Odds of beating the market over many years, a (very) naive model

Suppose you knew, or estimated, the probability of beating the market over a year. What then is the probability of beating the market over many years?

Enter the parameters below, then click "Calculate!" to calculate the probability of a portfolio beating the market the stated number of years (or, for bonus points -- the stated number of years, or more).

Assumptions:

Probability that the portfolio beats the market in a given year (0 to 100, as a percentage, so enter 20 for 20 percent):

Number of years to consider in total (1 to 50, as a number):

Number of years when the portfolio beats the market (0 to the number of years to consider in total, as a number):

 

Notes:

• This calculation is based on a fairly simple model, the binomial model.
• Answers and bonus answers are rounded. Keep this in mind.
• KEYBOARD INTERFACE SHORTCUTS (WINDOWS): Use the TAB, SHIFT+TAB, and ENTER keys to navigate the interface. TAB moves you forward from the first box to the second box and so on. SHIFT+TAB (that is, pressing the SHIFT key and TAB key simultaneously) moves you backward. Pressing the ENTER key while the focus is on the Calculate! button calculates the estimated return.
• A bit technical: JavaScript must be enabled for the above calculation to work.

The figure below shows the probability of beating the market from 0 to 12 years, when the probability of beating it each year is 20 percent. As you can see, the odds of beating the market 7, 8, 9, ... years out of the 12 approach amoeba-like small.

Let's add some sugar to this cup of tea. Let's increase the probability of success. The figure below shows the probability of beating the market from 0 to 12 years, when the probability of beating it each year is now 45 percent. The odds of beating the market now improve dramatically, but are still just 1 out of 1000 to beat the market 11 years out of 12, for instance.

In case you are wondering (and who isn't?), the maximum in these graphs is given by (roughly) the product of 12 times the probability. So, the maximum in the first graph is 2.4 (12 * 0.2, 2.4, rounded to 2) and the maximum in the second graph is 5.4 (12 * 0.45, 5.4, rounded to 5). You will see that the graphs nod approvingly.