Yield to maturity assumes you hold to the end and reinvest at the same yield. The holding period return uses the assumptions you actually expect.
Add the coupons, the interest earned on reinvesting them and the price you sell at, divide by the price you paid, and annualise over the holding period. An illustrative five-year 8.50 per cent bond bought at 94.00 and sold after three years at an unchanged yield ends at 124.05 per 100 of face bought at 94.00, a holding period return of 31.97 per cent, or 9.46 per cent a year: below the 10.06 per cent yield to maturity, because the coupons were reinvested at 4.00 per cent, not at 10.06.
Worked in full in The Credit Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Yield to maturity answers a narrow question: what a bond returns if it is held to maturity, never defaults, and every coupon is reinvested at that same yield. Most investors sell before maturity and park coupons in cash. The holding period return, also called horizon return or total return, replaces those assumptions with the ones you actually expect, and that is the number to compare with a fund, a loan or a bill over the same horizon.
| Input | Value |
|---|---|
| Coupon, paid semiannually | 8.50% (4.25 a half-year) |
| Price paid, on a coupon date | 94.00 |
| Maturity at purchase | 5 years |
| Yield to maturity at purchase | 10.06% |
| Holding period | 3 years, 6 coupons |
| Rate earned on reinvested coupons (Treasury bills) | 4.00% |
| Exit yield, base case | 10.06%, unchanged |
1. Coupons with interest on interest = 4.25 × [(1 + 2%)6 − 1] ÷ 2% = 26.81
2. Sale price = price of a two-year 8.50 per cent bond at the exit yield of 10.06% = 97.24
3. Total at the horizon = 26.81 + 97.24 = 124.05
4. Holding period return = 124.05 ÷ 94.00 − 1 = 31.97%
5. Annualised, bond-equivalent: 2 × [(124.05 ÷ 94.00)1/6 − 1] = 9.46%; effective annual: (124.05 ÷ 94.00)1/3 − 1 = 9.69%
In Excel: =FV(4%/2,6,-4.25) for step 1, =PRICE(Settle,Maturity,8.5%,10.06%,100,2) for step 2, and =2*((B3/94)^(1/6)-1) for step 5.
Quote the bond-equivalent figure when comparing with a yield to maturity, which is itself a semiannual rate doubled. The 10.06 per cent yield is 10.31 per cent on an effective annual basis; compare 9.69 with that, not with 10.06.
| Source | Amount |
|---|---|
| Six coupons of 4.25 | 25.50 |
| Interest on interest at 4.00% | 1.31 |
| Pull to par: 94.00 bought, 97.24 sold | 3.24 |
| Total gain | 30.05 |
| Total value at the horizon: 94.00 paid plus the gain | 124.05 |
Even with no change in yield, the price rises from 94.00 to 97.24, because a discount bond climbs towards 100 as maturity approaches. That pull to par is part of the return the yield to maturity promised. The shortfall against 10.06 per cent is entirely reinvestment: had the coupons earned 10.06 per cent, interest on interest would have been 3.43 rather than 1.31 and the return exactly 10.06. Reinvesting in bills costs 2.12 per 100, or 0.59 points a year.
On the book's line, which cost 11,750 for 12,500 of face, the base case ends at 15,507: a gain of 3,757 over three years.
| Yield at sale | Sale price | Total | Return a year (BEY) | Gain on 11,750 |
|---|---|---|---|---|
| 8.06%, 2 points lower | 100.80 | 127.61 | 10.45% | 4,202 |
| 9.06%, 1 point lower | 99.00 | 125.81 | 9.96% | 3,977 |
| 10.06%, unchanged | 97.24 | 124.05 | 9.46% | 3,757 |
| 11.06%, 1 point higher | 95.52 | 122.33 | 8.98% | 3,542 |
| 12.06%, 2 points higher | 93.84 | 120.65 | 8.50% | 3,332 |
| 14.06%, spread 4 points wider | 90.60 | 117.41 | 7.55% | 2,926 |
Three years in, the bond has only two years left, so its price is far less sensitive than at purchase: a two-point rise in yield costs 3.40 points of price, against 7.07 at purchase, and under one point of annual return. For the three-year return to fall to the 4.00 per cent a bill would have paid, the exit yield would have to reach 22.01 per cent, 11.95 points higher, with the bond sold at 79.05. At this horizon the yield move is not what decides the outcome. Default is: one default with a low recovery wipes out more than any of these rows, which is why the yield after expected losses is worked separately in a high-yield bond's yield after expected default losses.
How the 10.06 per cent is solved from the price is in how to calculate yield to maturity on a bond.
Holding period return = (coupons reinvested + sale price) ÷ price paid, annualised over the holding period. On this bond, 9.46 per cent a year against a 10.06 per cent yield to maturity, the gap being the rate earned on the coupons. Every line of the book's portfolio, quarter by quarter in the base case and each stress, is in the free workbooks for this case.
Yield to maturity assumes the bond is held to maturity and every coupon is reinvested at the yield itself. Holding period return uses your own horizon, sale price and reinvestment rate. A 94.00 bond yielding 10.06 per cent, sold after three years at the same yield with coupons in 4.00 per cent bills, returns 9.46 per cent a year.
Raise the ending value over the price to one over the number of periods and subtract one. With 124.05 per 94.00 over three years: (124.05 / 94.00)^(1/3) - 1 = 9.69 per cent effective annual, or 9.46 per cent on the bond-equivalent basis used for yields, two times the half-yearly rate.
On a high-coupon bond, a lot. Reinvesting the six coupons of an 8.50 per cent bond at 4.00 per cent instead of its 10.06 per cent yield cuts interest on interest from 3.43 to 1.31 per 100, and the three-year return from 10.06 to 9.46 per cent a year.
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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