The summary of their piece suggests that an additional percentage point of investment return at age 55 is nearly 4 times more powerful than at age 30 because of the increased asset base. It further suggests that "[P]lan sponsors should be thinking about how to construct a robust menu of investment choices that will facilitate an improved investing experience and outcome."

Wow!

There are some great phrases in there. I hate to disparage this summary more than others, but take a look at those words: "facilitate an improved investing experience and outcome." Isn't that really the same as saying that plan sponsors should have a fund menu that produces better returns on investment?

Duh!

Back to the whole point of this blog post, though, and I may get fairly technical for a moment, generally, when a pool of assets is invested in such a way as to increase the likelihood of generating higher investment returns, this is done by taking on more risk. At older ages, this goes against conventional wisdom which says to take less risk. Let's take a more geeky look.

For years, much of portfolio analysis has been done, oversimplifying somewhat, using a normal distribution (bell curve) of occurrences, or if you prefer, a mean-variance model. By way of example, this suggests that if you have an investment with a mean rate of return of 7% per year with a standard deviation (square root of variance) of 9%, then your average (and most likely) rate of return is 7% and that roughly 70% of the time, your annual rates of return will fall between -2% and 16% (+/- 1 standard deviation). Suppose you take on more risk for more return so that your mean is now 8% and your standard deviation 11%, then roughly 70% of the time, your annual rates of return will fall between -3% and 19%.

I want to look at three concepts now that perhaps direct added risk to the higher return portfolio (beware, these may not be for the faint of math):

- Geometric versus arithmetic returns
- non-normal distributions
- fat left tails

*Geometric versus arithmetic returns*

Let's consider three sets of annual rates of return, all of which average out to 7% per year (add up the 10 years and divide by 10) on an account balance of $100 (no new money coming in):

- .07, .07, .07, .07, .07, .07, .07, .07, .07, .07
- .115, .105. 095, .085, .075, .065, .055, .045, .035, .025
- .025, .035, .045, .055, .065, .075, .085, .095, .105, .115

In case 1, we wind up with an account balance of 196.72. In cases 2 and 3, we wind up with an account balance of 196.01. This is equivalent to an annual return of approximately 6.9614% which is less than 7.00%.

Suppose we add in annual contributions of $10 per year on the last day of that year. Let's see what happens then. Now, case 1 leaves us with an account balance of 265.80, case 2 with an account balance of 259.76, and case 3 with an account balance of 270.69. The average of cases 2 and 3 is an account balance of 265.23. From this, we can learn two things: 1) the additional risk decreases our expected final account balance; and 2) Wellington is correct that more leverage is attained from later investment returns, whether that leverage is positive or negative.

*Non-normal distributions*

So, you think I've been pretty geeky thus far. It's about to get more geeky.

Perhaps there have been two reasons that we have historically assumed that investment returns have been distributed normally; that is, they follow a normal distribution. First, it's simple. Second, and far more technically, there is this wondrous concept in probability called the Central Limit Theorem. Oversimplifying a whole bunch, it says that a probability distribution of a number of random occurrences that is really, really big (apologies to Ed Sullivan) will revert to a normal distribution. Hold on, occurrences of investment returns are not random. They have outside influences.

*Fat left tails*

So, if these returns are not normally distributed, how are they distributed? Well, we don't have an infinite number of occurrences to look at just yet, but real data suggests that the mode is somewhat right of center (the most common occurrence is a return higher than the mean), but that there are fat left tails (there are a significant number of occurrences (way more than 5%) that fall in the range that we would have expected under a normal distribution to be in the worst 5% of all annual returns.

So, if we return to our cases 1, 2, and 3 from above, there is probably more downside risk than our earlier calculations had demonstrated (I'm not going to re-do the math, just trust me on this one if you can't envision it). This implies that chasing higher returns later in life will give you a greater chance of retiring with a really big account balance, but on average, that same account balance will be lower. In fact, on average, it will be much lower.

*Back to common sense*

At the end of the day, you need to make your own investment decisions. If you think that you can increase your returns by taking on only tolerable additional amounts of risk, then this may be a good strategy. But, unless you really understand the risks that you are taking, don't go chasing after these additional returns late in your career just because you think you have a robust menu of investment choices that will facilitate an improved investing experience and outcome ... whatever that means.

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