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

income instruments.

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.

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