Returns compound: multiply, never add

Almost every mistake with returns comes from adding when the arithmetic calls for multiplying. This post is the multiplication, worked on five years of daily prices for SPY, AAPL, JPM, XOM and GLD (2019 to 2023, adjusted for dividends and splits), the same data the rest of this series uses.

a return, the observed valuethe quantity being computedrisk: the spread of returnsa loss

A return is a growth factor

A move from 100 to 110 is a return of +10%, written $0.10$ . The growth factor is $1 + 0.10 = 1.10$ : what one dollar became. Returns chain by multiplying growth factors, because each period starts from the last period’s ending value, not from the original one.

\key{R_{\text{total}}} = (1 + \good{r_1})(1 + \good{r_2})\cdots(1 + \good{r_n}) - 1

Case	Adding (wrong)	Multiplying	Result
+10% then −5%	5%	1.10 × 0.95 − 1	$\key{4.5\%}$
+50% then −50%	0%	1.50 × 0.50 − 1	$\bad{-25\%}$
+30% then −20%	10%	1.30 × 0.80 − 1	4%
1% a month for a year	12%	1.01¹² − 1	12.68%
0.04% a day for 252 days	10.08%	1.0004²⁵² − 1	10.60%

The second row is the one to remember. Up 50% takes 100 to 150; down 50% takes 150 to 75. The average of the two returns is zero and the account is down a quarter. The order does not matter: 0.5 × 1.5 is the same as 1.5 × 0.5, as long as no money goes in or out between the two.

On real data. Over the 1,257 trading days of 2019 to 2023, adding SPY’s daily returns gives +83.4%. Multiplying the growth factors gives $\key{+106.2\%}$ , which is what the account actually did. For AAPL the two answers are +188.6% and +408.1%. The gap is not an error term; it is the compounding of gains on gains.

One rate for many years: CAGR

The compound annual growth rate is the one steady yearly rate that gets you from the start value to the end value. Divide end by start, take the root for the number of years, subtract one.

\key{\text{CAGR}} = \left(\frac{V_{\text{end}}}{V_{\text{start}}}\right)^{1/\text{years}} - 1

Case	Wrong: total ÷ years	Right: the root	Result
$100 to $121 in 2 years	10.5%	√1.21 − 1	$\key{10\%}$ a year
$100 to $200 in 10 years	10%	2^(1/10) − 1	7.18% a year
SPY, +106.2% in 5 years	21.3%	2.062^(1/5) − 1	15.6% a year
AAPL, +408.1% in 5 years	81.8%	5.081^(1/5) − 1	38.5% a year

Dividing the total by the number of years overstates the rate, and the overstatement grows with the total: a fifth of AAPL’s +408% is 82% a year, more than double the real 38.5%.

Arithmetic mean, geometric mean, and the drag between them

Take SPY’s five yearly returns: +31%, +18%, +29%, −18%, +26%. Their plain average, the arithmetic mean, is 17.2%. The rate that actually compounds, the geometric mean, is the CAGR, 15.6%. The geometric mean is never above the arithmetic mean; they are equal only when every return is identical, and the gap grows with how much the returns swing. The gap has a name, volatility drag, and a close approximation:

\key{g} \approx \bar{r} - \frac{\risk{\sigma^2}}{2}

Long-run growth is about the average return minus half the variance. With the five yearly returns:

Ticker	Yearly returns	Arithmetic mean	Geometric mean	Gap	σ²/2
SPY	+31, +18, +29, −18, +26	17.2%	$\key{15.6\%}$	1.7%	1.7%
AAPL	+89, +82, +35, −26, +49	45.7%	38.4%	7.2%	8.5%
JPM	+45, −6, +28, −13, +31	17.0%	14.8%	2.2%	2.5%
XOM	+5, −36, +58, +87, −6	21.5%	$\key{13.1\%}$	8.4%	10.0%
GLD	+18, +25, −4, −1, +13	10.1%	9.5%	0.6%	0.6%

XOM is the lesson. Its arithmetic mean, 21.5%, is the highest of the five after AAPL; its geometric mean, 13.1%, is the lowest after gold. The swings between −36% and +87% cost it eight points a year. Two assets with the same average return and different volatility do not grow at the same rate, and the volatile one grows slower. The approximation is rough at the extremes (it overshoots for AAPL and XOM) and exact for the calmer ones; it is a guide, not a formula to report.

Recovery: a loss needs a bigger gain to undo

Lose 50% and you need +100% to get back, because the gain is earned on the smaller base.

\key{\text{gain to recover}} = \frac{1}{1 - \bad{L}} - 1

Loss	Gain needed
10%	11.1%
20%	25.0%
30%	42.9%
33.7%, SPY’s worst fall in these five years	$\key{50.9\%}$
50%	100%
80%	400%

The fall from a peak to a later low is a drawdown, and the table is why drawdown gets its own line in every fund report: a strategy that loses 50% once needs to double just to be where it was. The third panel of the chart draws this curve; the loss line is straight and the recovery line bends away from it.

Daily to yearly: returns use n, risk uses the square root of n

A daily return compounds over 252 trading days; a daily standard deviation grows only with the square root of 252. The mean and standard deviation post explains why. The two rules side by side, with the constants worth memorising:

Quantity	Daily to yearly	Monthly to yearly	Constant
Return	$(1 + r)^{252} - 1$ (or $\times 252$ for a rough mean)	$(1 + r)^{12} - 1$
Standard deviation	$\times \sqrt{252}$	$\times \sqrt{12}$	$\sqrt{252} \approx 15.87$ , $\sqrt{12} \approx 3.46$
Sharpe ratio	$\times \sqrt{252}$	$\times \sqrt{12}$	same as the standard deviation

Worked: a daily standard deviation of 1.2% is $1.2 \times 15.87 = 19.0\%$ a year, not $1.2 \times 252 = 302\%$ . A monthly standard deviation of 3% is $3 \times 3.46 = 10.4\%$ a year. Why 252: about 260 weekdays in a year, minus eight or so market holidays.

Rule of 72

Money at rate $r$ doubles in about $72 / r$ years, with $r$ in percent. At 6%, twelve years; at 9%, eight. The exact answer is $\ln 2 / \ln(1 + r)$ , and the rule is within a few months of it for rates between 3% and 12%. It also runs backwards: money that doubled in fifteen years grew at about 72 / 15 = 4.8% a year.

Rate	Rule of 72	Exact
3%	24.0 years	23.4
6%	12.0	11.9
9%	8.0	8.0
12%	6.0	6.1

Portfolios and shorts: the one place addition is right

Over a single period, a portfolio’s return is the weighted average of its holdings’ returns. Weights multiply returns, and the products add:

\key{r_p} = \sum_i w_i\,\good{r_i}

Sixty percent in an asset that made +10% and forty percent in one that made −5%: $0.6 \times 10 + 0.4 \times (-5) = 6 - 2 = 4\%$ . This is addition across assets at one moment, not across time; the next period still compounds on the new base.

A short position has a negative weight, so a falling price gives a positive contribution: short 20% of the book in a stock that fell 10% adds $-0.2 \times -0.10 = +2\%$ . Net exposure is the sum of the weights; gross exposure is the sum of their absolute values. Weights of 1.20, 0.30 and −0.50 are 100% net and 200% gross: fully invested on paper and levered twice over in fact.

Relative performance is a subtraction too: down 6% when the benchmark is down 10% is four points ahead. A negative return can beat its benchmark.

Traps

Trap	Truth
Add returns across time	Multiply growth factors
CAGR is the total divided by the years	Take the root
Annualise a standard deviation by × 252	× √252
A portfolio’s return is the average of its assets	Only with equal weights; otherwise the weighted average
A short in a falling stock hurts	Negative weight × negative return is positive
Weights that sum to 1 mean no leverage	Check gross exposure
A −6% year cannot beat the benchmark	It can, if the benchmark did worse
Same average return means same growth	The more volatile one grows slower

The code

import numpy as np
import pandas as pd

prices = pd.read_csv("prices.csv", index_col="date", parse_dates=True)   # adjusted closes
r = prices / prices.shift(1) - 1                                         # daily returns
spy = r["SPY"].dropna()

total = (1 + spy).prod() - 1                 # +106.2%: multiply growth factors
wrong = spy.sum()                            # +83.4%: adding, which no account does
years = (spy.index[-1] - spy.index[0]).days / 365.25
cagr = (1 + total) ** (1 / years) - 1        # 15.6% a year

yearly = prices["SPY"].resample("YE").last().pct_change().dropna()
arith = yearly.mean()                        # 17.2%
geo = (1 + yearly).prod() ** (1 / len(yearly)) - 1   # 15.6%, the rate that compounds
drag = yearly.var(ddof=0) / 2                # 1.7%, about the gap between them

recover = lambda loss: 1 / (1 - loss) - 1    # 0.337 -> 0.509

Summary

Idea	Formula	Remember it as
Growth factor	$1 + \good{r}$	what a dollar became
Total return	$\prod (1 + \good{r_t}) - 1$	multiply, never add
CAGR	$(V_{\text{end}}/V_{\text{start}})^{1/T} - 1$	the root, not the division
Volatility drag	$\key{g} \approx \bar{r} - \risk{\sigma^2}/2$	swings cost growth
Recovery	$1/(1 - \bad{L}) - 1$	−50% needs +100%
Annualise a return	$(1 + r)^n - 1$	$n$ = 252 or 12
Annualise risk	$\times \sqrt{n}$	15.87 or 3.46
Rule of 72	years to double ≈ $72 / r\%$	6% doubles in 12 years
Portfolio return	$\sum w_i \good{r_i}$	the one place addition is right

Mean and standard deviation: the average return and the typical swing around it, the two numbers every ratio in this series is built from.

References

Philip J. Romero and Tucker Balch, What Hedge Funds Really Do: An Introduction to Portfolio Management (Business Expert Press, 2014), chapter 2: returns, compounding, CAGR and the rule of 72
Ernest P. Chan, Quantitative Trading: How to Build Your Own Algorithmic Trading Business (2nd ed., Wiley, 2021), chapter 6: long-run growth as mean return minus half the variance