duration and convexity for fixed-income securities

Duration and convexity for Fixed-Income Securities

RES9850 Real Estate Capital Market

Professor Rui Yao

04/19/23 Professor Yao 2

Duration and convexity: Outline

I. Macaulay duration

II. Modified duration

III. Examples

IV. The uses and limits of duration

V. Duration intuition

VI. Convexity


A Quick Note

Fixed income securities’ prices are sensitive to changes in interest rates

This sensitivity tends to be greater for longer term bonds

But duration is a better measure of term than maturity Duration for 30-year zero = 30

Duration for 30-year coupon with coupon payment < 30

A 30-year mortgage has duration less than a 30-year bond with similar yield

Amortization

Prepayment option

NN

yyyP

)1(

C...

)1(

C

)1(

C2

21

10


I. (Macauly) duration

Weighted average term to maturity Measure of average maturity of the bond’s promised cash flows

Duration formula:

where:

is the share of time t CF in the bond price

and t is measured in years

0

10

)1/(

)(PVP

)(PV

P

yCF

CF

CFw

tt

T

tt

tt

T

ttm wtD

1

wt 1t 1

q


Duration - The expanded equation

For an annual coupon bond

Duration is shorter than maturity for all bonds except zero coupon bonds

Duration of a zero-coupon bond is equal to its maturity

NN

NN

T

t

tT

ttm

yyy

yN

yy

twtD

)1(

C...

)1(C

)1(C

)1(

C...

)1(C

2)1(

C1

PV(Bond)

)C(PV

22

11

22

11

11


IV An Example – page 1

Consider a 3-year 10% coupon bond selling at $107.87 to yield 7%. Coupon payments are made annually.

87.10779.8973.835.9bond of Price

79.89)07.1(

110)(

73.8)07.1(

10)(

35.9)07.1(

10)(

33

22

1

CFPV

CFPV

CFPV

Duration ( Dm) 1*9.35

107.87

2 *

8.73

107.87

3 *

89.79

107.87

2.7458


II. Modified duration (D*m)

Direct measure of price sensitivity to interest rate changes Can be used to estimate percentage price volatility of a bond

y

DD m

m

1*

P

P Dm

* y


Derivation of modified duration

So D*m measures the sensitivity of the % change in bond price to changes in yield

*

*

1

1

1

1)1(1

1

)1(

m

mm

N

tt

t

N

tt

t

Dy

P

P

PDPy

D

y

Ct

yy

P

y

CP


An Example – page 2

Modified duration of this bond:

If yields increase to 7.10%, how does the bond price change? The percentage price change of this bond is given by:

= –2.5661 .1%

= –.2566%

5661.207.1

7458.2* mD

yDP

Pm

*


An Example – page 3

What is the predicted change in dollar terms?

New predicted price: $107.87 – .2768 = $107.5932

Actual dollar price (using PV equation): $107.5966

2768$.

87.107$100

2566.100

2566.

PP

Good approximation!


Summary: Steps for finding the predicted price change

Step 1: Find Macaulay duration of bond. Step 2: Find modified duration of bond. Step 3: Recall that when interest rates change, the change in a bond’s price

can be related to the change in yield according to the rule:

Find percentage price change of bond Find predicted dollar price change in bond Add predicted dollar price change to original price of bond

Predicted new price of bond

yDP

Pm

*


V. Check your intuition

How does each of these changes affect duration?

1. Decreasing the coupon rate.

verify this with a 10-year bond with coupon rate from 5% to 15%, and ytm of 10%

2. Decreasing the yield-to-maturity.

verify this property with a 10-year bond with coupon rate of 10%, and ytm from 5% to 15%

3. Increasing the time to maturity.

verify this property with a par bond with a coupon rate of 10%, and term from 5 to 15 years


V. Dollar Duration We have derived the following relationship between duration and price

changes (bond returns):

Hence

Note the term on the RHS of the equation above measures the (absolute value of) slope of the yield-price curve, which is also called dollar duration

We can then predict price changes using dollar duration:

PDy

Pm

*

*mD

yP

P

yduryPDP m $)( *


Duration with intra-year compounding

In practice, lots of bonds do not pay annual coupon and we need to change the formula a bit to account for it

Some calculus (note: each step in summation is 1/m year so there are m*T terms in total)

So dollar duration

Duration

mT

n

n*m P

CPVn

myD

1

)(

/1

1

mT

nn

n

my

CP

1 /1

mT

nnm CPVn

myPD

1

* )()/1(

1

mmy

Cn

mymmy

Cn

y

P mT

nn

nmT

nn

n 1

/1)/1(

11

/1 111


V. Effective Duration– Numerical Approximation

Instead of calculating modified duration based on weighing the time of cash flow with the present value share of the CF, and then modify by dividing by (1+y), we can numerically approximate the modified duration from the slope of price-yield chart:

The slope of the yield-price curve is the dollar duration

Py

PDm

1*

PDy

Pm

*


V. Effective Duration – Numerical Approximation

We can approximate the slope of the graph / dollar duration by averaging the forward and backward slope (“central difference method”)

The duration is then

The modified duration then can be estimated as

This approach directly uses the idea that the duration measures price sensitivity to interest rate Can duration be negative?

y

PP

yy

PP

yy

PP

y

P

o

o

o

o

22

1

oPy

PPDm

1

2*

y

PP

y

P

2


VI. Duration and Convexity – Numerical Approximation

y°

y+y-

P-

Po

P+

yield

price


VI. Convexity

Duration is the first order approximation for percentage change in bond prices for a one percent change in yield to maturity For a fixed rate non-callable bond, duration underestimates change when

yield falls and overestimates when yield rises The difference is captured by convexity Convexity is typically positive for bond

Is this good or bad?

Mortgage is a difficult product to evaluate due to embedded call options Duration tends to become shorter when interest becomes lower as borrower

prepays mortgage Duration becomes longer when interest rate becomes higher as borrower

holds on to his mortgage Negative convexity

Opposite to the case of a typical bond


V. Convexity

The forecast of price response using dollar duration is

Essentially it is a linear projection using the slope measured at yo

However as soon as you move away from yo the slope will change

The rate of slope change is captured by dollar convexity

A little calculus yields (take first order derivative of dollar duration with respect to yield)

ydy

dPyPDPyDP mm **

T

tt

t

y

Ctt

ydy

Pd

1

222

2

)1()(

)1(

1

yoydy

Pd

2

2


V. Convexity – Numerical Approximation

What is the predicted dollar duration at y+ using dollar duration from yo and convexity measure at yo?

So the average of slopes at y+ and yo , which gives a better approximation of changes in prices when yield changes, is

ydy

Pddurdur

oyyyy

2

2

$$

ydy

Pddurdurdur

yoyyyyyyy oo

2

2

2

1$$$

2

1


V. Convexity

The forecast of price response using duration and convexity

In percentage term

The term

is referred to as convexity

22

2*

2

2

2

1

2

1$ y

dy

PdyPDyy

dy

PddurP

yoy

m

yoyyy o

22

2* 1

2

1y

dy

Pd

PyD

P

P

yoy

m

P

CPVtt

yy

Ctt

yPdy

Pd

Pt

T

t

T

tt

t

yoy

)()(

)1(

1

)1()(

)1(

111

1

22

1

222

2


V. Convexity with intra-year coupons

Dollar convexity with intra-year coupons

Convexity with intra-year coupons

mT

nn

n

mmy

Cnn

mydy

Pd

12

222

2 1

)/1()(

)/1(

1

21

22

21

222

2

1)()(

)/1(

1

1

)/1()(

)/1(

11

mP

CPVnn

my

mP

my

Cnn

mydy

Pd

PmT

n

t

mT

nn

n


V. Effective Dollar Convexity – Numerical Approximation

Instead of analytical formula, in practice $ convexity is frequently approximated using numerical methods based on price-yield relations

Numerical approximation is very useful when cash flow size and timing are uncertain due to built-in options in the bond payments

22

2

)(

2

y

PPP

y

y

PP

y

PP

dy

Pd o

o o


V. Homework

30 year T-bond has a yield to maturity of 3.0% and price at par, and coupon is paid annually.

1. Find out analytically the following measure at 3.0%: A. duration B. modified duration C. dollar duration D. dollar convexity E. convexity

2. Also calculate B, C, D, and E using numerical approximations using a step (delta y) of 1 basis point. How accurate is the approximation compared with analytical solutions from part 1?

3. Use dollar duration measure to predict price when yields change from 1% to 5% at 0.5% interval.

4. Use both duration and convexity to predict bond price when yields change from 1% to 5% at 0.5% interval.

note: you can use either the analytical duration/convexity or their numerical approximation for question 3 and 4.

5. What do you conclude when comparing results from 3 and 4? Which one is more accurate and why?

duration and convexity for fixed-income securities

Documents

bond price change

dollar price change

year bond

bonds price

modified duration of

macaulay duration of

price changes bond returns

year coupon