Somethin’ from Nothin’
A common catch-phrase used to explain the importance of diversification is “Don’t put all your eggs in one basket”. This anecdote of spreading out risk is intuitive: One basket might break, one might be stolen, and one might end up in cold storage with a lost key. The eggs and baskets do well to illustrate the power of diversification to eliminate risk, but I don’t think the analogy does the term justice.
By focusing only on distributing risk, we fail to capture most investors’ primary interest: seeking returns. With the following example, I’ll aim to uncover where the “free lunch” of diversification comes from — hopefully we won’t break too many eggs along the way.
To begin with, let’s invent an asset who’s long-term holding period return is zero — meaning that a dollar invested in this asset will still be worth a dollar at some indefinite point in the future. We’ll ignore inflation for the duration of this article. The second characteristic we’ll assign this asset is variance. By this, I mean to say that the value of the asset at any given point in time may drift higher or lower.
Here’s an example of an asset with varying valuation day to day but a long-run return that will converge back to $100:
So while the value of our investment will change, we know definitionally that its worth will converge back to the initial amount invested if we wait long enough. The first question to ask: Is this asset attractive?
Most would recognize that taking risk without a reward is irrational — especially if there is a more efficient way to earn that return. For example, if I can preserve the value of my investment in a savings account with zero risk, that choice minimizes my risk while preserving my return and is therefore most rational. While this is true narrowly speaking of this one imaginary asset, we will now introduce diversification to see what shenanigans we can unlock.
Instead of only one imaginary asset, we’ll expand the arsenal to 5 uncorrelated assets and see what happens when we put them all together. Let’s assume all 5 assets individually have the same characteristics as the first we mentioned. They each would preserve the value of your investment over long-periods of time as a stand-alone investment, but would also experience variance along the way. We’ll give each asset 1/5 of our investment to produce the following portfolio:
Admittedly, this portfolio feels like it’s headed nowhere fast, but let’s circle back to an important assumption about these assets. Recall that we assumed an investment in any one of these assets would retain it’s value over the long-run, resulting in a long-term return of 0%. That 0% return is referring to the compounded annualized growth rate, often shortened to “CAGR”. In other words, this return accounts for the fact that the current period’s return is impacted by the previous period’s return — and will either start the next period higher or lower depending on the current period’s return. This compounding effect helps when returns in the previous period were positive (higher starting balance) and also hurts when returns in the previous period were negative (lower starting balance). Additionally, the more variance in the return series of an investment, the more harmful to an assets CAGR. Recall that our 5 investments have significant variance, exhibiting returns 30% above average or 30% below average about 68% of the time in any given year. Said another way, the more we increase the volatility of an asset, the lower the CAGR will be, all else held equal. This concept is referred to as volatility drag — the reduction in compounded returns caused by the volatility of the time series. But this realization should give us pause… If these assets are so volatile — and volatility hurts CAGRs — how can we hack this characteristic to our benefit?
Let’s work backwards for a moment. If we know each asset has a CAGR of zero and also that each asset‘s return series is volatile, it follows that the return must be positive before accounting for the impact of volatility. We can calculate the return — excluding volatility drag — as the sum of each period’s return divided by the number of periods. In other words, the return excluding volatility drag is simply the arithmetic return. Since our imaginary assets don’t have real periodic returns to take the average of, we will use a handy formula for converting CAGR to arithmetic returns. Here’s the formula for those who were wondering:
Applying this formula to various imaginary assets reveals a pattern. For various assets with a CAGR of 0% each, the greater the volatility of the asset, the greater the arithmetic return of that asset:
Lest there be confusion, the higher volatility isn’t causing the arithmetic return to increase. Recall that we are working backwards, so the correct way to interpret this is flipped: The greater the volatility of an asset, the higher the arithmetic return needs to be in order to simply break even on the investment over time. This is the “drag” part of volatility drag. So, we now know something new about our investments; they each must have a positive arithmetic return in order to make up for the volatility of the assets. The below table adds this information to our imaginary assets:
As great as this new information is, it’s not yet actionable, since we can’t actually earn the arithmetic returns. The realized return for holding each of these assets will be impacted by volatility, so the only way we can achieve the arithmetic return is if we destroy the volatility. Hmm, if only there were a way to reduce the risk of an asset…
It’s about time we get ourselves some lunch. You’re paying right?
The way we reduce the volatility of assets is by combining them with other uncorrelated assets to reduce the portfolio risk. I like to compare this concept to noise canceling earphones. Sure, you can dampen the volume of the outside world with insulation, but it turns out that adding more noise actually reduces the perceived volume if the additional noise is out-of-phase. This is because the sound waves are able to cancel each other out. So let’s cancel out some noise in our portfolio.
Skipping over some annoying math, a portfolio with 5 equally-weighted assets that are uncorrelated, where each has a risk of 30%, will result in a portfolio risk of 13.42%. Moving to returns, we know each asset has an arithmetic return of 4.5%, so the portfolio also has an arithmetic return of 4.5% (the portfolio’s arithmetic return is just the weighted average of the assets’ arithmetic returns). But this is where diversification begins to do its magic. We’ve already seen that the portfolio has less than half the risk of investing in any of the individual assets. But recall that the volatility also impacts the CAGR. Using our new and improved portfolio risk level, we can calculate the CAGR of a portfolio with an arithmetic average return of 4.5% and a volatility of 13.42% to be…
3.6%
In other words, combining 5 volatile and uncorrelated assets that each have a CAGR of 0% annually results in a portfolio that is able to compound a positive return of 3.6% annually.
We’ve finally arrived at the other side of diversification that the egg basket analogy fails to capture. As great as reducing risk is through diversification, we shouldn’t overlook the very positive impact it can have on improving returns as well. While the above example may seem completely abstract with no real world corollaries, there are arguably dozens of such assets in the investing world. An often-overlooked asset class full of volatile instruments with low CAGRs are the individual precious metal, industrial metal, energy, and agricultural contracts within the commodities basket. Here, you can find plenty more than 5 uncorrelated assets with similarly high levels of volatility.
In a world that can demand so much and give away so little, it behooves us to reach out and grab what others won’t bother to pick up. Let diversification pay for your lunch — perhaps an egg-salad sandwich?
