FRM Exam Part I · Bond Yields and Return Calculations

Holding Period Return and Realized Bond Yield Explained

Updated 11 October 2026 · Fact-checked

Holding period return (HPR) is total gain over a period, price change plus income, divided by the starting price. Arithmetic return averages period returns, geometric return compounds them. Realized yield is the annualized return you actually earned, including reinvested coupons. Solve by finding ending value, then annualizing: (Ending ÷ Beginning)^(1/n) − 1.

Understand Return Calculations and Realized Yield

A return measures how much your money grew over a period. The holding period return (HPR) counts everything you received: the change in price plus any income such as a coupon or dividend. You divide the total gain by what you paid at the start.

Most questions give you several periods. Then you must choose how to summarize them. The arithmetic mean is the simple average of the period returns. The geometric mean is the constant per-period rate that gives the same final wealth. Because returns compound, the geometric mean is the right measure of the growth you actually earned. The arithmetic mean is always greater than or equal to the geometric mean. They are equal only when every period return is the same. The more volatile the returns, the bigger the gap.

A log (continuously compounded) return is ln(P1 ÷ P0). Log returns add across time, which makes them easy to sum over many periods. Simple returns do not add; you multiply (1 + R) terms. For small returns the two are close, but log return is always lower than the simple return for the same price move.

For a bond held to maturity, the yield to maturity assumes you reinvest every coupon at the YTM itself. In practice you rarely can. The realized compound yield uses the actual reinvestment rate. You compute the future value of the coupons (reinvested) plus the sale price or principal, then find the annualized rate that turns your purchase price into that total. If reinvestment rates fall below the YTM, your realized yield is lower. This is reinvestment risk.

Key formulas to remember

Holding period return (bond): HPR = (P1 − P0 + C) ÷ P0; P0 = purchase price, P1 = ending price, C = income received (coupons, plus reinvestment income if included). Use the full price (clean plus accrued) consistently.
Multi-period HPR: HPR = (1 + R1)(1 + R2)…(1 + Rn) − 1; Compounds the period returns. Do not add them.
Arithmetic mean return: R_A = (R1 + R2 + … + Rn) ÷ n; Best estimate of the expected return for a single future period. Overstates compound growth when returns vary.
Geometric mean return: R_G = [(1 + R1)(1 + R2)…(1 + Rn)]^(1/n) − 1; The compound average per period. R_G ≤ R_A.
Log return: r = ln(P1 ÷ P0) = ln(1 + R); Multi-period log return is the sum of single-period log returns. Include income by using total value instead of price if required.
Simple from log return: R = e^r − 1; Converts a log return back to a simple return.
Realized compound yield (annual): y = (Ending value ÷ P0)^(1/years) − 1; Ending value = sale price or principal + coupons + interest earned on reinvested coupons. For semiannual coupons, compute per half-year and then annualize as the question requires.
Future value of reinvested coupons: FV = C × [(1 + r)^n − 1] ÷ r; Ordinary annuity future value at the reinvestment rate r per coupon period, n coupons.

How to solve Return Calculations and Realized Yield questions

Use this sequence for any return or realized yield question. It keeps you from mixing up arithmetic, geometric and log measures.

1Identify what is asked: a single-period HPR, an average over several periods, a log return, or a realized yield with reinvestment.
2Write down the starting price P0 and every cash flow with its timing. Note the compounding period (annual or semiannual).
3For HPR, add the price change and income, then divide by P0.
4For several periods, convert each return to (1 + R). Multiply for total growth, or take the nth root for the geometric mean. Add and divide by n only if the question asks for the arithmetic mean.
5For log returns, use ln of the price relative and add them across periods. Convert back with e^r − 1 if a simple return is requested.
6For realized yield, compute the future value of coupons at the stated reinvestment rate, add the final price or principal, then take (Total ÷ P0)^(1/n) − 1.
7Match the answer to the requested frequency. If periods are half-years, annualize by doubling (bond-equivalent) or compounding, as the question states.
8Sanity check: the geometric mean must not exceed the arithmetic mean, and the realized yield should be below the YTM if reinvestment rates are below the YTM.

Quickest way: Growth-factor shortcut

When to use it: Use it when options are close together and you need a fast, reliable answer on a calculator.

Turn every return into a growth factor (1 + R). Work only with factors.
Multiply the factors in the calculator memory. Use y^x with exponent 1/n for the geometric mean.
For realized yield, find Total ending value first. Then one line: (Total ÷ P0)^(1/n) − 1.
Eliminate options: if one equals the arithmetic mean and the returns vary, it cannot be the geometric mean.
For log returns, use ln on the calculator and sum the values. No compounding is needed.

Common mistakes in Return Calculations and Realized Yield

Averaging returns arithmetically when asked for growth of wealth over several periods.
The simple average is quicker and feels natural.
Fix: If the question asks about compound or annualized growth, use the geometric mean. Check that R_G ≤ R_A.
Adding multi-period simple returns, for example +50% and −50% giving 0%.
Simple returns look additive, but they apply to different bases.
Fix: Multiply growth factors: 1.5 × 0.5 = 0.75, a 25% loss. Only log returns add.
Ignoring coupons in the holding period return.
Students focus on the price change and forget income.
Fix: Always put the coupon in the numerator: (P1 − P0 + C) ÷ P0.
Assuming coupons are reinvested at the YTM when the question gives a different reinvestment rate.
YTM is the usual default in bond pricing.
Fix: Use the stated reinvestment rate to grow each coupon. The realized yield then differs from the YTM.
Annualizing a semiannual realized return by simply doubling when compounding is required, or the reverse.
Quoting conventions are mixed up.
Fix: Read the wording. Doubling gives a bond-equivalent (nominal) yield. (1 + half-year return)² − 1 gives the effective annual yield.
Treating a log return as equal to a simple return for large moves.
They are nearly equal for small moves.
Fix: Compute ln(P1 ÷ P0) exactly. For a 20% gain, the log return is 18.23%, not 20%.

Worked examples

Example 1

An investor buys a bond for $950. One year later she receives a coupon of $60 and sells the bond for $970. Over the next two years, the bond's annual returns are +4% and −2%. First find the first-year HPR. Then find the geometric mean annual return and the arithmetic mean using the three annual returns: first year, +4% and −2%.

Show the solution

First-year HPR = (970 − 950 + 60) ÷ 950 = 80 ÷ 950 = 0.084211, or 8.42%.
The three annual returns are 8.4211%, 4% and −2%.
Growth factors: 1.084211, 1.04, 0.98.
Product = 1.084211 × 1.04 = 1.127579; × 0.98 = 1.105027.
Geometric mean = 1.105027^(1/3) − 1. The cube root of 1.105027 is about 1.03386, so R_G = 3.39%.
Arithmetic mean = (8.4211 + 4 − 2) ÷ 3 = 10.4211 ÷ 3 = 3.47%.
Check: 3.47% > 3.39%, consistent with R_A ≥ R_G.

Answer: First-year HPR = 8.42%; geometric mean ≈ 3.39% per year; arithmetic mean ≈ 3.47% per year.

Example 2

A 3-year bond with a face value of $1,000 pays an annual coupon of 6%. You buy it at par and hold it to maturity. You reinvest each coupon at 4% per year. What is the realized compound annual yield?

Show the solution

Coupon = 6% × 1,000 = $60 per year.
Reinvest at 4%: the first coupon grows for 2 years: 60 × 1.04² = 60 × 1.0816 = 64.896.
The second coupon grows for 1 year: 60 × 1.04 = 62.40.
The third coupon is received at maturity: 60.00.
Total coupons with interest = 64.896 + 62.40 + 60 = 187.296.
Ending value = 1,000 + 187.296 = 1,187.296.
Realized yield = (1,187.296 ÷ 1,000)^(1/3) − 1.
The cube root of 1.187296 is about 1.05898, so the yield is about 5.90%.

Answer: Realized compound yield ≈ 5.90% per year, below the 6% YTM because coupons were reinvested at only 4%.

Exam tips

When the question says average return over time or compound growth, think geometric. When it says expected return for next period, think arithmetic.
Check the reinvestment assumption in the wording. The YTM assumes reinvestment at the YTM, and realized yield replaces that assumption.
Use log returns when the question involves summing returns across periods or normal-distribution modelling of prices. Convert only if asked for a simple return.
Keep at least five decimals in growth factors until the last step. Rounding early can push you to a neighbouring option.
With four close options, estimate first. A realized yield with lower reinvestment rates must fall below the YTM, which often removes two options.

Practice questions from Bond Yields and Return Calculations

Return Calculations and Realized Yield: frequently asked questions

What is the difference between arithmetic and geometric return?

The arithmetic return is the simple average of period returns. The geometric return is the compound average rate that links starting and ending wealth. The geometric return is never greater than the arithmetic return, and the gap grows with volatility.

Why do log returns add over time but simple returns do not?

Because ln(P2 ÷ P0) = ln(P2 ÷ P1) + ln(P1 ÷ P0). Simple returns compound as (1 + R1)(1 + R2) − 1, so they do not add. This is why log returns are convenient for multi-period analysis.

What is realized compound yield on a bond?

It is the annualized return you actually earn from the purchase price to the final value. The final value includes the sale price or principal, the coupons, and the interest earned on reinvested coupons. It differs from the YTM when the reinvestment rate differs from the YTM.

What is reinvestment risk?

It is the risk that coupons must be reinvested at lower rates than expected. A lower reinvestment rate reduces the realized yield below the promised YTM. The risk is greater for high-coupon, long-maturity bonds held to maturity.