We see poorly diversified investment portfolios frequently. Sometimes they are the result of product-centric sales models, as Matt highlighted in *Anatomy of a Neglected Investment Portfolio*. Other times, ineffective diversification results when people (professionals as well as individuals) believe the wrong things about how investments work together. Practices which seem correct intuitively turn out to be detrimental, or at least limiting. Interestingly, some of these investing practices are based on mathematical myths.

In this blog we describe some of the most common math myths that individual investors and investment professionals believe and follow when diversifying. We illustrate how these myths have come to be prominent, and conclude by offering recommendations for effective, myth-free diversification. If math scares you, don’t fret: this article is written to be practical, not to impress algebra teachers.

#### Myth #1: Risk = Volatility

Unfortunately, and for reasons too complex to explore in this article, you’ve probably been coached to think of investment risk as exclusively price volatility. Granted, nobody likes it when stocks go down, but we’re all happy to have upward volatility. At a minimum, volatility measures should distinguish between good (upward) and bad (downward) volatility. Standard deviation – by far the most widely used investment volatility measure – describes both good and bad volatility.

We believe that there is a better way to define investment risk than volatility. **The best understanding of investment risk is*** the probability that your assets won’t meet your future needs*. Managing risk defined this way requires more work, but it also leads to far better results.

Another problem with volatility measures is scope: all things being equal, investors prefer lower volatility to higher volatility, but how precise is their sensitivity? Can investors tell the difference between, for example, 15% and 10% annual standard deviation? No. Unless a manager can produce stock-like returns, while driving portfolio volatility down to that of cash hidden in a mattress, investors aren’t likely to appreciate reductions in volatility.

With that as background, and because the association of risk and volatility is so pervasive among investors and advisors, we address two myths resulting from the “risk = volatility” framework.

#### Myth #2: More Investments Leads to Better Diversification

Correlation is a statistical measure that describes how two assets move in the same environment. The correlation matrix below shows how returns for Exchange Traded Fund (“ETFs”) representing several different asset classes relate to one another. A high and positive correlation indicates that the asset classes move very similarly, while a low, negative correlation says that they’re responses to the same environment will be very different. In theory, you can lower portfolio volatility by combining assets which have negative or low positive correlations.

Here’s how to read the chart: look at the box in row 2, column 3. Over the last 10 years, approximately 97% of the annual return for US Small-Cap stocks has been driven by the same factors that drive the returns for US Mid-Cap stocks, and vice versa. This makes sense, intuitively. At the other end of the correlation spectrum is the relationship between Short-term Taxable Bonds and US Small-Cap Stocks (row 3, column 9): in general they’ve tended to move reliably in opposite directions. Combining asset classes with low correlations and attractive return potential is a smart way to approach diversification.

As we wrote in Common 401(k) Pitfalls and How to Avoid Them, investors often allocate their retirement plans by simply dividing their investments equally among 10 or 20 different mutual funds. But over-diversifying isn’t limited to just individuals. We recently reviewed portfolios from three different investment advisors and were taken aback at the portfolio composition: each portfolio had more than 15 funds, some with positions as small as 1% of the total. Why so many funds?

This tendency may just reflect a lack of understanding. But among advisors we find a pervasive if seldom acknowledged belief that managing volatility – not earning attractive returns – is their key professional mandate. Setting aside our objection to the first myth above, we still wonder: **is there a mathematical justification for managing volatility by owning 20 funds?**

The chart below shows the returns for several simple asset-allocated portfolios. Using Large Cap Stocks as an example, the way to understand standard deviation is that while the average annual return was 13%, in roughly 2/3 of years, the expected range would be between 2% and 24% (+/- 11.0%).

Reviewing the above charts describing correlation, compound returns and volatility leads us to some simple observations:

**After you add bonds to an all-stock portfolio, you don’t pick up much (if any) volatility-reducing benefit by adding other asset classes.**- Some assets do out-perform others, and portfolio return can be enhanced by adding exposure to high-return assets. While it was not the case in this period, one example is small company stocks’ tendency to out-perform large company stocks over time. However,
- you also increase volatility along with higher returns when moving from moderate return assets to high return assets, and
- simply having high volatility does not guarantee higher returns. Most strikingly: while standard deviation for both a) 100 SPY and b) 60 SPY / 20 EEM / 20 IGSB was 11%, adding emerging market stocks actually lowered returns.

#### Myth #3: Lowering Volatility Creates More Wealth

Read that myth again. It sounds like a straw man argument, but it’s not. This is a real myth which financial institutions use to sell products, and we’ve been running into it for over a decade now. Hedge funds use it to sell their low-volatility strategies. We also saw the same line of thought while reviewing a package of financial planning software. The argument goes like this: if you invest in XYZ low-volatility fund instead of a higher-volatility competitor’s product, you’ll have more money in the end – even if both funds have the same return.

The operative phrase is “the same return.” **This myth exploits the difference between the simple average rate of return, and compounded return.** The chart below illustrates the issue:

In the example, Investment A returns 10% every year for 3 years, growing from $1,000 to $1,331. Investment B has wide variances in annual returns and grows from $1,000 to $1,296 over the same time frame. Both investments A and B had simple average annual rates of return of 10%. Unfortunately, simple average returns don’t tell the story correctly. The compounded rate of return is the only relevant economic comparison. On that basis, investment A returned more than investment B because compounded return was actually higher, not because the volatility was lower. **Bear in mind: return is just a measure of what you have at the end of the period vs. what you started with.**

#### Principles for Myth-free Diversification

Whether you’re a do-it-yourself investor or working with an advisor, here are some simple things you can do to achieve truly effective diversification.

1.**You don’t need more than 4 or 5 funds at most to minimize volatility.** If you own any more funds than that, it will become very hard to measure whether you or your advisor are adding value – which may be the reason for having 20 funds in the first place. Caveat emptor.

2. **Understand that if you start with an allocation to stocks, the single most effective volatility-reducing move you can make is to carve out a piece for bonds.** The bigger the carve-out, the lower the volatility but also the lower the return. Don’t believe that lowering volatility is your primary objective.

3. **Your main objective is to ensure that your future assets meet your future needs when you have them.** This is a far better way to understand your risk as an investor than mean variance measures. Simple college cost and retirement calculators with monte carlo simulators can provide a quantitative sense of how likely you are to succeed in the future.

_{Sources: Peloton Wealth Strategists Research, Y-Charts}