Calculate holding period, average, annualized, real and currency-adjusted returns.
HPR = (ending price minus beginning price + income) / beginning price. Buy at $50, receive a $1 dividend, sell at $54: HPR = (54 minus 50 + 1) / 50 = 10%. Total return includes income; price return does not. Over long periods reinvested dividends account for a large share of equity wealth, so compare funds and indexes on total return.
The arithmetic mean is the simple average of periodic returns and is the best estimate of the next single period. The geometric mean is the constant rate that produces the actual ending wealth:
A portfolio that gains 50% and then loses 50% has an arithmetic mean of 0% but ends at 75 cents on the dollar: geometric mean = (1.5 x 0.5)^(1/2) minus 1 = minus 13.4%. The geometric mean is always at or below the arithmetic mean, and the gap widens with volatility. Approximation: geometric mean = arithmetic mean minus half the variance.
Annual return = (1 + periodic return)^(periods per year) minus 1. A 1% monthly return is 12.68% a year, not 12%. Do not annualize periods shorter than a year when reporting performance.
(1 + real) = (1 + nominal) / (1 + inflation). A 7% nominal return with 3% inflation is a 3.88% real return.
Return in home currency = (1 + foreign asset return) x (1 + change in the foreign currency) minus 1. A Canadian holds a US stock that rises 10% while the US dollar rises 5% against the Canadian dollar: return = 1.10 x 1.05 minus 1 = 15.5%. Had the US dollar fallen 5%, the return would be 4.5%.
What the investor keeps is the return after fees, trading costs and taxes. A 2% annual fee on a 7% gross return takes more than a quarter of the return each year.
Over more than a century, global equities have earned a real return of roughly 5% a year and government bonds roughly 1% to 2%, with long stretches far above and below those averages (approximate figures from long-run studies such as the UBS Global Investment Returns Yearbook).
The geometric mean over several years is often reported as the CAGR:
$10,000 that grows to $16,105 in five years has a CAGR of (1.6105)^(0.2) minus 1 = 10.0%. Cumulative return (61%) and annual return (10%) are easy to confuse in marketing material; check which one is quoted.
An investor earns 5% interest in a taxable account, taxed at 40%, while inflation is 2.5%.
The investor's purchasing power barely grew. Tax applies to the nominal return, including the part that only compensates for inflation, which is why interest-paying assets in taxable accounts are so tax-inefficient (Module 9 on asset location).
The approximation 'geometric mean = arithmetic mean minus half the variance' quantifies how volatility erodes compound growth. An asset with an arithmetic mean of 10% and volatility of 20% compounds at about 10% minus 0.5 x 0.04 = 8%. Two assets with the same average return but different volatility end with very different wealth.
A fund gains 25% in year one and loses 20% in year two. Its arithmetic average is +2.5% a year, but 1.25 x 0.80 = 1.00: the investor is exactly where she started. Losses need larger percentage gains to recover, a point Manage Your Risk covers in depth.
For describing what an investment actually did, use the geometric mean. For a one-period forecast of expected return, the arithmetic mean is the better estimate. For projecting wealth over many years, use the geometric mean, or the arithmetic mean with explicit volatility in a simulation (RS 205, Module 10).
A nominal CAGR of 7% with inflation of 2.5% is a real CAGR of 1.07 / 1.025 minus 1 = 4.39%. Long-run planning should be done in real terms, because expenses in retirement will be in future dollars.
Download ten years of annual total returns for a broad stock index. Compute the arithmetic mean, the geometric mean, the growth of $10,000 and the real geometric return using CPI.
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