You are a student. You need money and fast. You have $1,000 cash on hand, and within 10 years, you want to place it on the investment of your lifetime. Open a recent newspaper and turn to the business section. Amidst the mess of worry and panic, you are bound to be met with a flurry of headlines on the next new big investment, companies that are expected to achieve great returns.
“BRLA has exceptional IPO; 237% projected growth – next Google?”
“Undiscovered investment gold mine, 23% expected returns, SIO”
“Anti-Lehman Brothers: High faith in OVI – 64% growth next year!”
So many opportunities! However, with the limited capital, you may only choose one of these companies within the 10-year span. Which opportunity will you take? When should you take it? How do you maximize the probability that you will pick the best possible investment? Impossible, you say! Indeed; in the way it’s described, an optimal strategy is not possible. With several assumptions however, mathematics provides us a way to be a savvy investor…
Let’s assume these conditions are met:
1) Headlines are released often and are accurate (if it says Company A will have a 30% return, it will)
2) Only one investment may be chosen in the entire 10-year span
3) There is no pattern to the yields that the headlines promise. In other words, past yields will not have an effect on the yield of future opportunities. For example, a string of ten 75% yields does not give us any information on future yields.
4) The total number of opportunities presented (in headlines) is known in this 10-year span
5) Each headline is presented to you one at a time. Once presented, you must decide whether or not you would like to invest in that company. A decision to invest or to ignore the headline cannot be reversed…
6) Each investment will have a different yield. The aim is to snatch the opportunity with the greatest yield.
The hardest part of this problem is the unpredictability of the future. Though a 157% yield may sound attractive now, how do you know if a streak of 250% yields won’t appear in the future? Likewise, a 1% yield may sound bad, but there is no guarantee that future yields will be any better!
This is called… the Secretary’s Problem.
Some Simple Analysis
Let’s examine a case of 1,000 headlines, each with yields that are ranked from 1 to 1000. If we were to always choose headline one we see to be the one we invest in, we have a 1/1000 or .1% chance of choosing the best headline.
Not so optimal…
Since we do not know the range of the yields, at any given moment, the only information about a yield is its rank relative to that of previous yields.
Consider this: what if we were to skip headline one? Or headline two? Or perhaps the first ten headlines? Why not the entire first half?
If we were to skip the first 500 headlines, as we draw future headlines, we have a general idea of which yields are “average” and which yields are not. Thus, if we were to draw a headline that has a yield greater than the first 500 headlines, there is a good chance that this might be the winning investment (it has yields greater than half of our entire data set)! Our strategy is this: skip the first 500 headlines and the first headline that has a yield greater than the first 500 will be the winning investment.
Does this give us a better probability?
Yes! The probability of a headline landing in a certain half is 50%. Observe that if a headline displaying the second-greatest yield is found in the first half, and the headline with the best yield is found in the second half, we would always choose the right investment because no other headline in the last 500 will be greater than this second-greatest yield. The probability of this occurring is 25%.
We jumped from .1% to 25%, from impossible to somewhat plausible. Yet, this is still not the most optimal strategy!
Solution
The optimal strategy, it turns it, is quite simple (the proof is quite elegant as well!). Given n headlines, if we were to skip the first n/e headlines, we would have a 1/e chance (around 36.78%) of choosing the right headline. In case you didn’t know, the letter e represents Euler’s number, which is approximately 2.71828. In our example of 1,000 headlines, using this strategy, we would skip the first 368 headlines, after which we choose the first occurrence of a greater yield. Our probability of succeeding is 36.78%!!
Other Applications
In the original Secretary’s Problem, n candidates are interviewed for the position of secretary. At the end of each interview, the interviewer must decide whether to hire the candidate. If rejected, the candidate may not be reviewed again. Thus, since this scenario is exactly parallel to the above financial problem, the probability of hiring the best candidate for the secretarial job is highest when we also use the same technique of skipping the first n/e secretaries.
The Secretary’s Problem solution does not only apply to financial headlines and unfilled job positions. Have a lifelong goal of finding true love? Want to find your soul mate before you’re forty? Maximize your chances! Use this strategy!
If you are twenty now, and meet one potential spouse every month, you are expected to meet 240 potential spouses before you are forty. Thus, by using the proven strategy and simply ignore the first 240/e (approximately 88) potential spouses, you may never have to waste time going on dates, stress over love, or feel the sadness of a breakup every again! After ignoring the first few potential spouses, you simply go for the one that is better than the rest! It’s mathematically proven! It can’t be wrong!!!
Of course, I’m just kidding. On a more serious note, the Secretary Problem is one of many problems in optimal stopping theory, an area of study in mathematics with great applications in finance and economics. This technique of viewing limited data, a sample of the entire data set, and ultimately forming conclusions about it, is the fundamentals of statistics.
This talk of limited data reminds me of another problem called the German Tank Problem. In World War 2, Allied leaders, worried about the strength of the German army, called two groups, one of intelligence-gathering agents and the other of mathematicians, to estimate the number of German tanks produced each month. The two set out to do their jobs…
Both groups returned few months later with their estimates, one estimate of 1,400 from espionage group and the other of 256 from analyzing samples of numbers found on the side of German tanks. The government, skeptical of the mathematicians’ smaller estimate, believed the spies. After the war however, documents concerning the number of tanks produced each month was found in Berlin. Guess which group was closer in their estimates? The true number was revealed to be 255. This incidence simply shows the power of mathematics and statistics. Point proven: mathematics can be applied to anything, in this case, war as well!
I enjoyed this digression. Expect a return to game theory soon. Stay tuned…