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The BARRA Brainteaser for Winter 1997Fall 1996 Brainteaser Solution
Against the Gods: The Remarkable Story of Risk
Figure 1
The BARRA Brainteaser
For Winter 1997
by Ronald N. Kahn
Sydney and Steve are quantitative researchers at the international investment firm I.B. Quant. As a first step in researching a new emerging markets strategy, they have flown to the South Pacific to investigate opportunities in the tropical islands. Unfortunately, the trip goes awry and they end up shipwrecked with little but an American flag and each other's company.
After some months of mind-numbing tedium, they realize they can remember just the first few digits of p. They argue whether 3.14 or 3.15 is a better approximation.
Given only their American flag and a stick of length L (equal to the width of a stripe on the flag), suggest a method to estimate p to sufficient accuracy to solve the argument. Make reasonable assumptions to estimate how long resolving the argument will require.
Responses for the BARRA Brainteaser can be sent to Ronald Kahn at BARRA via e-mail to ron.kahn@barra.com. You may also fax or mail your responses to his attention: fax 510.548.4374; mail to BARRA, 1995 University Avenue, Berkeley, CA 94704.
Brainteaser from Fall 1996
by Eugene ReznikMr. 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 growth. 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 m1 and standard deviation s, and that starting with the k + 1st period, the mean of the random walk abruptly changes from m1 to m2. Also assume that the probability of the above event happening in any given period is p; that is,![[ equation 1 ]](../images/BTSolEq1NL163.gif){kind=small}
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 false alarms exceeding pthreshold.
Solution
For notation purposes, it is more convenient to let k denote the first period with expected return m2, rather than the last period with expected return m1.With every strategy S, which at time vs ³ 1 will signal to sell the stock, we associate a risk RS given by:
where
is the probability of signaling a false alarm,
is the expected delay between the time the return distribution actually changes and the time the alarm is signaled, and c is a constant.
Mr. I.P.O. has to find a strategy S* such that for some value of c, and for all other strategies S,
while
Let
be the conditional probability that the mean of the return distribution has not changed from m1 to m2 during the first n periods. Then
In other words, for fixed values of p, s, m1, and m2, the distribution of future returns can be completely characterized by the statistic Pn. This, in turn, implies that the decision to hold or to sell the stock at the end of the first, and every successive period, is dependent only upon Pn!
Suppose r1 is the observed return over the first period. The risk of exiting the position is
Let rhold(P1) denote the risk associated with the best of all strategies which do not signal to exit the position. Clearly,
It can also be shown that rhold(P1) is concave. 1 Therefore, the curves rexit(P1) and rhold(P1) will intersect in exactly one point, which we denote by P* (see Figure 1).
We can conclude that the optimal behavior at the end of the first period has to be based on comparison of P1 to P*. If P1 £ P*, rhold(P1) will exceed rexit(P1), and it is best to exit the position; otherwise, the stock should be held for at least one more period. Similarly, after n periods the position should be held if Pn > P* and exited if the opposite is true.
After n periods, the problem will have changed only in that the distribution of future returns will be characterized by Pn rather than P1. Therefore, holding on to the stock if Pn > P*, and selling it otherwise, comprises the optimal strategy. (Note that P* is independent of n.1)
The conditional probability Pn can be calculated using the following recursive relation.
where f1 and f2 denote normal probability densities with standard deviation s and means m1 and m2, respectively. (The latter result is easily obtained using the definition of conditional probability.)
The threshold value P* depends only on the specified false alarm probability and can be obtained numerically.
Notes
1 A rigorous proof is available upon request.
