Duration slidesWhy does duration give us a measure of elasticity?

Peter Fraser

1(c) P Fraser 2010

Forget weighted averages?

Duration originally conceived as alternative to maturity.

More importantly used as sensitivity of price changes to interest rates.

Basic relationship:

(c) P Fraser 2010 2

*dP

MD Pdr

2

2...

1 1 1 1n n

C C NC N

r r r rD

P

Why? (mathematical interlude)

Price of a bond is discounted sum of cash

flows

(c) P Fraser 2010 3

2 3

1...

1 1 1 1 1n n

C C C CP

r r r r r

Differentiate to discover sensitivity to

rates dP

dr

Why? (cont…)

Use product and chain rule for each

term.

(c) P Fraser 2010 4

2 3 1 1

2...

1 1 1 1n n

dP C C NC N

dr r r r r

multiply both sides by

2

21 ...

1 1 1 1n n

dP C C NC Nr

dr r r r r

1 r

Why? (cont…)

(c) P Fraser 2010 5

1

1 *

*1

dPr

dr DP

dPr D P

dr

dP DP

dr r=-MD!

Right-hand side divided by price =

- duration. Divide both sides by price