What is “average return” (short answer)
– The average return is a simple summary measure that describes the typical periodic return in a series. The most basic form is the arithmetic average: add up each period’s simple returns and divide by the number of periods. It is quick to compute but ignores compounding and cash flows.
Key definitions
– Arithmetic average return: (Sum of periodic simple returns) ÷ (Number of periods). Use when you want the mean of independent period returns or short-run expected return estimates.
– Geometric average return (a.k.a. compounded or time-weighted return): The nth root of the product of (1 + each period return), minus 1. It reflects compounding and is the correct long-run average growth rate of a single invested amount over n equal periods.
– Time-weighted rate of return (TWR): A return measure that chains subperiod returns (geometric method) so that external cash flows do not distort the performance figure; commonly used to evaluate an investment manager.
– Money-weighted rate of return (MWRR): A return that accounts for the timing and size of investor cash flows (deposits/withdrawals). Mathematically equivalent to the internal rate of return (IRR), the discount rate that makes the net present value of cash flows equal zero.
– Simple growth rate: (Ending Value − Beginning Value) ÷ Beginning Value. This gives the percent change over one period.
Why the distinction matters (short)
– Arithmetic average ignores compounding and any cash flows into or out of the account. It can overstate the long-run average when returns are volatile.
– Geometric average incorporates compounding and is the correct measure of an investment’s long-term growth rate when no external flows distort returns.
– MWRR is appropriate when you, the investor, control timing/size of cash flows and want a return reflecting your experience.
– TWR is appropriate to compare managers because it neutralizes client cash-flow timing.
Formulas (clean)
– Arithmetic average return = (R1 + R2 + … + Rn) / n
– Geometric average return = (∏(1 + Ri))^(1/n) − 1 [product over i = 1 to n]
– Simple growth rate = (EV − BV) / BV
– MWRR = discount rate r such that Σ (CFt / (1 + r)^t) = 0 (IRR equation; CF0, CF1… include contributions/withdrawals and final value)
Worked numeric examples
1) Arithmetic vs. geometric (five annual returns)
– Annual returns: 10%, 15%, 10%, 0%, 5%
– Arithmetic average = (0.10 + 0
.15 + 0.10 + 0 + 0.05) / 5 = 0.40 / 5 = 0.08 = 8.0%
Geometric average (compound annual growth rate for these returns)
Step 1 — convert each return to 1 + Ri: 1.10, 1.15, 1.10, 1.00, 1.05
Step 2 — multiply: 1.10 × 1.15 × 1.10 × 1.00 × 1.05 = 1.460075
Step 3 — take the 5th root and subtract 1: (1.460075)^(1/5) − 1 ≈ 1.0786 − 1 = 0.0786 ≈ 7.86%
Interpretation: arithmetic = 8.00% (simple average of yearly numbers). geometric = 7.86% (annualized compound return over the five-year sequence). Geometric is always ≤ arithmetic when there is volatility; it reflects compounding.
Simple growth rate (total over period)
If beginning value (BV) = 100 and ending value (EV) = 146.0075 (the product from above), then
Simple growth rate = (EV − BV) / BV = (146.0075 − 100) / 100 = 0.460075 = 46.01% total growth over five years.
Note: this
Note: this total growth (46.01% over five years) is not the same as an annualized compound rate. To express multi‑period performance as an annualized rate you must account for compounding.
Formulas and steps (definitions first)
– CAGR (compound annual growth rate): the annualized geometric return. Formula: CAGR = (EV / BV)^(1/n) − 1, where EV = ending value, BV = beginning value, n = number of periods (years).
– Continuously compounded return (instantaneous or log return): r_cont = ln(EV / BV) / n. Exponentiating r_cont gives the same CAGR as geometric compounding: exp(r_cont) − 1 = CAGR.
– Arithmetic mean (simple average): sum of period returns divided by n. Use for expected return in a single upcoming period when returns are independent.
– Time‑weighted return (TWR): removes external cash flows to measure manager performance. Calculate by chaining period returns between cash flows.
– Dollar‑weighted return (internal rate of return, IRR): the investor’s actual rate of return accounting for the timing and size of cash flows.
Worked numeric example (use the EV and BV from above)
– BV = 100, EV = 146.0075, n = 5 years.
– CAGR = (146.0075 / 100)^(1/5) − 1 = (1.460075)^(0.2) − 1 ≈ 0.07862 = 7.862% per year.
– Continuous return = ln(1.460075) / 5 ≈ 0.37845 / 5 ≈ 0.07569 = 7.569% per year (as a continuously compounded rate). Converting: exp(0.07569) − 1 ≈ 7.862% (matches CAGR).
– Arithmetic average of the five annual simple returns (the numbers used earlier) was 8.00% — higher than the CAGR
That arithmetic average (8.00%) exceeds the compound annual growth rate (CAGR, 7.862%) is normal: the arithmetic mean treats each period independently and ignores compounding and volatility. The geometric mean (CAGR) captures multi‑period compounding, so volatility “drags” the long‑run compounded return below the arithmetic average unless every period return is identical.
Why arithmetic ≥ geometric (short explanation)
– Arithmetic mean = (1/n) Σ R_i, where R_i are simple period returns. It is the expected single‑period return if you sample a period at random.
– Geometric mean (CAGR) = (Π(1+R_i))^(1/n) − 1. It is the constant per‑period rate that compounds to the same cumulative result.
– Only when all R_i are equal do arithmetic and geometric means match. Any variability makes the geometric mean lower.
Worked numeric illustration (volatility drag)
1) Two‑year example with large volatility:
– Year 1: +50% → 1.50
– Year 2: −50% → 0.50
– Cumulative factor = 1.50 × 0.50 = 0.75 (a 25% loss over two years)
– Arithmetic average = (50% + (−50%))/2 = 0%
– Geometric (annualized) = (0.75)^(1/2) − 1 ≈ 0.866025 − 1 = −13.40% per year
Interpretation: although the average of the two yearly returns is 0%, compounding causes a negative multi‑year result.
2) Small returns, variance approximation (useful rule of thumb)
– For relatively small returns, ln(1+R) ≈ R − R^2/2. Averaging logs gives:
mean log ≈ mean R −
(continuing)
mean R − 1/2 · Var(R), where Var(R) is the variance of the simple returns. More precisely:
– For small returns, ln(1 + R) ≈ R − R^2/2.
– Taking expectations and using E[R^2] = (E[R])^2 + Var(R) gives E[ln(1 + R)] ≈ E[R] − 1/2 · Var(R) − 1/2 · (E[R])^2.
– If E[R] is small (so (E[R])^2 is negligible), the common approximation is:
E[ln(1 + R)] ≈ μ − 1/2·σ^2,
where μ = E[R] and σ^2 = Var(R).
Exponentiating the mean log return gives the geometric (compounded) mean:
– Exact (discrete) geometric mean over n periods: R_geo = (Π_{i=1}^n (1 + R_i))^{1/n} − 1.
– Using the log approximation, the annualized geometric mean ≈ exp(μ − 1/2·σ^2) − 1. For small μ and σ this further approximates to μ − 1/2·σ^2 (the volatility drag).
Worked numeric example (step‑by‑step)
– Inputs: arithmetic mean μ = 8% = 0.08, standard deviation σ = 15% = 0.15.
– Compute variance: σ^2 = 0.15^2 = 0.0225.
– Volatility drag (½·σ^2) = 0.5 × 0.0225 = 0.01125 =
= 0.01125 = 1.125%.
– Compute μ − ½·σ^2: 0.08 − 0.01125 = 0.06875 = 6.875%.
– Annualized geometric mean (log‑exponential approximation): exp(0.06875) − 1 ≈ 0.07117 = 7.117% (rounded 7.12%).
– Quick comparison:
– Arithmetic mean: 8.00%
– Geometric (log‑exponential): ≈ 7.12%
– Linear approx (μ − ½σ^