# Financial Geekery

Some of you are doubtless reading this and thinking, "It's simple: does my 401(k) balance go up every year?" Sadly, it's not that simple. Not by a long shot. Here are some things you need to know.

Risk v. reward

First and foremost, there are two primary characteristics to measure your investment's performance: return and risk. Annualized returns are important, but risk is equally important.

Let's say you have two imaginary investments: one returns a flat 4% every year, and one returns an average of 6% each year, but has a standard deviation of 15%. If you browse r/personalfinance for any length of time, you'll find that the common advice would be to favor the 6% investment, e.g. invest in your portfolio rather than pay off your mortgage. The numbers vary, of course, but the advice is depressingly consistent: go with the higher return, and ignore volatility.

It seems like simple math, right? 6% > 4%, and therefore is a better investment. Not so -- and for multiple reasons!

First: arithmetic and geometric returns are not the same. Arithmetic returns are straightforward to calculate: if my investment returned 1% one year, then 4%, then 0%, then 15%, the arithmetic return is the average of all those: (1+4+0+15)/4 = 5%.

Geometric returns, however, work a little differently: they take the end portfolio value and calculate the average return that would be necessary to get there from the beginning portfolio value. For example, let's say you have a portfolio worth $1,000 that goes through the four years of volatility mentioned above. You'd end up with $1,208. However, the rate of return that gets you from $1,000 to $1,208 in four years isn't 5% -- it's 4.84%.

What's the deal? The answer is volatility. If there were no volatility, geometric and arithmetic returns would be the same. However, whenever there's nonzero volatility, geometric returns will always be less -- and the greater the volatility, the greater the loss.

The takeaway here is this: always pay attention to whether you're dealing with arithmetic or geometric returns. In most cases, geometric returns will be more valuable in your calculations, especially if you're using straight-line projections.

Second: your "sequence of returns" matters. In the example above, I used a particular sequence of returns: 1%, then 4%, then 0%, then 15%. No matter how you order them, the arithmetic return is 5%, and the geometric return is 4.84%. But your portfolio doesn't exist in a vacuum -- you're (ideally) putting money into it each year, and in retirement you'll be taking money out. If the "bad year" (0% in this example) occurs the year you retire, the end results can be dramatically different than if it occurs the year you kick the bucket.

This is precisely where Monte Carlo simulations come into play. By running projections on a thousand different sequences of returns, they can give you a good sense of the range of possibilities.

(Side note: Monte Carlo simulations are one of the exceptions to the comment I made earlier about using geometric v. arithmetic returns. Because they separate expected returns from volatility, you should use the expected arithmetic return for their input, rather than geometric. Otherwise, you're effectively double-counting volatility.)

Back to the original question of whether the volatile 6% or flat 4% investment is better: Monte Carlo can help you make a better apples-to-apples comparison. By taking volatility and your specific financial context into account, they can help you determine whether your goals are best served by a lack of volatility (thus decreasing the number of scenarios where bad timing wrecks your plan) or increased returns (thus decreasing the number of scenarios where low returns wreck your plan).

**The Sharpe ratio**

The Sharpe ratio is a cool little metric that can be used to measure performance relative to risk. The idea is relatively simple: you take the arithmetic return above and beyond the "risk-free rate" (often defined as the return on US Treasury bills) and divide it by its volatility, as measured by standard deviation. So if T-Bills are returning 2% and you have an investment with an arithmetic return of 7% and a standard deviation of 10%, your Sharpe ratio is (7%-2%)/10% = 0.5.

"Awesome!" you say. "Now I can make an apples-to-apples comparison between two investments without all this Monte Carlo stuff!" Well...sometimes.

For example: the Sharpe ratio can be a useful tool to determine whether Mutual Fund A has higher returns than Fund B simply because it's taking on more risk, or because it's efficient at maximizing risk-adjusted returns (see this post on DFA for an example).

This doesn't mean that maximizing the Sharpe ratio should be the end-all be-all for your portfolio! Take a look at this risk/reward curve:

You'll notice that there's a point of diminishing returns, where increasing risk gets you less and less incremental gain. If your goal is just to maximize the Sharpe ratio, you'd pick a portfolio that's right on that inflection point. That doesn't take your financial context into account, though; depending on your situation, you may just want those higher returns or that lower volatility, even it's not maximally efficient as far as the Sharpe ratio is concerned.

And by the way: this is a trap investment managers can fall into, as well. We can become hyper-focused on risk/reward efficiency, to the point where we undervalue investments on the far left and right hand of the spectrum. When you're talking to your investment manager, make sure they remember that their client is you, not your portfolio!

**Max drawdown**

Another characteristic to bear in mind is this: in the case of a market downturn, how much of my investment can I expect to lose? The Great Recession was terribly painful...but it was by no means unfathomable in the realm of investment theory. When looking at an investment and measuring your risk tolerance and risk capacity, ask yourself this: how did it perform during the Great Recession, and could I handle that if another one occurred this year?

"Past performance may not be indicative of future results"

We see this disclaimer everywhere, but I'm convinced that we don't fully understand it. I've seen ostensible fiduciaries such as 401(k) committees and church endowment funds use 5-year returns as their basis for determining investment managers. And while that's better than using the past year's return...no. Just no.

Why do stocks generally outperform bonds? Because there's more risk, more volatility. But volatility means that I have to include the word "generally" in that sentence, because stocks do not, in fact, always outperform bonds, even over rolling 10-year periods!

In other words, volatility means we have to accept the fact that the future is uncertain. We have to change our way of thinking, to accept the fact that the best we can do is tilt the odds in our favor.

Given that this is true, why in the name of Markowitz would we use a single 5-year return datapoint to pick a mutual fund or investment manager? That would be like walking into a casino, finding a blackjack player who got 21 by hitting on an 18, and saying, "That's a great strategy! I'm always going to hit on 18!" (And then being thoroughly shocked when you bust a majority of the time.)

Also, people have a habit of comparing their portfolio against the S&P500. Which is understandable, given the fact that the financial media throws that benchmark around quite a lot, but...let's dig a bit. A main goal of portfolio design is creating an efficient mix of non-correlated asset classes (and if that phrase was gibberish to you, follow the link). If this is true, then an efficient portfolio is likely going to contain, within itself, the companies that make up the S&P500...and small companies, and international companies, and intermediate-term bonds, and short-term bonds, and likely other asset classes as well. Why? Because various asset classes outperform other various asset classes at various times. And if this is true, then by the very nature of what we're doing, there will be times that the S&P500 outperforms the rest of the portfolio as a whole!

"Wait a minute!" I hear you cry. "Are you telling me that I have to wait 15 or 20 years before I know whether I have an outperforming strategy -- and even then I can't be 100% sure?" I hate to break it to you, but: yes, I am. This is why I recommend systematic, robust, research-backed strategies based on decades of data, whose benefits won't vanish once everyone knows about them.

The hard truth

So, I'm sad to say, it's not as simple as seeing whether your portfolio is worth more than it was last year, or whether mutual fund A had higher returns than mutual fund B. To truly evaluate an investment, you need an array of tools: an understanding of arithmetic v. geometric returns, Sharpe ratios, an appreciation of uncertainty, Monte Carlo simulators, and others.

And that's why I post here: to bring you those tools. Hopefully the links in this article have given you plenty to chew on, but if you find yourself with a question not answered, then by all means: send me an e-mail. Most of my articles are inspired by client and colleague conversations!

Britton is an engineer-turned-financial-planner in Austin, Texas. As such, he shies away from suits and commissions, and instead tends towards blue jeans, data-driven analysis, and a fee-only approach to financial planning.

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