In 1738, Daniel Bernoulli devised a simple model of risk aversion ( English translation here ). Nobel Prize winner and author of Thinking Fast and Slow Daniel Kahneman criticizes Bernoulli’s theory extensively, describing it as “Bernoulli’s Error.” I disagree with Kahneman. I think Kahneman misunderstands Bernoulli’s claims. Bernoulli’s theory of decision-making is best described with some examples. He claims that doubling your net worth is as positive as dividing it by 2 is negative. So, if your net worth is $200,000, receiving another $200,000 is as positive as losing $100,000 is negative. Bernoulli applies the same type of rule to smaller changes as well. Going from $200,000 to $250,000 is multiplying by 1.25. Going from $200,000 to $160,000 is dividing by 1.25. So, winning $50,000 is as good for you as losing $40,000 is bad for you. (For the more mathematically-inclined, the utility of your net worth is proportional to its logarithm.) Kahneman’s extensive research h...

Larry Swedroe has long advocated adding a small amount of commodities to a portfolio to boost risk-adjusted returns. The theory says that while the commodities lower the overall expected return, they more than make up for this with their lowering of the portfolio risk (standard deviation). I’ve been resistant to this idea despite Swedroe’s numerical examples of improved risk-adjusted return. I think I can finally explain my reluctance when it comes to commodities. In Swedroe’s most recent book (see my review here ), he gives a clear example showing the effect of adding some commodities to a particular portfolio based on historical returns from 1975-2011. The portfolio begins with a 60/40 split between various equities and 5-year treasury notes, and has the following characteristics. Before Commodities Annualized Return: 12.4% Annual Standard Deviation: 11.8% He then replaces part of the equities to give the portfolio a 4% exposure to commodities. After Commodities Annu...

We now understand well that investors consistently make certain types of behavioural mistakes, such as selling stocks after the market drops, buying back in when stocks are soaring, and chasing last year’s hot mutual fund. How best to adjust portfolios to deal with these cognitive errors is a subject for debate. One approach is to take the way an investor feels about gains and losses and construct an asset allocation that maximizes the investor’s perceived net gain. This is the approach taken by Morningstar with their star ratings that bake in an assumed average level of risk aversion . However, this can lead to extremely conservative portfolios, which is not in the best interests of most investors. The real answer is to evaluate potential asset allocations based on two criteria. The first is how beneficial the allocation would be for the investor (as opposed to how comfortable the investor would be). The second is how likely the investor is to stick with the allocation throug...

It is well known that people often make irrational financial decisions even in fairly simple situations where they have all the information they need to make a good decision. I have an idea about why we are this way that is so simple that it is very unlikely to be original, but I couldn't find this idea in other writings in a quick search. One simple model of the value (or utility) of money is that each doubling of your savings has the same incremental value. So, if you start with $100,000, dropping to $50,000 is as detrimental as doubling to $200,000 is beneficial. For small gains and losses, the sizes of steps of equal utility differ by less. For example, a loss of only $1000 is as detrimental as gaining $1010 is beneficial. However, throughout most of human evolution, great wealth for a single individual did not exist. Before the advent of storing food, a large kill would only last until the meat rotted or was taken by other hungry people or animals. We are simply not ...

This is a Sunday feature looking back at selected articles from the early days of this blog before readership had ramped up. Enjoy. Most commentators advise investors to shift money from equities to fixed income as they age, and this makes sense. We may disagree on the exact asset allocation percentages and exactly how soon before (or after) retirement to start lightening up on equities, but it seems clear enough that the average 40-year old should have more in equities than the average 80-year old. However, there is a body of academic work that argues that investors should maintain a constant asset allocation regardless of their age. This work is based on what is known as constant relative- risk aversion (CRRA). I’ll show the problems with CRRA in an example below. One consequence of the CRRA assumption is that the optimal asset allocation percentages remain constant regardless of the length of the investor’s investment horizon. Paul A. Samuelson advocates this view in his...

This is a Labour Day edition of the usual Sunday feature looking back at selected articles from the early days of this blog before readership had ramped up. Enjoy. Your reaction to a game show scenario can reveal important information about your attitudes as an investor. It can even tell you what mix of investments are appropriate for your portfolio. Let’s get right to it. You are playing Deal or No Deal and you are down to two amounts: one penny and a million dollars! What is the minimum offer you would accept from the banker? For those not familiar with this game show, here is the situation. You are about to toss a coin. If it comes up tails, you get nothing (and lose nothing). If it comes up heads, you get a million dollars. Just before you toss, someone offers you a sum of money to give up your chance to toss for the million dollars. What is the minimum such offer you would accept? It turns out that your answer depends on how rich you are. Bill Gates would likely ...

Suppose you’re given a chance to bet on the toss of a coin. If it comes up heads you win $20, and if tails you lose $10. Would you take this bet? Given a chance to do it 100 times in a row would you do it? This experiment was actually performed in a coffee shop in Westwood, Los Angeles, as explained by Jason Zweig in his book, Your Money & Your Brain . The average outcome is to win $5 every time you take this bet. But, if you do it only once, you could lose $10 instead of winning $20. What about doing it 100 times? The expected outcome is to win $500. The odds that you’ll actually lose money are less than 1 in 2000. The odds of losing $200 or more are less than one in a million. The odds of winning more than $200 are over 97%. This is an incredibly good bet. Amazingly, two out of three people accepted the one-time bet, but only 43% said they would be willing to repeat the bet 100 times. What are the possible explanations for the 57% of people who turned this down? ...