Holding period return
An investor may buy a bond at issue or later in the secondary market, and may hold it to maturity or sell earlier. The return earned for the period actually held is the holding period return. Three things make it up: the coupons received, the interest those coupons earn when reinvested at whatever rate prevails, and the gain or loss on sale when the sale price differs from the purchase price. Add the three and express the total as a percentage of the cost price.
HPR = (Coupons + Reinvestment income on coupons + (Sale price - Purchase price)) / Purchase price
- Buy at 104, receive a coupon of 8, reinvest it at 7% for the year, sell at 110 after one year.
- HPR = (8 + 8 x 7% + (110 - 104)) / 104 = (8 + 0.56 + 6) / 104 = 14.00%.
HPR is a crude figure: the excess earned over the initial investment, in percentage terms, for the horizon held, which is often longer than a year. It ignores compounding. Whether the bond was held one year or five, the formula is the same, so an HPR must never be equated with or compared to an annualised return. The annualised, compounded version is the realised yield, below.
Current yield
The simplest measure: divide the coupon by the current market price and express the result as a percentage. The 8.24% government security trading at 104 has a current yield of 8.24 / 104 = 7.92%. Current yield ignores all future cash flows from the bond, which is its biggest drawback and the reason it is not widely used. It is the bond equivalent of a stock's dividend yield.
Yield to maturity
YTM is the more comprehensive and widely used measure. It takes every future cash flow from the bond, the coupons, the reinvestment income on those coupons and the redemption value, and finds the rate at which their present value equals the bond's market price. The rate that equates today's outflow (the price paid) with the present value of the future inflows is the YTM. It is the internal rate of return of the bond investment, found by trial and error or with a spreadsheet function such as XIRR.
First, that the investor holds the bond to maturity; sell earlier and the YTM is not what you earn. Second, that every coupon is reinvested for the remaining tenor at the same rate, the YTM itself. That implies interest rates stay unchanged for the whole life of the bond and are the same for every tenor: a static, flat yield curve. The assumptions make YTM impractical as a forecast of what an investor will actually earn. Its advantage is that it is simple and quick to compute, which is why everyone uses it.
A 9.70% bond issued on 19 July 2007 and maturing on 19 July 2017, paying interest annually, is bought at 103 on 1 November 2014. Its YTM is the single rate that discounts the remaining annual coupons and the redemption value on 19 July 2017 back to exactly 103 on 1 November 2014. Because the price is above par, the YTM comes out below the 9.70% coupon: part of each year's coupon is offset by the 3 rupees the holder loses when only 100 is redeemed.
Realised yield
YTM applies only if the bond is held to maturity with coupons reinvested at the YTM. When an investor holds for a shorter period and sells, at a gain or a loss, the return actually achieved is the realised yield, and it is annualised.
RY = [ (Coupon x ((1 + reinvestment rate)^holding period - 1) / reinvestment rate + Sale proceeds) / Purchase price ]^(1 / holding period) - 1
- The numerator accumulates all coupons at the reinvestment rate to the end of the holding period and adds the sale proceeds.
- Dividing by the purchase price gives the total growth multiple; the root annualises it.
Invest 1,000 today in a bond with ten years to maturity and a 12% annual coupon. Hold five years, reinvesting each 120 coupon at 14%, then sell for 1,050. The coupons accumulate to 120 x ((1.14^5 - 1) / 0.14) = 120 x 6.610 = 793.2. Add the sale proceeds: 1,843.2. Divide by 1,000 and take the fifth root: 1.8432^(1/5) = 1.130. Realised yield is 13.0% a year.
Three rates appear in a realised yield question: the coupon (what the bond pays), the reinvestment rate (what the coupons earn once received) and the answer (the realised yield). They differ. The most common slip is compounding the coupons at the coupon rate.
- Holding period return adds coupons received, interest earned by reinvesting them and the gain or loss on sale, and divides by the purchase price. It ignores compounding, so it must not be compared with an annualised return.
- Current yield is coupon divided by current market price. It ignores every future cash flow, is comparable to a stock's dividend yield, and is not widely used.
- Yield to maturity is the rate that equates the bond's price to the present value of all future cash flows: the bond's internal rate of return. It is earned only if the bond is held to maturity and coupons are reinvested at the YTM, which implies a static, flat yield curve.
- Realised yield is the annualised return over a holding period shorter than maturity: accumulate the coupons at the reinvestment rate, add the sale proceeds, divide by the purchase price, take the holding-period root, subtract one.