How to calculate YTM in case coupon payments are reinvested at a different rate than the bond's coupon rate?

Question

I know that calculations of yield to maturity(YTM) assume that all coupon payments are reinvested at the same rate as the bond's current yield and take into account the bond's current market price, par value, coupon interest rate, and term to maturity.

I was reading a book where the author goes like this - Consider the example of a five-year 10 percent bond paying interest semi-annually which is purchased at par value of 100. If immediately after the bond is purchased, interest rates decline to 5 percent, the bond will initially rise to 121.88 from 100. The bond rises in price to reflect the present value of 10 percent interest coupons discounted at a 5 percent interest rate over five years. The bond could be sold for a profit of nearly 22 percent. However, if the investor decides to hold the bond to maturity, the annualized return will be only 9.10 percent. This is because the interest coupons are reinvested at 5 percent, not 10 percent.

Now, I don't understand this line - "if the investor decides to hold the bond to maturity, the annualized return will be only 9.10 percent". How to calculate and verify this ? Is there any formula/equation for doing that ?

Answer 1

In my opinion it's a flawed argument because there is no reinvestment assumption in the ytm computation. Investopedia is not a reliable source generally.

It is a common fallacy to state the reinvestment assumption. Some papers trying to address this problem.

• The controversial reinvestment assumption in IRR and NPV estimates: New evidence against reinvestment assumption; Arjunan and Kannapiran
• The IRR, NPV and the Fallacy of the Reinvestment Rate Assumptions; Lohmann
• The Internal Rate of Return and the Reinvestment Fallacy; Keane
• Yield-to-Maturity and the Reinvestment of Coupon Payments; Shawn M. Forbes, John J. Hatem, and Chris Paul
• The Reinvestment Rate Assumption Fallacy for IRR and NPV: A Pedagogical Note; Magni, Carlo Alberto and Martin, John D.

All NPV of a bond (and internal rate of return: IRR) does is to discount cashflows. It is just trying to measure the return offered by a project taking into account the timings of cash flows. The discount rate that matches the quoted NPV of a bond is the YTM. As soon as scenarios on how the interim cash flows might be used are included in the calculation of NPV (YTM or IRR), you would be calculating the NPV (or IRR) of a different set of cashflows. Hence, there is no separate accounting for reinvested cashflows or other stuff.

The confusion comes from the observation that the YTM (IRR) is not equal to the total profit expressed in percent. However, a 5% bond also just pays a 5% coupon rate every year. If YTM is also 5% it is priced at par. Yet, if your bond has a maturity date in 5 years, you do not get 100*(1,05)^5 ~ 127,63 but simply 5*5+100 = 125 if interest is paid annually.

YTM just expresses the bond coupon (instead of the 5%) in a comparable manner, taking the price of the bond into account.

Answer 2

The assumption (not a requirement) of the IRR method (which is how yield is calculated) is that all cash inflows and outflows are borrowed/invested at the same constant rate to end up with the same amount of cash at the end of the stream. IRR (Yield) is simply that constant rate at which all cashflows are borrowed/reinvested.

Yes if the bondholder keeps that bond to maturity and reinvests the coupons at the current yield at that time (5% instead of 10%), then the cash held at the end of the bond will be less than the original amount that was projected by the YTM (10%) because the investor will get less return off of the reinvested coupons.

How to calculate and verify this ?

If the bond was held to maturity, and coupons were reinvested at 5%, the amount of cash you'd have at the end would be 155.26, which is an annualized return of 9.2% (155.26/100) ^ (1/5). I believe the author is using semiannual compounding instead of annual, but the results are close enough for an illustration.

I agree with AKdemy that this is not really a "requirement" of the YTM formula but an assumption to explain the actual meaning of the yield. Certainly yields change over time, and as yields go down, bond values go up, so the bond could be sold for a "profit", but that "profit" could only be reinvested at the current (lower) yield (if invested in an equivalent bond), so it's not like the profit is a windfall - they would be getting cash now and investing it in lower coupon bonds.

Meaning, if the investor took that 121.88 and bought a different bond trading at par with a 5% coupon, and reinvested the coupons at the same 5%, they'd end up with exactly the same amount of cash in the end as if they had kept the bond until maturity (155.26).