Distinguish gross realized returns from net realized returns and calculate a bond's realized return over a holding period, reinvestments included.
The idea
Gross realized return measures what the bond position itself earned over a holding period: the ending price plus every coupon received, plus the interest earned by reinvesting those coupons until the horizon, relative to the purchase price. Total realized return = (ending price + coupons + reinvestment income − purchase price) ÷ purchase price. To compare holding periods of different lengths, the annualized figure comes from Annualized return = (1 + total)^(1/years) − 1. Net realized return deducts the cost of holding the position, mainly the financing cost of money borrowed to buy the bond, so it is below gross when the position is leveraged. Reinvestment income depends on the rates actually available when each coupon arrives, so it is known only afterwards. Typical errors are dropping reinvestment income, using face value instead of the price paid as the base, averaging instead of compounding, and reporting gross when net is asked.
Formulas
Total gross realised return over a holding period
R = (Pe + C + RI − P0) ÷ P0
- Pe
- the price at the end of the holding period
- C
- the coupons received
- RI
- interest earned by reinvesting coupons to the horizon
- P0
- the purchase price paid
Base it on the price paid, not face value, and include reinvestment income. Net realised return then deducts the financing cost.
Annualised realised return
Ra = (1 + R)^(1 ÷ n) − 1
- R
- the total return over the holding period
- n
- the length of the holding period in years
Compound, do not divide: R ÷ n overstates the annual figure for multi-year periods.
A practice question
Which statement correctly distinguishes a bond position's gross realized return from its net realized return over a holding period?
- Gross counts price change, coupons and reinvestment income; net also deducts the cost of financing the position. - correct
- Gross includes the financing costs of the position, whereas net is measured before them.Why not: Reverses the two: financing costs come out only in the net figure.
- Gross counts only the price change; net adds the coupons and reinvestment income.Why not: Treats coupons and reinvestment income as net-only items; both already belong in the gross return.
- Gross is the annualized figure and net is the cumulative figure over the holding period.Why not: Confuses gross versus net with annualized versus cumulative; either can be stated either way.
Gross realized return is what the bond position earned: ending price plus coupons plus interest on reinvested coupons, relative to the purchase price. Net realized return subtracts the cost of financing the position, so it is lower when the position is funded with borrowed money.
A harder one
A bond with face value 100 and a 5% annual coupon is bought at 96.00 with three years to maturity. It is sold exactly two years later at 99.20, just after the second coupon is paid. The first coupon is reinvested for one year at 3%, and the second coupon is received at the sale date. What is the total gross realized return over the two-year holding period, closest to?
- 13.75%Why not: Leaves out the interest earned on reinvesting the first coupon, so the realized return is understated.
- 8.70%Why not: Drops the second coupon that arrives at the sale date.
- 13.91% - correct
- 3.33%Why not: Counts only the price gain and ignores coupons and reinvestment income.
Ending value = 99.20 + 5 x 1.03 + 5 = 109.35. Total gross realized return = (109.35 - 96.00) / 96.00 = 13.91%. Reinvestment income of 0.15 is the only part that depends on the reinvestment rate.
The hardest tier
A 4-year bond with face value 100 and a 6% annual coupon is bought at par and sold three years later at 97.50, just after the third coupon. Rates have risen, and the first two coupons are reinvested at 10% until the sale date (two years for the first coupon, one year for the second). What is the annualized gross realized return, closest to?
- 5.48% - correct
- 4.92%Why not: Leaves out the reinvestment income, so the ending value is too low.
- 5.79%Why not: Divides the cumulative return by three (arithmetic) instead of taking the geometric root.
- 5.25%Why not: Reinvests the first two coupons at the 6% coupon rate instead of the stated 10%.
Ending value = 97.50 + 6 x 1.10² + 6 x 1.10 + 6 = 117.36. Annualized = (117.36/100)^(1/3) - 1 = 5.48%. Reinvesting at 10% rather than 6% adds about 23 bp a year.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Bond Yields and Return Calculations
- Define a bond's spread, interpret it, and explain its derivation from the bond price together with a term structure of rates.
- Explain what a bond's yield to maturity (YTM) is, interpret it, and use it in pricing the bond.
- Explain the way a bond's YTM is calculated when its structure and price are known.
- Determine the price of an annuity as well as that of a perpetuity.
- Explain how spot rates and YTM are related.
- Explain what the coupon effect is and how coupon rate, YTM and bond prices relate to one another.
- Break a bond position or portfolio's profit and loss (P&L) down into separate drivers, namely carry roll-down, rate change and spread change effects, and explain each.
- Calculate carry roll-down under the interest rate assumptions most commonly made for it, and outline what those assumptions are.