How Macaulay Duration Matches a Bond Investment Horizon
Summary
The document derives the horizon at which a bond’s future value is locally insensitive to a small yield change. It writes future value as the sum of each cash flow compounded or discounted to the investment horizon, then differentiates with respect to yield. Setting that sensitivity to zero gives a horizon equal to the present-value-weighted average payment time, which is Macaulay duration under the stated constant continuously compounded yield convention.
The explanation interprets the result as an offset between two effects: yield changes alter the value of cash flows still ahead, while changing the accumulation value of cash flows already received. A numerical illustration uses a three-year annual coupon bond and reports a duration of 2.5 years; it says a one-percentage-point yield shift produces the smallest future-value change at that horizon. The result is local and relies on the specified yield and reinvestment assumptions. It does not imply immunity to large rate moves or nonparallel yield-curve changes.
Key ideas
- Future value at a chosen horizon can be expressed by carrying each bond cash flow to that date.
- Setting the future-value sensitivity to yield to zero gives the Macaulay-duration horizon.
- At that horizon, the effects on past and future cash flows offset for a small yield change.
- The numerical illustration shows a minimum value change at duration under its stated assumptions.
- The result is local and does not guarantee protection from large or nonuniform rate moves.
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Full text
# Macaulay Duration - Liability matching
# Macaulay Duration - Liability matching
Can someone provide a detailed example to prove the following statement: "When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk."
Source: Understanding Fixed-Income Risk and Return, CFA Program 2022
## Answer by Pontus Hultkrantz (score 3, accepted)
https://quant.stackexchange.com/a/71170
> Can someone provide a detailed example to prove the following statement: "When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk."
The future value at investment horizon time $t$ of a set of $n$ cashflows is $$ FV(t;y) = \sum_{i=1}^n CF_i \cdot D(t,t_i) = \sum_{i=1}^n CF_i \cdot \exp\left(-y\cdot (t_i-t)\right), $$ where $D(t,t_i)$ be the discount factor between time $t$ and time $t_i$, and $y$ is the constant continuously compunded interest rate or yield.
For what value of $t$ is the future value immune to a small change in yield? \begin{align} \frac{\partial FV(t;y)}{\partial y} &= -\sum_{i=1}^n CF_i \cdot (t_i-t) \exp\left(-y\cdot (t_i-t)\right), \end{align} setting equal to zero and solving for $t$ gives us \begin{align} t^* = \frac{\sum_{i=1}^n t_i \cdot CF_i \cdot \exp(-y\cdot t_i)}{\sum_{i=1}^n CF_i \cdot \exp(-y\cdot t_i)}, \end{align} which is identical to the Macaulay duration.
Relative to horizon $t$, cashflows before time $t$ are compounded forward (increasing in value), and cashflows after time $t$ are discounted back (decreasing in value). A small positive increase in yield $y$ will cause future cashflows to be less valuable (price risk) at time $t$ wheareas past received cashflows be more valuable (investment risk). If the horizon is $t^*$, which is equal to the Macaulay duration, these two effects will cancel out.
Consider a 3y annual paying coupon bond with notional $N=100$, yield $y=19.5\%$ , coupon $c=21.5\%$, $t_i=i$ where $i=1,2,3$. The cashflows are $CF_i=N\cdot c=21.5$ for $i<3$ and $CF_3=Nc+N = 121.5$.
Performing the above exercise you will see that the Macaulay duration is $t^*=2.5$.
The change in the future value to a small change $\epsilon$ in yield is given by $\Delta FV(t;y) = FV(t;y+\epsilon) - FV(t;y)$. With e.g. $\epsilon=1\%$ you will find that the change in value is minimized when $t=t^*$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.