# Risk Parity Portfolios:

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2 an abstract concept until a loss occurs. When that happens, managers and clients alike always want to know what contributed to the loss. Going back to our previous example of a 60/40 portfolio, the table below displays the average contribution to losses with three different thresholds. TABLE 1: AVERAGE LOSS CONTRIBUTION FOR THE 60/40 PORTFOLIO BASED ON THE RUSSELL 1000 AND LEHMAN AGGREGATE BOND INDICES: Loss> Stocks Bonds N 2% 95.6% 4.4% 44 3% 100.1% 0.1% 25 4% 101.9% 1.9% 14 For losses above 2%, stocks, on average, contributed 96% of the losses. This is very close to the risk contribution of 93% we calculated earlier. For losses greater than 3% or 4%, the contributions from stocks are higher, above 100%. Although the data in this table is influenced by sampling error (N is the number of monthly returns for a given threshold) and higher tail risks from stocks, it provides empirical evidence for the economic interpretation of risk contribution; it approximates the expected loss contribution from underlying components of the portfolio. This is true when we use variances and covariances to calculate risk contribution. Risk Parity Portfolios It can now be understood why a 60/40 portfolio is not a well-diversified portfolio. When a loss of decent size occurs, over 90% is attributable to the stocks. To put it differently, the diversification effect of bonds is insignificant in a 60/40 portfolio. Conversely, this would imply that any large loss in stocks will result in a loss of similar size for the whole portfolio. This is hardly diversification. How can we use these insights to design a portfolio that limits the impact of large losses from individual components? This can be accomplished if we make sure the expected loss contribution is the same for all components. The concept of risk contribution and its economic interpretation thus leads us to the development of Risk Parity Portfolios that allocate risk equally among asset classes. TABLE 2: RETURN CHARACTERISTICS OF INDICES AND PORTFOLIOS: Russell 1000 Lehman Agg 60/40 Parity Parity (L) Average 8.3% 3.7% 6.4% 4.7% 8.4% Standard deviation Sharpe ratio 15.1% 4.6% 9.6% 5.4% 9.6% While Risk Parity Portfolios can utilize many asset classes, 6 it helps to illustrate their potential benefits using the stock/ bond example. For a fully invested portfolio, an allocation of 23% in the Russell 1000 Index and 77% in the Lehman Aggregate Bond Index would have equal risk contribution from stocks and bonds. Table 2 shows some of the return characteristics of this Risk Parity Portfolio and a leveraged version denoted by (L), along with those for the underlying indices as well as for the 60/40 portfolio, all measured by excess return over three-month Treasury bills. As the data show, the Russell 1000 had the highest average return at 8.3%, but also a much higher standard deviation. And, as a result, it has the lowest return-risk, or Sharpe ratio, at The bond index had lower average return as well as a lower standard deviation, and its Sharpe ratio, at 0.80, is better. For the 60/40 portfolio, both the average return and the standard deviation are between those of stocks and bonds. More importantly, its Sharpe ratio, at 0.67, is lower than that of bonds, which is an indication of poor diversification: The overall portfolio s Sharpe ratio is lower than one of its components. In contrast, the Risk Parity Portfolio s Sharpe ratio of 0.87 is higher than those of stocks and bonds, representing the benefits of true diversification. Note that the Risk Parity Portfolio s Sharpe ratio is substantially higher than that of 6 They include eight asset classes: U.S. large-cap equity, U.S. small-cap equity, international equity, emerging-market equity, government bonds, corporate bonds, TIPS, and commodities.

3 the 60/40 portfolio. 7 For a fair comparison of average returns, the leveraged version is adjusted so that it has the same level of risk as the 60/40 portfolio. We shall have more to say on the subject of leverage later. What about the loss contribution of the various components of the Risk Parity Portfolio? Table 3 shows that for a loss of 2% or more, stocks contributed 48% and bonds 52%; for a loss of 3% or more, stocks contributed 45% and bonds 55%. These numbers are close to parity, given the limited numbers of sample points. TABLE 3: AVERAGE LOSS CONTRIBUTION OF THE PARITY PORTFOLIO Loss> Stocks Bonds N 2% 48.4% 51.6% 17 3% 45.4% 54.6% 6 The optimality of Risk Parity Portfolios In contrast to the investment approach of most traditional asset allocation portfolios, which typically involves forecasting long-term asset return and employing mean-variance optimization, Risk Parity Portfolios are based purely on risk diversification. The question remains: Why should risk diversification, especially parity risk contribution, lead to efficient portfolios? The reason is that Risk Parity Portfolios are actually meanvariance optimal if the underlying components have equal Sharpe ratios and their returns are uncorrelated. Are these realistic assumptions? First, equal Sharpe ratios imply that the expected return is proportional to the risk for each asset class. This is theoretically appealing because it means that assets are priced by their risk. In practice, we can derive the implied return of an asset class from its Sharpe ratio and assess whether the implied return is realistic. For instance, if we assume a Sharpe ratio of 0.3, then the implied excess returns (over a risk-free rate) would be 4.5% for stocks and 1.5% for bonds. Second, the actual correlation between stocks and bonds, while not zero, is quite low. These considerations lead us to believe that the Risk Parity Portfolios are efficient, not only in terms of allocating risk, but also in the classical mean-variance sense under the assumption we just tested. Risk Parity Portfolios have additional benefits. First, each asset is guaranteed to have a non-zero weight in the portfolios. Second, the weights are influenced by asset return correlations in a desirable way: Assets that have exhibited higher correlations with other asset classes will have a lower weight and those that have exhibited a lower correlation with other asset classes will have a higher weight. Commodities, for instance, would have a significant weight due to its low correlations with both stocks and bonds. Targeting risk/return level with appropriate leverage While the Risk Parity Portfolio in Table 2 has a high Sharpe ratio, the unleveraged version has lower return than the 60/40 portfolio due to much lower risk. An investor may not be able to achieve his or her return objective simply by creating a portfolio with a high Sharpe ratio. One solution is the use of leverage to achieve higher levels of return. For instance, the leveraged Risk Parity Portfolio in Table 2 has a leverage ratio of 1.8:1. Table 4 shows one additional leveraged Risk Parity Portfolio, whose risk level is the same as that of stocks. It outperformed the Russell 1000 by close to 5% per year with a leverage ratio of 2.8:1, and the second portfolio outperformed the 60/40 portfolio by 2% per year with a leverage ratio of 1.8:1. TABLE 4: COMPARISON BETWEEN PARITY PORTFOLIOS WITH THE STOCK INDEX AND THE 60/40 PORTFOLIO Russell 1000 Parity (L) 60/40 Parity (L) Average 8.3% 13.2% 6.4% 8.4% Standard deviation Sharpe ratio 15.1% 15.1% 9.6% 9.6% One way to interpret the Sharpe ratio is the return in percentage points for every 1% of risk taken. For example, for every 1% of risk taken, the 60/40 portfolio returns 0.67% while the parity portfolio returns 0.87% per annum.

4 Many investors have grown to accept the fact that some degree of leverage is necessary and beneficial in investing. For example, investing in common stocks has inherited leverage since companies issue debts to grow their business. And, many hedge fund strategies employ leverage to enhance returns while controlling risks. In the case of Risk Parity Portfolios, leverage is necessary and relatively easy to implement with futures when the risk level is above 5%. While use of leverage entails some short-term risk, our analysis shows that it is appropriate and reasonably safe to employ leverage in the Risk Parity Portfolios over the long run. Since bonds have much lower risk than stocks, the Risk Parity Portfolios leverage bonds such that they would have the same risk contribution as stocks. As a result, the leveraged version of the Risk Parity Portfolio maintains the high Sharpe ratio but also has higher returns. Using the Risk Parity Portfolios Our backtests 8 show that the Risk Parity Portfolios had a Sharpe ratio of 1.1 over the period from 1983 to 2004, translating to excess returns of 4.5%, 11.3%, and 22.6%, respectively, for the three strategies. We believe that the Risk Parity Portfolios are well suited to the needs of institutional investors today. Given the current challenge posed by relatively low returns from most asset classes, investors must seek better alpha sources as well as extract higher return from their existing market exposure. For many investors, the beta risk actually represents the majority of their total risk budget. The Risk Parity Portfolios provide a more efficient alternative to traditional asset allocation: They limit the risk of overexposure to any individual asset class, while simultaneously providing ample exposure to all of them. With Risk Parity Portfolios, investors can reap the benefits of true diversification: Their eggs are placed evenly and safely in many baskets. Risk Parity Portfolios can be used as stand-alone beta products. They can also be combined with alpha strategies to further increase returns. As a beta strategy, Risk Parity Portfolios can be used in the following three ways: An unleveraged version with 4% 5% risk, similar to that of the Lehman Aggregate Bond Index A leveraged version with a leverage ratio of about 2:1 and a risk target of around 8% 10%, similar to that of domestic or global balanced portfolios A global macro strategy with 16% 20% risk and leverage of 4:1, similar to that of a typical hedge fund

6 Copyright 2011 PanAgora Asset Management, Inc. All Rights Reserved PanAgora Asset Management 470 Atlantic Avenue Boston, MA United States of America /11

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