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 nearly six years of daily prices for SPY, NVDA, GS, XOM and GLD (2020 to 2025, 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

**Table 1.** Adding returns against multiplying growth factors
Case	Adding (wrong)	Multiplying	Result
+10% then −5%	5%	$1.10 \times 0.95 - 1$	$\key{4.5\%}$
+50% then −50%	0%	$1.50 \times 0.50 - 1$	$\bad{-25\%}$
+30% then −20%	10%	$1.30 \times 0.80 - 1$	4%
1% a month for a year	12%	$1.01^{12} - 1$	12.68%
0.04% a day for 252 days	10.08%	$1.0004^{252} - 1$	10.60%

The second row of Table 1 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: $0.5 \times 1.5$ is the same as $1.5 \times 0.5$ , as long as no money goes in or out between the two.

On real data. Over the 1,466 trading days of 2020 to 2025, adding SPY’s daily returns gives +95.4%. Multiplying the growth factors gives $\key{+128.4\%}$ , which is what the account actually did. For NVDA the two answers are +435.6% and +3291.0%. The gap is not an error term; it is the compounding of gains on gains.

Figure 1. Same average, different ending; sum against product; the gain a loss needs

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

**Table 2.** CAGR by the root against total divided by years
Case	Wrong: total divided by years	Right: the root	Result
$100 to $121 in 2 years	10.5%	$\sqrt{1.21} - 1$	$\key{10\%}$ a year
$100 to $200 in 10 years	10%	$2^{1/10} - 1$	7.18% a year
SPY, +128.4% in 5.83 years	22.0%	$2.284^{1/5.83} - 1$	15.2% a year
NVDA, +3291.0% in 5.83 years	564.6%	$33.91^{1/5.83} - 1$	83.0% a year

In Table 2, dividing the total by the number of years overstates the rate, and the overstatement grows with the total: NVDA’s +3291% divided by 5.83 years is 565% a year, almost seven times the real 83.0%.

Arithmetic mean, geometric mean, and the drag between them

Take SPY’s six yearly returns, counting the ten months of 2025 as a year: +17%, +29%, −18%, +26%, +25%, +17%. Their plain average, the arithmetic mean, is 16.0%. The rate that actually compounds, the geometric mean, is 14.8% (the CAGR above is 15.2%; the two differ only because the sixth year is short). 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. The variance, $\risk{\sigma^2}$ , is the square of the standard deviation, a measure of how far returns swing around their average; the mean and standard deviation post explains both. Table 3 checks the rule on each ticker’s six yearly returns, 2020 to 2025, each a percentage change in price over the calendar year, with the gap between the two means next to the half-variance that predicts it:

**Table 3.** Arithmetic and geometric means of yearly returns by ticker, 2020 to 2025
Ticker	Year by year (%)	Arithmetic mean	Geometric mean	Gap	$\sigma^2/2$
SPY	+17, +29, −18, +26, +25, +17	16.0%	$\key{14.8\%}$	1.3%	1.3%
NVDA	+118, +125, −50, +239, +171, +51	109.1%	79.9%	29.1%	41.6%
GS	+15, +48, −8, +16, +52, +40	27.2%	25.3%	1.9%	2.3%
XOM	−37, +58, +87, −6, +11, +9	20.3%	$\key{13.3\%}$	7.1%	8.4%
GLD	+24, −4, −1, +13, +27, +52	18.4%	16.9%	1.5%	1.8%

XOM is the lesson. Its arithmetic mean, 20.3%, is four points above SPY’s 16.0%; its geometric mean, 13.3%, is the lowest of the five, a point and a half below SPY’s. The swings between −37% and +87% cost it seven points a year, so an average return above the index’s turned into slower growth than the index. 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 NVDA and XOM) and close 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

**Table 4.** Gain needed to recover a loss
Loss	Gain needed
10%	11.1%
20%	25.0%
30%	42.9%
33.7%, SPY’s worst fall of 2020 to 2025	$\key{50.9\%}$
50%	100%
80%	400%

The fall from a peak to a later low is a drawdown, and Table 4 is why drawdown gets its own line in fund reports: a strategy that loses 50% once needs to double just to be where it was. The third panel of Figure 1 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. Table 5 has the two rules side by side, with the constants worth memorising:

**Table 5.** Annualising returns, standard deviation and Sharpe ratio
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 (return per unit of risk, its own post)	$\times \sqrt{252}$	$\times \sqrt{12}$	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 Table 6 shows the rule 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.

**Table 6.** Years to double by the rule of 72 against the exact answer
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 (selling borrowed shares, to profit if the price falls) has a negative weight, so a falling price gives a positive contribution: short 20% of the portfolio 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, but holding positions worth twice the money, which is leverage (holding more than you own, with borrowed money or borrowed shares).

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

Table 7 is the list of mistakes this post is about, with the fix beside each.

**Table 7.** Traps with returns and what is true instead
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 $\times 252$	$\times \sqrt{252}$
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 $\times$ 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

The SPY numbers above. The script reads the blog’s price file from the examples repo, so it runs as pasted (pip install pandas). prices holds the five tickers’ daily closes, one column each; r the daily returns; spy SPY’s column of them.

import pandas as pd

URL = "https://raw.githubusercontent.com/ankurtrapasiya/examples/main/data/prices.csv"
prices = pd.read_csv(URL, index_col="date", parse_dates=True)   # adjusted closes, 5 tickers
r = prices / prices.shift(1) - 1                 # daily returns: today / yesterday - 1
spy = r["SPY"].dropna()

total = (1 + spy).prod() - 1                     # multiply growth factors
wrong = spy.sum()                                # adding, which no account does
years = (spy.index[-1] - spy.index[0]).days / 365.25
cagr = (1 + total) ** (1 / years) - 1            # the steady yearly rate
print(f"total {total:.1%}, added {wrong:.1%}, CAGR {cagr:.1%} a year")   # 128.4%, 95.4%, 15.2%

px = prices["SPY"]
year_end = px.resample("YE").last()                          # the last close of each year
start = year_end.shift(1, fill_value=px.iloc[0])             # each year starts from last year's close
yearly = year_end / start - 1                                # 2020 starts from the first close of 2020
print((yearly * 100).round(0).tolist())                      # 17, 29, -18, 26, 25, 17
arith = yearly.mean()
geo = (1 + yearly).prod() ** (1 / len(yearly)) - 1   # the rate that compounds
drag = yearly.var(ddof=0) / 2                    # half the variance of the yearly returns
print(f"arithmetic {arith:.1%}, geometric {geo:.1%}, half the variance {drag:.1%}")

fall = -(px / px.cummax() - 1).min()             # the worst fall from a previous peak
recover = lambda loss: 1 / (1 - loss) - 1        # a small one-line function
print(f"worst fall {fall:.1%}, needs {recover(fall):.1%} to recover")   # 33.7%, 50.9%

It prints the total (128.4%), the added-up figure (95.4%) and the CAGR (15.2%); then the six yearly returns; then the arithmetic and geometric means and half the variance (16.0%, 14.8%, 1.3%); then the worst fall and the gain it needs (33.7%, 50.9%). Two lines need a word:

resample("YE").last() keeps the last close of each calendar year. Each year’s return then starts from the previous year’s close, and 2020 from the first close in the file; without that start, 2020 would be dropped. The last “year” is the ten months of 2025 the file holds.
px.cummax() is the highest close so far on each day, so px / px.cummax() - 1 is how far below its peak the price sits; the most negative value is the worst fall.

Summary

Table 8 collects the formulas.

**Table 8.** Formulas worth remembering
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 $\approx 72 / r\%$	6% doubles in 12 years
Portfolio return	$\sum w_i \good{r_i}$	the one place addition is right

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 (Wiley, 2009), chapter 6, “Money and Risk Management”: long-run growth as mean return minus half the variance, $g = m - s^2/2$