The BARRA Brainteaser

by Eugene Reznik

Mr. I.P. Oltman strongly believes in investing in the stock of companies that have just gone public. His experience leads him to conclude that before an abrupt decline in its expected returns, every newly issued stock experiences a period of constant grow th. Over the years, Mr. I.P.O. has been able to "beat" the market by holding the recently offered stocks until he believes they have reached the end of their initial growth period.

To be successful, Mr. I.P.O. has to have an edge when determining whether the company has ended its initial growth. In making his decision to exit the position on a particular stock, Mr. I.P.O. does not rely on any of the fundamental principles. In fact, all he has available to him are the returns on the stock since its issuance.

Assume that Mr. I.P.O. is correct in his fundamental assumption about the price behavior of new stocks. Further assume that for the first k periods, the returns on a typical stock are normally distributed with mean µ and standard deviati on , and that starting with the k + 1^st period, the mean of the random walk abruptly changes from µ₁ to µ₂. Also assume t hat the probability of the above event happening in any given period is p; that is,

Devise an optimal strategy for Mr. I.P.O. to detect the change in the stock's behavior. This strategy must minimize the expected delay between the jump in the stock's expected return and the time Mr. I.P.O. exits the position, without the probability of f alse alarms exceeding p_threshold.

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.

Summer 1996 Brainteaser and Solution

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 te chnologies.

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 e ither 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 re gulatory 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 doubl e-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

The real challenge of the problem is to answer the most general question: "What should the size of the investment be when one of the companies needs exactly j, and the other exactly k successful quarters to win the market?"

Before we proceed, we need to establish one essential fact about the "at the money" options that are available in each period. Because the stock price can move up or down by the same amount, and because we assumed zero interest rates, these options provid e a double-or-nothing return.

The problem can be solved by constructing a binary tree, each node of which will contain the portfolio value and the amount to be invested in options. The root of the tree has an initial portfolio value of $1000 (see Figure 1). A move to the right (left) in the tree corresponds to MI (VI) gaining market share and, accordingly, the return on the options being +100 percent (-100 percent).

Figure 1

The entries in the nodes of the tree have to satisfy some simple relations. In order to formulate these, we label the nodes using two variables, m and k, where m denotes the maximum number of periods before the market is capitalized a nd k is the number of victorious quarters which separate MI from reaching 50 percent market share.

Using our notation, we have the trivial relations (also shown in Figure 1):

Combining these, we have:

(1)

Note that the net gain (loss) of losing one and winning one of two consecutive bets should not depend on whether the first bet was won and the second lost, or vice versa.

This yields:

Solving for we get:

(2)

The value of the portfolio at the terminal nodes of the tree must equal $0 or $2000, depending on which company has won the market. This, in turn, implies the following boundary conditions for bets:

(3)

The boundary conditions along with either (1) or (2) enable us to fill out the entire tree, working our way backward from the boundaries. The results for the case n = 4 are shown in Figure 2.

Figure 2: The first number in any node is the accumulated wealth; the second is the size of the investment in options in dollars.

We are now in a position to answer parts (b) and (c) of the problem: the investments to be made at each node are uniquely determined, and are independent of the probability of moving to the right or to the left in the tree structure. (This result is simil ar to option price being independent of the probability of an up move in the binomial pricing model.)

Figure 3: Pascal's Triangle

To answer part (a), note that construction of the tree of bets using (2) and (3) is very similar to the construction of the classic Pascal's triangle (see Figure 3). The entries in Pascal's triangle are the values of — is, the number of ways to select k out of n distinguishable objects. These satisfy:

(4)

(5)

(6)

Comparing (2) to (4) and (3) to (5), we obtain the final result:

(7)

The factor of arises from the fact that in constructing the "bet tree," we are averaging two previous entries instead of summing them (as is done in constructing Pascal's triangle). T he factor of 1000 simply propagates from the boundary conditions.

An alternative solution is to obtain a closed-form expression for VALUE_m,k:

Using this result, we have:

which is the same as the previous result!

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