# Yield-to-Maturity and its assumption

Reading about Yield-to-Maturity (YTM) I found out that two assumptions have to be made:

• the bond holder must keep the bond until maturity;
• coupons must be reinvested at the same YTM. Violating those hypotheses causes the bond holder incurring in two types of risks: the price risk and the reinvestment risk.

So I was wondering: since interest rates are ever-changing, will a bond holder be automatically subject to the reinvestment risk? It is hard for me envisaging that he can still reinvest his coupons at the same YTM.

I kindly ask you where my reasoning fails.

• Yes, the bond holder will have reinvestment risk because X could be anything on a coupon date. If you want no reinvestment risk you could buy a zero coupon bond.
– dm63
Apr 14, 2017 at 12:49

It's simpler to just think of the yield to maturity as the internal rate of return of the bond given the current price. It's like the discount rate you would apply to the final payout and coupons, such that the result is the market price.

A short paper by Forbes, Hatem, and Paul explains that yield to maturity ignores reinvestment. Strictly speaking, yield to maturity is an internal rate of return, not the return you would get at the horizon. The return you get at the horizon depends on the reinvestment policy. The return at the horizon only matches the yield to maturity if the coupons are invested at the same yield as the yield to maturity.

For instance, assume a \$1000 bond with \$50 annual payments and 2 years until maturity and a 10% yield to maturity. The current price is \$913.22. The sum of the return and the coupons is \$1100. Ignoring reinvestment, the return at the end of two years is 20% cumulatively (1100 / 913.22 - 1) or 9.75% annualized. However, if the \$50 from year 1 is re-invested at a 10% rate, then the investor would now have$1105, generating a 21% return cumulatively (1105 / 913.22 - 1) or 10% annualized.

Long story short, the yield to maturity is a bond's internal rate of return given its current price. It is only the return you would earn if you held the bond to maturity if you reinvest at that same rate.

Yes, its called yield to maturity because you hove to hold it till maturity.

For the reinvestment risk, suppose the day you get the coupon the coupon the day bond is trading at x%. Now with the money you get from the coupon, you can buy these bonds and realize x%. So the reinvestment risk is eliminated to the extent of the liquidity in the market.

• Fine. However, the probability that the percentage x% matches the YTM is low, isn't it? That's what I want to understand. Mar 15, 2017 at 11:41
• right, so you can invest at the current YTM which might not be equal to the yield at which was the bond was issued ! Mar 15, 2017 at 12:12

Just to add two things:

• in fixed income world bonds are typically quoted by YTM instead of prices but if we go back to prices, then an analogous question would be: what conditions need to hold so that an investor who bought a bond at price X will realize the rate of return (YTM) of Y%?

• I see YTM only in terms of a measure that allows me to compare similar bonds - the realized rate of return is a different thing

• also: YTM assumes a flat yield curve so this is quite far from reality

• Two things: 1) Most bond markets are actually quoted on a price basis and yields are derived from price, not the other way around. 2) If you plot subsequently realized bond return vs starting yield levels, they track each other extremely well, even if you're rolling your bonds monthly. So YTM is actually a very good expected return measure for default-free bonds. Apr 15, 2017 at 19:20

I think for not too large yield movements , you would earn the ytm by holding for bond duration (rather than till maturity) and reinvesting coupons in bond

• i think what i wrote is correct... it can be checked in a spreadsheet Apr 25, 2017 at 14:10