# How to get to this answer on Macauley duration?

Can you explain why the answer to the following question is approximately 4.5%:

An investor buys a bond that has a Macaulay duration of 3.0 and a yield to maturity of 4.5%. The investor plans to sell the bond after three years. If the yield curve has a parallel downward shift of 100 basis points immediately after the investor buys the bond, her annualized horizon return is most likely to be:

Duration (of which Macaualy is one type) is only a linear approximation of how the bond value will change with a small change in yield.

• can you explain how the shift does not effect it – user3022875 Nov 17 '15 at 22:54
• because she plans to hold for a time period equal to the duration – Alex C Nov 17 '15 at 22:57
• @AlexC I think he's looking for a more detailed answer on that point. – SRKX Nov 18 '15 at 8:49

If the bonds yield goes down by $100 \text{bps}$ and the duration is $3$, the bond price will increase by approximately $3\%$.

Without any subsequent movement over the next three years, the bond should yield 3.5% p.a. after the yield rate movement.

The return during the total holding period of three years would be approximately:

$$3\% \text{(yield rate shift)} + 3\cdot 3.5\% \text{(annual bond yield)} = 13.5\%.$$

Lets approximate again and divide that by three to get the annual value: $4.5\%$.

Please be aware that there are several assumptions and simplifications made here.

First of all, the Macaulay Duration is only a linear approximation of the bonds price sensitivity to yield price changes.

The other is that we simply assume all given interest rates to be continuously compounded values when we add, multiply and divide them to scale the returns over time. This is only a good approximation for small rates $r$, where $\ln (1+r) \approx r$. In particular, this definitely an approximation for the duration impact term because the duration formula gives us the price impact in terms of discrete returns.

We also assume that there either are no coupons at all or they can be reinvested at exactly the yield of the bond (without transaction costs or even market impact).

The third assumption whe have to make is that after the initial yield movement, the bonds yield will remain constant during the holding period.

I went on a rant below, but this is actually a trick question.

If the time to maturity of the bond is 3 years, if its current yield to maturity is 4.5%, and if you hold the bond to maturity, then the annualized horizon return will be 4.5%, assuming all interim cash flows can be reinvested at the 4.5% yield. If cash flows cannot be reinvested at 4.5%, then the holding period return will drift away from 4.5% somewhat.

Given that the position is held to maturity, the instantaneous 100 bp change in yield is irrelevant, since any unrealized losses/gains because of the yield shift will be subsequently offset. Consider the simpler scenario of a zero coupon bond which has no coupon payments and only a principal payment on the maturity date. If yield increases substantially, you may encounter a substantial unrealized loss initially. But at maturity, if the bond doesn't default, you'll still get back the full principal amount, and that's the only quantity relevant for computing the total return over this time period.

However, the question uses a "Macauley duration" of 3. The only case where mac duration equals time to maturity is for zero coupon bonds and when yields are continuously compounded. In other cases, this 4.5% annualized return is a further approximation.

So the approximation comes from two fronts: 1) cash flows may not be reinvested at 4.5%, and 2) the time to maturity may not be the same as the mac duration.

Original rant below:

This is in my mind an extremely misleading question.

First of all, Macauley duration as a concept has historical value, but virtually no value in practical applications. By contrast, modified duration describes the percentage change in price for a small change in yield. Modified duration can be computed easily given Macauley duration and yield to maturity: $$\text{modified duration} = \frac{\text{Macauley duration}}{1 + \text{yield} / \text{compounding frequency}}.$$ In this case, assuming compounding frequency is semiannual (as is the case for virtually all bonds issued int he US), then the modified duration is $$D_\text{mod} = \frac{3.0}{1 + 4.5\% / 2} = 2.93$$. So if yield changes by 100 bp, the linear approximation of percentage change in bond price should be 2.93% -- not approximately 4.5% and not 3%.

As to the "approximately" part, this is because bond price is not a linear function of yield, rather a convex function. A better approximation requires "convexity":

$$\text{percentage change in price} = -\text{mod duration}\times \Delta y + \frac{1}{2} \times \text{convexity} \times (\Delta y)^2.$$