Monte Carlo Analysis - A Tool for Evaluating Investment Returns

hile standard deviation is certainly the most popular measure of risk, there is plenty we can learn about an investment by using other methods to examine possible returns. In this article we focus on a statistical technique called Monte Carlo. We would like to illustrate how this technique can be used to gain some insight on the qualitative behavior of an investment, by identifying some of the non-traditional measures of investment performance.

While measures like expected return and volatility are very common, they may only offer limited insight on an investment, and none between investments that share these two values.

As a simple example, consider the following. Suppose we have three different investments each held over a 3 month period with three equally likely outcomes for the three scenarios listed in Table 1 below.

Table 1

Each of these investments has an expected return of 0%, and a risk as measured by the standard deviation of returns of 20%. Yet measured by other objective investment measures we can see a different picture as Table 2 illustrates:

Table 2

Monte Carlo analysis provides a very effective way to picture the range of possible investment outcomes. This technique is also very useful because it allows one to analyze not just option transactions, but more and less complicated transactions as well.

For example, the same analysis that is applied to a simple stock investments can be applied to very complicated 'exotic' option investments and therefore places them for comparative purposes, in the same perspective.

The basic idea of Monte Carlo is to randomly sample stock prices through the holding period of an investment and observe exactly what cash flows occur. As we chose more and more sample paths, a distribution begins to appear illustrating the range and frequency of possible outcomes for the investment.

The important part of this technique is in selecting the paths in the simulation. In other words, what constitutes an appropriate mechanism of selecting the paths from which to determine the distribution of likely results? The basic idea is to use the assumptions that are embedded in the Black-Scholes methodology which assumes that returns follow a normal distribution, with an expected return equal to the risk-free rate and a standard deviation that can be calculated from the movements of the stock underlying the considered investments.

Table 3 illustrates below what a representative set of paths would look like using this approach. Each path follows a random path for a three month period starting from the initial stock price.

Table 3

On the final date, the stock prices form a distribution that looks like Table 4. Extreme returns, both positive and negative, are less likely, while returns closer to zero are the most likely outcomes.

Table 4

In addition to displaying the data as a histogram of return possibilities, we can also graph the cumulative distribution of returns (see Table 5) which looks like the following:

Table 5

Several features of the investment can be read from this graph. The 50th percentile, or median, investment return is 0%, and extreme returns are -60% and 60%. Additionally, the inner two-thirds of the investment, from the 17th to 83rd percentile is -20% to 20%. These values are a result of the assumptions that were used in generating the paths for the simulation.

We can generate similar analysis for the most complicated of investments as well, but for now let's consider a simple option strategy. Rather than investing $100 in a stock which has the outcomes illustrated above, suppose one were to purchase an at-the-money call for three-months at a cost of $8. The cumulative return is illustrated in Table 6 below compared with the stock investment as well. In this manner, we can view the investment profile of both investments at the same time and visualize their respective properties. We can see how often the option strategy underperforms the stock investment and by how much.

Table 6

As another example, suppose we compare a collar strategy, where in addition to holding stock, an investor buys a put that is out-of-the-money and sells a call that is also out-of-the-money. In this manner, some of the upside is sold off to purchase protection in the event that the stock declines. Table 7 below illustrates the distribution of outcomes for this strategy.

Table 7

Diagrams like these are helpful in comparing investments or even in gaining a better understanding of a single investment.

Monte Carlo analysis presents an investor with a more complete understanding of an investments likely range of returns as well as extreme (best-case and worst-case) possibilities.