Lecture 3

Duration

Learning outcomes

• By the end of this lecture you should:

– Be comfortable with the concept of duration and know how to calculate it and what affects it

– Be able to use duration to measure and think about interest rate risk in bond portfolios

– Be familiar with the concept of convexity and know why it’s an interesting characteristic of a bond

– Be able to immunize a liability by constructing an appropriate bond portfolio

Duration as a maturity measure

• The maturity of a bond is the time of its last

cash flow (the FV)

• But some cash flows, the coupons, mature

earlier

• We would like a measure that takes this into

account

Duration as a maturity measure

• Recall that the price of a bond is the sum of the present

values of its future cash flows

• The part of a price that is due to the CF at a certain

maturity says something about how important that

maturity is

• Let’s use that as weights and calculate a weighted

average of the maturity of all the bonds cash flows

• We call this “average time to cash flows” the (Macaulay)

duration, D, of the bond:

( )( )

( )

1 1 1

1t

T T Tt t

t

t t ts

s

PV CF CF yD w t t t

PV CF P= = =

+= = =

∑ ∑ ∑

∑

Example

• Consider a five-year, 10%-coupon bond with a

face value of $100 and a 10% YTM

0

10

20

30

40

50

60

70

80

1 2 3 4 5

$

Present value of cash flows

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Weights of cash flows

Year 5

Year 4

Year 3

Year 2

Year 1

Example

• Let’s calculate the weights

• Since YTM = C, we know that the bond trades

at par

• Let’s calculate the weights:

( )( ) ( )

Py

CF

CFPV

CFPVw

t

t

t

t

tt

+==

∑ 1

%30.681001.1

110

%83.61001.1

10

%51.71001.1

10

%26.81001.1

10

%09.91001.1

10

55

44

33

22

1

==

==

==

==

==

w

w

w

w

w

Example

• How long is our average weight for a (present

value) dollar?

• You can visualize the duration as the fulcrum

point when stacking the discounted cash flows

on a line:

16.456830.040683.030751.020826.010909.0 =⋅+⋅+⋅+⋅+⋅=D

0

10

20

30

40

50

60

70

80

1 2 3 4 5

$

Present value of cash flows

What affects the duration?

• Yield– The higher the yield, the more we discount cash flows

that are far away. The weights on distant cash flows decrease and so does duration.

• Coupon rate– A higher coupon rate means larger close cash flows.

When the coupon rate is zero, i.e. for a zero-coupon bond, the entire price is made up of the face value and duration equals maturity.

• Time to maturity– The further away in time our final cash flows are, the

longer is the average maturity. This is almost trivial.

The coupon rate effect

0

20

40

60

80

100

1 2 3 4 5

$

Present value of cash flows, C = 10 %

0

20

40

60

80

100

1 2 3 4 5

$

Present value of cash flows, C = 20 %

0

20

40

60

80

100

1 2 3 4 5

$

Present value of cash flows, C = 0 %

Duration as a yield sensitivity measure

• Consider a 2-year and a 20-year zero-coupon bond. Their pricing equations are:

• The cash flow is further away for the 20-year bond and the yield “discounts over a longer period”. Therefore the price is more sensitive to changes in the yield.

( )

( )20

2

1

1

y

FVP

y

FVP

B

A

+=

+=

Modified duration

• We are interested in how much the price of a

bond changes when the interest changes

• Recall that the price of a bond is the sum of

the present values of its future cash flows:

• Let’s take the first derivative of this expression

with respect to the yield:

( )∑= +

=T

tt

t

y

CFP

1 1

( )( )∑∑

=

−

=

+∂

=+∂

=∂

∂ T

t

t

t

T

t

t

t yCFyy

CF

yy

P

11

11

1

1

Modified duration

• We recall that the derivative of a sum is the

sum of the derivatives:

• This is starting to look familiar. Let’s

manipulate this expression to get duration on

the RHS:

( )( ) ( )

ty

CF

ytCF

y

P T

tt

tT

ttt ⋅

+−=

+−=

∂

∂∑∑

=+

=+

1

1

1

1

11

1

( )( )

DPty

CF

y

P

P

y T

t

t

t −=

⋅

+−=

∂

∂+∑

=1 1

1

Modified duration

• Since we are interested in the (percentage)

change in the bond price, let’s keep that on

the LHS:

• This is a nifty expression. So nifty that we give

D/(1+y) a special name and call it modified

duration, D*

( )yDy

y

D

P

P∂−=∂

+−=

∂ *

1

Modified duration

• The relationship is only true for very small

changes in the yield

• If we consider non-trivial changes in the yield

we are in effect making a first-order Taylor

approximation:

yDP

P∆−≈

∆ *

Duration approximation

Convexity

• We always underestimate the price, since we ignore the second derivative and the price-yield relationship is convex

• In principle, we could get a better estimate with a second (or higher) order Taylor approximation

• The severity of the error depends on where we are on the Price-Yield curve

• Recall that the YTM is specific to each bond

• Convexity gives rise to a favorable asymmetry in the price effects of yield changes

• Since investors value this asymmetry, convexity is priced

Asymmetric price effect due to

convexity

Portfolio duration

• We can think of coupon bonds as a portfolio of zero coupon bonds

• In the end we only care about the cash flows

• By the same logic, we may add up the cash flows from a bond portfolio and think of it as one bond

• Our pricing and duration formulas still apply

• Specifically, the duration of our portfolio will be a weighted average of the durations of its components

Portfolio duration (optional “proof”)

• Suppose we buy one bond A and one bond B

• Our portfolio will be worth

• The PV of our total CF at some time t will be

• The portfolio duration will be:

( )

( ) ( )[ ]

( ) ( )

( ) ( )

BBAAP

T

t B

B

t

BA

B

T

t A

A

t

BA

AP

T

t

B

t

BA

T

t

A

t

BA

P

T

t

B

t

A

t

BA

P

T

t P

P

tP

DDD

tP

CFPV

PP

Pt

P

CFPV

PP

PD

tCFPVPP

tCFPVPP

D

tCFPVCFPVPP

D

tP

CFPVD

ωω +=

⋅+

+⋅+

=

⋅+

+⋅+

=

⋅++

=

⋅=

∑∑

∑∑

∑

∑

==

==

=

=

11

11

1

1

11

1

BAP PPP +=

( ) ( ) ( )B

t

A

t

P

t CFPVCFPVCFPV +=

Asset-liability matching

• When we buy a bond we are in effect postponing cash flows (saving)

• We may want to do this if we know that we have some future liabilities, such as tax debts or foreseeable investments, that we must meet

• If we buy a zero-coupon bond whose maturity matches that of the liability, we don’t have to worry about reinvestment or liquidity risks

Immunization

• What if there is no such bond?

• We would be exposed to changes in the yield

• Higher yields are beneficial if we have taken on reinvestment risk, since we can reinvest our coupons at better interests

• Lower yields are beneficial if we have taken on liquidity risk, since we can sell our bonds at a higher price if the discount rate is lower

• We can construct a bond portfolio that balances these effects against each other to match our liabilities

Immunization

• Recall that we can think of duration as a

measure of the sensitivity of our bonds to

changes in their yields

• If we set the duration of our bond portfolio

equal to the duration of our liabilities, price

changes resulting from (parallel) yield

movements will cancel out

• This process is known as immunization

Immunization

• Given a liability with duration DL and two

bonds with durations DA and DB chose

portfolio weights XA and (1 - XA) such that:

• Chose the size of the investment so that the

present value of the portfolio is equal to the

present value of the liability

( )

BA

BLA

LBAAAP

DD

DDX

DDXDXD

−

−=

=−+⋅= 1

Example

• Suppose we have a liability of $1000 at t = 5

• There are only two bonds on the market: A 3-year, 10 %

coupon bond and a 10-year zero-coupon bond. They both

have a FV of $100. Let’s assume a flat term structure with y =

5%. The price of Bond A is $61.39.

L

Bond A

Bond B

10 10 110

100

-1000

T = 3

T = 10

T = 5

Example

• Bond A carries liquidity risk since we have to sell it at t = 5

• Bond B carries reinvestment risk since we have to reinvest the coupons and FV

• Strategy: Calculate the durations of the liability and bonds and choose portfolio weights so that the durations of our assets and liabilities are equal

• The liquidity and reinvestment risks will cancel out

Example

• The durations of the liability and bond A are trivially 5

respectively 10 years

• We calculate the duration of Bond B as above:

• Now assume we invest some fraction XA in bond A and the

rest, (1 – XA), in bond B

• Our portfolio duration is:

75.2362.113

05.1/1102

62.113

05.1/101

62.113

05.1/10

62.11305.1

100

05.1

11

05.0

10

32

33

=⋅+⋅+⋅=

=+

−=

B

B

D

P

( )BAAAP DXDXD −+⋅= 1

Example

• Choose X so that the durations (yield

sensitivities) are equal:

• We should invest 31% in bond A and 69% in

bond B

( )

31.075.210

75.25

1

=−

−=

−

−=

=⋅−+⋅

=−+⋅=

BA

BLA

LBABAA

LBAAAP

DD

DDX

DDXDDX

DDXDXD

Example

• We want to invest enough money in our bond

portfolio to meet the liability

• That means the present value of our portfolio

has to be equal to the present value of our

liability

• The PV of the liability is straight forward to

calculate:

( ) 53.78305.1

1000

5==LPV

Example

• Our total investment should be $783.53

• 31% should be invested in bond A

• We buy bond A

• 69% should be invested in bond B

• We buy bond B

96.339.61

53.78331.053.78331.0=

⋅=

⋅

AP

76.462.113

53.78369.053.78369.0=

⋅=

⋅

BP

Rebalancing

• The portfolio are chosen to equalize the

durations given the yield and maturities that

we see today

• These parameters will change constantly

• To keep the portfolio immune we must

constantly recalculate the appropriate weights

and adjust our portfolio accordingly