Portfolio management
Conflict between rationality and common-sense
If a particular stock is thought too risky to invest in, can we reduce our risk by investing in a number of similar but different stocks?
Most everyone has heard of the Law of Large Numbers. It was first proved by Daniel Bernoulli in the early 18th century. In an old paper written by Paul Samuelson titled Risk and Uncertainty: A Fallacy of Large Numbers, he suggested that in a betting and investing context the Law of Large Numbers is often given an invalid interpretation.
Who is Paul Samuelson?
Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he “has done more than any other contemporary economist to raise the level of scientific analysis in economic theory”. (Wikipedia)
A test of valor
The paper is quite short and worth reading. It begins with what Samuelson calls A TEST OF VALOR.
Samuelson relates how “S. Ulam, already a distinguished mathematician when we were Junior Fellows together at Harvard a quarter century ago, once said: “l define a coward as someone who will not bet when you offer him two to one odds and let him choose his side.”
Cowards, loss aversion and common sense
In the short paper Samuelson tells how he “offered some lunch colleagues to bet each $200 to $100 that the side of a coin they specified would not appear at the first toss. One distinguished scholar -who lays no claim to advanced mathematical skills – gave answer: I won’t bet because I would feel the $100 loss more than the $200 gain. But I’ll take you on if you promise to let me make 100 such bets.’
Is this what Ulam defined as cowardice? The first part of the response to the lunch bet offer is an example of what behavioural psychologists call the notion of loss aversion; feeling a $100 loss more than a $200 gain. That is, it is viewed as a normal human response to risk and is not cowardice.
But the real story here is whether the willingness to make 100 such bets reduces the risk of the single bet game, or in some way makes it more bearable or manageable. When does common sense override loss aversion?
Irrationality of compounding a mistake
Samuelson’s argument proceeds as follows: “The virtual certainty of making a large gain must at first glance seem a powerful argument in favor of the decision to contract for a long sequence of favorable bets. But should it be, when we recall that virtual certainty cannot be complete certainty and realize that the improbable loss will be very great indeed if it does occur?”
He suggests the rule should be expressed as follows: “Each outcome must have its utility reckoned at the appropriate probability; and when this is done it will be found that no sequence is acceptable if each of its single plays is not acceptable.” (Emphasis added)
Model of rational choice
Put another way, it can be expressed in terms of expected value. An expected value is the potential value times the chances of it occurring. So, Samuelson’s argument is that if the expected value of the first bet in too low to justify the risk, the expected value of the sequence of bets will also be too low to justify the risk. This is the model of rational choice.
Rationality and common-sense conflict
We can let Daniel Kahneman sort this out for us. As for common sense, Kahneman points to calculations done by Matthew Rabin and Richard Thaler that: “the aggregated gamble of one hundred 50-50 lose $100/gain $200 bets has an expected return of $5,000, with only a 1/2,300 chance of losing any money and merely a 1/62,000 chance of losing more then $1,000.” (Kahneman, D. (2011). Thinking, Fast and Slow) p337
Samuelson’s model of rational choice, his utility theory, posits that if you won’t take on the single bet, you shouldn’t take on the 100-bet game. But, with these Rabin/Thaler calculations in mind, it would be perfectly sensible to take on the 100-bet game even if you didn’t want to take on the single bet game. The Law of Large Numbers has made the 100-bet game so favorable that playing it is common sense.
Kahneman’s conclusion was that “if utility theory can be consistent with such a foolish preference under any circumstances, then something must be wrong with it as a model of rational choice.” (Kahneman, 2011) p337 (Emphasis added)
Kahneman is telling us that an ordinary person might, without being thought silly or irrational or cowardly, reject the single bet but decide to take on the 100-bet game.
Daniel Kahneman was a Professor of Psychology at Princeton University who received the Nobel Prize in Economics for his pioneering work in decision making in face of uncertainty. He challenged the so-called rational model of judgment and decision making. He thought of humans are neither fully rational nor fully selfish.
Where this leaves us
Remember, we started with this question: If a particular stock is thought too risky to invest in, can we reduce our risk by investing in a number of similar but different stocks?
It’s fascinating (in the context of a 15-stock portfolio, for example) to reflect on whether 15 simultaneous bets would offer the same Law of Large Numbers advantage as 15 sequential bets. Investing in a portfolio of 15 companies simultaneously is not precisely like investing in one company at a time over 15 time periods. There are differences. For example, the amounts at stake each investment might be entirely different.
So, how does investing compare with the single bet game and the 100-bet game? Can we gain any insights into investing from this discussion?
Let’s look at a number of common-sense investing rules that allow us to take advantage of the Law of Large Numbers.
- First and foremost, diversification and balance work. Diversification is called the only free lunch in investing. Of course each and every stock is risky. That is why the returns to stocks offer a risk premium over bonds. If a particular stock offers a reasonable risk/reward profile, a diversified portfolio of comparable stock will have a lower risk overall. This is a mini application of the Law of Large Numbers.
- One corollary is that one would never put all one’s investments into a single stock, no matter how sound the company. Any single company can cause investors a catastrophic idiosyncratic total loss. In fact, under-diversification carries the similar heightened risk. That said, in my view, a concentrated portfolio of a dozen or more stocks with proper balance does offer good diversification. It is a basket one has the time and energy to watch carefully. Over a certain number of stocks in a portfolio, the Law of Large Numbers adds so little that the portfolio becomes a dog from every village.
- Behavioral psychology has taught us that human beings are prone to loss aversion. First step is for investors to learn about it. When we know about it, we can address it. Second, the remedy is to learning to think like an investor. This means looking at things over the long haul and in the context of a portfolio, broad framing. Broad framing is framing with the Law of Large Numbers in mind.
- One of the rules of diversification and balance is that the businesses of the stocks in the portfolio should be as independent as possible. By this I refer to business activities in independent sectors of the economy. Thus, investing in ten different car parts companies or ten banks or ten mining companies or even ten tech companies does not provide diversification. Independence is one of the key drivers of the Law of Large Numbers.
- Some companies are wild long shots where the chances of a win are miniscule. These are stocks comparable to a lottery. Investing in dozens of these companies does not improve the chances of making money. This is the classic case of where it makes no sense to invest in the first bet let alone invest in 100 more like it. That is, no sequence is acceptable if each of its single plays is not acceptable.
Conclusion
The answer to the question is this. If a stock is too risky to buy, it should not be bought and buying a number of comparable but different stocks will not decrease the risk. If the risk reward for the stock is positive but one is reluctant to buy because of loss aversion, one simply has to get over loss aversion and the stock should be bought. Buying a diversified portfolio of such stocks does reduce risk. This is the free lunch of diversification which itself is a product of the Law of Large Numbers.
Simple games of chance like Paul Samuelson’s lunch colleague challenge can help focus investors’ minds. What is it that helps us manage the risk of common stock investing? As we have seen, utility theory and Samuelson’s model of rational choice don’t always fit with common sense. A lot of the insights of behavioral psychology around decision making in face of uncertainty are useful. The Law of Large Numbers can be our friend. We can also rely on common sense and our learned experience.
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Other posts on investment psychology
This post is part of a series. Readers are invited to read Investment psychology explainer for Mr. Market – introduction This will give you a better understanding of some of the terms and ideas and give you links to other posts in the series.
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