What Bond Yields Measure and Why Yield to Maturity Has Limits

# Meaning/importance of "yields" (bonds)


# Meaning/importance of "yields" (bonds)












After reading many articles on bond yields (yield-to-maturity) I'm still not getting what they are used for by investors. I understand the math behind its evaluation, but, say, what exactly I can tell when I look at yields of 2 bonds? If the yield of bond A is higher then it's a more attractive investment? Put it differently, what do investors get from those numbers on bond yields that The Wall Street Journal publishes every day? What's their thought process in this respect?

## Answer by Dom (score 3, accepted)

https://quant.stackexchange.com/a/58437

Basically you are right to be skeptical about the use of the yield to maturity as a metric for comparing investments. It is useful, but imperfect, and it is important to understand its limitations.

The simplest measure of bond return is the current yield

$y_c = c/P$

which is the coupon divided by the price. If there was one coupon left, this might make sense. However this measure does not take into account the fact that for a typical bond, these coupons will be received over multiple years in the future and so their present value today is not $c$.

The yield-to-maturity $y$ is the next simplest yield measure that can be calculated that actually takes into account the timing of the cashflows and hence the time-value of money.

The value of $y$ is then the annualized period return that sets the total return of investing the price of the bond until the bond matures to holding the bond (with coupon frequency $f$) and reinvesting each of the received coupons until bond maturity.

$P (1+y/f)^N = \frac{c}{f} \sum_{i=1}^N {(1+y/f)^{N-i}} + 1$

On the left we have the (full) price of the bond invested for N coupon periods resulting in the calculated amount at maturity. On the right we have the sum over the coupons plus par, with each payment being re-invested at yield $y$ until maturity. Clearly this formula assumes that we can re-invest the coupon payments at the rate of return $y$ until bond maturity.

This formula can be rearranged to give

$P = \frac{c}{f} \sum_{i=1}^N \frac{1}{(1+y/f)^i} + \frac{1}{(1+y/f)^N}$

The key conclusion is that $y$ is only equal to the bond's total rate of return if it does not change over the life of the bond.

The assumption of an unchanging reinvestment rate means that this measure of return is flawed unless the term structure of interest rates is actually currently flat (at best you might like to think of it as some "average" rate of return). If it is not flat, then we should be using the current term structure of interest rates to determine at what forward rate we can reinvest each of the future coupons.

What does make the $P(y)$ formula useful is that given a price $P$, we can solve for $y$ and vice-versa. As a result, the yield is widely used as a more intuitive way of quoting bond prices. Traders will agree on a yield and the price follows uniquely.

To reiterate, as a measure of return the yield-to-maturity is flawed unless it remains constant (or the curve is flat). If not, it cannot even be used to compare the return of two bonds with different coupons and same maturity. Coupon effects may mean that two bonds with the same maturity priced using the same full term structure can have different yields. So it's only a rough measure of yield - in an ideal world a full term structure of interest rates would be constructed and used.

A great many risk measures (duration and convexity) are calculated by measuring the sensitivity of the bond price to changes in this yield. What these have in common is that they assume that the term structure of interest rates is moving in a parallel fashion - the same sized bump at every maturity. The weakness of these measures is somewhat mitigated by the fact that a trader hedging a 5 year bond will use a 6 year-bond and so the assumption of a parallel shift is not so bad as we would expect 5 and 6 year bond yields to move by about the same amount.

Anyone wanting to do better risk management would need to use more sophisticated measures of risk that involved perturbing zero rates, using some parametric yield curve fitting approach like Nelson-Siegel or key rate durations.

## Answer by Sharad (score 2)

https://quant.stackexchange.com/a/58455

This should really be a comment to Dom's excellent answer (or to your question) but I don't have enough reputation to do so. For a textbook treatment that essentially covers the same points as Dom, see Chapters 1-3 of Fixed Income Securities by Tuckman et al (3rd edition). Somewhat unconventionally, the book starts by talking about Spot and Forward rates before Yields, but from a pedagogical standpoint it makes the point about the yield calculation implicitly relying on a flat term structure of rates more self-evident (other shortcomings of the yield calculation are also covered). The meat of the discussion can be found on pp. 99-114.