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Duration A measure of the effective maturity of a bond

The weighted average of the times (periods) until each payment is received, with the weights proportional to the present value of the payment

Duration is equal to maturity for zero coupon bonds

Duration of a perpetuity is (1+r)/r

Duration Formula

(YTM) rateinterest

maturity

t'' periodin flowcash

occurs flowcash when period

11*

11

i

T

CF

t

i

CF

i

CFtDUR

t

T

tt

tT

tt

t

Duration FormulaAnother Perspective

Pi

CF

wi

CFP

where

twtP

i

CF

i

CF

ti

CF

DUR

t

t

tT

t t

t

T

tt

T

t

t

t

T

t T

t t

t

t

t

1 ,

1

* *1

1

*1

1

111

1

Workout Problem-Duration

Calculate the duration of an asset that makes nominal payments of $120 one year from now, $140 two years from now, and $160 three years from now. Assume the YTM is 10%. Calculate the duration of another asset that makes nominal payments of $160 one year from now, $140 two years from now, and $120 three years from now, also with an YTM of 10%.

– Spreadsheet

Duration Properties

The longer the term to maturity of a bond, everything else being equal, the greater its duration.

When interest rates rise, everything else being equal, the duration of a coupon bond falls. (convexity)

The higher the coupon rate on the bond, everything else being equal, the shorter the bond’s duration.

Duration is additive: The duration of a portfolio of securities is the weighted average of the durations of the individual securities, with the weights reflecting the proportion of the portfolio invested in each.

Algebraic Duration Relations

Where D* is modified Duration, D* = D/(1+y) But, using some algebra

P-D

Py1

D - y P/

*

y D- P/P *

Immunization Example

Insurance co must make $19,487 in 7 years, market rates are 10%, PV of payment is $10,000. Using 4 year zero coupon bonds and perpetuities, immunize this obligation against interest rate risk.

Duration of Liabilities = 7 yearsDuration of zero-coupon bonds = 4Duration of perpetuities = 1.1/.1 = 11

Solve: x*4 + (1-x)*11 = 7

x = 57%, therefore buy $5,700 worth of zero coupon bonds and $4,300 worth of perpetuities

Pricing Error from Convexity

Price

Yield

Duration

Pricing Error from

Convexity

Correction for Convexity

)(21 2* yConvexityyD

P

P

Modify the pricing equation:

Convexity is Equal to:

Convexity

How does convexity affect the approximation error of the bond return when we match only the modified duration?

As an investor, do we like convexity?

In general, the higher the coupon rate, the lower the convexity of the bond.

Example

Annual coupon paying bond– matures in 2 years, par=1000, – coupon rate =10%, y=10%

Price=$1000 Time when cash is received:

– t1=1 ($100 is received), t2=2 ($1100 is received)

Find approximate percentage change in bond price using both duration and convexity if yields increase by 100bps

Example

Using D* only (D* = 1.7355)

Using both D* and convexity (4.6583)

Differences can be meaningful

Convexity of a Portfolio

The convexity of a portfolio is the weighted sum of the convexity’s of each bond in the portfolio where weights are the fraction of your investment equity in each

Therefore, we can match convexity of portfolio similar to matching modified duration

1 1 2 2 ...p j jC w C w C w C

Active Bond Management: Swapping Strategies

Substitution Intermarket Rate anticipation Pure yield pickup Tax Others

Interest Rate Swaps

Interest rate swap basic characteristics– One party pays fixed and receives variable– Other party pays variable and receives fixed– Principal is notional (not exchanged)

Growth in market– Started in 1980– Estimated over $60 trillion today

Hedging applications – Banks Speculative applications – Fixed Income Asset

Management

A and B Transform Assets

A B

LIBOR

5%

LIBOR+0.8%

5.2%

Existing Asset

Existing Asset

Swap{

Assume Portfolio manager A thinks interest rates are going up while manager B thinks interest rates will stay level or go down

Swap Example:Financial Institution is Involved

A F.I. B

LIBOR LIBORLIBOR+0.8%

4.95% 5.05%

5.2%