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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.
Next Section: Computing Probability of Profit
Previous Section: A Practical Introduction to Mathematical Probability
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