# 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."

• You want a mathematical proof or just an intuitive explanation? Jun 5 at 13:43
• And what have you tried, and what went wrong? Jun 5 at 14:37
• I understand the idea behind this statement (higher yield -> lower bond price but I can reinvest the coupons at a higher yield and it offsets each other if my investment horizon = the Mac Duration). I would like a simple example with numbers to convince myself that this statement is actually true. Thanks. @PontusHultkrantz Jun 5 at 19:04
• Note: the assumption is that yields-to-maturity increase by the same amount at all maturities (a so called parallel shift) which is not the way markets actually move or how the zero coupon curve is usually modelled. It is a true statement but of limited value IMO. See also quant.stackexchange.com/questions/41164/… Jun 8 at 6:59
• @nbbo2: good point. I was thinking the implied assumption is also that the yield changes shortly after the bond is purchased such that all received cash flows before horizon is compounded at this new yield. Why would the yield only change close to inception? For calculating PV it makes more sense as it would be a one day risk. Jun 8 at 18:22

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."

##### Proof:

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.

##### Example:

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 $$t=t^*$$.