Randomness Is a Tax – Until It Isn't

When is randomness an asset vs a tax? | Sidhant Bansal

Click Activate above to run the code cells in this post. This post is a companion to the Sharpe post and is meant to dig deeper into scenarios where randomness helps vs hurts. Let’s formulate a lot of real-world scenarios as: You get a random variable X, which you might have some control over, say its mean or distribution. Life then passes it through some function f, and the outcome you observe is f(X). A simplistic worldview, but one that captures a surprising number of situations. Take 3 different scenarios Case 1 $f$ is linear. Say you flip a fair coin for \$1, or choose not to play at all. If your utility is just money itself, i.e. $f(X) = X$, then: \[E[X] = 0\] in both cases. Nothing interesting happens. Case 2 Now, let’s say you have a \$10k a non-trivial amount, and two investment options: (i) gives guaranteed 10% return year-on-year, (ii) gives 50% return with 0.5 probability and -30% return with 0.5 probability. Which one would you prefer if you had to stay invested long-term. If investing just for one year, then the expected returns match-up, since first option gets you to \$11k with certainty, while the second option gets you to \$15k with 0.5 probability and \$7k with 0.5 probability, so the expected value is also \$11k. But we already said, you are in it for the long-game. Think through this exercise, before reading up. Open a calculator, do a simulation in your head if needed, and see which one you would prefer. Suppose your wealth after $T$ years is: \[W_T = W_0 \prod_{t=1}^{T}(1+r_t)\] The important thing to notice is that returns don’t add, they multiply. So instead of looking at $E[r]$, let us take logs: \[\log W_T = \log W_0 + \sum_{t=1}^{T}\log(1+r_t)\] So the long-run growth rate is governed by: \[E[\log(1+r)]\] For option (i): \[\log(1.1) \approx 0.0953\] For option (ii): \[\frac{1}{2}\log(1.5) + \frac{1}{2}\log(0.7) \approx 0.0244\] So even though both have the same arithmetic expected return of 10%, their long-term compound growth rates are very different. More generally, for small returns: \[\log(1+r) \approx r - \frac{r^2}{2}\] Taking expectations: \[E[\log(1+r)] \approx E[r] - \frac{1}{2}E[r^2]\] If $\mu = E[r]$ and $\sigma^2 = Var(r)$, then: \[E[r^2] = \mu^2 + \sigma^2\] So: \[E[\log(1+r)] \approx \mu - \frac{1}{2}\mu^2 - \frac{1}{2}\sigma^2\] Ignoring the small $\mu^2$ term: \[E[\log(1+r)] \approx \mu - \frac{1}{2}\sigma^2\] This is the volatility drag. The variance quietly taxes you. Here, the relevant function was $f(x) = \log{x}$ and the randomness turned out to be a tax. Case 3 Suppose there is a call option on a stock. Say that stock will end up at \$80 with 0.5 probability or \$120 with 0.5 probability tomorrow. Now what would be the payoff of a call option with strike price \$100 (in effect at the “fair” price of the stock right now)? Crude intuition could be, that it is worthless, why will I pay to buy a option at a price equal to the expected price of the stock tomorrow? But, compare the expected payoff if you bought the option: you make \$20 with 0.5 probability and 0 with 0.5 probability, so the expected payoff is \$10. So the option is worth \$10. If you see it for cheaper in the market, then you should buy it. Now also notice that if the stock was even more volatile, say 70/130 or 60/140, then the expected payoff of the option would be even higher, so the more volatile the stock, the more valuable the option is. This is why being long options means you are long volatility. Here, the relevant function was: $f(S) = \max(S-100,0)$, where $S$ is the stock price at expiry, and the randomness turned out to be an asset. The pattern At this point we have seen randomness: does nothing hurts us helps us What changed? In Case 1, $f$ was linear. $f(x) = x$ There is no curvature. Randomness neither helps nor hurts. Only the mean matters. In Case 2, $f$ was concave. $f(x) = \log x$ Bad outcomes hurt more than good outcomes help. Randomness becomes a tax. In Case 3, $f$ was convex. $f(S) = \max(S-100,0)$ Bad outcomes are capped at zero, while good outcomes keep going. Randomness becomes an asset. This phenomenon is captured by Jensen’s inequality. For convex functions: $E[f(X)] \geq f(E[X])$ For concave functions: $E[f(X)] \leq f(E[X])$ Jensen-demo import numpy as np import matplotlib.pyplot as plt

# Same random variable throughout # Think of X as tomorrow's stock price x_vals = np.array([80, 120]) probs = np.array([0.5, 0.5])

EX = np.sum(probs * x_vals)

functions = [ ("Linear: $f(x)=x$", lambda x: x), ("Convex: call option $f(x)=\\max(x-100,0)$", lambda x: np.maximum(x - 100, 0)), ("Concave: $f(x)=\\log(x)$", lambda x: np.log(x)),

x_grid = np.linspace(60, 140, 400)

fig, axes = plt.subplots(1, 3, figsize=(15, 4))

for ax, (title, f) in zip(axes, functions): y_grid = f(x_grid) y_vals = f(x_vals)

EfX = np.sum(probs * y_vals) fEX = f(EX) gap = EfX - fEX

ax.plot(x_grid, y_grid) ax.scatter(x_vals, y_vals, s=90, color="red",...

Randomness Is a Tax – Until It Isn't

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