

x

1  100 94 .

26  0 .

97  

x

2

x

1  16 .

97  94 .

26  29 .

13 

x

2  16 .

97   

x

1

x

2  1 .

021  0 .

221 Modified duration Barbel portfolio consists of very short and very long bonds.

Bullet portfolio consists of bonds with similar maturities.

Which of them has higher convexity?

Zvi Wiener Fabozzi Ch 4 slide 70

FRM-98, Question 18

A portfolio consists of two positions. One is long $100 of a two year bond priced at 101 with a duration of 1.7; the other position is short $50 of a five year bond priced at 99 with a duration of 4.1. What is the duration of the portfolio?

A. 0.68

B. 0.61

C. -0.68

D. -0.61

Zvi Wiener Fabozzi Ch 4 slide 71

Zvi Wiener

FRM-98, Question 18

D



D

1

P P

1  101 1 .

7 101  49 .

5

D

 2

P

2 

P

 4 .

1 101 49 .

5  49 .

5   0 .

61 Note that $100 means notional amount and can be misunderstood.

Fabozzi Ch 4 slide 72

Useful formulas

1 

a



a

2 

a

3   

a N

 1 

a N

 1 1 

a P



cF

1 

y



cF

( 1 

y

) 2   

cF

( 1 

y

)

T



c y F

  1  ( 1  1

y

)

T

   ( 1 

F y

)

T

 ( 1 

F y

)

T



Zvi Wiener Fabozzi Ch 4 slide 73

Volatilities of IR/bond prices

Price

volatility in % Euro 30d Euro 180d Euro 360d Swap 2Y Swap 5Y Swap 10Y Zero 2Y Zero 5Y Zero 10Y End 99 0.22

0.30

0.52

1.57

4.23

8.47

1.55

4.07

7.76

End 96 0.05

0.19

0.58

1.57

4.70

9.82

1.64

4.67

9.31

23.53

slide 74

Duration approximation

 

P P



D

*   ( 

y

) What duration makes bond as volatile as FX?

What duration makes bond as volatile as stocks?

A 10 year bond has yearly price volatility of 8% which is similar to major FX.

30-year bonds have volatility similar to equities (20%).

slide 75 Zvi Wiener Fabozzi Ch 4

Volatilities of yields

Yield

volatility in %, 99  (  y/y) Euro 30d Euro 180d Euro 360d Swap 2Y Swap 5Y Swap 10Y Zero 2Y Zero 5Y Zero 10Y 45 10 9 12.5

13 12.5

13.4

13.9

13.1

 (  y) 2.5

0.62

0.57

0.86

0.92

0.91

0.84

0.89

0.85

0.74

slide 76