232 The Management of Bond Investments and Trading of Debt
As has been already explained, in the general case but only in the general case, bonds
with short maturities and high coupons have low duration. By contrast, bonds with long
maturities and low coupons have high duration. This can be best appreciated if one
recalls that weights, in the weighted average, are present values of each cash flow
expressed as a percentage of the present value of all the bond’s cash flows – and,
therefore, they are underpinning the bond’s price. Moreover, Macaulay’s algorithm is
linked to the price volatility of a bond, and a modified form of it can help in accounting
for the transaction price (purchase price) of a bond. This is presented in Appendix 10.D.
The duration algorithm, which accounts for purchase price of the debt instrument,
is a function of interest payment in period t; number of payment periods in the year;
annual yield of the bond, both in absolute value and divided by the number of
payments in the year; and transaction price of the bond. Among themselves, these are
the ingredients which help in making a factual and documented pricing of fixed
In conclusion, Macaulay’s algorithm is so useful because it defines a proxy that
incorporates the weighted average of maturities of a bond’s coupon and principal
repayment cash flow. It should be remembered that the weights are fractions of the
bond’s price represented, in each timeframe, by the cash flow – and that in modern
finance cash flow is king.
Unlike maturity, duration is a measure of the sensitivity of a debt security’s price to
changes in yield. Because a low duration bond will show a lesser change in price for a
given change in yield than a high duration bond, floating rate debt instruments usually
have very small interest rate sensitivity: hence, low duration.
Macaulay’s duration algorithm is also a useful tool in the calculation of intrinsic value
of an investment, since it is largely based on the discounted cash flow of the coupon
stream. That is the amount by which the bond’s price is changing for a unit change (one
basis point) in interest rates. This is the sense of the reference made to theta.
10.6 Practical applications of duration and convexity
In sections 10.2 to 10.5, duration has been examined along the lines of a calculation
based on the timing of future cash flows. As the careful reader will recall, duration
has been considered as the life in years of a notional zero-coupon bond whose fair
value would change by the same amount as the real bond in response to a change in
market interest rates. Hence the similitude to theta.
The usefulness of information about the duration of a bond, or a portfolio, might be
enhanced by also disclosing the convexity. Convexity is the extent to which duration
itself changes as prices change – capturing the curvature of the price movement – and
it arises for several reasons. Prior to looking into these reasons, however, it is proper to
define convexity – as well as concavity, its opposite.
Concavity and convexity can be defined in different ways. The simplest is a geo-
metric approach. In Figure 10.1, the polygon OABCD constitutes a convex set of
points: given any two points in this polygon, the segment joining them is also in the
polygon, which is not the case with concavity. An extreme of a convex set is any
point, in that set, which does not lie on a segment joining some other two points. For
instance O, A, B, C, D are extreme points.