Frequently Asked Questions In Quantitative Finance

Chapter 5: Models and Equations 283

This is the slope of the price/yield curve. The quantity

−

1

V

dV

dy

is called theMacaulay duration.(Themodified dura-
tionis similar but uses the discretely compounded
rate.) The Macaulay duration is a measure of the aver-
age life of the bond.

For small movements in the yield, the duration gives a
good measure of the change in value with a change in the
yield. For larger movements we need to look at higher
order terms in the Taylor series expansion ofV(y).

Convexity The Taylor series expansion ofVgives

dV

V

=

1

V

dV

dy

δy+

1

2 V

d^2 V

dy^2

(δy)^2 +···,

whereδyis a change in yield. For very small movements
in the yield, the change in the price of a bond can be
measured by the duration. For larger movements we
must take account of the curvature in the price/yield
relationship.

Thedollar convexityis defined as

d^2 V

dy^2

=(T−t)^2 Pe−y(T−t)+

∑N

i= 1

Ci(ti−t)^2 e−y(ti−t).

and theconvexityis

1

V

d^2 V

dy^2

.

Yields are associated with individual bonds. Ideally we
would like a consistent interest rate theory that can be
used for all financial instruments simultaneously. The