# Bond Overall Return vs Yield to Maturity

I've been working on what I had hoped to be a simple model demonstrating that a bond "returns" its yield-to-maturity over its life. However, whatever data I use, I end up with a return that is a little higher than the yield-to-maturity.

My process has been to:

• Take a relevant yield curve
• Value the bond today at that yield curve
• Calculate the implied yield curve at future points in time, and revalue the bond at those future yield curves taking into account coupons paid
• Assume that coupons are reinvested at prevailing rates on the yield curve, at each time received

Doing this, I receive a "return" (ending total value vs. initial bond price) that is a) a little higher than the yield-to-maturity, and b) equal to the point on the yield curve of the bond's maturity (as if it were a zero coupon bond).

I've always learned that a bond should "earn" its yield-to-maturity if its coupons were reinvested and the yield curve moved as implied in the initial term structure. Am I missing something critical in my reasoning?

Let's take a look at the price-yield formula (for simplicity, we'll use a 2-year bond):

$$P = \frac{c/2}{1+y/2} + \frac{c/2}{(1 + y/2)^2} + \frac{c/2}{(1 + y/2)^3} + \frac{100 + c/2}{(1 + y/2)^4}.$$

Multiplying both sides by $$(1 + y/2)^4$$ gives:

$$P \left(1 + \frac{y}{2}\right)^4 = \frac{c}{2} \left(1 + \frac{y}{2}\right)^3 + \frac{c}{2} \left(1 + \frac{y}{2}\right)^2 + \frac{c}{2} \left(1 + \frac{y}{2}\right) + \left(100 + \frac{c}{2}\right).$$

The right hand side includes: 1) the future value of the first coupon payment, reinvested yield to maturity for 1.5 years; the second coupon payment, reinvested at yield for 1 year, etc. So the RHS is the future of all cash flows, assuming each can be reinvested at today's yield to maturity.

The left hand side is the today's price, compounded at yield to maturity semiannually for two years.

So you're mostly right about yield to maturity being a proxy for future compounded returns, except that the cash flows must be reinvested at today's yield to maturity, not the implied forward yield.

• This is exactly it, thank you. I "simplified" the model to reinvest at the YTM instead of the prevailing term structure and return then = initial YTM,
– MJR
Dec 13, 2018 at 10:03