Origin of the $-\frac{1}{P}$ in Macaulay Duration?

Question

Changes in the yield curve affect the total return of a coupon bond instrument, hence I want to compare different bond instruments in how sensitive they are to $y$.

Well, I just take the derivative, nice and elegant. $$P_{c} = \sum_{i}^{n}(c_{i} \cdot e^{-y_{i}T_{i}})$$ $$\frac{d}{dy}(P_{c}) = \sum_{i}^{n}(-T_{i} \cdot c_{i} \cdot e^{-y_{i}T_{i}})$$

But wait, everyone seem to be using using Macaulay duration and it involves this $-\frac{1}{P}$ term. Why?

$$\frac{d}{dy}(P_{c}) = \sum_{i}^{n}(-\frac{1}{P_{c}} \cdot -T_{i} \cdot c_{i} \cdot e^{-y_{i}T_{i}})$$

Answer 1

We want the duration $D$ to satisfy $$\mathrm{d}P=-PD\mathrm{d}y,$$ i.e. it tells us the proportional change in the bond price if the interest rate (yield) changes. The minus is due to the inverse relationship between bond yield and bond price. Thus, $$D=-\frac{1}{P}\frac{\mathrm{d} P}{\mathrm{d} y}.$$ Duration can be seen as a linear approximation to the bond price’s sensitivity towards changes in the interest rates, i.e. it is the approximate change in the bond price given a 1% change in its yield to maturity. A second order approximation leads to the notion of convexity.

If you consider continuous compounding as you did in your question, then $$D=-\frac{1}{P}\sum_{i=1}^n c_{t_i}t_ie^{-y\cdot t_i}$$ which gives rise to the explanation as ``weighted sum of coupon payment dates''. However, if you use discrete compounding, the derivatives are not as nice and in order to get $\mathrm{d}P=-PD\mathrm{d}y$, you need to introduce the modified duration. This is only needed for discrete compounding though.

Answer 2

Here is some historical context.

Macaulay introduced the duration concept as in the ‘duration of cash flows’ sense, in a way to measure the effective term of the loan. For the weights, he considered (or more like debated) alternatives, but then concluded to use present value. So Duration as per Macaulay is a present value weighted average of cash flows time. And to cast your mathematical sensitivity into this definition, you will need to divide it by the P, and multiply it by minus 1 to make it positive. Hence the $\frac{-1}{P}$. Note: in simple compounding, there is an extra adjustment because the derivative reduces the power of $1+y$ by one, so you will need to multiply it by $1+y$ to convert it to Macaulay Duration.

Later on, independently, Hicks amongst others developed the concept in the elasticity sense, which the economists define as:

$\frac{d \ln P}{d \ln \left(1+y\right)}=\frac{1+y}{P} \frac{dP}{dy}$

This is the same definition that one sees in the price elasticity of demand. Usually the economists add minus sign so that higher magnitude (say -4) means more responsiveness than a lower magnitude (say -2). The minus sign does not add much because in normal cases one knows the number will be negative, so ignoring it produces a more intuitive measure.

And it is easy to see that the Macaulay’s duration and the elasticity represent the same thing.

Note: this elasticity definition is in terms of simple compounding but you can adjust for the continuous compounding.