The BARRA Brainteaser

by Eugene Reznik

The year is 2000 A.D. Mars Inc. (MI) and Venus Inc. (VI) have just been granted patent rights for two radically different technologies for building commercial antigravity vehicles. Both are about to penetrate the market with new products based on their technologies.

During each quarter, either MI or VI will gain 20 percent of the total potential market. (At the present time, the entire market is up for grabs.) Despite the beliefs of the founders of MI and VI, neither of the technologies is superior to the other, so either one is equally likely to gain popularity in any given period. The battle is expected to continue until the majority of the market subscribes to one of the two technologies, at which point the other rapidly vanishes. Clearly, this will take at least three and at most five quarters.

Below is one possible sequence of events.

Date	MI	VI	Unpenetrated
January 1, 2000	0%	0%	100%
April 1, 2000	20%	0%	80%
July 1, 2000	20%	20%	60%
October 1, 2000	20%	40%	40%
January 1, 2001	20%	60%	20%
		Battle over. VI has won!

The owners of Mars Inc. have decided to cash in on their invention by taking their company public. The analysts forecast that the returns on the MI stock will be +/-10 percent per quarter depending on whether MI or VI gain market share during that quarter. Should VI beat MI to the 50% mark, MI's stock in the subsequent quarter will go down to zero. In the example above, if S₀ is the IPO price on January 1, 2000, MI's stock will take the following path:

Mr. F. X. Variance, a client of the money managing firm you work for, strongly believes in MI's technology. After making a profit of $1,000 in his fixed income portfolio, he now wants to use this money to gamble on the long-term success of MI. Because regulatory issues prevent him from owning MI's stock directly, he has asked you to use options to replicate a "double-or-nothing" bet on the victory of MI over VI. Mr. Variance wants to exactly double his $1,000 if MI is the first to get over 50 percent of the market. Otherwise, as in the example above, he wants to lose it all. (Since MI and VI are equally likely to take over the market, let's assume zero interest rates to keep Mr. Variance's bet fair.)

Given this task, you decide that three-month "at the money" calls on MI's stock maturing at the end of each quarter will best suit your needs. (Such options will be available as long as MI is in business.)

(a) What should be your initial investment (in dollars) in such options, and the subsequent strategy? In other words, how much should you bet, now and in each subsequent quarter until the market is monopolized by one of the companies, to guarantee a double-or-nothing return on the $1,000?

(b) Is the answer to part (a) unique?

(c) Suppose that MI is able to increase the probability of winning the market share in any given period by investing the money it raised through the IPO in advertising. How would this affect the size of your initial and subsequent investments?

Suppose that in any given quarter, rather than 20 percent, one of the companies gains 100/(2n - 1) percent of the total potential market. Thus the competition could last as few as n and as many as (2n - 1) quarters. Answer questions (a) through (c) under the above generalization.

Hint/Example: For the case n = 2 the investments should be: $500 on 1/1/2000; $500 on 4/1/2000; and if the market is not yet monopolized, the last investment on 7/1/2000 should be $1,000. Responses for the BARRA Brainteaser can be sent to Eugene Reznik at BARRA via e-mail to eugene.reznik@barra.com. You may also fax or mail your responses to his attention; fax: 510.548.4374 or mail to BARRA, 1995 University Avenue, Berkeley, CA 94704.

Solution to Spring 1996 Brainteaser

We start with 100 managers. N have skill, and 100 - N are unskilled. We will assume that far less than half of them have skill, i.e., N << 50.

(a) Probabilities

For the skilled managers, their probability of winning (being above median in performance) is 100 percent.

For the 100 - N unskilled managers, 50 of them will be losers (they make up all 50 losers) and the rest (50 - N) will be (lucky) winners. So for these unskilled managers:

(1)

(2)
where P _L is the probability of losing and P_W is the probability of winning.

Note that P_L + P_W = 1.0. Also note that these probabilities make sense in the limit of no skilled managers

(N = 0, P_L = P_W = 0.5)

and half skilled managers

(N = 50, P_L = 1.0, P_W = 0.0).

(b) Analyzing the contingency table

Let's focus on the consistent losers. Only unskilled managers lose.
Expected two-time losers =

(3)

We observe 27 two-time losers:

(4)

Solving Equation 4:

N = 7.4

(5)

Since we seek an integer solution, we expect seven skilled managers. Note that this is just an expectation, and given the not statistically significant contingency table, the probability exceeds 5 percent that statistical fluctuations in a world with no skilled managers led to this estimate.

(c) How do we find the skilled managers? Since they always win, the key is to run multiple trials. After two trials, 7/27 = 26 percent of the two-time winners are skilled. After T trials, we expect:

(6)

Then, the fraction of T time winners with skill is:

(7)

Using N = 7, to have at least 50 percent of consistent winners be skilled requires at least T = 4 trials. To increase the ratio above 95 percent requires eight trials.

John Kattar suggested a clever improvement on this idea. Run each successive trial on only the repeat winners. In this simple model, that will lead to the winners much more quickly.

In the real world, multiple trials require many years, raise the question of whether even truly skilled managers can maintain their performance over long periods, and limit investor choices to managers with very long track records. An alternative approach: using shorter time periods raises a different problem. Over short periods, even skilled managers may underperform.