Assistant Professor atWarwick Business SchoolAP atWBS

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Market Timing

The market timing game simulates a stock market index that follows a random walk with drift. The model assumes 8% annual drift (typical equity return) and 20% annual volatility (typical market risk). Each day, the index moves up or down randomly, but with a slight upward bias over time. You are guessing whether the index will be higher or lower after three months.

The Underlying Process

The index price evolves according to the stochastic differential equation:

dS_t = \mu S_t dt + \sigma S_t dW_t \quad \text{with} \quad \mu = 0.08, \; \sigma = 0.20

where the index value $S_t$ follows a geometric Brownian motion $W_t$ , with drift $\mu$ and volatility $\sigma$ .

Confidence Intervals

Computing Confidence Intervals

For geometric Brownian motion, the index log returns are normally distributed:

\log\left(\frac{S_{t+\Delta t}}{S_t}\right) \sim \mathcal{N}\left(\left(\mu - \frac{\sigma^2}{2}\right)\Delta t, \sigma^2 \Delta t\right)

This allows us to construct confidence intervals for future index values. For a 95% confidence interval:

S_{t+\Delta t} \in \left[ S_t \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)\Delta t - 1.96\sigma\sqrt{\Delta t}\right), S_t \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)\Delta t + 1.96\sigma\sqrt{\Delta t}\right) \right]

The $-\sigma^2/2$ term is the volatility drag that reduces the median path below the mean. Note: These are pointwise intervals—while 95% of paths lie within the intervals at any given time, many paths will exit the intervals at some point during their trajectory.

Can You Beat the Market?

Expected Returns

The expected log-return over a period $\Delta t$ is:

E\left[\log\left(\frac{S_{t+\Delta t}}{S_t}\right) \;\bigg|\; \mathcal{F}_t\right] = \left(\mu - \frac{\sigma^2}{2}\right) \Delta t

The expectation depends only on the time horizon (Markov property), not on past history. With our parameters, the 3-month expected log-return is $(0.08 - 0.02) \times 0.25 = 1.5\%$ .

Variance of Returns

The log-return over a period $\Delta t$ has variance:

\text{Var}\left[\log\left(\frac{S_{t+\Delta t}}{S_t}\right) \;\bigg|\; \mathcal{F}_t\right] = \sigma^2 \Delta t

The variance grows linearly with time, independent of the price path taken. With our parameters, the 3-month standard deviation is $\sigma\sqrt{3/12} = 20\% \times 0.5 = 10\%$ , making both wins and losses quite likely!

The Verdict

You cannot systematically beat the 8% arithmetic drift. Under geometric Brownian motion (constant $\mu$ and $\sigma$ ), future index movements are independent of past history (Markov property), and the best predictor of future values is simply the current level adjusted for drift.

Always going long yields an expected log return of

\left(\mu - \frac{1}{2}\sigma^2\right) \Delta t = (8\% - 2\%) \times 0.25 = 1.5\%

over 3 months, and no timing strategy can systematically improve this expected return. The expected log return is lower than the drift $\mu = 8\%$ due to volatility drag, the compounding effect of losses requiring larger gains to recover.

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