A minimum variance portfolio seeks an allocation that is able to withstand stock market crashes as a result of acknowledging the limitations of standard measures of risk. Does the minimum variance portfolio make sense for you as an investor? It depends on a number of factors, but even if you opt for more traditional portfolio construction, understanding the purpose of the minimum variance portfolio and having a better understanding of the flaws of the typical ways we measure risk will both make you a better investor overall. So, read through our article on this concept, absorb the information and then decide to what degree you wish to implement some of the methods based on your own personal situation and preference.

## A recent stock market crash

The U.S. stock market realized the coronavirus threat on February 20^{th}. A month later, the S&P 500 Index had lost about 33% of its value. If you use the standard deviation to calculate the probability of that loss, you’d conclude it would happen only once in several million years. It would be called a “black swan.”

But 2020 didn’t host the first zero-probability market selloff. They occurred during the global financial crisis, the recession surrounding 9/11, and “Black Monday” in 1987 when the market lost 22% in one day, among others. Something’s wrong here because the probability doesn’t match reality. Conventional statistics got something wrong.

Conventional investment theory got something else wrong too. It’s the risk-return relationship. Convention claims a positive relationship between risk and return. But minimum variance (minimum risk) portfolios have produced greater returns than higher-risk portfolios.

Investors willing to reject convention can construct portfolios designed for high returns and greater stability (less risk) during market selloffs, selloffs whose probability has been understated by old ways of measuring risk. Nobody blames you if you think this sounds too good to be true. The rest of this article illustrates the problem with the conventional metric, gives detailed steps on how to responsibly construct and maintain minimum variance portfolios, and provides sources.

## How the Standard Deviation Masks Large Losses

Investors often use the standard deviation statistic to measure risk and calculate probabilities like the probability of the depth of the coronavirus market selloff. The statistic works well when the distribution of returns is normal. But it works like an average to cancel extreme values. The cancellation effectively masks large gains and losses. You can’t have a risk metric mask extreme values like this because risk is about the probability of extreme values.

The figure titled “Historical Returns Compared to Normal Distribution of Returns” helps explain how the standard deviation masks large losses and why it’s important for investors. The histogram with narrow vertical bars represents S&P 500 Index historical returns. The solid bell-shaped curve represents a normal distribution.

The bell curve doesn’t represent real returns. They were simulated. The simulation code prescribed the distribution’s normal form. It also prescribed a mean and standard deviation to match the S&P 500 Index historical returns. The idea was to make the two distributions identical except for the distribution form. This would reveal the flaw in the standard deviation as an investment risk metric.

Many observations cluster near the middle of the histogram, and too many fall outside the range of the normal distribution. Therefore, the histogram does not distribute in normal form. It is these disproportionate allocations – too many in the middle and too many in the tails – that cause the standard deviation to give a bad reading.

The standard deviation can’t detect the shape of the distribution. It simply measures the average size of deviations from the mean; it assumes the distribution is normal.

- The excessive number of observations near the mean of historical returns leads to
*an excessive number*of small deviations from the mean. - This overwhelming number of small deviations cancels out the extremely large deviations from the mean.

The statistic can’t tell the difference between these two distributions even though the risk, represented by extreme returns in the S&P 500 Index, is greater than in the normal distribution.

This result is based on a sample of 1,008 non-intersecting 23 trading-day periods since 1927.

Here’s another way to look at it. A numerical table of the returns simulated in the normal distribution reveals the normal distribution’s greatest loss equals 16.36%. But the S&P 500 Index historical distribution has 15 observations with even greater losses. The standard deviation calculated on the historical distribution completely misses 15 extreme losses because they were masked by the cluster of observations close to the mean. You can see the loss representing the coronavirus loss in the histogram. It’s the left-most observation that falls to the left of the unlabeled -0.3 line.

If the standard deviation masks extreme losses, how can it reliably measure investment risk? It can’t. But when Harry Markowitz wrote the seminal article for Modern Porfolio Theory (MPT) in the early 1950s and when Drs. Sharpe, Litner, and Mossin published the associated Capital Asset Pricing Model in the 1960s they used the standard deviation because it was the best total risk metric available. These are still the dominant frameworks investors use to manage, measure, and interpret investments.

## Risk and Return

The conventional formulation of the investment risk-return relationship holds that risk and return are positively related. The formulation is important because it says if investors use investment portfolios with more risk, their portfolios can earn greater returns.

By the first decade of the 21^{st} century, research had convincingly revealed a flaw in this conventional risk-return relationship formulation. Researchers had discovered lower risk portfolios can earn higher returns. This finding created a tsunami for conventional thinkers, but it presented a valuable opportunity for investors to reduce portfolio losses during selloffs like the coronavirus selloff. By reducing portfolio losses, minimum variance portfolios get a head start when markets start to recover.

## The Minimum Variance Portfolio

“Variance” is one of the other statistics MPT uses to measure volatility risk. “Low risk portfolio” is an equivalent descriptor. But minimum variance doesn’t mean an investor should only use low-risk investments in the portfolio. It’s more flexible than that.

- Even high-risk investments can be suitable minimum variance portfolio assets. Minimum variance portfolios differ from riskier portfolios based on how they allocate assets, not because they reject high-risk investments.
- Minimum variance portfolio allocation is not driven by fixed percentage allocation like the popular 60% stock and 40% bond allocation. Whereas fixed percentage allocation is driven by relative investment value, minimum variance allocation is driven by the risk the assets contribute to total portfolio risk.
- Rebalancing is an essential portfolio maintenance step aimed at keeping portfolio variance at a minimum.

### Minimum Variance Portfolio Construction

As you read about constructing a minimum variance portfolio, a spreadsheet or calculator will be useful. For investors who don’t want to do the arithmetic, retail minimum variance portfolios are available. The last section titled “Minimum Variance Funds” lists some of these funds.

Please keep in mind the calculations here are based on historical returns and risk. They can change, so today’s minimum variance portfolio might not be tomorrow’s. That’s why rebalancing is so important. Rebalancing is like preparing for the future while looking in the rearview mirror. It’s a miserably imperfect process, but it’s better than buy & hold.

The method described here limits the number of details and choices so you can focus on the main steps. The example assumes only four portfolio assets and five possible allocations. Also, this method is not the only way to construct a minimum variance portfolio. A quick search reveals lots of websites with advice.

This example uses four exchange-traded funds (ETFs) to illustrate minimum variance portfolio construction: iShares Core S&P 500 (IVV), SPDR Dow Jones REIT (RWR), Schwab Emerging Markets (SCHE), and iShares 20+ Year Treasury Bond (TLT). These examples are not meant to suggest a real portfolio should be constructed with these assets. They weren’t randomly picked, either. IVV replicates S&P 500 Index performance. Also, RWR’s REIT investment returns often do not change in the same direction at the same time as IVV returns. This attribute suggests a low correlation which sets up potential for greater portfolio diversification. SCHE invests in less-develop markets; the returns also can have low correlations with IVV. TLT invests in U.S. Treasury bonds that produce returns with low correlations to stocks.

Here’s what you can expect in the next few paragraphs:

- Determine allocation permutations.
- Calculate weighted average portfolio risk.
- Calculate total portfolio risk.
- Calculate allocations that would result in an equal contribution of risk by each asset. This is called the “Equal Risk” portfolio.
- Calculate the weighted average portfolio risk and the total portfolio risk of the Equal Risk portfolio.
- Select the minimum variance portfolio from the calculation results.

### Step 1. Allocation Permutations for a Four-Asset Portfolio.

Four allocations or permutations are listed in the table, below:

These allocations appear in the rows labeled “Allocation” and a number one through four. Allocation 1, for example, assigns 0% of value to IVV, 25% to RWR, 25% to SCHE, and 50% to TLT. These four permutations were selected for the illustration from the total of 19 permutations because the combination of the four permutations gives equal weight to each asset.

### Step 2. Risk

Free historical price and dividend data can be downloaded from Yahoo Finance, Nasdaq.com, and others. A rough approximation of semivariance might be the easiest risk to calculate. The method calculates the standard deviation as described in “Stock Market Volatility: What it is, how it’s measured and how to manage it.” However, instead of calculating the standard deviation for an entire historical distribution, the method only calculates the statistic using negative returns.

A better approach might be “downside deviation,” which enables the user to prescribe a “minimum acceptable return.”

As a last resort, you can look up the three-, five-, and ten-year standard deviations for most ETFs and mutual funds at Morningstar.com. These Morningstar standard deviations suffer the flaw of masking large losses.

The best risk metric is expected shortfall (ES), also known as conditional value at risk. This is the metric used in the table. ES values represent the average worst-case scenario, and the “worst-case” is prescribed by the user. Users can apply a closed-form equation or find ES values by simulation.

In this table, all prescribed worst losses fall below the worst 10% of observations. For example, if you have 100 observations of gains and losses, then these ES values would be the average of the 10 worst returns.

There’s one other valuable property of the ES statistic: Investment returns rarely distribute like a normal distribution, so the user can prescribe the distribution form. Examples of normal distribution alternatives include Cauchy, logistic, and Gumbel.

The minimum variance portfolio construction method described here calculates ES risk from two sources. The first source is the risk each asset imposes. Values in the “Risk by Asset” row of the table are the risk values calculated from the last three years of daily returns for each asset. Three years is a minimum for this process. In the IVV column, -0.0455 is the average worst *daily *return IVV should suffer *if* prescribed worst losses fall below the 10^{th} percentile.

The method uses these individual asset ES values to calculate the average weighted ES by allocation. This is a weighted average of the risk imposed by each portfolio asset. For example, the Allocation 1 weighted average of -0.0414 is calculated by multiplying -0.0455 by 0%, -0.0569 by 25%, -0.0480 by 25%, and -0.0304 by 50%. The four products are summed to give the weighted average of -0.0414. (Rounding error could give a slightly different result). These values appear in the column titled “Average Weighted Risk by Allocation.” This risk source is important because later it is used to make comparisons with a benchmark portfolio, the Equal Risk portfolio to be explained later.

The second risk source is total portfolio risk. The ES value is calculated from the daily returns each allocation produced the same way ES was calculated for each asset. These values appear in the column titled “Total Portfolio Risk by Allocation.” This risk source is important because it is used for comparisons with the Equal Risk portfolio too.

### Step 3. The Equal Risk Portfolio

The Equal Risk portfolio allocates each asset so each contributes the same amount of risk. Here are the steps to calculate the allocation to each asset:

- Sum the risk values for each asset.
- Divide each asset’s risk value by the sum calculated in step 1. The sum of this result equals one.
- Divide 1 by each result in step 2.
- Sum the results in step 3.
- Divide each result in step 3 by the sum in step 4. The sum of this result equals one. These are the allocation percentages for each asset in the Equal Risk portfolio.

All of the values you need to execute steps 1 through 5 are in the table.

### Step 4. Calculate Average Weighted Risk and Total Portfolio Risk for Equal Risk Portfolio

These are the same calculations described in Step 2 applied to the Equal Risk portfolio.

### Step 5. Select the Minimum Variance Portfolio

Selection of the minimum variance portfolio compares the two risks of the Equal Risk portfolio with the two risks of the four considered allocations:

- Identify allocations with average weighted risk
*greater*(more negative) than the average weighted risk of the Equal Risk portfolio. The emboldened average weighted risk values in the table satisfy this criterion. - Identify allocations with total portfolio risk
*less*(less negative) than the total portfolio risk of the Equal Risk portfolio. The emboldened total portfolio risk values in the table satisfy this criterion. - Select the allocation that satisfies both comparison criteria.

In this illustration, Allocation 2 satisfies both criteria. If no allocation satisfies both criteria, then the Equal Risk portfolio serves as the minimum variance portfolio. If more than one allocation satisfies both criteria, then choose the allocation with the lowest total portfolio risk.

If this last three-step process seems arbitrary, it’s because an explanation is needed for the selection criteria. First, the Equal Risk portfolio is nothing more than a useful low-risk benchmark. Some research suggests “risk parity” portfolios can produce high risk-adjusted returns. The Equal Risk portfolio is not technically a risk parity portfolio, but it shares the equal risk allocation property of a risk parity portfolio.

The reason for selecting the allocation with the least total portfolio risk – the second step in the selection process – is probably self-evident. The average weighted risk criterion – the first step in the selection process – potentially serves two purposes:

*Stretching*the difference between average and total portfolio risk can increase portfolio diversification. This conclusion is based on the work of two diversification pioneers, Yves Choueifaty and Yves Coignard. One of the conclusions from their work is that diversification resides in the difference between the two kinds of risk; the bigger the difference, the greater the diversification.- Maximizing average weighted risk presents the potential to increase returns.

## Minimum Variance Portfolio Funds

This is for investors who don’t want to do the math. ETFs usually impose lower costs and provide greater tax efficiency than mutual funds. Most available minimum variance funds are ETFs:

- Fidelity Low Volatility Factor ETF (FDLO)
- First Trust Dorsey Wright Momentum & Low Volatility ETF (DVOL)
- Invesco S&P Minimum Variance ETF (SPMV)
- IQ S&P High Yield Low Volatility Bond ETF (HYLV)
- iShares MSCI USA Min Vol Factor USA ETF (USMV)
- Legg Mason Low Volatility High Dividend ETF (LVHD)
- SPDR SSGA U.S. Large Cap Low Volatility Index ETF (LGLV)

For a more complete list and additional information about minimum variance ETFs, ETF.com can be a useful resource. Another minimum variance ETF resource is ETFDB.com.

## Conclusion

The standard deviation masks risk when historical return distributions stray from the normal form. Part of the solution uses risk metrics investors didn’t have in the middle of the last century when investors started widespread use of the standard deviation. Now we know extremely large market selloffs should be expected. Minimum variance portfolios can help investors preserve value during extreme selloffs, so their portfolios are well-positioned to gain value when markets recover.

[1] This result is based on a sample of 1,008 non-intersecting 23 trading-day periods since 1927.

[2] “REIT” stands for real estate investment trust. A REIT holds the stock of companies invested in commercial real estate.