The BARRA Brainteaser

by Ron Kahn

Lincoln and Douglas have just left a large consulting firm to set up their own shop. They plan to help sponsors with manager selection and are considering running a fund of funds. However, this venture is marred by an old debate the two have never resolved: can active managers add value?

Lincoln thinks the answer is "yes," and he is very eager to run the fund of funds. He is convinced not only that active managers add value, but that he can find which skillful managers to invest in.

Douglas, ever the skeptic, doesn't believe a word of it. "I'll prove that you can't find the skillful investors." So, using BARRA's Style Analyzer product, he divides 100 mutual funds into winners and losers (based on whether their information ratio is above or below the median) in two separate periods, then generates the following contingency table:

		Period 2
		W	L
Period 1	W	27 54%	23 46%
	L	23 46%	27 54%

"The probability of winners repeating is only 54 percent,¹ not statistically significant. You can't find managers who consistently add value."

Lincoln, a skilled debater, immediately responds by saying, "Based on your findings, I not only believe that there are some skillful active managers, but I can tell you how many there are. And, I can devise a strategy to find them."

How many skillful managers are there in Douglas's database?

Use the following simple model to answer this question and the following questions. Out of 100 managers, N have skill (they are always in the top half), and 100 ­ N have no skill. The unskilled managers are (approximately) randomly in either the top or bottom half, though they are more often in the bottom half, because on average they lose to the skilled managers.

Given this model and this number of skilled managers, how could you use Douglas' general approach to find them?

Solution to Winter 1996 Brainteaser

1. We can approximate the problem in the following way:

(a) Assume that the stocks are boxes.
(b) Assume that the investors are balls.

The problem now becomes a classic occupancy problem—we are randomly distributing the balls amongst the boxes. This problem can be solved by assuming the balls are distributed amongst the boxes by a Poisson process.

In general,

(1)

where lt is the average number of balls (investors) per box (stock) and N(A) is the number of balls in box A.

So the probability that a particular box is empty is

(2)

and the probability that a particular box is not empty is

(3)

To solve our particular problem, we need to optimize Y, for some lt:

(4)

Plugging in equations (2) and (3) to equation (4), differentiating Y with respect to lt, and then solving for lt, we get that the optimal number of investors is 281.

2. This is now a simple exercise of plugging in different values to equation (1).

The probability that at least one investor has invested in all 100 stocks given 1000 investors is

Where lt is 10 in this case. The solution is 0.995.

The probability that at least two investors have invested in all 100 stocks given 1000 investors is

The solution here is 0.95.

¹ Out of 50 Period 1 winners, only 27, or 54 percent, were winners in Period 2.