   # The Role of Probability in Modeling Investments

The probability modeling of investments focuses on modeling the seemingly random movements of a security's price; stock or index. We project future price probabilities by examining the security's history. Most significantly, we examine the volatility of the price. Price volatility is the day to day changes in closing price as a percentage of the security's price. Price volatility determines the horizontal spread of the lognormal probability distribution. The above graph shows the lognormal probability distributions of a \$100 security with 20% and 40% annual volatilities. You can see that probability of the security doubling (\$200) is much higher with 40% volatility than 20% volatility; and that this probability is equal to the probability of the security price decreasing to half (\$50).

Probability modeling used on this web site attempts to extend historical volatility trends into the future. The site does not model significant specific security or market events. Company scandals and war are examples of events that are beyond the scope of modeling, but can cause significant price movement. We also assumes that a security's volatility is constant. This type of modeling is most useful for analyzing short term investments.

## The major trend (risk neutral growth rate)

The random walk of a security is not completely random, but rather it is guided along a trend line. We refer to this trend line as the "risk neutral" growth rate. We use the term "risk neutral", because the probability of performance above this rate should equal the probability of performance below this rate. It is often quoted that the market (S&P 500) has averaged about a 10% per year increase since the 1920's. This is obviously a major trend. Shorter term indicators are also useful. One can look at the S&P 500's 200-day moving average, a stock's 200-day moving average, or even a stock's 20-day exponential moving average (a moving average that weights recent history most heavily). Instead of examining past trends you could also consider earnings and P/E ratio predictions to establish this trend line. You should determine which indicator best predicts the major growth trend for the security that you are analyzing and specify it as a parameter to the analysis. Most literature I have read recommends that you enter a "risk free interest rate" as this parameter (the interest rate for the U.S. Treasury bill with a maturity closest to the term of your investment). I have found this to be far too conservative if your bullish and far to optimistic if your bearish. You will find that the results for short term investments (less than 90 day) are not very sensitive to this growth rate parameter. Volatility dominates for short term investments.
Numbers that work for me are:
 Predicted Trend Risk Neutral Growth Rate Rocketing 15% Normal 10% Weak 4% Slow leak -2% Crashing -7%
Once you have found a good investment using the above numbers, to be safe, perform a sensitivity test, do two analysis, one using a more optimistic risk neutral rate of growth, the other a more pessimistic rate of growth.

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