We can derive the relationship between a change in the yield to maturity and the change in the market value of a standard fixed-income bond using a bit of algebra and calculus. Equation 6.1 is a general bond pricing equation very similar to equation 3.9 in Chapter 3.

(6.1)

The periodic coupon payments (PMT) and the principal (FV) to be redeemed in full at maturity are discounted at the yield per period (y). The settlement date is t days into the T-day period and there are N periods to maturity counting from the beginning of the current period. Here the present value of the future cash flows is the market value (MV) of the bond, that is, the full (or “dirty”) price. The risk statistics for the bond are concerned with cash value, independent of how that amount is broken down for accounting into the flat (or “clean”) price and accrued interest.

Yield duration and convexity entail estimating the change in market value, denoted dMV, caused by an instantaneous change in the yield to maturity per period, dy. A useful way of obtaining this estimation is with a Taylor series expansion. Technically, this assumes that the basic relationship in equation 6.1 is continuous and differentiable with respect to the yield. This Taylor series can go out to any number of terms depending on the required degree of precision. All we need for bond math are the first two, as shown in equation 6.2.

(6.2)

In words, the change in market value is estimated by the first partial derivative of the bond pricing formula times the change in the yield plus one-half of the second partial derivative times the change in the yield squared. The first term is the essence of yield duration; the second term is the essence of yield convexity. The partial derivatives are calculated holding the other variables (PMT, N, FV, t/T) constant when the yield changes.

At this point we can define a number of versions of the yield duration statistic.

(6.3)

This expression produces the famous statistic first described by Frederick Macaulay, a Canadian economist, in his study of U.S. railroad bond yields and stock prices between 1857 and 1936. The minus sign is part of the definition so that the duration will be a positive number. That's because the first derivative is negative due to the usual inverse relationship between the bond yield and its market value. We see some circumstances in Chapters 7 and 8 when the derivative actually is positive – we call that phenomenon negative duration.

Closely related and more commonly used is modified duration.

(6.4)

We'll see that modified duration relates directly to the percentage change in the market value of the bond. Sometimes we want to estimate the change in value in terms of money, usually for a certain amount of par value.

(6.5)

In the U.S., this statistic is often called the dollar duration, but I prefer money duration because bond math is ecumenical in spirit.

The convexity statistics for the bond similarly can be defined with respect to the second partial derivative.

(6.6)

(6.7)

These are the definitions for convexity that I like to use. Some textbooks divide them by two, thereby combining the one-half term in equation 6.2 with the second partial derivative. Also, we see later in the chapter that the convexity statistic reported on Bloomberg divides the expression in equation 6.6 by 100.

To integrate these definitions, divide both sides of equation 6.2 by MV.

(6.8)

This connects the percentage change in the market value (dMV/MV) to the change in the yield to maturity. Now substitute the definitions given in equations 6.4 and 6.6 into 6.8.

(6.9)

The percentage change in the market value of the bond is approximated by the modified duration times the change in the yield to maturity, plus one-half the convexity statistic times the change in the yield squared. The latter is known as the convexity adjustment to duration. Similarly, the change in market value in money terms is approximated by the money duration and convexity statistics.

(6.10)

So far we have related the instantaneous change in the yield per period, dy, to the change in market value, dMV. In practice, bond yield statistics invariably are annualized. Therefore, a more useful expression is to estimate the change in market value, either on a percentage basis or in money terms, to the change in the annual yield. Let Y be the annual percentage rate and PER the number of periods in the year (i.e., the APR and its periodicity). Then Y = y * PER, dy = dY/PER, and (dy)2 = (dY)2/PER2. Substitute those into 6.9 to get an expression relating the percentage change in market value to the change in the annual yield to maturity.

(6.11)

Modified duration divided by PER is the annual modified duration and the convexity divided by PER squared is the annual convexity.

(6.12)

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