Types of Financial Returns

A single investment can be described by many different return figures: a holding-period return, an average, an annualized rate, a log return, a figure before or after fees and taxes, and a return on borrowed money. Each answers a different question. This module shows how to calculate each type, how they relate to one another, and which one to use when comparing investments or reporting results.

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What you will learn

1. Calculate holding-period returns, including income, and link returns from consecutive periods into a cumulative return.

HPR = (ending price - beginning price + income) / beginning price.
Link consecutive returns by multiplying growth factors (1 + R) and subtracting 1.
Ending value = beginning value x product of growth factors; the order of returns does not change the result.
Adding period returns overstates gains and understates losses; averaging gives a per-period figure, not a cumulative one.
Linking assumes income is reinvested and no external cash flows occur.

2. Calculate and compare arithmetic, geometric and harmonic means of returns or prices, including trimmed and winsorized means, and choose the mean that fits the purpose.

Arithmetic mean: average single-period return; standard estimate of next period's expected return.
Geometric mean: compound growth rate that reproduces ending wealth; always <= arithmetic mean.
Harmonic mean: average cost per unit under equal-amount (cost-averaging) purchases.
For positive, unequal values: harmonic < geometric < arithmetic.
Trimmed mean removes extremes; winsorized mean replaces them with the nearest remaining values.

3. Annualize returns measured over different horizons, and calculate and interpret continuously compounded returns.

Annualized return = (1 + HPR)^c - 1, where c = number of such periods in a year.
Continuously compounded return r = ln(1 + HPR) = ln(P1 / P0); convert back with HPR = e^r - 1.
Log returns are lower than HPRs for gains and more negative for losses.
Log returns add across periods; ordinary HPRs must be multiplied.
Additivity makes log returns the standard input for statistical and simulation models.

4. Distinguish gross from net, pre-tax from after-tax, and nominal from real returns, and calculate the return on a leveraged position.

Gross return: before management and administrative fees, after trading costs; net return: after those fees.
After-tax returns depend on the type of income, tax rates and the timing of realized gains.
Real return = (1 + nominal) / (1 + inflation) - 1.
Leveraged return R_L = R_p + (V_B / V_E)(R_p - r_D).
Leverage magnifies gains when R_p > r_D and magnifies losses when R_p < r_D.

Module at a glance

Link consecutive returns by multiplying (1 + R) factors; never add them.
Arithmetic mean for next-period expectations, geometric mean for compound growth, harmonic mean for average cost.
Trimmed means drop extreme observations; winsorized means replace them.
Annualize with (1 + HPR)^c - 1; c is the number of periods in a year.
Log return r = ln(1 + HPR); log returns add over time, and HPR = e^r - 1.
Compare gross with gross and net with net; taxes and inflation change what investors keep.
Leveraged return = R_p + (V_B / V_E)(R_p - r_D): leverage magnifies gains and losses.

This free module has 4 lessons, 13 practice questions with full explanations.

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