Risk parity – the benefits of a conditional approach

Risk parity is a meaningful and robust approach for building well-diversified portfolios, but it relies on historical volatility estimates, which penalises upside risk as well as downside risk and leads to a massive overweighting of bonds versus equities, even in a low yield environment. The authors from EDHEC Risk-Institute build the case for an alternative method.

 

Risk parity has become an increasingly popular approach for building well-diversified portfolios within and across asset classes.

In a nutshell, the goal of the methodology is to ensure that the contribution to the overall risk of the portfolio will be identical for all constituent assets, which stands in contrast to an equally-weighted strategy that would also recommend an equal contribution but instead simply be expressed in terms of dollar budgets as opposed to risk budgets.

While intuitively appealing and empirically attractive, this approach suffers from two major shortcomings.

Firstly, typical risk parity portfolio strategies used in an asset allocation context inevitably involve a substantial overweighting of bonds with respect to equities, which might be a problem in a low bond yield environment, with mean-reversion implying that a drop in long-term bond prices might be more likely than a further increase in bond prices.

Secondly, standard approaches to risk parity are based on portfolio volatility as a risk measure, implying that upside risk is penalised as much as downside risk, in obvious contradiction with investors’ preferences.

In recent research supported by Lyxor Asset Management as part of the “Risk Allocation Solutions” research chair at EDHEC-Risk Institute, we develop a conditional approach to risk parity, which contrasts with standard unconditional risk parity portfolios based on historical volatility estimates, with an attempt to alleviate the two aforementioned concerns.

In a first step, we recognise that duration is a decreasing function of the bond yield for a coupon-paying bond.

As a result, a decrease in bond yield levels should lead to an increase in bond duration, and as such, an increase in bond volatility should lead, all else being equal, to a decrease in the allocation to bonds for a risk parity portfolio.

By using a robust econometric procedure that explicitly relates unobservable bond volatility levels to observable bond yield levels, we actually obtain an explicit response to the risk parity bond allocation as a function of changes in yield levels.

In a second step, we define a new class of conditional risk parity (RP) portfolios with respect to downside risk measures such as semi-variance, Value-at-Risk (VaR) or expected shortfall.

In all of these cases, conditional estimates of expected returns on stocks and bonds actually impact the estimated risk levels, so that an economic environment where the risk premium is historically high for stocks and low for bonds would lead to a further decrease in the allocation to bonds with respect to an unconditional risk parity technique, but also with respect to a conditional risk parity technique relying solely on volatility as a risk measure.

In a final step, we analyse a competing approach that explicitly accounts for changes in risk premium levels, as opposed to having them indirectly impact the risk parity portfolio through their impact on the portfolio downside risk.

From a formal perspective, this last extension is based on the recognition that risk parity is equivalent to a Sharpe ratio maximisation program under the assumption that all asset classes have the same risk premium and the same correlations.

While this constant risk premium assumption can be regarded as a reasonable, or at least agnostic, prior from an unconditional perspective, it is hardly defendable from a conditional perspective (that is for all possible market conditions), particularly when bond yields are low and dividend yields are high.

In this context, it would be preferable to adjust the risk budgets dynamically around the long-term 50 per cent/50 per cent risk parity target so as to better reflect the current market environment, and we provide an analytical expression for the dynamic adjustment to risk budgets for each asset class that optimally reflects changes in market conditions, by defining these time-varying risk budgets as the set of risk budget targets that makes the risk budgeting portfolio a maximum Sharpe ratio portfolio given current estimated levels of volatility and Sharpe ratios for equities and bonds, while assuming an identical long-term Sharpe ratio for the two asset classes.

 

Duration-based volatility is an instantaneous and observable volatility bond measure that avoids the sample dependency and overweighting of bonds in a low interest rate environment, which inevitably follow from the use of historical volatility measures in the construction of risk parity portfolios.

In response to the two major problems identified with historical volatility, namely sample dependency and backward-looking bias, we introduce in our research an alternative bond volatility measure, which we refer to as “duration-based volatility” (in short, DUR volatility).

This measure is suggested by the model-free approximation of the return on a bond portfolio as the product of the negative of duration times the yield change: thus, DUR volatility equals duration times the yield volatility.

Duration is readily observable, both for a single bond and for a bond portfolio, where it is given as the weighted average of constituents’ durations, but yield volatility has to be estimated.

In this sense, DUR volatility is quasi-observable, rather than fully observable, but unlike rolling-window (RW) and GARCH volatilities, it reacts instantaneously to changes in bond yields, which translate into changes in bond duration.

As such, DUR volatility also helps to address the issue of bond overweighting in a low-interest rate context. Indeed, duration for a coupon-paying bond can be shown to be a decreasing function of the bond yield.

At the index level, where multiple constituents are involved, making the analysis less explicit, the negative relationship between duration and yield is still confirmed empirically.

The consequence on bond volatility estimates over the past 30 years is shown in Exhibit 1: while RW and GARCH volatilities have been stable or even decreasing slightly, the DUR volatility has followed an increasing trend due to the decreasing trend in bond yield levels, and has eventually exceeded the former measures.

Hence, at the end of our sample period, in December 2012, a risk parity portfolio of stocks and bonds has a lower allocation to bonds when DUR volatility is used instead of RW or GARCH estimates.

To see that this “CRP-VOL-DUR” strategy is also more responsive to changes in market conditions than “CRP-VOL-GARCH” or “URP-VOL-RW”, one can look at the percentage of months in the sample over which bond allocation and bond yield moved in the same direction: Exhibit 2 indicates that this concordance rate is substantially higher for CRP-VOL-DUR. The results shown here refer to portfolios of stocks and bonds, but the same conclusions remain valid when we introduce commodities as a third asset class.

Exhibit 1: Comparison of bond volatility measures; January 1973 – December 2012.

This figure shows the yield of the Barclays US Treasury index, together with its volatility estimated by three methods: a five-year rolling window (VB-RW), a GARCH(1,1) model (VB-GARCH), and a volatility measure proportional to the duration (VB-DUR). Duration is computed using the approximation of Campbell et al. (1997).

 

Risk parity portfolios constructed with a downside risk measure show a higher degree of sensitivity with respect to market conditions, and lead to further decreases in bond allocation in a low yield environment compared to risk parity portfolios constructed on the basis of volatility as a risk measure.

As discussed above, the choice of volatility as the reference risk measure in the definition of risk parity portfolios can be criticised for not capturing investors’ concern over downside risk.

As a result, assets with substantial downside risk can be overweighted as long as they have low volatility, which would be the case for bonds in a low yield environment, when mean-reversion back to higher yield levels would imply that most of the risk faced by investors is on the downside.

A natural idea is to penalise such assets by introducing a downside risk measure in the definition of risk parity portfolios.

The only mathematical requirement is that the measure should be homogeneous of degree 1 in portfolio weights, because this condition is needed to decompose the portfolio risk as a sum of contributions from constituents.

Semi-volatility, defined as the volatility of negative returns, and Value-at-Risk (VaR), defined as the quantile of order of the portfolio loss distribution, both satisfy this condition.

To obtain explicit expressions for these measures as functions of portfolio weights, and to compute the risk contribution of each asset and then to solve for the set of weights that equates these contributions, one needs to make an assumption on the distribution of returns.

A mathematically convenient choice is the Gaussian distribution, but it is hardly appropriate for an analysis of asymmetric and fat-tailed return distributions. In this context, we propose to use the Cornish-Fisher expansion to obtain a correction to the Gaussian VaR that accounts for the presence of non-zero skewness and excess kurtosis levels.

One drawback of these dissymmetric risk measures is that they depend on expected returns, which are notoriously hard to estimate.

Empirical research, however, provides useful guidance with respect to the relevance of state variables such as dividend yields as proxies for the time-varying expected return on equities.

Similarly, intuition suggests that the current yield level can be taken as a predictor for expected bond returns.

Indeed, the price of a bond is a decreasing function of the yield, which exhibits mean reversion. In the current environment, when bond yields have reached historically low levels, bonds appear to be expensive relative to historical standards, thus implying that their expected return is low.

In this spirit, we construct our estimates for expected returns on stocks and bonds in two steps. First, we compute an expected return forecast from a predictive regression, the predictor being the dividend-price ratio for the stock index and the current yield for the bond index. Second, we shrink the forecast towards a prior, which is itself obtained by assuming that the risk premium on the asset equals the current volatility times the Sharpe ratio measured over a very long sample (1926-2012).

 

It appears from Exhibit 2 that the CRP-NGVAR99 strategy, which equates the contributions to the non-Gaussian VaR at 99 per cent, displays a concordance rate (again defined as the percentage of months in the sample over which bond allocation and bond yield moved in the same direction) that is higher compared to other risk parity strategies, notably the one relying on the duration-based volatility measure.

For Gaussian downside risk measures (not shown in the Exhibit), we find that concordance rates are between those of the Cornish-Fisher VaR and the volatility, which suggests that the highest degree of responsiveness to changes in market conditions is obtained with the risk parity portfolio strategy based on the non-Gaussian VaR risk measures.

Interestingly, our results also indicate that the ratio of stock risk to bond risk tends to be substantially lower with this VaR measure compared to volatility or Gaussian risk measures, especially when bond yields are particularly low, suggesting an increase in downside risk for bond portfolios.

As a result, the CRP-NGVAR99 strategy tends to lead to a lower bond allocation than other risk parity strategies tested in our research, especially by the end of the sample period when bond yield have reached unusually low levels.

 

Exhibit 2: Concordance rates of risk parity strategies invested in stock and bond; January 1973 – December 2012.

The concordance rate is defined as the percentage of months in the sample in which the yield and the bond weight moved in the same direction. URP-VOL-RW is the standard risk parity strategy that relies on historical volatilities; CRP-VOL-DUR is a risk parity strategy that uses duration-based volatility as the bond volatility measure; CRP-NGVAR99 is a strategy that equates the contributions of constituents to non-Gaussian VaR at 99%; MSR-SAME-SR is a maximum Sharpe ratio strategy that uses the same prior Sharpe ratio for both constituents; and MSR-DIFF-SR uses different priors.

 

The maximum Sharpe ratio portfolio, which can be interpreted as a risk parity portfolio under some conditions (identical pairwise correlations and identical Sharpe ratios), is a less robust alternative to incorporating expected returns in the construction of a well-diversified policy portfolio.

Under some conditions, the maximum Sharpe ratio (MSR) portfolio can be interpreted as a risk parity portfolio.

Indeed, in “The Properties of Equally Weighted Risk Contribution Portfolios” Journal of Portfolio Management, the authors establish that the two portfolios coincide if all assets have the same pairwise correlations and the same Sharpe ratio.

In the case of two assets, this condition reduces to the equality of Sharpe ratios, an assumption which can be regarded as a reasonable agnostic long-term prior, but which is unlikely to be satisfied in all market conditions, in particular if bond yields are extremely low and dividend yields are extremely high.

Alternatively, one can also show that the MSR portfolio achieves risk parity for a risk measure defined as volatility with a penalisation for expected return.

In our research, we test two MSR portfolios, which differ through the priors used for expected returns. Exhibit 2 shows that the concordance rates are close to those of the risk parity strategies constructed from duration-based volatility or Gaussian VaR.

Hence, our results suggest that there are no additional benefits to expect from MSR strategies in this dimension. Another non-negligible drawback of these portfolios over the risk parity portfolio based on the Cornish-Fisher approximation, which uses the same expected return estimates, is that they involve a much higher level of turnover.

This property reflects the well-known fact that MSR weights have high sensitivity to expected returns, and, as a consequence, are more impacted by estimation errors in these parameters. Overall, one key difference between MSR portfolios and conditional risk parity portfolios based on downside risk measures is that even if they both rely on expected return estimates, risk parity strategies treat these estimates as (directional) risk measures as opposed to pure expected return inputs as in the case of the MSR strategy, implying a lower sensitivity to estimation risk.

 

Conditional risk parity strategies display better performance in a scenario of increasing bond yields.

The concordance rates and bond weights give indications on the behaviour of the various risk parity strategies.

In order to see the benefits of the conditional approaches in a more material way, we simulate an increase in interest rates as of December 2012 until December 2017.

The use of Monte-Carlo simulations is almost an obligation here as the historical sample that was available for this study (1973-2012) is mostly characterised by a long decrease in interest rates starting in the 1980s, with no periods of sustained decreases in yields.

To test for the effectiveness of improved risk parity strategies in responding to increases in interest rates, we have simulated two economic scenarios: in the first one, the reversion of bond yields from their current level back to their long-term mean value is complete after five years (on average), while the second scenario assumes a more rapid increase in interest rates, with a mere two-year time period needed to reach the same value. Exhibit 3 provides an overview of the results.

The main observation is that the conditional risk parity strategies perform better over the period of increases in interest rates, if only because they start with a lower bond allocation.

In particular, they avoid the severe losses displayed by the URP-VOL-RW strategy in the first year of the increasing trend for interest rates.

On the other hand, they also have a higher stock exposure, so they are more impacted by the performance of the stock market.

To give a sense of this impact, we have assumed that in the first scenario, the equity market is not impacted, while in the second one, the rapid increase in rates results in a strong market downturn after two years.

We note that the CRP-NGVAR99 strategy displays the worst performance in this year, due to a higher stock allocation, but it recovers in the next two years, where it appears to be the top performer.

 

Exhibit 3: Simulated performance of risk parity strategies invested in stock and bond.

(a)   Scenario 1 (slow increase in interest rates)

	2013	2014	2015	2016	2017
Yield (av.)	0.02	0.036	0.047	0.054	0.059
Bond index	-0.097	-0.053	-0.034	-0.015	-0.005
Stock index	0.061	0.062	0.054	0.059	0.061
URP-VOL-RW	-0.061	-0.021	-0.006	0.011	0.020
CRP-VOL-DUR	-0.048	-0.012	-0.002	0.013	0.020
CRP-NGVAR99	-0.022	0.003	0.008	0.019	0.025

 

(b)   Scenario 2 (rapid increase in interest rates)

	2013	2014	2015	2016	2017
Yield (av.)	0.031	0.055	0.063	0.065	0.066
Bond index	-0.184	-0.047	-0.01	0.007	0.01
Stock index	0.061	0.007	-0.102	0.036	0.102
URP-VOL-RW	-0.127	-0.029	-0.029	0.021	0.039
CRP-VOL-DUR	-0.11	-0.026	-0.031	0.021	0.038
CRP-NGVAR99	-0.069	-0.018	-0.041	0.026	0.05

This table shows the simulated average bond yield (first line) and the simulated expected excess return of risk parity strategies in a period of interest rate increase. Scenario 1 corresponds to a slow increase with no impact on the equity market, while Scenario 2 is characterised by a rapid increase, and an equity bear market in 2015. Strategies are defined in the legend of Exhibit 2.

 

Conclusion

Overall, our analysis suggests that it is possible to construct alternative forms of risk parity strategies that alleviate some of the concerns posed by the standard approach based on historical volatility measures.

The risk parity strategy relying on duration-based volatility, which is not as dependent on past bond returns as rolling-window or GARCH volatilities, is a first step towards the introduction of an instantaneous cheapness indicator in a portfolio construction methodology that otherwise solely focuses on risk management.

An improved responsiveness to changes in bond yield levels can be achieved through the use of a downside risk measure, such as the Cornish-Fisher VaR, and this increased sensitivity would result in better performance in a period of sharp increases in interest rates.

While the implementation of such strategies requires an estimate for expected return parameters, these strategies show a substantially higher degree of robustness to errors in such estimates compared to standard mean-variance portfolio construction techniques.

The impact of estimation errors on out-of-sample performance and the benefits of correcting weights for parameter uncertainty have been extensively studied in the literature on mean-variance optimal portfolios, but a similar work for risk parity portfolios remains to be done.

 

Lionel Martellini, professor of finance, EDHEC Business School, and scientific director, EDHEC-Risk Institute; Vincent Milhau, deputy scientific director, EDHEC-Risk Institute; and Andrea Tarelli, senior research engineer, EDHEC-Risk Institute.

The research from which this article was drawn was supported by Lyxor Asset Management as part of the “Risk Allocation Solutions” research chair at EDHEC-Risk Institute.