Yield to Call and Yield to Worst

Quote the yield to maturity on a callable bond and you're describing a cash-flow stream the bond may never actually pay. If the issuer calls the bond early — typically because rates have fallen and it can refinance more cheaply — the investor gets principal back years ahead of schedule and stops collecting the coupon that yield-to-maturity assumed would run to the final maturity date.

Yield to call

Yield to call (YTC) is calculated exactly like yield to maturity, except the "maturity" date is swapped for a specific call date, and the "face value" is swapped for the call price on that date. It answers: if this bond gets called on this particular date, what annualized return would I have earned?

$$P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{K}{(1+y)^n}$$

In words: the bond's price today equals the coupons discounted up to the call date, plus the call price $K$ discounted back from that same date, all at the same yield $y$ — solve for $y$ and that is the yield to call for that date.

A bond typically has several call dates (many callable bonds can be called on any coupon date after a lockout period, or on a schedule of specific dates with different call prices), so there isn't one YTC — there's a YTC for every date the issuer is allowed to call.

Yield to worst

Yield to worst (YTW) is simply the minimum across yield-to-maturity and every yield-to-call:

$$YTW = \min\big(YTM,\ YTC_1,\ YTC_2,\ \ldots,\ YTC_n\big)$$

In words: compute the yield assuming the bond is redeemed on each possible date, and take whichever scenario gives the investor the least return — that is the number to plan around, since the issuer will naturally choose to call whenever it's in the issuer's favor, not the investor's.

Worked example

A 10-year, 5% annual-coupon bond trades at 104 ($1,040 per $1,000 face). It is callable at 102 in year 5 and at 101 in year 7, and matures at par in year 10.

1. Yield to maturity: solving $1040 = \sum_{t=1}^{10} \frac{50}{(1+y)^t} + \frac{1000}{(1+y)^{10}}$ gives approximately $y = 4.42\%$.
2. Yield to call, year 5: solving $1040 = \sum_{t=1}^{5} \frac{50}{(1+y)^t} + \frac{1020}{(1+y)^{5}}$ gives approximately $y = 3.99\%$.
3. Yield to call, year 7: solving $1040 = \sum_{t=1}^{7} \frac{50}{(1+y)^t} + \frac{1010}{(1+y)^{7}}$ gives approximately $y = 4.31\%$.
4. Yield to worst: the minimum of 4.42%, 3.99%, and 4.31% is 3.99%, the year-5 call — that is the number a buyer should use to judge the bond, not the 4.42% headline YTM.

What this means in practice

Yield to worst is the standard quoted yield on callable corporate and municipal bonds precisely because it protects an investor from over-optimistic assumptions. Trading desks, index providers, and fund managers all report YTW rather than YTM for callable universes, and it's the figure that determines whether a premium bond (priced above par, likely to be called) actually looks attractive once the call risk is priced in.