Yield to maturity and yield to worst
A coupon is what a bond pays; its yield is what you earn at today's price. If the issuer can repay early, quote the yield you can count on.
- Calculate a bond's current yield and yield to maturity
- Calculate the yield to call and pick the yield to worst
- Price a bond from the yield on comparable credits
- Say when the call, rather than maturity, is the worst case
The intuition
You lend a friend $100 for five years at $6 a year. A year later you sell the loan to someone else for $95. They still get $6 a year and $100 at the end, but they paid only $95, so they earn more than 6%: the coupons plus the $5 they gain when the loan is repaid. Had they paid $105, they would earn less.
A bond's yield to maturity (YTM) is the single discount rate that makes its coupons and repayment worth today's price. Many high-yield bonds are callable: the issuer can repay early at a set call price. It will do that when refinancing is cheaper, which is exactly when it is worse for the holder, so credit desks quote the yield to worst: the lower of the yield to maturity and the yield to call.
Price = Σ coupon ÷ (1 + y)ᵗ + redemption ÷ (1 + y)ⁿ. YTM solves it to maturity, redeeming at 100; the yield to call solves it to the call date, redeeming at the call price. Yield to worst = the lower of the two. Current yield = coupon ÷ price.
Why it works
- The conventions here: annual coupons, prices per 100 of face value, whole years to maturity and to the first call date, no accrued interest. A yield is found from the price by trial and error (the site uses Newton's method); a price from a yield is a direct sum.
- Below 100, YTM is above the coupon and the current yield, because the pull back to 100 adds return. Above 100 it is below both.
- A call caps the upside. The richer the bond, the more attractive calling it is for the issuer, and a called holder loses the coupons that would have come after the call date.
- Yield to worst is the conservative quote. It assumes whichever redemption date is worse for the holder.
- Price and yield move opposite ways. A lower yield on comparable credits means a higher fair price.
- The worst case can switch. As the price rises, the yield to call falls faster than the yield to maturity, so the richer the bond, the more likely it is quoted to the call.
| Current yield: 6 ÷ 104 | 5.77% |
| Yield to maturity: solve 104 = Σ 6 ÷ (1 + y)ᵗ + 100 ÷ (1 + y)⁵ | 5.07% |
| Yield to call: solve 104 = 6 ÷ (1 + y) + 108 ÷ (1 + y)² | 4.83% |
| Yield to worst: the lower | 4.83%, to the call |
| Fair price if comparables yield 7%: Σ 6 ÷ 1.07ᵗ + 100 ÷ 1.07⁵ | 95.90 |
A quick check on the YTM: (6 + (100 − 104) ÷ 5) ÷ ((100 + 104) ÷ 2) = 5.10%, close to the exact 5.07%.
The formulas
Every cash flow, discounted at one rate.
What you earn if you hold it to the end.
What you earn if the issuer calls.
The return you can count on.
Income only, ignoring the pull to 100.
Worked example
Solve the yield twice: to maturity at 100, and to the call date at the call price. Then take the lower.
See it move
Same bond. Change its price, its coupon, the years to maturity, the call date and price, and the yield on comparable credits.
- Raise the price. Every yield falls, and the yield to call falls faster than the yield to maturity.
- Raise the call price. The yield to call rises; the yield to maturity does not move.
- Raise the yield on comparables. The fair price falls.
- Raise the coupon. The fair price rises.
Run it backwards
Same bond, reversed: comparable credits yield a set rate. What should this bond trade at?
Discount each coupon and the final 100 at the comparables' yield and add them up. That is the price at which this bond would yield the same as its peers.
If the market price is below it, the bond yields more than its peers: either the market sees extra risk in the name, or the bond is cheap.
Traps
Say it in the interview
“What's the difference between yield to maturity and yield to worst?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- A yield solves price = Σ cash flows ÷ (1 + y)ᵗ.
- Below par, yield is above the coupon; above par, below it.
- Yield to worst = the lower of the yield to maturity and to call.
- The richer a callable bond, the more likely it is worst to the call.