# Fixed Income Question

An investor buys a bond that has a Mac duration of 15.0 and a modified duration of 14.5. If the rate of return on reinvested coupon income is 4.0% and the bond is sold after 3 years, the investors annualized holding period return is most likely to be:

A) equal to the bond’s YTM at the time of purchase.

B) less than the bond’s YTM at the time of purchase,

C) greater than the bond’s YTM at the time of purchase.

It would have been nice if they’d given you the frequency of coupon payments, but let’s assume that it’s semiannual. In that case:

DMod = DMac / (1 + YTM/2)

1 + YTM/2 = DMac / DMod = 15 / 14.5 = 1.0345

YTM/2 = 3.45%

YTM = 6.9%

In that case, because the reinvestment rate is 4%, and assuming that the bond is sold with the same YTM, the realized annual return will be less than the YTM.

Note that if the bond paid coupons annually, then the YTM would be 3.45%, and you would conclude that the realized annual return is higher than the YTM. And if the bond were sold at a YTM different from that when it was purchased, you cannot conclude anything.

In short, there’s a lot of information missing in this question. The author thought himself more clever than he actually is.

Would you automatically assume semi-annual pay unless explicitly told it was annual.

Hopefully there will be no such ambiguity on the actual exam

This is an interesting question… managed to figure out the YTM using modD=MacD/(1+r) but didn’t semi annual coupon it. Even then I still wouldn’t be able to interpret it with a high level of confidence, just plug and chug really.

I think it would be good if the sources of these questions were stated.

Hm, I’ve also chosen B although my logic was the different: annual YTM = 3,45%, interest rates have increased (since you can reinvest coupon payments at 4%), investment horizon < Mac Duration => realized return is less than bond’s YTM (because the bond’s price drops more than you will received from higher reinvested income and investment horizon isn’t long enough to offset these effects). Where I am wrong?

jammer89, as far as I remember, the question is from CFAI March Mock.

“B Based on the relationship between mac and modified duration, the bond’s YTM is 3.45%. If the rate of return on reinvested coupon is higher than 3.45%, the YTM increased after investor bought the bond. Over an investment horizon shorter than the mac duration, an increase in YTM decreases the bond’s market price by more than it increases reinvestment income. Therefore, the investor’s annualized HPR is less than the YTM at issuance”

Can someone explain the implication behind the bolded statement?

gunner.21, there are two graphics in the CFAI cirriculum, book 5, page 511 which helped me a lot to understand this concept. I would strongly recommend to look at them.

When interest rates increase, bond prices decrease, but reinvestment income increases.

The Macaulay duration is the point of indifference: if you hold the investment for exactly the Macaulay duration (15 years, in this case), then the total amount of your portfolio (bond plus coupons plus reinvestment income) will be the same as it would have been had interest rates not increased.

If you hold the investment for a period shorter than the Macaulay duration, your total portfolio’s value will be less than it would have been had interest rates not risen; if longer, more.

So, compare it to the mac duration and if the holding period is less than the mac duration, then the HPR < YTM? (Considering interest rate are rising)

Yes (assuming you meant annualized HPR).