Games of Chance (II): Probability and the Practice of Finance

In my previous post, I introduced Brownian Motion (and its discrete version, the Random Walk), summarized the history of its discovery and briefly mentioned the close link between Brownian motion and the idea of “fair games” in Finance [1]. In today’s post, I will use an example in Finance to illustrate the practical value of Brownian Motion.

Finance theory assume (rightly in most cases) that people are risk-averse, which roughly means they hate losing money in risky gambles, and investment in things like stocks is a risky gamble. It turns out that with a bit of Probability mathematics, and the assumption that stock prices follow the jiggly random path of a Brownian Motion, beautiful equations can be derived to answer important questions like” “what is the probability at losing least x% of your original capital over an investment holding period of y years?

The Case Study

John, aged 65, has just retired with a portfolio of stocks and bonds. He plans to draw down (progressively sell) these assets to provide him with retirement income for 20 years. Therefore, he can’t afford a big loss over this 20-year period. John’s problem is one that many retirees face. With life span ever increasing, seniors will need to continue growing their money even in retirement, contrary to the standard advice that retirees should withdraw from taking any form of risks. Yet, investment carries the risk of loss, and retirees, of all people, are least able to afford huge haircuts on their hard-earned money. What to do? While there are no easy solutions to the problem of balancing risk and return under these constraints, mathematics can provide concrete guidance on how to avoid financial potholes.

One way to think about risk is the probability of your wealth falling below a “critical” amount during some point over your holding period. For example, it is reasonable to think that most retirees will squirm at the thought of losing 30% of more of their initial capital over a 20-year holding period. A loss as big as this is hard to recover, especially for someone with no more regular savings!

To assess the likelihood of going below a critical level requires the mathematics of stochastic processes. A stochastic process models how a random variable (such as the price of a stock or a portfolio of stocks) fluctuate over time. There are many types of stochastic processes ranging from simple to extremely complicated. But one that I will assume in this blog is lo and behold – the ubiquitous Brownian Motion!

Historically, the long run average returns of stocks is positive. Therefore, it makes sense to incorporate this fact into our formula, and the tool to do that is to use a special version of the Brownian Motion called the Geometric Brownian Motion or GBM.

To convince you that the GBM can approximate the trajectory of real stock prices well, consider the two graphs below show. The top graph is a plot of the Dow Jones index of leading US stocks, and the bottom graph is a simulation of this index based on the GBM. The resemblance of the GBM to the path of the real stock price is uncanny.

A Formula for Quantifying Loss Probability

Let P(loss) represent the probability an investment will penetrate a critical value at some point during a T-year period. Using a set of mathematical results [see note 2] called First Passage Time (or First Hitting Time), we can write a formula for this probability as follows:

\displaystyle P(loss) = N \bigg[\frac{ln(C/S) - \mu T}{\sigma \sqrt T} \bigg] \times (C/S)^{\frac{2\mu}{\sigma^2}} +N \bigg[\frac{ln(C/S) + \mu T} {\sigma \sqrt T} \bigg]


N[.] is the cumulative normal distribution function (in Excel, the function is NORMSDIST)
ln is the natural logarithm
C = critical value
S = Starting value
\mu = mean return of investment (in natural log)
\sigma = standard deviation of investment
T = number of years in the investment horizon

Suppose \mu = 0.08 and \sigma = 0.18 , which are typical numbers to represent the average return and volatility of stock indices in developed markets. Suppose T = 20 years. For simplicity, let S = $1 and C = $0.70. Then, we are talking about the loss of 30% at some point over a 20-year horizon. Note: the starting value can be values other than $1. If C is fixed, the ratio C/S will remain the same.

Plugging these numbers into the formula, we have P(loss) = 17%. That is, there is a 17% chance of seeing your initial wealth melt to 70% or below at some point over the 20-year horizon. Note that this formula (the second term after the + sign) already takes account the well-known upward drift of stock prices over the “long run” as mentioned earlier.

Is 17% something to worry about? Only you can answer that question; the formula simply gives you a ballpark estimate of the loss probability. It may be tempting to try to beat the odds by taking more risk. This may or may not pan out the way you expect. Suppose you switch to more volatile (say, emerging market) stocks in the hope of capturing a higher mean return. To be concrete, suppose the mean from such stock is 10% but the volatility (standard deviation) is also higher at 24%. Using the formula, P(loss) is now equal to 29%. A loss probability as high as this will probably scare off many investors!

What about taking less risk by investing in assets with lower volatility. Suppose you switch to stocks with a 6% mean return and volatility of 15%. Then, the formula gives a P(loss) of 14% which is in line with our intuition: taking less risk implies a lower likelihood of loss.

In summary, the P(loss) formula is a beautiful application of stochastic theory to finance. First, the formula is elegant. There are no nasty infinities to deal with, for instance and because the distribution of Brownian Motion is Gaussian (bell-shaped), there are just two parameters for asset returns: the mean rate of return and volatility. Second, as the case study has demonstrated, the formula provides investors with practical insights on how to strike a balance between seeking returns and controlling risk.


[1] There’s a nice Youtube tutorial explaining the properties of Brownian motions. At

The mathematicians who have intensively studied Brownian motions belong to the scientific Hall of Fame. They include Albert Einstein, who first derived a mathematical theory of this random phenomenon and related it to the heat equation used in physics. Some years later, the MIT mathematician Norbert Wiener derived the probability distribution of a Brownian motion, which paved the way for the advance analysis of more general stochastic processes. In honor of Wiener’s contributions, the standard Brownian motion is now also known as the Wiener process. Brownian motions were further studied by the famous mathematician Paul Levy who published his work in an influential book in 1948. I should also mention the great French mathematician, Pierre Laplace (1749 – 1827). Although Laplace did not prove theorems on Brownian motions (they weren’t discovered yet!), his famous discovery, the Laplace Transform, proved to be a highly versatile tool for many branches of science. In particular, using the Laplace Transform simplifies the derivation of the expression for the First Passage Time (FPT) of Brownian motions.

[2] First Passage Time (FPT) distributions for diffusion processes are also known as boundary crossing distributions, first hitting time or first exit time distributions. Other than finance, these distributions have wide scientific applications in field such as engineering, ecology and health science. For example, FPT is used to calculate expected lifetimes of devices in engineering and patients (in medicine). Expected lifetime is determined if we can compute the time to breakdown of a device (when it reaches a certain threshold state) or when time to a patient’s death. For more technical discussions of FPT distributions, see Samuel Karlin and Howard Taylor, A First Course in Stochastic Processes (2nd ed), Academic Press, New York, 1975. Another reference is: Luigi Ricciardi and Shunsuke Sato, “Diffusion processes and first-passage time problems”, in Luigi Ricciardi (ed), Lectures in Applied Mathematics and Informatics, Manchester University Press, 1990.

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