Interpreting Macaulay Duration as Weighted Cash-Flow Timing and Rate Risk

# Interpretation of Macaulay Duration


# Interpretation of Macaulay Duration












I am having a difficulty conceptualizing the meaning of "Macaulay duration" - I want to note I completely understand the math, this isn't the issue. Modified duration & Efficitive Duration make total sense to me as they are refer to a first order approximation of a change in yield on the price of a bond (eg, a 100 bp change in yield causes price to increase/decrease 110 bp). But Macaulay duration is usually quoted in years/time. How does one interpret this? What does it mean for a bond to have a duration of 6 yrs? Can someone help enlighten me? Thank you!

## Answer by zuiqo (score 9)

https://quant.stackexchange.com/a/4716

Maccauly Duration means nothing else than that after the given amount of years, you will have your capital investment back as nominal amount.

If you have \$100 invested, and you have a duration of two years, after two years you will have gotten \$100 repaid, not directly dependent of interest rate or payment scheduling (indirectly they are of course!).

I found this image helpful:

In this example, the bond will be valued at 130.46, which is the sum of PV's of all CF's. After the MD - 1.78 years - you will have received exactly your capital investment, which was the nominal amount of the bond. You are investing the nominal amount and this series of cashflows is valued higher (which is economically reasonable, due to risk exposure!). So while the value of the bond is 130.46, after 1.7775 years you will have exactly 100 in your hands, and receive the remainder - 30.46 - in the remaining lifetime of the bond (here thats 0.22 years).

You can see how the small payments sum up to the invested amount, and why the Macaulay Duration is always shorter than the period of payments of the bond.

Of course this number is not exact! You will not have 100 on your account after 1.78 years, but less. You will have to wait for the coupon payment after the MD to actually surpass the 100\$ (in the example thats the final payment.)

## Answer by Matt Wolf (score 3)

https://quant.stackexchange.com/a/4719

The simple but accurate answer should be that Macaulay Duration is the weighted average maturity of cash flows (in years). That is how it is defined in almost every text book and looked at by most market practitioners. That is why its quoted in years and it gives an indication of when, on a weighted basis, cash flows are paid out (mature). For example, in the image of phi, the pv of cash flow at t1 (9.61) is paid out/matures at t1. In his example MD is 1.78 meaning the bulk of the maturity of cash flows occurs close to t2, simply because the last coupon is paid at t2 plus the par value is returned to the investor. I would not make it more complicated than it really is.

Edit: The following link may make it clearer in case there is still confusion out there: http://www.econ.ohio-state.edu/jhm/ts/duration.htm

Just keep in mind the MD of a zero-coupon bond equals the maturity of the bond.

## Answer by Stephen Ge (score 3)

https://quant.stackexchange.com/a/69964

There are many ways to understand the Macaulay Duration, one of them is from "the interest rate risk" point of view.

For a fixed coupon bond, there are two risks that is caused by the change of interest rate, one is the bond price risk and the other is the coupon reinvestment risk.

If interest rate rises, then bond price drops but the coupon reinvestment return increases.

If interest rate drops, then bond price rises but the coupon reinvestment return declines.

Clearly, there is a trade-off between bond price & coupon reinvestment risks. By setting the investment horizon equals to the Macaulay Duration, the two effects, the bond price effect and the coupon reinvestment effect, cancel each other.

To see this, let us consider a bond with periodic annual fixed coupon payments $C$ and the par value is $M$; the bond will mature at time $T$, with continuously-compounded interest rate $r$.

The price of bond $P$ at the beginning is: \begin{equation} P=\sum_{t=1}^T Ce^{-rt} + Me^{-rT}. \end{equation} Let us select a time $t_d\in[0,T]$, the coupon reinvestment income at $t_d$ is given by \begin{equation} R_{t_d}=\sum_{t=1}^\tau Ce^{r(t_d-t)}, \end{equation} where $\tau$ is the lastest coupon payment date before $t_d$. If we decide to sell the bond at $t_d$, the bond price will be: \begin{equation} P_{t_d}=\sum_{t=\tau+1}^T Ce^{-r(t-t_d)} + Me^{-r(T-t_d)}. \end{equation}

By taking the first derivative of $R_{t_d}$ and $P_{t_d}$ with respect to the interest rate $r$, we could observe how changes in interest rate affect the bond price & the coupon reinvestment income:

\begin{equation} \frac{d R_{t_d}}{dr}=\sum_{t=1}^\tau (t_d-t)C e^{r(t_d-t)} = e^{rt_d} \sum_{t=1}^\tau C(t_d-t)e^{-rt} \end{equation} and \begin{equation} \begin{aligned} \frac{d P_{t_d}}{dr}&=-\sum_{t=\tau+1}^T (t-t_d)Ce^{-r(t-t_d)} - M(T-t_d)e^{-r(T-t_d)} \\ &= -e^{rt_d}\sum_{t=\tau+1}^T C(t-t_d)e^{-rt} - M(T-t_d)e^{rt_d}e^{-rT} . \end{aligned} \end{equation}

By letting $\frac{d R_{t_d}}{dr} + \frac{d P_{t_d}}{dr} = 0$, two effects cancel each other, hence \begin{equation} \begin{aligned} \sum_{t=1}^\tau C(t_d-t)e^{-rt} - \sum_{t=\tau+1}^T C(t-t_d)e^{-rt} - &M(T-t_d)e^{-rT} = 0 \\ t_d\left(\sum_{t=1}^\tau Ce^{-rt} + \sum_{t=\tau+1}^T Ce^{-rt} + Me^{-rT}\right) & = \sum_{t=1}^\tau tC e^{-rt} + \sum_{t=\tau+1}^T tCe^{-rt} + TMe^{-rT} \\ t_d P & = \sum_{t=1}^T tC e^{-rt} + TMe^{-rT}. \end{aligned} \end{equation} Thus, we have \begin{equation} t_d = \frac{1}{P}\times \left[\sum_{t=1}^T tC e^{-rt} + TMe^{-rT} \right] = -\frac{1}{P}\times \frac{dP}{dr}, \end{equation} which exactly equals to the mathematical definition of the Macaulay Duration.

## Answer by user19338 (score 1)

https://quant.stackexchange.com/a/23136

Perhaps there is another way to arrive at the "weighted average maturity of cash flows". Suppose that we have a coupon paying bond with a continuously-compounded yield $y$ which pays a coupon of value $C_i$ at time $t_i$ for $1 \leq i \leq n$. What would be the maturity of a zero-coupon bond with the same yield $y$ which has the same present value as the coupon-paying bond?

Let $X$ be the face-value of such a zero-coupon bond and let $t$ be its maturity date so that the zero-coupon bond has present value $Xe^{-yt}$. It follows that $Xe^{-yt} = \sum_{i=1}^n C_ie^{-y t_i}$. If one differentiates with respect to $y$ one sees that $t = \frac{\sum_{i=1}^n t_iC_ie^{-rt_i}}{\sum_{i=1}^n C_ie^{-y t_i}} = \sum_{i=1}^n \omega_i t_i$ which is the weighted weighted average maturity of cash flows with $\omega_i$ equal to the proportion of the present value of the coupon-paying bond associated to the cash flow at time $t_i$.

## Answer by Stephen Ge (score 0)

https://quant.stackexchange.com/a/69980

Considering a $T$ years bond with periodic annual fixed coupon payments $C$ and par value $M$ with continuously-compounded interest rate $r$. The most common way to explain the Macaulay Duration of such coupon bond is to replicate it using a bond portfolio that contains $T$ zero-coupon bonds with different maturities.

The overall price of this bond portfolio is equal to the fixed coupon bond price $P$, such that \begin{equation} \begin{aligned} P_1 & = Ce^{-r} \\ P_2 & = Ce^{-2r} \\ & \vdots\\ P_T & = (C+M)e^{-Tr} \end{aligned} \end{equation} and \begin{equation} P = \sum_{t=1}^{T} P_t \end{equation}

Because the duration of a zero-coupon bond equals its maturity, we would like to know the average duration or maturity of our bond portfolio. Because different cash flows have different time values, we cannot do the simple average of all maturities, instead, we use percentage of total capital invested $P$ as the weight, such that \begin{equation} w_t := \frac{P_t}{P} \end{equation}

So the Macaulay Duration of a coupon bond is defined as the weighted average maturity of all zero-coupon bonds inside the equivalent bond portfolio: \begin{equation} \text{MacD} := \sum_{t=1}^{T} w_t t = \frac{1}{P}\times \left[\sum_{t=1}^T tC e^{-tr} + TMe^{-Tr} \right]. \end{equation}