Yield to maturity, yield to each call and the minimum, worked on one illustrative callable high-yield bond, with the price at which the worst date changes.
Yield to worst is the lowest of the bond's yield to maturity and its yield to each call date, each solved from the same price with the call price in place of par. For an illustrative six-year bond with a 7.25 per cent coupon, priced at 104.50 and callable from year two, the yield to maturity is 6.34 per cent but the yield to worst is 5.97 per cent, to the par call in year four, not to the first call. That 37 basis point gap is the part of the quoted yield the issuer can take back.
Worked in full in The Credit Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A callable bond does not have one set of cash flows. If the issuer can refinance more cheaply, it repays early at the call price, and the investor's coupons stop. Each possible redemption date gives a different yield from the same purchase price. The convention is to quote the lowest, because the issuer holds the option and will exercise it when it suits the issuer, which is exactly when it hurts the holder. Yield to worst is the lowest yield the investor can earn, held to redemption, if the issuer does not default, whatever it does with the call.
The case below is illustrative: a BB bond bought on a coupon date, so there is no accrued interest and every period is a whole half-year.
| Input | Value |
|---|---|
| Coupon, paid semiannually | 7.25% |
| Coupon per half-year, per 100 of face | 3.625 |
| Price, per 100 of face | 104.50 |
| Years to maturity | 6 |
| Call in year 2 at | 103.625 |
| Call in year 3 at | 101.8125 |
| Call in year 4 and after at | 100.00 |
The schedule is the usual high-yield shape: non-call for two years, then a premium of half the coupon stepping down to par.
Step 1: the yield to maturity. Solve for the rate that discounts twelve coupons of 3.625 and 100 at maturity back to 104.50.
104.50 = Σ 3.625 / (1 + y/2)k for k = 1 to 12, + 100 / (1 + y/2)12
y = 6.34%. Excel: =RATE(12,3.625,-104.5,100)*2
Step 2: the yield to each call. Replace the number of periods with the half-years to the call date, and the 100 at the end with the call price. To the first call that is four coupons and 103.625, 118.125 of cash in all for 104.50 paid.
Year 2 call: =RATE(4,3.625,-104.5,103.625)*2 = 6.54%
Year 3 call: =RATE(6,3.625,-104.5,101.8125)*2 = 6.14%
Year 4 call: =RATE(8,3.625,-104.5,100)*2 = 5.97%
Step 3: take the minimum. Here it is the year four call at par, 5.97 per cent. Excel's YIELD function does the same job with dates and a redemption value; the yield to worst is then =MIN() of the four results.
| Redemption | Price received | Half-years | Yield |
|---|---|---|---|
| First call, year 2 | 103.625 | 4 | 6.54% |
| Second call, year 3 | 101.8125 | 6 | 6.14% |
| Par call, year 4 | 100.00 | 8 | 5.97% |
| Maturity, year 6 | 100.00 | 12 | 6.34% |
The intuition is the premium. The investor paid 4.50 over par. Called at 103.625 in two years, only 0.875 of that premium is lost, about 0.438 a year. Called at par in year four, all 4.50 is lost over four years, 1.125 a year, against 0.75 a year if the bond runs to maturity. The worst date is the one where the premium paid is written off fastest, net of whatever call premium comes back.
The current yield, coupon over price, is 6.94 per cent. It is the highest number on the screen and the least relevant: it ignores the premium entirely.
The worst date is not fixed. It moves with the price, and that is the reason the measure exists.
| Price | Call yr 2 | Call yr 3 | Call yr 4 | Maturity | Yield to worst | Worst date |
|---|---|---|---|---|---|---|
| 96.00 | 11.20% | 9.35% | 8.45% | 8.11% | 8.11% | Maturity |
| 100.00 | 8.95% | 7.80% | 7.25% | 7.25% | 7.25% | Maturity |
| 102.00 | 7.86% | 7.05% | 6.67% | 6.84% | 6.67% | Call yr 4 |
| 103.00 | 7.33% | 6.69% | 6.39% | 6.64% | 6.39% | Call yr 4 |
| 104.50 | 6.54% | 6.14% | 5.97% | 6.34% | 5.97% | Call yr 4 |
| 106.00 | 5.77% | 5.61% | 5.56% | 6.04% | 5.56% | Call yr 4 |
| 108.00 | 4.76% | 4.92% | 5.02% | 5.66% | 4.76% | Call yr 2 |
Below par, the worst date is maturity: no issuer pays a call premium to retire debt it could buy back more cheaply in the market, and yield to worst equals yield to maturity. That is the case of the 8.50 per cent bond at 94.00 worked in how to calculate yield to maturity. At par every call at 100 yields the coupon, 7.25 per cent. Above par the par call takes over; the year three call becomes the worst only from about 106.71, and the first call from about 106.98. The first call yield falls below the yield to maturity from about 105.13.
The common mistake is to rank premium bonds by yield to maturity. Here it overstates the guaranteed yield by 37 basis points, and on a bond trading at 108.00 the gap is 90. A portfolio screen sorted on yield to maturity systematically favours bonds that are about to be called.
The second mistake is to treat yield to worst as the expected return. It is still a promised yield: it assumes every coupon is paid. On the book's assumptions a BB issuer defaults 1 per cent a year with 40 per cent recovery, an expected loss of 0.60 per cent a year. The bond's expected yield is then 5.37 per cent, not 5.97, and not the 6.34 a yield-to-maturity screen shows. The method is the one in yield after expected default losses, and the full chain from displayed to earned yield on a seventeen-line portfolio is in the free workbooks for the book's case.
Two technical points. A make-whole call during the non-call period is priced at a spread over government bonds and almost never exercised for refinancing, so it is usually ignored in the yield to worst. And on a bond bought between coupon dates, solve on the full price including accrued interest, which YIELD does for you.
It is never higher, because yield to maturity is one of the yields it takes the minimum of. Below par the two are equal: an issuer will not pay a call premium to retire debt it can buy back more cheaply. On the illustrative 7.25 per cent bond, at 96.00 both are 8.11 per cent, while at 104.50 yield to maturity is 6.34 per cent and yield to worst 5.97.
Use RATE with the half-years to the call date, the semiannual coupon, the price as a negative and the call price as the future value, then double it. For a 7.25 per cent bond at 104.50 callable in two years at 103.625: =RATE(4,3.625,-104.5,103.625)*2 gives 6.54 per cent. With settlement and call dates, YIELD does the same and handles accrued interest.
Because the premium paid is written off over the time to the call, net of the call premium received. At 104.50 the first call at 103.625 loses only 0.875 points over two years, a yield of 6.54 per cent, while the par call in year four loses all 4.50 points, 5.97 per cent. The first call only becomes the worst above about 106.98.
This article is one calculation from The Credit Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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