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### Simple Math and the Power of Compounding

January 18, 2019 | Posted in: Investing

We spend a lot of time talking to our clients about diversification, downside protection, and the long-term value of reducing short-term volatility.  Those concepts are important to us, part of our DNA as a firm and as investment professionals.  But why?  In simple terms, if over the long term markets go up (which they do, or else who would invest in them?), why do we care about diversification and dampening risk?  There are two parts to the answer.

The first relates to the tricky crossroads between human behavior and investing. To be blunt, the vast majority of humans aren’t very good at investing largely because human psychology often leads us toward the wrong decisions at the wrong times.  Herd mentality, natural biases, pride, and good old fear and greed get in the way of independent thinking, objectivity, discipline, and holistic decision-making.  If you’re trying to time the markets with individual stock picks or wild swings from cash to equities and back again, repeated correct (or even just correct-ish) buy and sell decisions must be made, and that’s pretty tough to do with an imperfect toolbox.  No, humans and investing are a difficult marriage, so where you can create an all-weather investment plan that allows you to sleep comfortably at night and reduces the opportunity to make emotional point-in-time investment decisions, you should.  And diversification and risk management come in handy in that practice.

The second part to the answer is more straightforward and critically important: MATH.

“Compound interest is the eighth wonder of the world.”

“Compound interest is man’s greatest invention.”

Many of us have heard one of these quotes attributed to Albert Einstein even though there’s a great deal of debate as to whether he ever actually said either of these things.  Either way, the point is a very good one: there is power in compounding.  Let’s do some simple math exercises to test that out.

Exercise #1: You invest \$100 for two periods and can choose to receive either (1) +10% returns for both periods or (2) +20% the first period and 0% the second. Which do you choose?  Your first instinct may be that it doesn’t matter, since in both cases your average arithmetic return is +10% per period.  Let’s see if that’s right.

Option 1: \$100 gains 10% and becomes \$110; \$110 gains 10% and becomes \$121.
Option 2: \$100 gains 20% and becomes \$120; \$120 earns 0% and stays \$120.
So that first instinct was wrong, and it’s wrong because of compounding. A \$1 difference might not seem like a lot, but extrapolate that out over 20 periods, or 200 periods, and the gap between the steady return stream and the volatile return stream widens significantly.

But what happens when returns are negative? Exercise #2: You invest \$100 for two periods and can choose to receive either (1) -10% returns for both periods or (2) -20% the first period and 0% the second.  In both cases your average arithmetic return is -10%.  Which do you choose?  Given the results of the first exercise, you might think that compounding would work against you now that we’re heading south.  Let’s see.

Option 1: \$100 loses 10% and becomes \$90; \$90 loses 10% and becomes \$81.
Option 2: \$100 loses 20% and becomes \$80; \$80 earns 0% and stays \$80.
This time compounding spared you a little bit of pain. This means the steadier return stream worked out better both in a positive return environment and a negative.  Sounds pretty good!  But we haven’t yet looked at the most striking scenario: where the markets move up and down.

Exercise #3: You invest \$100 for two periods and can choose to receive either (1) +5% the first period and -3% in the second, or (2) +20% the first period and -18% in the second. In both cases your average arithmetic return is +1%.  Once again, which do you choose?  You’re onto me now, but let’s do the math anyway.

Option 1: \$100 gains 5% and becomes \$105; \$105 loses 3% and becomes \$102.
Option 2: \$100 gains 20% and becomes \$120; \$120 loses 18% and becomes \$98.
I use this example to illustrate two things. First, look closely at Option 2.  We gained a big percent, then lost a lesser percent, yet ended up with losses overall.  This is really important.  For every percent of gain I earn it takes less than a percent of loss to take me back to even.  And it doesn’t matter in what order the gains and losses come, the results are the same: e.g., start with \$100, lose 18% down to \$82, then gain 20% up to \$98.  So on the flipside, for every percent of loss I sustain, it requires more than a percent of gain to get back to even.  This is why protecting on the downside is so important: losses are much tougher to make up and can destroy portfolio value much faster than gains can build it.

Second, in the real world the market moves up and down, so a scenario with both positive and negative volatility is the most applicable to an actual portfolio. And here the less volatile return stream shows even more value than in the first two exercises.  The difference between the end points in Exercise #3 is meaningful (gains vs. losses) for just two periods.  To give you an idea of how wide a gap that can become over the longer term, if you start with \$100 and repeat each return series over a 50-period window, you end up with the following:

Option 1: \$158
Option 2: \$45
That’s huge. This is why you’ve probably heard us say, at some point, that less volatile short-term returns compound to a better long-term result, all else being equal.  We say it because it’s true.   And it’s true because of simple math, specifically the power of compounding.