Choosing between Mean Variance, Kelly and Risk
Choosing between Mean Variance, Kelly and Risk Parity in Asset Allocation In this report we will be comparing Mean Variance Optimization5, Kelly “bet-sizing”1,2,3,4 and Risk Parity6 in the context of asset allocation. Much has been claimed in recent years about the superiority of each method. In this report, we will document the results of our study about which is better. Our study shows that the relative performance of these allocation methods is a function of the accuracy of the expected returns model used. It is not the case that any one of them is better than others in all circumstances. Without a very good expected returns model, risk targeting methods like Risk Parity are a better choice. If one has an above average accuracy in expected returns model, the allocation method of Kelly-bet-sizing produced better back-tested results in our experiment. Only if someone has a stellar expected returns model, where expected Sharpe Ratios are in excess of 5, should one choose the Mean Variance Optimization5 as the allocation method. In this paper, we will first introduce the three methods and show an illustrative example of the nature of these allocation methods. Then we will show the results of our study and on what factors a prospective portfolio manager should choose between these three allocation methods. Read the next section, Introduction to asset allocation methods being studied here to understand the nature of these methods. For those who are familiar with these allocation methods, directly go to the section Study on which allocation method is best.
Introduction to asset allocation methods being studied here The three most popular methods of quantitative asset allocation are Mean Variance Optimization5, Kelly1 bet-sizing and Risk Parity6. To our knowledge, at this time the question of which one of these is the best for asset allocation is yet to be answered. We have attempted to answer this question here. In our study, we vary the accuracy of the expected returns model and see the results in each case. Before we get to the results, let’s look at a brief overview of these three methods of deciding allocation weights. We have also shown illustrative asset allocations that these methods would produce if used on a set of 40 ETFs. The details of the data used to compute these allocations is in Appendix A - “Quantitative inputs used to compute the illustrative allocations”. These are expected returns and risk computed on historical data of one year, on Jan 1st of 2008.
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Mean Variance Optimization (MVO) MVO tries to periodically choose an asset allocation to maximize expected return, while keeping the expected risk within mandated thresholds. This relies heavily on the model that we are using to compute expected returns. One can add constraints to keep the portfolio within mandated parameters like target volatility, concentration limits and deviation from benchmark. However, since all those are just constraints, the dominant factor in MVO is the expected return model.
Exhibit 1 - Sample Asset allocation using Mean Variance Optimization
Notes: The portfolio that Mean Variance Optimization would output based on the inputs in Appendix A. Among 40 ETFs it could have allocated to, MVO allocates almost everything to three securities. MVO bets mostly on the few securities that have the best expected returns in the model, since it just tries to maximize the expected returns. It does not worry about diversification or inaccuracies of the expected returns.
Kelly bet sizing
The method tries to periodically set asset allocation to each asset class in proportion to the ratio of its expected return to expected risk. This also relies on the expected return model but to a much lesser extent. Suppose there are two asset classes with expected returns of 4.1% and 4.0% respectively. MVO would assign all the allocation to the first one, whereas Kelly would allocate roughly equal amounts to them. All investments carry risk. This material is for informational purposes and should not be considered specific investment advice or recommendation to any person or organization. Past performance is not indicative of future performance. Please visit our website for full disclaimer and terms of use.
This method was first published by John L. Kelly Jr. in “A new Interpretation of Information Rate”4. It was based on Shannon’s work on information theory7 and was perhaps the first attempt at walk-forward analysis of an asset allocation algorithm. One way to think about it is that it gives equal importance to the expected risk and expected return. The amount of consideration to differences in expected returns of two asset classes is taken in context of their expected volatilities.
Exhibit 2 - Sample Asset allocation using Kelly bet-sizing
Notes: The portfolio that Kelly bet-sizing would have based on the inputs in Appendix A. Unlike MVO, Kelly bet-sizing isn’t an optimization method. Betting proportional to risk-adjusted returns helps in achieving a more balanced portfolio, since risk-adjusted returns are more comparable to each other as opposed to returns. As a result of these two factors, Kelly bet-sizing is a lot more diversified than MVO.
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Risk Parity In this method of asset allocation, one would seek to assign weights to asset classes to achieve the desired risk level in the most diversified manner. To that end, this allocation strategy tries to equalize risk contribution to each asset class. In that light, this might be more accurately termed as a risk-targeting asset allocation strategy. The reason it has been popularized as Risk Parity and not Risk Targeting is that if we are allowed to leverage the portfolio, there could be multiple portfolio allocations whose estimated risk will be close to the target-risk. Among them, the Risk Parity allocation will try to have equal risk contributions from all constituents. Risk Parity relies more on the expected risk model than on the expected return model. It still uses an expected return model but only to filter out asset classes to allocate to. A risk parity approach would use a slow moving version of the expected return model only to filter out which asset classes to be long in and which ones to not hold. In a portfolio that is allowed to go short, we could use the expected return model to choose which securities to go long in and which ones to go short. However, the position sizing is done based on the expected volatility model. Since expected volatility is more stable, the allocations by the Risk Parity method would be more stable than the other two approaches. All investments carry risk. This material is for informational purposes and should not be considered specific investment advice or recommendation to any person or organization. Past performance is not indicative of future performance. Please visit our website for full disclaimer and terms of use.
Exhibit 3 - Illustration of asset allocation output by Risk Parity
Notes: The portfolio that Risk Parity would have based on the inputs in Appendix A. Risk Parity is by far the most diversified. Almost every ETF in the eligible set is well represented. The largest allocation is less than 25% higher than the smallest allocation in this case.
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Study on which allocation method works best
Our intuition was that MVO does not work unless we have a stellar expected return model. To validate this, we tried to compare the performance of the three allocation methods in terms of Sharpe Ratio with expected returns models with varying accuracy. Our intuition was validated in the study. With poor expected returns, a Risk Parity approach, or equal risk contribution approach to asset allocation had the best performance on back-tested data. When the predictive accuracy of the expected return indicator is 5% or higher, a Kelly bet sizing approach would have achieved a higher Sharpe Ratio. Kelly Breiman approach continues to do the best even when the predictive accuracy is as high as 20%. Only at a predictive accuracy of 50% do we begin to see MVO doing better than other methods. However, even then MVO does not outperform Kelly Breiman by much. Our study shows that one would need a really really good model with expected Sharpe Ratios comfortably above 4, for MVO to achieve better results than Kelly bet-sizing. On the other hand Kelly-bet sizing achieves better returns than Risk Parity as long as the expected correlation of the odel is greater than 1%. That would roughly correspond to an out of sample Sharpe Ratio of 1. If you don’t believe your model will achieve an out of sample Sharpe of 1, you are better off using the Risk Parity allocation method. We have shown the same data in graphical and tabular form below.
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Exhibit 4 - Best allocation method changes based on the accuracy of the model
Predictive correlation of the expected return model
Mean Variance Optimization
Kelly-Breiman
Risk Parity
-50.00%
-2.72
-2.93
0.95
-20.00%
-0.36
-0.20
0.95
-10.00%
0.17
0.30
0.95
-5.00%
0.52
0.61
0.95
-1.00%
0.64
0.79
0.95
0.00%
0.73
0.91
0.95
1.00%
0.83
0.92
0.95
5.00%
0.87
1.15
0.95
10.00%
1.53
1.60
0.95
20.00%
1.77
2.12
0.95
50.00%
6.60
5.72
0.95
Notes: Sharpe Ratios of allocation methods corresponding to varying accuracy of the expected returns model All investments carry risk. This material is for informational purposes and should not be considered specific investment advice or recommendation to any person or organization. Past performance is not indicative of future performance. Please visit our website for full disclaimer and terms of use.
The reason why MVO needs a really good expected return model is that the amount of importance assigned to expected return model in computing allocations is highest in MVO. Kelly does not need as high an accuracy in the expected returns model and Risk Parity is most forgiving. Similarly the importance to the risk or volatility model is least in MVO, higher in Kelly and highest in Risk Parity. It is thus no surprise that MVO performs best if the expected return model is very very good. If the expected return model is not very accurate, Kelly and Risk Parity perform better than MVO. For most real world situations of model accuracy, I think Kelly bet sizing would achieve better performance than Mean Variance Optimization. From the allocation charts in Appendix A, visually, it is quite clear that a method like MVO, which tries to optimize expected returns, produces allocations that are very concentrated. Out of around 40 products, the MVO allocation just chooses three products. Kelly bet-sizing is less concentrated. However, three products still account for about fifty percent of the allocation. Risk Parity is very well diversified. The diversification is visually apparent from the allocation pie chart above. Another way to compare these methods is to see how much the allocation of the same product changes. In the charts in Appendix B, we have singled out and charted the allocation of the same product over time according to the three allocation methods. Looking at the charts below, we can see that allocations are stable over time for Risk Parity. Kelly bet sizing has significantly less consistent allocations and MVO allocations to each security are very sporadic. All investments carry risk. This material is for informational purposes and should not be considered specific investment advice or recommendation to any person or organization. Past performance is not indicative of future performance. Please visit our website for full disclaimer and terms of use.
Conclusion No allocation method is the best in all circumstances. While Mean Variance Optimization is probably the most common allocation method used, it will probably not yield the best returns if the portfolio manager does not have a very high accuracy model. Besides returns, portfolio managers should also consider the following factors when choosing the allocation method. Feature
Mean Variance Optimization
Kelly bet-sizing
Risk Parity
How diversified is the allocation
Least
Decent
Highest
Responsive to market changes
Highest
Decent
Low
Accuracy needed from the expected returns model
Higher than a Very high Sharpe Ratio of 1 Most forgiving accuracy needed needed
Ability to achieve target volatility
Good
Usually under achieving in volatility
Great
Scalability
Low, unless trading frequency is reduced
Good
Highest
Sensitivity to trading costs
Highest
Medium
Low
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Appendix A Quantitative inputs used to compute the illustrative allocations To generate the sort of asset allocation these methods come up with, we have used expected returns, volatility and correlation from a one year lookback model as of Jan 1 2008.
Exhibit 5 - Expected returns per security used as inputs to MVP, Kelly and Risk Parity
Notes: Expected returns that we used in the asset allocation decisions above
Notes: Expected risk that we used in the asset allocation decisions above
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Notes: Expected Returns/Risk that we used in the asset allocation decisions above.
Appendix B Showing the stability of allocation in each of the methods Exhibit 6 - Allocation to one security over time in each of the allocation methods
Notes: MVO allocation is very sporadic
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Notes: Kelly Breiman allocation is somewhat sporadic, however it is still responsive
Notes: Risk Parity allocation is quite stable. It is very marginally responsive. The dip in the crisis is mostly due to systematic risk management that we have applied due to drawdown and not because of the allocation method.
All investments carry risk. This material is for informational purposes and should not be considered specific investment advice or recommendation to any person or organization. Past performance is not indicative of future performance. Please visit our website for full disclaimer and terms of use.
Markets snapshot December 2017
Asset Class
December
YTD returns
Annual Volatility
December Return / Annual Volatility
qplum
0.98%
11.4%
4.9%
20.2%
US investor index
0.37%
6.2%
3.5%
10.7%
US Stocks
1.17%
15.8%
7.0%
16.7%
Non-US Stocks
2.08%
17.6%
8.0%
26.0%
Emerging Market Stocks
-0.34%
14.9%
11.7%
-2.9%
US Bonds
0.33%
2.7%
2.5%
13.2%
US Real-Estate
-0.15%
4.8%
9.5%
-1.6%
Non-US Real-Estate
3.18%
17.5%
7.7%
41.4%
BlackRock Global Allocation Fund
0.62%
9.2%
3.6%
16.9%
Global Low Risk Income Fund
0.55%
5.4%
2.0%
28.3%
Managed Futures (AQR)
0.33%
0.0%
6.7%
4.8%
Risk Parity (AQR)
2.13%
12.7%
6.3%
33.7%
HFRX Global Index
3.70%
17.1%
11.2%
32.9%
All investments carry risk. This material is for informational purposes and should not be considered specific investment advice or recommendation to any person or organization. Past performance is not indicative of future performance. Please visit our website for full disclaimer and terms of use.
References 1. Kelly-Breiman and “How much should I bet”? 2. Breiman, L., "Optimal Gambling Systems For Favorable Games," Jerzy Neyman, Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1, 65-78, 1961. 3. Gottleib, G., "An Optimal Betting Strategy For Repeated Games," Journal of Applied Probability ( 22), 787-795, 1985. 4. Kelly, J.L. Jr.. "A New Interpretation of Information Rate," Bell Systems Technical Journal, 35, (1956), 917-926. 5. Mean Variance Optimization - “Modern Portfolio Theory” 6. Risk Parity 7. Shannon, C, “A Mathematical Theory of Communication” 8. Understanding the Sharpe Ratio
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All investments carry risk. This material is for informational purposes and should not be considered specific investment advice or recommendation to any person or organization. Past performance is not indicative of future performance. Please visit our website for full disclaimer and terms of use.
