Can you decipher this?

One of the courses this term that I have been really enjoying is Prof. Uday Damodaran's Portfolio Management. A change from the somewhat theatrical to a definitely more structured course being one important parameter that has grabbed our attention, there are several other reasons why there is almost full attendance in his classes, even if they are held at 7 in the morning.

Anyways, talking about the teaching methodology of Dr. Damodaran was not the purpose of writing this post. Ever since the last class, when we played the game that is popularly called St. Petersburg Paradox and subsequently analyzed it, I have been wondering if the explanation was so simple that even a lay person with absolutely no idea about finance or economics can understand the same. I will try to enumerate what took place in the class room and leave it for the readers of this blog to decide if they can decipher what I am talking about.

The game went like this: the class was divided into groups of six with one person in each group acting as the gaming house and the other five enacting the roles of players. There were five rounds in total with each round starting with closed bids by the players being submitted to the gaming house. The gaming house, based on the received bids, chose the player it wants to play with. Ideally, the player chosen should be the one with the highest bid but in case of ties, it depends on the gaming house's discretion. Irrespective of whether a player's bid has been accepted or not, he/she has to forfeit the amount that he/she has bid on the round to the gaming house.

Once the player is chosen, the game begins. The game involves tossing a coin till a head comes up. The number of times that the coin has to be tossed before the head turns up decides the payoff for the player who gets 2 raised to the power n where n is the number of tosses before a head appears. So the minimum that a player will get out of this game is 1 and the maximum can be infinity. Similar methodology is followed for the other four rounds.

Once the game was over, an average was taken of the amount that the players had bid for (including successful and unsuccessful bids) and for our class, it turned out to be nearly 7. Next up, Dr. Damodaran presented another game which is popularly known as the fair game. In this game, the player gives 1 Re. to play and a coin is tossed. If the coin turns head, the player gets Rs.2 and he/she gets 0 if it turns tail. When asked about how many of us would play this game, the majority said they will.

This, Dr. Damodaran said, disproved the basic assumption of finance and economics that we had been trying to understand for the past sessions of the course. The probability that head occurs is 0.5 and that of tail appearing is also 0.5. The expected payoff from the game, therefore, would be 0.5*2 + 0.5*0 (that is, a summation of probability of event multiplied by payoff from the event) which turns out to be 1. Thus, a person playing this game would be foregoing a certain 1 Re. for an uncertain 1 Re. (return from the game is only an expected payoff, remember but the money that the player puts into the game is a certain Re. 1). This is certainly not rational investor behavior but something that most people would claim to have when asked about it, primarily because of the positive connotations associated with risk-takers.

However, a similar payoff vs. investment analysis exercise carried out with the St. Petersburg game shows that the reality is quite different. Here, probability of getting head in the first toss (with payoff of 1) is 1/2. Head in the second toss has the probability (1/2) squared and the payoff is 2 raised to the power 1. For the head to appear in third toss, probability is (1/2) to the power 3 and the payoff is 2 to the power 2. Going on in similar fashion, we find that for each event (where event means the number of times that the coin has to be tossed before a head comes up), the product of probability and payoff is 1/2 in each case. Thus, expected payoff from the game would be a summation of 1/2 infinite times, which comes to infinity.

For an infinitely paying game, therefore, the average amount that our class bid was only Rs. 7. And we said we are risk lovers when asked about the fair game...just goes on to show how there is much more to behavioral finance than just asking the basic questions and trusting the investors' answers blindly. If put in a guise, the real picture does come up. There was a further follow up to this discussion, explaining matters in much greater detail but perhaps, I will save that for some other post to avoid an overdose and of course that post will come up given this one makes at least some sense.