*This post is available as a PDF download here. *

# Summary

- Risk parity portfolios attempt to diversify across asset classes and strategies by risk contribution as opposed to dollar allocation.
- Implementing a risk parity strategy requires making a number of important construction decisions. A key question we have to answer is “How are we going to measure risk?”
- One approach is to use historical data to estimate risk. When using this approach, we have to decide how much historical data to use for the calculation. This parameter is often referred to as the lookback period.
- We test lookback periods ranging from one quarter to twenty-five years and conclude that risk parity results are robust to the lookback period selected. This is good news as it means we don’t have to worry too much about choosing the “right” parameter. A simple approach given these results is to use a range of lookback periods when calculating volatilities and correlations for a risk parity implementation.

Generally speaking, risk parity portfolios attempt to diversify across asset classes and strategies by risk contribution as opposed to dollar allocation. The classic argument for risk parity typically begins by noting that many traditional portfolios (e.g. the 60/40 or “balanced” portfolio) are dominated by equity risk. A risk parity approach, on the other hand, can lead to portfolios that are more appropriately diversified across risk factors.

**Dollar Allocation vs. Risk Allocation for 60/40 S&P 500/Barclays Aggregate Bond Portfolio**

**Dollar Allocation vs. Risk Allocation for Equal Risk Contribution Portfolio of S&P 500 and Barclays Aggregate Bond**

Risk parity checks off all of our main boxes for strategy selection. It is backed by both strong empirical evidence and sound economic theory. Just as importantly, risk parity can be implemented through a simple, transparent, and systematic investment process.

When constructing a risk parity portfolio, there are a number of questions that need to be answered, including:

- What is my investment universe?
- How am I going to measure risk?
- How do I account for correlations?
- Am I willing to use leverage? If so, how do I set my total notional exposure? Or framed differently, what is my target volatility?
- How do I account for real world complexities like trading, transaction costs, and taxes?

With respect to question #2, many practitioners use historical data to calculate risk. When using the historical approach, we have to select both the risk metric(s) to use and the period length(s) over which to measure the selected metrics. For example, Salient’s risk parity white paper uses rolling two-year periods of daily returns to calculate volatilities and correlations.

In this commentary, we are going to explore the importance of the lookback period. Namely, does the length of lookback period matter? And if it does, are we better off using as much data as possible (i.e. long lookback periods) or focusing on just the more recent past (i.e. short lookback periods).

**Building a Simple Risk Parity Strategy**

To answer these questions, we have to build a risk parity strategy. For simplicity, we will use ETFs for our analysis even though most real world risk parity strategies implement via futures. We consider ETFs in four asset classes:

- Equities
- Fixed Income
- Real Assets (Commodities and TIPs)
- Credit/Currency (investment grade and high yield corporate bonds, dollar-denominated emerging market bonds, local currency emerging market bonds).

Our target risk allocation is presented in the following pie chart. Within each asset class, we distribute the risk budget either equally or by market-capitalization. For chart readability, we only show the risk budget at the asset class level (full risk budget targets are available on request).

Note that the credit/currency bucket gets half the risk budget of the other three asset classes. This is a byproduct of using ETFs instead of futures contracts. In a real-life implementation, we could truly isolate the credit and currency exposure by using credit default swaps and currency forwards, respectively. However, with ETFs we are forced to get the exposure via bonds. These bonds have embedded interest rate risk in additional to the credit/currency components. Therefore, we lower the risk budget for this bucket to avoid being too structurally overweight interest rate risk.

We examined lookback periods ranging from 63 trading days (approximately one quarter of data) to 6300 trading days (25 years). For each lookback period, we computed two hypothetical portfolios. One portfolio was managed using simple volatilities and correlations, while the other used exponentially-weighted volatilities and correlations. The decay factor for the exponentially-weighted metrics were set so that the half-life was equal to half of the lookback period[2].

For each combination of lookback period (1 quarter to 25 years) and calculation type (simple or exponentially-weighted), we created a hypothetical portfolio. Each of these portfolios was rebalanced monthly using the following process:

- Estimate volatilities and correlations using the given lookback period and calculation type. Volatilities were estimated using daily data. Correlations were estimated using weekly data.
- Compute the portfolio weights such that the risk contribution of each positon equals the target risk budget. We normalize these weights so that they sum to 1.
- Scale the portfolio weights to a total notional value such that we target a volatility of 10%.

**Evaluating the Allocations**

From an allocation perspective, the results aren’t all that surprising. Below, we present a box-and-whisker plot for the total notional exposure though time. The data points represented are the minimum, 25^{th} percentile, median, 75^{th} percentile, and maximum total exposure. We see more variation in allocations for shorter lookback periods. For example, a lookback period of 63 trading days – or one quarter- leads to total notional exposure that ranges from a low of 47% to a high of 552% when simple volatilities are used. On the other hand, the range is only 137% to 263% for a 10-year lookback period.

We also see that median exposure tends to decline as the lookback period is expanded. The median exposure is 214% for the 63-day lookback and 149% for the 10-year lookback.

A natural byproduct of more reactive allocations is higher turnover. We do see that turnover declines very quickly as the lookback period is expanded. When using simple volatility, annualized turnover is 337% when one quarter of data is used, 155% when two quarters are used, 107% when three quarters are used, and 79% when 1 year is used. Turnover drops below 50% once at least two years of data is incorporated.

In terms of asset classes, we see the average allocation to real assets and fixed income increasing and the average allocation to equity and credit/currency decreasing as the lookback period is extended. This makes sense given the market path during the period studied. The longer lookback periods have more “memory” of the volatility spike experienced by many “risk-on” asset classes during the global financial crisis. These asset classes are therefore penalized for 2007-09 even during the more benign risk environment of the current equity bull market. The trends are similar, but less pronounced, when we use exponentially-weighted statistics.

**Evaluating Performance**

The Sharpe Ratios generated by the various strategies are remarkably stable. They are all clustered between a minimum of 0.61 (two-year lookback, simple volatility) and 0.72 (1-quarter lookback, exponentially-weighted volatility).

When simple volatilities are used, Sharpe Ratios initially decline as the lookback period is expanded through two years. From two years to five years, Sharpe Ratios recover before stabilizing for 5+ years of data at a level that is almost identical to the Sharpe Ratio for a one-quarter lookback.

Using exponentially-weighted statistics seems to eliminate much of this noise since, excluding the one-quarter lookback, all of the exponentially-weighted Sharpe Ratios are between 0.676 and 0.681.

Using the Sharpe Ratio comparison methodology outlined in Opdyke (2007)[3], we were able to verify that these Sharpe Ratio differences are not statistically significant when we account for skew and kurtosis.

We do see slightly more significant outperformance for shorter lookbacks when we evaluate the strategies using their alpha relative to a 60/40 U.S. stock/U.S. bond benchmark. For example, the 1-quarter exponentially-weighted lookback has annualized alpha of 6.0% versus a median of 4.8% across all exponentially-weighted lookbacks.

From a risk perspective, performance is likewise pretty clustered. The maximum drawdown across all strategies tested ranges from 25% to 31%. We do see somewhat better risk mitigation for shorter, especially shorter exponentially-weighted, lookbacks.

**Evaluating the Objective**

As we stated above, we built the simple risk parity strategies to target a 10% volatility level. Therefore, we can also evaluate the strategies based upon how they delivered on this risk objective, both on average across the whole period and through time.

The next chart graphs the annualized volatility for each of the strategies relative to the 10% target. We see that both of the shortest lookback period strategies overshoot the target by more than 1%. For both types of volatility measures, we generally see the strategies undershoot the volatility target for shorter lookbacks and overshoot it for longer lookbacks.

For reference, a 95% confidence interval for the annualized volatility would be approximately 9.8% to 10.2% or a difference of +/- 0.2%.

We can also look at rolling volatilities through time. For simplicity, we will just look at the 0.25-ewm, 5-ewm, and 25-ewm portfolios. We will compare them to the rolling volatility for the 60/40 benchmark.

All three risk-parity portfolios outperform the 60/40 when it comes to maintaining a relatively constant volatility. Comparing the different lookback periods, we see that shorter lookback periods generally do a better job.

**Blending Signals**

Based on the analysis so far, there is good news and bad news. The bad news is that there is no clear winner when it comes to the optimal lookback period. In the backtests, shorter lookback periods delivered better risk-adjusted performance and were more effective in delivering a consistent volatility profile. However, this came at a cost as these strategies experienced much higher turnover than their slower-reacting brethren. In addition, the shortest lookback periods were furthest from achieving the 10% target volatility over the full period.

The good news is that the overall results were very robust to the choice of lookback period. In other words, we don’t have to worry too much about selecting the “right” lookback period. No matter what we choose, we can be reasonably comfortable that we will capture the main features of the risk parity style.

In these types of scenarios, we typically favor blending multiple signals. There are a few ways we could do this. One would be to simply to pick a few different lookbacks, calculate each portfolio independently, and then average the portfolios together. For example, we could pick three lookbacks:

- A short lookback to maximize adaptability (i.e. a one-quarter lookback).
- A medium lookback to capture most of the market cycle, providing for more stable estimates while still allowing for longer term regime changes (i.e. a 5-year lookback).
- A long lookback to take into account as much information as we can (i.e. a 25-year lookback).

When we blend these three portfolios equally, we end up with a Sharpe Ratio of 0.71 compared to 0.72, 0.68, and 0.68 for the short, medium, and long-term lookbacks, respectively. Furthermore, we end up relatively close to the target volatility (10.2%) and much lower turnover than the short lookback period alone.

**Areas for Additional Study**

This is just the tip of the iceberg when it comes to exploring this topic. There is nearly endless literature on the topic of better volatility estimators alone. Other areas to look into include adaptive volatility measures that automatically adjust the lookback period over time[4] and the impact of lookback periods on alternative risk measures (e.g. value-at-risk, conditional value-at-risk, conditional drawdown-at-risk).

**Conclusion**

Risk parity portfolios attempt to diversify across asset classes and strategies by risk contribution as opposed to dollar allocation.

Implementing a risk parity strategy requires making a number of important construction decisions. One key question we have to answer is “How are we going to measure risk?” One approach is to use historical data to estimate risk. When using this approach, we have to decide how much historical data to use for the calculation. This parameter is often referred to as the lookback period.

We tested lookback periods ranging from one quarter to twenty-five years. We found that there was no clear winner. Shorter lookback periods delivered better risk-adjusted performance and were more effective in delivering a consistent volatility profile. However, this came at a cost as these strategies experienced much higher turnover than their slower-reacting brethren. In addition, the shortest lookback periods were furthest from achieving the 10% target volatility over the full period.

While this might appear frustrating at first, the results are in fact very good news. Risk parity is very robust to the choice of the lookback period. In other words, we don’t have to worry too much about selecting the “right” lookback period. No matter what we choose, we can be reasonably comfortable that we will capture the main features of the risk parity style. In these situations, a simple and effective approach is to use multiple lookback periods. We show that this type of blended approach can help to isolate the best attributes of both shorter and longer lookback periods.

[1] For one such argument, see *Leverage Aversion and Risk Parity *by Asness, Frazzini, and Pederson (2012) which can be found at https://www.aqr.com/library/journal-articles/leverage-aversion-and-risk-parity.

[2] As an example, consider the 63-day lookback period. The exponentially-weighted metrics were computed using alpha equal to 0.022 (or in RiskMetrics terminology a decay factor of 0.978). This leads to the exponentially-weighted volatilities having a half-life of 31.5 days.

[3] http://www.datamineit.com/Sharpe%20Ratio%20Comparisons%20-%20J.D.%20Opdyke%20-%20preprint%20-%20Journal%20of%20Asset%20Management,%20Vol8(5),%202007.pdf

[4] See here for an example from David Varadi.

## 3 Pingbacks