M.9.3 Multiplicative Growth

Basic Prerequisites: 02 · Ensemble vs Time Averages and basic calculus (Taylor expansion, Jensen).

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1. Intuition & Practical Objective

Page 02 said: when wealth multiplies, ensemble and time averages split, and the split is the Jensen gap. This page computes that gap and shows it is not abstract - it is a real, permanent tax on every volatile portfolio, called the volatility drag (or variance drain, or geometric-mean discount).

The practical objective: stop adding returns and start adding log returns. A portfolio’s compounded growth rate is the average of its log returns, and log returns have a built-in penalty for volatility. Two assets with the same arithmetic mean return can have very different compounded outcomes if their volatilities differ. The quieter one wins over time.

Concretely: a stock with arithmetic mean return $\mu$ and volatility $\sigma$ compounds at approximately $\mu -\frac{1}{2}{\sigma }^2$. At $\sigma =15\%$ that is a drag of $1.125\%$/year - free money you lose simply for having wiggles. At $\sigma =50\%$ the drag is $12.5\%$/year, and a strategy that “averages $+8\%$” can compound negative.

Takeaway. Compounded wealth lives in log space, where returns are additive and ergodic. Volatility is a cost, not a free-spending lottery ticket: it lowers the typical (geometric) outcome below the arithmetic mean by $\sim \frac{1}{2}{\sigma }^2$.

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2. Mathematical Ground Truth & Derivations

2.1 From simple to log returns

For a one-period simple return $R_t$, the log return is $r_t=\ln (1+R_t)$. A multi-period product becomes a sum:

$$1+R_t[k]=\prod _{j=0}^{k-1}(1+R_{t-j})⟹r_t[k]=\sum _{j=0}^{k-1}{r}_{t-j}.$$

So log returns are additive across time, restoring the ergodic structure of page 02. The geometric mean return is $\exp (\overline{r})-1$, where $\overline{r}=\frac{1}{k}\sum r_{t-j}$ is the time average of log returns - the $g$ of page 01.

2.2 The volatility drag (Jensen gap)

Let $R$ have mean $\mu$ and variance ${\sigma }^2$, small enough that $|R|<1$. Expand

$$\ln (1+R)=R-\frac{1}{2}{R}^2+\frac{1}{3}{R}^3-\cdots ⟹\mathbb{E}[\ln (1+R)]=\mu -\frac{1}{2}({\mu }^2+{\sigma }^2)+\cdots$$

Collecting terms, to second order,

$$\boxed{g=\mathbb{E}[\ln (1+R)]\approx \ln (1+\mu )-\frac{{\sigma }^2}{2(1+\mu {)}^2}\approx \mu -\frac{{\sigma }^2}{2}}$$

The ensemble growth rate is $\ln \mathbb{E}[1+R]=\ln (1+\mu )$, so the drag separating typical from ensemble growth is exactly

$$\ln (1+\mu )-g\approx \frac{{\sigma }^2}{2(1+\mu {)}^2}.$$

This is the Jensen gap of page 02, now quantified: volatility, not direction, determines how far the typical path lags the average.

2.3 The continuous limit: geometric Brownian motion

For $dS=\mu Sdt+\sigma SdW$, Itô gives $S_t=S_0\exp ((\mu -\frac{1}{2}{\sigma }^2)t+\sigma W_t)$, so

$$\mathbb{E}[S_t]=S_0{e}^{\mu t}\text{(ensemble)},\mathrm{med}(S_t)=S_0{e}^{(\mu -\frac{1}{2}{\sigma }^2)t}\text{(typical)},$$

and the time-average growth rate is $g_{\infty }=\mu -\frac{1}{2}{\sigma }^2$. With a risky fraction $f$ and riskless rate $r$, the continuous growth rate is the Kelly growth function

$$g_{\infty }(f)=r+f(m-r)-\frac{1}{2}{s}^2{f}^2.$$

This is a downward parabola in $f$, concave, with a unique maximum - the object maximised on page 04. The quadratic penalty $-\frac{1}{2}{s}^2{f}^2$ is the volatility drag made lever-aware.

2.4 The S&P number

Thorp’s continuous approximation with historical inputs $m=11\%$, $s=15\%$, $r=6\%$ gives the risk-adjusted drag explicitly; the ensemble-per-annum object is $e^m=27.11\times$ over 30 years while the typical (median) outcome is $e^{m-\frac{1}{2}{s}^2}=19.35\times$. A factor of $\sim 1.4$ in terminal wealth, from volatility alone.

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3. Computational Implementation - the drag, measured

Stdlib only. We price the same GBM two ways: the ensemble mean and the typical (geometric) path, and confirm the drag is $\frac{1}{2}{\sigma }^2$.

The ensemble mean ($27.11\times$) and the geometric mean / median ($19.35\times$) differ by exactly the accumulated drag $e^{\frac{1}{2}{\sigma }^{2}T}=e^{0.3375}=1.40$. The typical investor captures $19.3\times$, not $27.1\times$ - and the $+11\%$ “expected return” is really a $+9.88\%$ compounding rate.

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4. Failure Modes & First-Principles Breakdowns

1. Averaging simple returns across periods. The arithmetic mean of $+50\%$ and $-40\%$ is $+5\%$; the compounded result is $-10\%$. Cross-period averages must be taken on log returns.
2. Ignoring the drag when levering. The drag grows like $f^2{\sigma }^2$: doubling exposure quadruples the variance penalty. This is why leverage can turn a positive edge into a negative growth rate long before it causes a margin call (page 04).
3. Trusting the second-order approximation in the tail. $\mathbb{E}[\ln (1+R)]\approx \mu -\frac{1}{2}{\sigma }^2$ fails for large moves; real returns are fat-tailed (excess kurtosis, Tsay Ch. 1) so the true drag is larger and jumps make the log expansion invalid. Use the exact $\mathbb{E}[\ln (1+R)]$ where the distribution is known.
4. Confusing “expected return” with “compounding rate” in a backtest. Reporting $\mathbb{E}[R]$ overstates the achievable CAGR by $\sim \frac{1}{2}{\sigma }^2$; funds that quote arithmetic means systematically overstate what an account would have compounded.

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5. References

• Tsay, Ruey S.: Analysis of Financial Time Series (3rd ed.)
• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• Shreve, Steven E.: Stochastic Calculus for Finance I
• Peters, Ole: The Ergodicity Problem in Economics, Nature Physics 15 (2019)

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6. Connected Graph Bridges

• Back: 02 · Ensemble vs Time Averages
• Forward: 04 · Kelly Criterion · Index Hub
• Theory: Itô Integral & Doeblin (the $\frac{1}{2}{\sigma }^2$ correction) · Econometrics & Time Series (log returns, fat tails)
• Applications: Mean–Variance Optimization · Extreme Value Theory & Fat Tails