One equation, solved by trial and error or by RATE(). The bond's coupon and its current yield are both lower, and the return actually earned can be far lower still.
Yield to maturity is the single discount rate that makes the present value of a bond's remaining coupons and principal equal to the price you pay. For an illustrative bond with an 8.50 per cent semiannual coupon, five years to run and a price of 94.00, that rate is 10.06 per cent: above the 8.50 coupon and above the 9.04 per cent current yield, because the 6.00 points of discount are earned on the way to par.
Worked in full in The Credit Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
There is no closed-form answer. You either solve it by trial and error, which is what Excel does behind RATE(), or interpolate between two guesses by hand. Both are shown below, on a bond that is cheap for a reason.
| Input | Value |
|---|---|
| Coupon, paid semiannually | 8.50% |
| Coupon per half-year, per 100 of face | 4.25 |
| Price per 100 of face | 94.00 |
| Years to maturity | 5 (10 half-years) |
| Face held | 12,500 |
| Cost of the holding | 11,750 |
Price = Σ (C/2) ÷ (1 + y/2)k + 100 ÷ (1 + y/2)n, for k = 1 to n
Here C is the annual coupon per 100, n the number of half-years and y the yield to maturity, quoted as twice the half-yearly rate. Find the y that makes the right-hand side equal 94.00. In Excel: =RATE(10, 4.25, -94, 100)*2 returns 10.06 per cent. With real dates, =YIELD(settlement, maturity, 8.5%, 94, 100, 2) does the same and handles accrued interest between coupon dates.
Step 1: current yield, as a lower bound. The coupon over the price is 8.50 / 94.00 = 9.04 per cent. The bond also repays 100 for every 94.00 paid, so the yield to maturity must be higher.
Step 2: price the bond at trial yields. Discount the ten cash flows of 4.25, the last one 104.25, at each guess:
| Trial yield | Price | Against 94.00 |
|---|---|---|
| 9% | 98.02 | Too high: the yield must be higher |
| 10% | 94.21 | Slightly high |
| 11% | 90.58 | Too low |
Step 3: interpolate. Between 10 and 11 per cent the price falls by 3.63. The target sits 0.21 below the 10 per cent price, so the yield is about 10 + 0.21 / 3.63 = 10.057 per cent. The exact solution, iterated to convergence, is 10.0562 per cent, or 5.0281 per cent a half-year.
Step 4: check the split. At 10.06 per cent the ten coupons are worth 32.77 today and the principal 61.23. Together they make 94.00, which is the definition of the yield.
The quick textbook approximation, coupon plus discount spread over the life, divided by the average of par and price, gives (8.50 + 1.20) / 97.00 = 10.00 per cent. It is close here and drifts further off for long bonds or deep discounts. Use it to sanity-check, not to quote.
On 12,500 of face bought for 11,750, the bond pays 531.25 every half-year and 12,500 at maturity. The yield that links those flows to the price is 10.06 per cent, a bond-equivalent rate. Restated as an effective annual rate, (1 + 0.050281)2 − 1, it is 10.31 per cent. Compare it with a 4.00 per cent Treasury bill and the spread is 6.06 points.
| Measure | Value | What it includes |
|---|---|---|
| Coupon rate | 8.50% | Coupon over par |
| Current yield | 9.04% | Coupon over price |
| Yield to maturity | 10.06% | Coupons and the pull to par, discounted |
| Pull to par | 1.01 | Yield to maturity less current yield, points |
The lower the price, the more of the yield comes from the pull to par rather than the coupon. The same bond at other prices:
| Price | Current yield | Yield to maturity |
|---|---|---|
| 90.00 | 9.44% | 11.16% |
| 92.00 | 9.24% | 10.60% |
| 94.00 | 9.04% | 10.06% |
| 96.00 | 8.85% | 9.52% |
| 98.00 | 8.67% | 9.01% |
| 100.00 | 8.50% | 8.50% |
Each two points of price is worth a little over half a point of yield on a five-year bond. At par the three measures coincide. Across an illustrative high-yield sleeve the pattern holds: a BB bond with a 6.75 per cent coupon at 99.00 yields 6.99, a B bond with an 8.00 per cent coupon at 97.00 yields 8.75, and the CCC bond at 94.00 yields 10.06, with a pull to par six times that of the BB.
Yield to maturity is an internal rate of return on promised cash flows: 10.06 per cent for an 8.50 per cent bond at 94.00, found by RATE() or by interpolating between two trial prices. It is the right number for comparing bonds and the starting point for duration, but it is a promise, not a forecast. The Credit Investor follows a whole portfolio from the yield on the screen to the return actually earned, and the free workbook for this case computes this yield both with RATE and from the explicit cash flows.
Current yield is the annual coupon divided by the price and ignores the gain or loss to par at maturity. Yield to maturity includes it. An illustrative bond paying 8.50 per cent and priced at 94.00 has a current yield of 9.04 per cent and a yield to maturity of 10.06 per cent; the 1.01 point gap is the pull to par, the 6.00 points of discount earned over five years.
With RATE on the coupon periods, multiplied by the number of periods a year: for five years of semiannual coupons of 4.25 per 100 and a price of 94, =RATE(10, 4.25, -94, 100)*2 returns 10.06 per cent. With dates, =YIELD(settlement, maturity, 8.5%, 94, 100, 2) gives the same answer when settlement falls on a coupon date.
Only if every coupon and the principal are paid on time and each coupon is reinvested at the same yield. Reinvest the coupons of an illustrative 10.06 per cent bond at a 4.00 per cent bill rate and the realised yield is 9.08 per cent. Allow for a 15 per cent annual default probability with 40 per cent recovery and the expected yield falls to 1.06 per cent.
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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