## Monday, June 2, 2014

### A Financial Quiz

I enjoy taking financial quizzes, even the type that ask dumb questions like “is taking a vacation financially irresponsible?” Of course, the answer is “it depends,” and people get to argue about it pointlessly. I have a very short quiz with more objective answers.

1. Siblings Amy and Brad inherited \$50,000 and contributed the money to their RRSPs. Amy invests her money in North American stock index ETFs with low fees. Brad chooses his favourite 5 stocks each year and invests all his money in them. If Brad’s choices are essentially random, what is the probability that he will have less money than Amy after 40 years?

A) Less than 50%.
B) 50%
C) More than 50%.

2. A third sibling, Charlie, also received \$50,000 and put it in his RRSP. Charlie decides each month which way the stock market is going and either invests fully in North American low-cost stock index ETFs or he parks it in short-term government debt. What fraction of the time does Charlie have to guess right for him to end up with more money than Amy?

A) Less than 50%.
B) 50%
C) More than 50%.

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1. The correct answer is C. Brad’s chances of ending up with less money than Amy with his stock-picking is greater than 50%. Under one set of assumptions (*), Brad ends up with less money than Amy with a probability of about 65%. It might seem like the odds should be 50% because Brad is choosing randomly from the same set of stocks that Amy owns, but this isn’t the case. For the probability to be above 50%, it might seem like there must be some missing money, but this isn’t the case, either.

To understand why, imagine that there are many Brads all stock picking. Most Brads will lose to Amy and a minority will beat Amy. But the winning Brads tend to beat Amy by more than the losing Brads lose to Amy. This is the nature of taking greater risk; it tends to concentrate winnings in fewer hands. It takes two losing Brads who end up with \$100,000 less than Amy to make up for one winning Brad who ends up with \$200,000 more than Amy.

Investors with a rational level of risk aversion should prefer to invest like Amy. Investing like Brad only makes sense if Brad can pick stocks that are sufficiently above average that it compensates for the extra risk Brad is taking.

2. The correct answer is C. Charlie has to be right with his market timing guesses more than 50% of the time to keep up with Amy. In one experiment, Charlie needed to be right 60% of the time to get the same returns as Amy. In this case, the explanation is simpler. Over the long run, stocks returns are higher than short-term interest rates. If Charlie’s guesses are purely random, he will be out of the stock market half the time and, on average, will lose money compared to Amy who remains fully invested. Charlie has to be right more than half the time just to keep pace with Amy.

The theme of this quiz is that the feeling that stock pickers and market timers are as likely to win as they are to lose is incorrect. Both will lose more than half the time if their guesses are random. They have to have above-average skill among the professional investors who dominate modern stock markets.

* Assumptions: many lognormally-distributed stocks with identical expected returns, 50% standard deviation, and identical correlation coefficients of 0.16, so that the overall market has standard deviation 20%.

1. For question 1, why is it that most random Brads will lose to Amy, and only a minority will beat her? I would have thought that, with random picks, it would be 50:50. I got the same answer as you though, based on the fact that at least Brad would not be paying the (small) MER. I do note that he would have a much higher volatility, with a concentrated portfolio.

1. @Anonymous: I've tried to answer your question a number of different ways on this blog. Let's try a new way. Suppose that Amy averages 5% per year for 40 years, and that some losing Brads average 3%, and some winning Brads average 7%. Amy ends up with \$352,000. The losing Brads end up with \$163,000 (\$189,000 less than Amy), and the winning Brads end up with \$749,000 (\$397,000 more than Amy). Even though the odds are about 50/50 in one year, it takes about two losing Brads to make up for one winning Brad over 40 years.

2. The math seems to make perfect sense. I question the conclusion, though. If the odds are 50/50 in any one year, shouldn't the results be that for every loser, there is a winner? My conculsion would be then that the winning percentages and the losing ones don't follow an arithmetic mean, but a geometric one, so that the NUMBER of winners and losers balance out.

2. Wait a minute Michael, are you saying that if I pick individual stocks the amount I'm likely to gain is more than the amount I'm likely to lose? I'm off to sell my index funds now :)

To be a bit more serious, one of the reasons I've stayed away from active mutual funds is that I've always had the impression that the ones that "beat the index" often did it by a small amount like 0.5% while the ones that fell behind could inflict permanent (relative) losses of 30% or more. But even if that isn't the case I have more than enough reasons to stay the course.

1. @Richard: The same logic works for the lottery. If you win, your winnings are very likely to be more than the cost of the ticket.

The statistics of mutual funds relative to the index are affected greatly by fees. If we could enter a magical world where there are no fund fees, the outperformance of the lucky funds would look a lot better. Back in the real world it's still common for a few small funds to beat the market soundly over 5 years (although we can't identify them in advance).

3. An anonymous comment seemed to get lost:

"The math seems to make perfect sense. I question the conclusion, though. If the odds are 50/50 in any one year, shouldn't the results be that for every loser, there is a winner? My conculsion would be then that the winning percentages and the losing ones don't follow an arithmetic mean, but a geometric one, so that the NUMBER of winners and losers balance out. "

The total number of dollars will balance out, but not the number of winners and losers. The reason for this comes from the fact that everyone with the same investments (but different amounts invested) gets the same percentage return, not the same dollar return. So once an investor gets a lead, that lead tends to grow; winners win by more than the amount losers lose.