Are retail investors taking more risk than they realize? The following discussion is directed towards individual investors who are saving for long term goals such as retirement, primarily using a buy-and-hold approach in accounts like a 401k. The conventional wisdom says that the more time until you need your principal, the more risk you can afford to take, and therefore you should invest more in stocks. What is the logic behind this? Let’s say someone offers you a $100 “investment” based on a coin flip. Heads, you earn 33%, tails, you lose 25%. So after one flip you’ll either have $133 or $75. Since each outcome is equally likely (50/50), on average your $100 investment will return $104, or 4%. That’s not too bad. Let’s imagine that each coin-flip in this example is equal to one year of investing in the stock market. Here’s what a simple payoff tree looks like:

However, you can make this investment multiple times. Every time you flip the coin, there is still a 50/50 chance of heads/tails, and the payoffs remain the same. Since the coin is random, you figure that the more times you flip it, the more likely it is that the total number of heads versus tails will even out. This is equivalent to investing over multiple years. Your financial adviser confirms your intuition and tells you that historically, the variation of stock market returns decreases as you invest over longer period of times. It’s a reasonable-sounding (but flawed) assumption that the more times you flip, the less likely you are to get a bunch of heads or tails in a row. But you believe what your financial adviser tells you, so you expect the risk of this investment to go down if you’re willing to flip it four times instead of just once. Here’s what the payoff tree looks like now:

The average ending portfolio value (probability weighted average) is now $117, which is your 4% expected return compounded 4 times. However, something strange has happened with the ending values. The dispersion between the best and worst cases has increased, from $313 to $31. You also notice that there’s a 70% chance (6% + 25% + 38%) you don’t do any better than break even, and a 31% chance (6% + 25%) that your $100 investment is worth $56 or less after 4 flips. If this money was needed to support you in retirement, would you consider these to be good odds? Is your coin-flip investment (stocks) really better than a less risky alternative investment that might guarantee you an ending value of $106 over four years but with no downside risk? This graph from the Wall Street Journal shows that anyone who started investing after 1996 would’ve been better off in a no-risk money market account than in a broad market index:

Consider the coin flip investment similar to investing in stocks. As you increase the number of investment periods above 4, the trends illustrated in this overly simplified example only continue to increase. The best case gets much better, and the worst case gets much worse. The risk of a substantial shortfall increases as you flip the coin more times. This is contrary to the traditional wisdom that a longer investment horizon reduces the risk of volatile assets. I’ve written about the myth of time diversification before, but this example clearly illustrates how the risk to your portfolio’s final value increases, rather than decreases, over time. At the end of the day, the final value is what matters – standard deviation and expected returns are great, but you can’t eat them.

My point is not that investors should avoid stocks entirely, but that they should be aware of the fact that investing for the long term does not eliminate the risk of stocks. Riskier assets have a higher expected return, but not necessarily a higher realized return. You are not guaranteed a higher return just because you took more risk. If investors are going to be compensated for taking risk, then there has to be a real risk of loss over the long term or else why would the risk premium exist? If you still don’t believe me, take a look at a graph of the Nikkei over the last 25 years:

If you think I’m being overly pessimistic, or if you think the Japanese economy is sufficiently different from the US economy, then how about the returns on the S&P 500 from 1966-1982?

## 3 comments:

Interesting post, but wrong :)

The numbers you picked are equal, in the sense that the most likely outcome is H=T so the 33% and -25% returns all cancel each other out. In any string of flips you expect the outcome to be break-even, not $117. Do the simulation and you'll see this is the case. There is something wrong with the 4% expectation you mention.

Of course in that case we expect equal chances of being above and below the break-even. Your modeled investment is terrible in the sense that you'll get your breakeven a lot more easily by doing nothing. It's a lot more risky, so it's good that you conclude it to be so!

Also, you consider the spread out to the tails as the "risk". In reality businesses can all go bankrupt and the value fall to zero so the downside is a lot worse than you contemplate. But businesses or portfolios don't go bankrupt that often.

Your car could blow up, but you don't consider that a real probability. If you did you would say "If I drive my car I could experience every outcome from a fatal crash to arriving safetly, with the likely expectation to be something worse than arriving safely." But you still drive.

Regarding risk premium: Risk premium is NOT something that you "get" from risky investments. Risky investments with bad returns are just bad. You should be thinking that you are demanding a risk premium, not accepting it. If you are offered a risky investment then think "I'm going to need another 10% before I go for this over something safer."

I like your post though, and the illustrations and analysis.

You're correct in that the most likely (mode) outcome over 4 flips is $100, but to compute the expected (mean) terminal value, you have to take the ending values multiplied by the probability of each scenario. That's why the expectation is $117, not $100. Those probabilities come from Excel's BINOMDIST function.

As for the exact numbers, this is a simplified model to make a point - I don't think the stock market actually returns +33% or -25% with a 0.5 probability each year.

I also don't expect that the entire economy is going to collapse and all firms are going to go bankrupt. But a few years of negative returns is certainly possible, and I wanted to illustrate the effect this would have on a portfolio.

I see where I went wrong. The most likely outcome is zero by far, but the expected value is $117.

In my opinion, questioning risk statistically is a bit strange because the definition of risk in finance is related to the movement of prices. It's something you measure.

It's like saying "The true length of your finger is not what you measured with a ruler." Umm. Yes it is :)

The definition of risk for common people might be different than the simple financial definition. Catastrophic loss might cost far more than just the loss on the security for example.

As well, stocks do not really behave normally. Prices over time have way too many "statistically unlikely" extreme events.

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