*This post is available as a PDF here. *

# Summary

- Under a simple model of asset prices, expected returns and volatilities can be used to calculate expected maximum drawdowns over a given timeframe.
- However, these expected drawdowns do not line up with the drawdowns investors have experienced. Simple models have underestimated drawdown risk in equities, low volatility equities, and income strategies, and overestimated historical drawdown risk in trend following strategies.
- Incorporating autocorrelation can improve drawdown estimates, resulting in a more transparent assessment of the drawdown risk, which can lead to portfolios that are more closely in line with investor goals and risk tolerances.

Volatility is the most ubiquitous measure of risk. This likely stems from its importance in Modern Portfolio Theory as *the *measure of risk.

Yet, a volatile asset is not necessarily risky in the traditional sense. We have covered this in a number of commentaries, including *Volatiliy and “Crashing Up”* [1] and *Volatility: Good, Bad, or Indifferent*?“[2]

Upside volatility is fine and is often desirable. It’s the downside that hurts.

When analyzing an investment, one option is to break out the downside volatility and use that as a measure of risk (e.g. in Sortino ratios). Another option is to focus on drawdowns, which more closely align with what investors experience emotionally.

In the case of drawdowns, the intuition is that higher volatility implies higher drawdowns. This intuition is correct, but there is an important caveat that may be critical in portfolio construction.

**The Model: A Refresher on Geometric Brownian Motion **

Asset prices are often modeled by a type of random walk: the geometric Brownian motion (GBM), which is a convenient way to simplify calculations (e.g. the Black-Scholes model and many Monte Carlo simulations). However, GBM makes some assumptions about the structure of asset class returns:[3]

- They are log-normally distributed – There is evidence of fat tails in returns, but this is often mitigated by working with monthly, quarterly, or annual data.
- They are identically distributed – The parameters (i.e. expected returns and volatility) are assumed to be constant. In the real world, expected returns morph over time (e.g. as valuations change), and volatility tends to cluster.[4]
- They are independent – This means that one return does not influence the following returns, which we know is not entirely true because mean revision and momentum are evident in many asset classes.

All in all, GBM is a simple model that works in many cases as long as we remember that: “All models are wrong; some models are useful.”

From this model, we can determine the distribution of drawdowns that we would expect given a set of parameters (expected return, volatility, and time frame). From the distribution, we can then calculate the expected maximum drawdown.

**The Relationship Between Volatility and Drawdowns **

In the world of mathematical finance, deriving a deterministic formula can be a wonderful thing. But the formula for the distribution of maximum drawdowns given a set of parameters is…complicated.

We could spend a page listing equations full of trigonometric and hyperbolic functions, improper integrals, infinite sums, and numerically solved eigenvalues.[5] Or we could utilize a quick Monte Carlo simulation to show the distribution.

We will choose the latter.

*Maximum Drawdown Distributions for a Range of Volatility Levels (Expected return = 5%, T = 120 months)*

*Calculations by Newfound Research.*

As volatility increases, the distribution of maximum drawdown gets flattened out, and the expected drawdown increases. As the volatility increases, the ratio of volatility to expected maximum drawdown also increases.

*Expected Maximum Drawdowns for a Range of Volatility Levels (Expected return = 5%, T = 120 months)*

*Calculations by Newfound Research. *

To build on the intuition, we can also look at how drawdowns change for different values of expected return and length of observation timeframe.

*Maximum Drawdown Distributions for a Range of Expected Return Levels (Volatility = 15%, T = 120 months)*

*Calculations by Newfound Research. *

* *

*Maximum Drawdown Distributions for a Range of Timeframes (Expected return = 5%, Volatility = 15%)*

*Calculations by Newfound Research. *

As expected, higher expected returns improve the drawdown profile and larger drawdowns become more likely over longer timeframes.

**Putting the Model to Use **

In our presentation on constructing an unconstrained sleeve, we included to following strategies, which have common data since 1992:

- Equity (MSCI World index)
- Equity – Trend (Invested in MSCI World index or short-term U.S. Treasuries based on a simple trend following rule)
- Equity – Minimum Volatility (MSCI World Minimum Volatility Index)
- Macro – Trend (Salient Trend index)
- Macro – Risk Parity (Salient Risk Parity index)
- Macro – Contrarian (30% MSCI World index, 30% Vanguard Total Bond Market fund (VBMFX), 40% in either of the two assets based on a simple mean reversion rule)
- Macro – Income (80% in the HFRI Relative Value Fixed-Income Corporate index and 20% in a 7-Year Constant Maturity U.S. Treasury index)
- Bond – Intermediate-Term U.S. Treasury (7-Year Constant Maturity U.S. Treasury index)

Over the nearly 25 years of data, we have the following parameters to plug into the drawdown model along with the realized maximum drawdowns that we hope to match.

*Strategy Parameters from December 1992 – June 2017*

*Source: MSCI, HFRI, Salient, and St. Louis Federal Reserve. Calculations by Newfound Research. Performance is backtested and purely hypothetical. Past performance is not a guarantee of future results. Performance is gross of all fees. Returns include the reinvestment of dividends, capital gains, and other distributions. Data from December 1992 to June 2017.*

All asset classes had strong positive average returns over the period, and all of the unconstrained strategies outperformed equities on a risk-adjusted basis, with most besting them on an absolute basis as well.

Maximum drawdowns in the other asset classes were also substantially lower than that of equities, in general.

The question is: if we had known the expected returns and volatilities *a priori*, how accurately could we have forecasted the maximum drawdowns that were actually realized?

Could we have predicted, for instance, that equities could lose 54% and that Income could be down nearly 20% despite less volatility and a higher return than intermediate Treasuries?

*Actual and Expected Maximum Drawdowns*

*Source: MSCI, HFRI, Salient, and St. Louis Federal Reserve. Calculations by Newfound Research. Performance is backtested and purely hypothetical. Past performance is not a guarantee of future results. Performance is gross of all fees. Returns include the reinvestment of dividends, capital gains, and other distributions. Data from December 1992 to June 2017.*

* *

The chart shows that the drawdown model most notably underpredicted drawdowns for Equities, Minimum Volatility, and Income and overpredicted drawdowns for Bonds and Trend Following.

**The Assumption of Independent Returns **

This brings us back to our initial GBM model assumptions. The shortcoming in this case is with the third assumption, the independence of returns.

If our returns are independent, volatility can be calculated by taking the standard deviation of monthly returns and annualized by multiplying by the square root of 12. If the returns are not independent – that is, if there is momentum or mean reversion – then annualizing in this simple way is inaccurate.

The dependence of returns from period to period can be measured by *autocorrelation*, which is similar to correlation except that it takes the correlation of an asset with itself, merely shifted by a specified amount called the *lag*.

The chart below shows the autocorrelation function for the MSCI World index.

*Autocorrelation in the MSCI World Index*

*Source: MSCI. Calculations by Newfound Research. Performance is backtested and purely hypothetical. Past performance is not a guarantee of future results. Performance is gross of all fees. Returns include the reinvestment of dividends, capital gains, and other distributions. Data from December 1992 to June 2017.*

* ** *

The zero-lag bar is always equal to 1 since it represents the correlation of the returns with themselves, so from here on out, we will leave it off the charts.

The other autocorrelation values appear small, but with a 90% significance cutoff of ±0.1, the lag 1 autocorrelation is significant. Since it is positive, it represents a momentum effect.

**Autocorrelation and Volatility **

The effect of autocorrelation on volatility can be approximated by the following equation:

where is the *i*^{th} lag autocorrelation and *k* is the number of lags we are considering. When all the autocorrelations are 0, this reduces to the familiar square root of 12 rule. The approximation is valid when the number of data points (in our case 295 months) is much larger than *k*.[6]

Focusing on lags of 6 months or less as in Lo (2002)[7], we can see that many of the strategies presented before exhibit significant autocorrelation – positive in the case of Income, Equities, and Minimum Volatility Equities and negative for the trend following strategies.

*Autocorrelation through Lag 6 for Each Strategy*

*Source: MSCI, HFRI, Salient, and St. Louis Federal Reserve. Calculations by Newfound Research. Performance is backtested and purely hypothetical. Past performance is not a guarantee of future results. Performance is gross of all fees. Returns include the reinvestment of dividends, capital gains, and other distributions. Data from December 1992 to June 2017.*

The totals of the autocorrelations show the combined effects across the lags.

*Sum of Autocorrelations through Lag 6 for Each Strategy*

And the chart below shows the standard estimate of volatility compared to the adjusted estimate that accounts for autocorrelations.

*Volatility with and without Autocorrelation Adjustments*

Finally, we arrive at our initial chart of drawdowns with the adjusted estimate included.

*Actual and Expected Maximum Drawdowns with and without Autocorrelation Adjustments*

For all strategies, with the exception of risk parity, the adjusted drawdown estimate is closer to the actual maximum drawdown over the period.

The Equities, Minimum Volatility Equities, and Income estimates have increased to a level much more representative of what investors actually experienced. The trend following strategies and bonds were not penalized as severely after they were given credit for their negative autocorrelations.

**Conclusion: Takeaways from the Historical Analysis and the World Going Forward**

From this historical analysis, we can identify five important takeaways:

- Positive autocorrelation in equities (momentum) can lead to large drawdowns. Managing this risk through diversification and prudent risk management strategies is critical to avoid, or at least mitigate, scenarios like 2008.
- Low volatility is not low risk. Accounting for the autocorrelation captures some of this effect, but the realized drawdown in low volatility equities in 2008 was still larger than the model predicts.
- Significant positive autocorrelation in income assets can produce large drawdowns. Downside protection is important to manage this risk.
- Negative autocorrelation in trend following strategies can lead to an overestimation of drawdown risk, which may lead to either under allocation or exclusion from a portfolio. These strategies may be better at managing risk than their volatility indicates.
- The flight to safety nature of bonds is not captured by accounting for autocorrelation. Including bonds within a portfolio can make sense as an equity hedge regardless of the interest rate environment. Even with rising interest rates, the speed at which bonds react to large sell-offs in equities has historically provided a ballast to a portfolio during market crises.

Since much of the autocorrelation in specific strategies is either driven by behavioral biases (e.g. momentum) or structural features (e.g. trend following[8]), these points will likely be relevant going forward.

As the timeframe continues to expand into the future, as we showed in the probability distribution graphs at the beginning, the expected maximum drawdown increases.

Asymptotically, the expected maximum drawdown grows at a rate of log(*T*), so it never stops increasing, in theory, although it does slow down.

Perhaps the biggest issue is that low growth expectations in equities can magnify drawdowns, making diversification and risk management even more important.

Using the “Yield & Growth” capital market assumptions from Research Affiliates, which assume that there is no valuation-driven mean reversion (i.e. valuations stay the same going forward), the adjusted average nominal returns for U.S. equities is 5.3% compared to the historical value of 9.0%.

Over a 10-year period, keeping volatility at 15%, the expected maximum drawdown under the historical return scenario is 48%. Under the forward return assumptions, it is 57%. If we factor in autocorrelations at their historical levels, this expected maximum drawdown increases to 67%.

While 48% and 67% drawdowns might not seem that different – both are pretty bad – the 67% drawdown requires over *double* the return to climb back to the pre-drawdown level (92% vs. 203%)!

On the surface, the fact that a portfolio steeply declines in value may be the primary concern, and for retirement and other spending events, this is indeed a very important consideration. But large drawdowns create strong emotional responses even for investors who can, at least on paper, weather them. These emotional times are when well-intentioned plans are often abandoned, locking in losses and putting the portfolio on a rough road going forward.

By more accurately assessing risks and employing disciplined approaches to investing in alternatives and constructing portfolios, we hope to smooth out the investing ride by developing a plan that can be adhered to.

[1] https://blog.thinknewfound.com/2015/02/volatility-crashing/

[2] https://blog.thinknewfound.com/2015/07/weekly-commentary-volatility-good-bad-indifferent/

[3] GBM also assumes that prices follow a continuous path. In reality, there are often jumps, especially when looking at a single stock, but this is not as much of a problem when working with diversified portfolio or ETFs.

[4] We outlined a method for dealing with this by shocking the covariance matrix: https://blog.thinknewfound.com/2016/10/shock-covariance-system/

[5] If you are a mathematical masochist – like we are – the full derivation of the maximum drawdown distribution and the equations for the expected value are presented in this paper (On the Maximum Drawdown of a Brownian Motion) by Atiya, et. Al. (2003).

[6] For the exact equation, each autocorrelation is multiplied by a factor of (n-i)/n, where n is the number of returns.

[7] Lo, Andrew W., The Statistics of Sharpe Ratios. Financial Analysts Journal, Vol. 58, No. 4, July/August 2002. Available at SSRN: https://ssrn.com/abstract=377260

[8] Burghardt, G. and Liu, L. Autocorrelation Effects on CTA and Equity Risk Measurement, The Journal of Alternative Investments (Summer 2013)

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