Compute simple and continuously compounded returns, distinguish them and convert from one to the other.

FRM® Part I · Quantitative Analysis · Measuring Returns, Volatility, and Correlation · FRM1.QA.12.a

The idea

A simple return is the percentage change in value over one period, including any income D. Simple return = (P1 + D − P0) ÷ P0. The continuously compounded (log) return is the natural log of the gross return. Log return = ln((P1 + D) ÷ P0). Writing R for the simple return and r for the log return, r = ln(1 + R). To go back, use Simple return = e^r − 1. For small moves the two are close, but a log return is below the simple return when positive, and the gap widens as the move grows. The practical difference is aggregation. Log returns add across periods, so a two-year log return is the sum of the two annual figures, whereas simple returns must be compounded. Two-year return = (1 + R1) × (1 + R2) − 1. Never add simple returns across time as though they were log returns. Models mostly use log returns for this additivity, but convert back to a simple return when reporting the actual gain or loss on a position.

Formulas

Continuously compounded return from prices

r = ln((P₁ + D) ÷ P₀)

P₀: the price at the start
P₁: the price at the end
D: any income paid during the period

Log returns add across periods. Income belongs in the numerator; leaving it out understates the return.

Simple return from a log return

R = eʳ − 1

r: the continuously compounded return

The inverse of r = ln(1 + R). Convert back to a simple return before reporting the actual gain or loss on a position.

Simple return over two periods

two-period return = (1 + R₁) × (1 + R₂) − 1

R₁, R₂: the simple returns of the two periods

Simple returns compound; only log returns may be added across time.

A practice question

From the question bank, with the reason each wrong option is wrong.

A bond fund earned a simple return of 8.00% over the year. Its continuously compounded return is closest to:

7.70% - correct
8.33%Why not: Applies e^R - 1, which converts a log return to a simple one, in the wrong direction.
8.00%Why not: Treats the simple and continuously compounded returns as equal.
7.41%Why not: Uses R / (1 + R) instead of ln(1 + R).

r = ln(1 + R) = ln(1.08) = 0.0770, or 7.70%. The log return is lower than the simple return for a positive move.

A harder one

A stock bought at USD 50 is worth USD 56 a year later and paid a USD 1 dividend during the year. The continuously compounded return for the year is closest to:

14.00%Why not: This is the simple total return, (56 + 1 - 50) / 50, not the log return.
11.33%Why not: Takes ln(56 / 50) and leaves out the dividend.
12.00%Why not: Simple price-only return: ignores both the dividend and the log conversion.
13.10% - correct

Total gross return = (56 + 1) / 50 = 1.14, so r = ln(1.14) = 0.1310 or 13.10%. Income must be included in the numerator.

The hardest tier

A fund's price rises 60% in year one and falls 40% in year two, both simple returns. Its continuously compounded return over the two years is closest to:

-4.08% - correct
-4.00%Why not: This is the two-year simple return, 1.6 x 0.6 - 1, not the log return.
10.00%Why not: Averages the two simple returns, (60% - 40%) / 2, and ignores compounding.
-2.04%Why not: Gives the average per-year log return rather than the two-year total.

ln(1.6) = 0.4700 and ln(0.6) = -0.5108, which sum to -0.0408, or -4.08%. Equivalent check: ln(0.96) = -0.0408.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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