Risk Parity

Reverse Engineering AQR’s Risk Parity Strategy

I’m going to start this post by saying that it makes no sense for anyone to pay management fees to get a return stream that is highly correlated to any existing asset class. Unfortunately, many actively managed funds fall in this category.

There’s two reasons for this. One, you can replicate this return stream by just investing in that asset class yourself, likely through low-cost ETFs. Two, even if the fees are low enough that you wouldn’t save that much money by just doing it yourself, adding a highly correlated return stream to something that already exists in your portfolio isn’t going to add anything beneficial to your risk adjusted returns.

With this in mind, I tried to reverse engineer AQR’s Risk Parity Fund using low-cost ETFs that are accessible to retail investors. The idea is to find a way for retail investors to replicate a risk parity strategy for their own portfolio. If you’re unfamiliar with risk parity, you can take a look at my post on how to implement a risk parity strategy.

Why AQR’s Risk Parity Fund Can Be Reverse Engineered

AQR is one of the few fund managers that offer their risk parity strategy as a mutual fund. This makes their daily returns and portfolio holdings easily accessible since mutual funds are held to a higher level of disclosure compared to hedge funds. In addition, AQR has published a number of articles which provide a little bit of insight into exactly how they implement their risk parity strategy.

Moreover, the core concepts of risk parity are well understood. It’s going to invest in a wide range of asset classes across all geographies. At it’s core, it could be described as a fully long, passive investment strategy. It’s passive in the sense that it’s not trying to implement any market views but rather trying to harvest the risk premium of multiple assets. This makes it ripe for replication. AQR does adjust asset weightings based on their risk forecasts and other proprietary measures though, so it’s not purely passive.

Alright, so why attempt to reverse engineer AQR’s Risk Parity Fund in the first place? The fund has an expense ratio of around 1% and the investment minimum for individual investors is $1 million. So this isn’t something that a normal person has access to, and it’s not even clear you would want to given the high fees. I also wanted to empirically determine their portfolio allocations to various asset classes.

AQR’s Portfolio Holdings

First, a look at AQR’s portfolio holdings. I provide the holdings in a table sorted by weight at the end of this blog post. I encourage you to take a look at the full list if you’re interested.

Some observations: AQR invests in a wide range of assets across all geographies, including equities, sovereign bonds, inflation linked bonds, credit spreads, commodities, and currencies. Fixed income have high weight and equities have low weight as is expected for risk parity strategies. Leverage is harnessed through futures, swaps, and forwards — again no big surprise there.

Exposure to credit spreads is through shorting credit default swap indexes. Exposure to inflation linked bonds is through buying the physical bonds because futures for inflation linked bonds do not exist. Equities (rank 6) and commodities (rank 23) only appear once in the top 25 holdings.

Replicating AQR’s Risk Parity Strategy

This is the methodology I used. First, I identified low-cost ETFs for each of the major market segments (I wrote about this before here) that are represented in AQR’s Risk Parity Fund.

Second, I ran a regression on the daily returns of AQR on the daily returns of the various ETFs, subject to the constraint of having a zero intercept and making the sum of the coefficients be close to one. The interpretation is that I wanted the regression to empirically determine the weights of the various ETFs for an unleveraged portfolio and to try to make it match the returns of the leveraged portfolio.

The correct way to do this would be to implement a learning algorithm where the cost function incorporates these constraints, but I just did it using regular linear regression and trial and error to get an acceptable solution informed by my intuition.

Here are the ETFs that I selected along with the model output. The coefficients under the “Estimate” column are the portfolio weights which sum to 100%.

Call:
lm(formula = AQRNX ~ 0 + GLD + SCHP + SPY + TLO + EFA + VWO + EMB + GSG, data = data)

Coefficients:
     Estimate Std. Error t value Pr(>|t|)    
GLD  0.034      0.007      4.903 1.05e-06 *** (Gold)
SCHP 0.265      0.034      7.684 2.75e-14 *** (US TIPS)
SPY  0.163      0.017      9.389  < 2e-16 *** (US Equities)
TLO  0.122      0.015      7.936 4.03e-15 *** (US Sovereign Bonds)
EFA  0.028      0.015      1.928    0.054 .   (Developed Equities)
VWO  0.076      0.011      6.731 2.39e-11 *** (Emerging Equities)
EMB  0.221      0.020     11.078  < 2e-16 *** (Emerging Sovereign Bonds) 
GSG  0.092      0.007     13.207  < 2e-16 *** (Commodities)

This is what this synthetic risk parity strategy looks like. It’s done pretty well at replicating AQR’s strategy with a correlation of 96% and a R-squared of 69%. And the weighted average expense ratio for this portfolio is only 0.20%.

1-014-aqr-risk-parity

There are a few limitations that prevent this synthetic strategy from matching AQR’s strategy more closely. AQR’s strategy is obviously levered, with total exposure around 300% of their net asset value, while the synthetic strategy is unlevered. This explains the roughly 5% difference in cumulative return over this time period.

Usually risk parity funds are levered to an arbitrary level of volatility, so just think of this synthetic strategy as a less volatile version of AQR’s strategy. AQR targets an annualized volatility of 10%.

To try to get more volatility from the unlevered ETF portfolio, I selected an ETF for US sovereign bonds with a 25 year average maturity (TLO). AQR’s exposure to sovereign bonds for developed markets is the 10 year futures contract, but selecting bonds with longer maturity allows for more volatility and more return. It’s a way to simulate the leverage that’s in risk parity strategies. I think using the long maturity ETF explains why there is only a 12% weight for US sovereign bonds and a 26% weight for US TIPS.

The synthetic strategy also has no exposure to credit spreads. This is more of a limitation with the current universe of ETF offerings which are mostly US focused. None of the broad market corporate credit ETFs I tried produced good results in the regression. I think it might have something to do with the fact that AQR gets exposure to credit spreads by shorting credit default swaps — kind of hard to replicate that.

Overall, this model yields pretty reasonable weights and provides good replication. The code underlying this post can be viewed at my Github repository. If you enjoyed reading this post, please consider following Signal Plot via email.

Below you can find the position weights for AQR’s Risk Parity Fund.

Asset ClassSecurity DescriptionWeight
1Fixed IncomeU.S. 10 Yr Treasury Note Future45.8%
2CreditCDX.NA.IG26.6%
3CreditiTraxx Europe26.5%
4Fixed IncomeEuro Bund 10 Yr Bund Future25.3%
5CreditiTraxx Europe Crossover13.7%
6EquityE-Mini S&P 500 Index Future13.6%
7Fixed IncomeSingapore Interest Rate Swap9.3%
8Fixed IncomeSouth Korea Interest Rate Swap9.1%
9TIPSTII 0.125 04/15/208.0%
10TIPSTII 0.125 04/15/197.3%
11CreditCDX.NA.HY5.9%
12Fixed IncomeSouth Africa Interest Rate Swap5.1%
13CurrencyTRY vs USD Forward4.6%
14Fixed IncomePoland Interest Rate Swap4.6%
15Fixed IncomeHong Kong Interest Rate Swap4.5%
16TIPSTII 0.375 07/15/254.2%
17TIPSTII 0.250 01/15/253.9%
18Fixed IncomeOSE Japan 10 Yr Bond Future3.8%
19CurrencyKRW vs USD Forward3.6%
20CreditCDX.EM3.5%
21CurrencyBRL vs USD Forward3.5%
22CurrencyMXN vs USD Forward3.2%
23CommodityGold Future3.1%
24CurrencyPLN vs USD Forward3.0%
25TIPSTII 0.125 07/15/242.9%
26TIPSUKTI 0.125 03/22/242.8%
27CurrencyHUF vs USD Forward2.8%
28TIPSDBRI 0.100 04/15/262.7%
29CommodityBrent Oil Future2.3%
30Fixed IncomeHungary Interest Rate Swap2.3%
31EquityHSCEI China Index Future2.3%
32CurrencyINR vs USD Forward2.3%
33TIPSUKTI 0.125 03/22/262.2%
34CommodityLME Aluminum2.2%
35Fixed IncomeCzech Republic Interest Rate Swap2.2%
36CurrencyZAR vs USD Forward2.1%
37TIPSDBRI 1.750 04/15/202.1%
38EquityOSE Japan Topix Index Future2.0%
39TIPSFRTR 2.250 07/25/202.0%
40EquityFTSE100 Index Future1.7%
41EquityDJ Euro Stoxx 50 Future1.6%
42CommoditySilver Future1.6%
43CommodityWTI Crude Future1.5%
44EquityS&P Mid 400 E-Mini Index Future1.4%
45Fixed IncomeCanada 10 Yr Bond Future1.4%
46CommoditySugar Future1.4%
47CommodityCorn Future1.3%
48EquityRussell 2000 EMini1.2%
49EquityKOSPI 200 Index Future1.2%
50TIPSDBRI 0.100 04/15/231.2%
51EquityMSCI Taiwan Index Future1.1%
52EquityDAX Index Future1.0%
53TIPSFRTR 0.250 07/25/241.0%
54CommoditySoybean Future0.9%
55TIPSFRTR 0.100 03/01/250.9%
56CommodityLive Cattle Future0.9%
57Fixed IncomeAustralia 10 Yr Bond Future0.9%
58EquityBovespa Index Future0.9%
59CurrencyCNH vs USD Forward0.8%
60CommodityLME Copper0.8%
61TIPSFRTR 1.100 07/25/220.8%
62CommoditySoy Oil Future0.8%
63CommodityGas Oil Future (100MT)0.7%
64TIPSFRTR 0.100 07/25/210.6%
65CommodityCoffee Future0.6%
66EquityHang Seng Index Future0.6%
67CommodityUnleaded Gas RBOB Future0.6%
68CommodityCotton No. 2 Future0.5%
69CommodityHeating Oil ULSD Future0.5%
70CommodityNickel Future0.5%
71EquitySwiss Market Index Future0.5%
72EquitySGX CNX Nifty Index Future0.5%
73CommodityNatural Gas Future0.5%
74EquitySPI 200 Index Future0.5%
75CommodityZinc Future0.4%
76CommodityLean Hog Future0.4%
77CommodityLead Future0.3%
78CommodityWheat Future0.3%
79EquityS&P/TSE 60 Index Future0.3%
80EquityTAIEX Futures0.3%
81EquitySouth Africa Top 40 Index Future0.3%
82EquityCAC40 Index Future0.2%
83CommoditySoy Meal Future0.1%
84CommodityFeeder Cattle Future0.1%
85EquityIBEX 35 Index Future0.0%
86CommodityCocoa Future0.0%
87CommodityWheat Future (KCB)0.0%
88CurrencyEUR vs USD Forward-12.8%

About

I am interested in building investing systems, and this blog contains my research and analysis on this topic. I previously worked as an analyst at Bridgewater Associates, a hedge fund that utilizes a systematic, global macro investing style.

  1. Minor point but by the looks of that graph, you calculated your 0.96 correlation using prices instead of returns (should be using returns correlation).

    • You’re right, I had thought about this point some time after I wrote it, and I think correlation of the daily or weekly returns is more appropriate. The correlation should still be pretty high but less than 0.96.

  2. Burrito Dan

    You are comparing your fees to AQR’s without taking into account leverage (for which you pay interest, and fees over a higher AUM), and without taking into account trading costs and slippage. Once you add these up, how do your fees compare?

    • The synthetic strategy I created is using ETFs and no leverage is being used. That partially explains why the synthetic strategy has less absolute return than AQR’s leveraged strategy. So the synthetic strategy does not incur any costs for leverage. There are trading costs, but risk parity is at it’s core a buy and hold strategy, so trading costs are minimal too. The largest cost is still the ETF’s expense ratio.

  3. Nice attempt, but you are missing a number of features in AQR’s product.

    First, like many RP managers they optimize to sharpe ratios and then lever up to the desired risk level. For fixed income, this typically results in clustering around the 5Y note. You can’t do that because your volatility would be too low. So you extend durations causing your sharpe ratio to drop while RP managers simply lever up using futures. To achieve the same effect you would have to use margin – why not do that in your backtest assuming you account for the interest charges?

    Second, they have procedures in place to control drawdowns. That would have come into play in 2008-2009 but your backtest doesn’t extend that far back.

    Nick de Peyster
    http://undervaluedstocks.info

    • If I recall correctly, AQR’s exposure to developed world fixed income is at the 10 year note. For most developed countries, the 10 year rate is considered the “benchmark” long rate.

      You’re right that this approach will likely have a lower sharpe ratio and you could use margin to achieve a similar effect, but I didn’t test for this because margin borrowing costs depend on your broker and change over time. The margin costs for a retail investor probably make this form of leverage not worthwhile.

      Thank you for your comment and sorry for the delay in responding.

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  6. Very interesting. But isn’t it a bit odd to implement a Risk Parity strategy with no leverage? In your previous article, for example, you had 27% in equities and 106% in bonds, for a total exposure of 133%, i.e 33% leverage.

    • So you datamined the weights of the AQR strategy using historical data? Clearly, your pnl should be close to that of AQR’s. But how do you trade your replicated version going forward? You still haven’t identified the weight generating mechanism of their risk parity strategy

      • Good question and thanks for commenting. When I was writing this post, I thought about putting in some caveats about within sample testing versus out-of-sample testing. So you’re right about the shortcomings of this approach, but I thought it would be an interesting two hour analysis.

        The thing about industry implementations of risk parity is that no one knows what other people are doing because that information is proprietary. But the theory states that high vol assets like equities will get low weight and low vol assets like bonds will get high weight. Sure the weightings can change at the margin, but there’s not going to be huge swings in the asset weightings. And you know that risk parity will always be fully long in a basket of securities that won’t change.

        I did just run a regression on the first half of the data versus the second half, and the weightings remained very consistent (within 3% or so for almost all the assets), so for at least since inception, AQR’s weightings haven’t shifted that much. I’d love to hear ideas on how to identify their weight generating mechanism.

    • It’s true that implementing risk parity without leverage is odd since the expected returns of an unlevered portfolio will probably be too low for most institutional investors, but I wanted something simple that was more practical for individual investors.

      It’s hard to use leverage if you’re an individual investor since the contract sizes for futures contracts are too big for most people unless you have a lot of money. You could always just use margin, but some custodians prohibit margin for retirement accounts.

      • Let me elaborate on my comment. I went to a talk by Frazzini of AQR some time ago, where he said leverage is essential to their strategy. He believes people are reluctant to buy low-beta stocks and leverage them up to a beta of 1, and this “mistake” everyone makes is he claims the source of the strategy’s profit. I have no idea if he is right and/or is telling the truth but if he is, then building a portfolio with a large number of ingredients like Zinc Futures and Czech Interest Rate Swaps will enable a close fit to past returns but is not getting at the essence of what supposedly makes this work.

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