An Introduction to Risk Parity Hossein Kazemi

Transcription

1 An Introduction to Risk Parity Hossein Kazemi In the aftermath of the financial crisis, investors and asset allocators have started the usual ritual of rethinking the way they aroached asset allocation and risk management. Academic/Practitioner journals are full of articles that are suosed to show investors what went wrong and how they can adjust their models and theories in order to rotect themselves against substantial losses next time equity and credit markets exerience significant losses. Most of these recommendations should be viewed with a great deal of sketicism as they are bound to incororate a healthy dose of data snooing and over fitting biases. For examle, both Barclay Caital Global Bond Index and MSCI World Equity Index have earned about 7% annual nominal return since 990, with volatility of the bond index being about /3 of the volatility of the equity index. Clearly, going forward it is all but imossible for the bond index to reeat the erformance of the last 0 years. Therefore, any model that would recommend a significant allocation to fixed instruments should be carefully analyzed and its assumtions should be questioned. The so-called risk arity aroach to asset allocation has enjoyed a revival during the last few years because such a ortfolio would have outerformed the normal ortfolios with their tyical significant allocations to equities. In this note, we discuss the risk arity aroach to asset allocation and examine its underlying assumtions. The central idea of the risk arity aroach is that in a well-diversified ortfolio all asset classes should have the same marginal contribution to the total risk of the ortfolio. For examle, as shown below, in a tyical 60/40 ortfolio, equity risk accounts for almost 90% of the total risk of the ortfolio, which is significantly higher than its weight, 60%. Under the risk arity aroach, there is generally a significant allocation to low risk asset classes and allocations to equities and other risky assets are tyically below what we normally observe for most diversified institutional quality ortfolios. Therefore, we want to know if this aroach is based on sound economic and financial reasoning or is it just another attemt to extraolate the results of the last ten years into the future. Basics of the Risk Parity Aroach The risk arity aroach defines a well-diversified ortfolio as one where all asset classes have the same marginal contribution to the total risk of the ortfolio. In this sense, a risk arity ortfolio is an equally weighted ortfolio, where the weights refer to risk rather than dollar amount invested in each asset. This aroach highlights three different issues. First, to aly the risk arity aroach, we need a definition of the total risk of a ortfolio. Second, we need a method to measure the marginal contribution of each asset class to the total risk of the ortfolio. Third, to emloy this aroach, we do not need an estimate of exected returns to imlement the risk arity aroach. The last oint is one of the advantages of this aroach because as we have seen during the last two decades, forecasting returns is a risky business. On the other hand, the risk arity aroach requires accurate estimates of volatility and other measures of risk, which have been to shown to be relatively stable and therefore can be redicted with a good deal of accuracy. Total risk is tyically measured by the volatility of the rate of return on the ortfolio. This means that risk arity works within the same framework as Harry Markowitz s mean-variance aroach. Alternatively, one could use VaR as a measure of total risk. The advantage of using VaR as a measure of total risk is that one can incororate skewness and kurtosis in the measure of In theory, bonds could still offer significant returns in real term if one were to assume that a eriod of significant deflation lies ahead.

2 total risk. For the urose of this introductory note, we will use standard deviation as a measure of total risk. Once we have decided to use standard deviation as the measure of total risk, the contribution of each asset class to the total risk of the ortfolio is well defined and can be easily calculated. The general definition of marginal contribution of an asset class to the total risk of a ortfolio is given by the following exression: MCi Total Risk of = ( Weight of Asset Class i) Weight of Asset Class i Here, MCi is the marginal contribution of asset class i to the total risk of the ortfolio. The last term determines the change in the total risk of the ortfolio if there is a very small change in the weight of asset class i. It turns out that the total risk of the ortfolio is then equal to the sum of the marginal contributions. That is, if there are N assets in the ortfolio, then Total Risk = MC + MC + L + MC N To see how this works, let us consider the case of only two risky assets. The rate of return and the standard deviation of the rate of return on this ortfolio, E R ] and σ R ], are: [ ] [ ] E R = w E R + w E R [ ] [ ] [, ] σ R = w σ R + w σ R + w w Cov R R Where, w and w are the weights of the two assets (they add u to one), [ ] E [ R ] are exected returns on the two assets, σ [ R ] and σ [ R ] rates of return on the two assets, and [, ] [ [ E R and are standard deviations of the Cov R R is the covariance between the two assets. The marginal contributions of the two assets to the total risk of the ortfolio are: [ ] [ ] MC w in σ R w wσ R + w Cov R, R = = in w σ R [ ] [ ] MC w in σ R w w σ R + w Cov R, R = = in w σ R The following table rovides all the information we need to calculate the marginal contributions of Barclay Caital Global Bond Index and MSCI World Equity Index to the total risk of a ortfolio consisting of 60% in equity and 40% in fixed income.

3 MSCI World Index Barclays Caital Global Aggregate 60/ Monthly Standard Deviation 4.50%.6%.95% Covariance Between the Two 0.0% Given the above table, the marginal contributions are: MC MSCI ( ) 60% 4.50% + 40% 0.0% = 60% =.64%.95% MC BarCa ( ) 40%.6% + 60% 0.0% = 40% = 0.3%.95% We can see that equity contributes.64% to the total risk of.95%, while the rest, 0.3%, is contributed by fixed income. In addition, we can see that although the weight of equity is 60%, its contribution to the total risk is 89.34% (.64%/.95%). Given the oor erformance of equities during the last 0 years, one may wonder if it is sensible to allocate so much of a ortfolio s total risk to equity risk. The general formula for calculating the marginal contribution of each asset to the total volatility of a ortfolio when there are more than two assets is: MC i = w i N j= wjcov Ri, R j σ R β σ = wi i R The second line is a rather simle method for calculating the marginal contribution of an asset class. It states that the marginal contribution is equal to the weight of the asset times the beta of the asset with resect to the ortfolio times the total risk of the ortfolio. Here beta is defined as: Cov Ri, R βi = σ R where Cov Ri, R is the covariance between the ortfolio and the rate of return on asset i. In the revious examle, the betas of equity and fixed income assets with resect to the ortfolio are.49 and 0.7, resectively. For instance, the marginal contribution of equity is then equal to:.64% = 60%.49.95% 3

4 To create a ortfolio using the risk arity aroach, we need to adjust the weights until the marginal contributions of the two asset classes are equal. Using trial and error or an otimization ackage such as Microsoft Excel s Solver, one can show that when 6.45% is allocated to equity and 73.55% to fixed income, risk arity is achieved. MSCI World Index Barclays Caital Global Aggregate Total Risk of Risk Parity Weights 6.45% 73.55%.9% Marginal Contribution in Risk Parity Port 0.955% 0.955% As exected, risk arity requires a significant allocation to fixed income and as stated in the introduction, this ortfolio would have erformed very well during the last 0 years with an annualized rate of return of 7.%. This is roughly equal to the annualized rate of return on the 60/40 ortfolio with a volatility that is 50% smaller than that of the 60/40 ortfolio. Given such an imressive result, it is no wonder that several risk arity based investment roducts have recently aeared in markets. Economic Foundation of Risk Parity Aroach As discussed above, risk arity ortfolios make relatively large allocations to low risk asset classes. Notwithstanding the erformance of such ortfolios over the last 0 years, it is safe to say that going forward a ortfolio with a monthly standard deviation of.9% is not likely to rovide a rate of return required by most investors. Given this, is there a reason to use this aroach to asset allocation? It turns out that if one is willing to use leverage, there is a rather strong economic reason to exect a risk arity ortfolio to erform rather well and even outerform a tyical ortfolio where relatively large allocations are made to risky assets. To see this, we need to go back to the fundamental results of Modern Theory and secifically, the results reorted by Markowitz and then later by Share and others. According to Markowitz s original results, if investors care only about mean and variance of their ortfolios, then they should invest only in ortfolios that lot on the efficient frontier. These ortfolios have the lowest risk for a given level of exected return. The following figure dislays the familiar efficient frontier. This can be done using Solver tool of Microsoft Excel. 4

5 Mean Return Efficient Frontier X M Standard Deviation According to Markowitz, investors should ick a ortfolio that falls on the line segment MX. Investors who are willing to take some risk will ick a ortfolio close to X and those who are more risk averse would select a ortfolio close to M. Even though the 60/40 ortfolio is not likely to be on the efficient frontier, it is likely to be closer to X than to M. On the other hand, the risk arity ortfolio is likely to be closer to M. Again, there is no reason to believe that the risk arity ortfolio is an efficient ortfolio. In the following grah, we have lotted hyothetical ortfolios M, X, 60/40 and risk arity. We have lotted the riskless rate as well. Mean Return Z 60/40 X M RP Riskless Rate Standard Deviation The tangent line originating from the riskless rate is known as the caital market line. It identifies a set of ortfolios that can be constructed as the combination of two ortfolios/assets: (a) the riskless asset and (b) the efficient ortfolio that lies on the tangency oint. For examle a ortfolio that lies between oints RP and the riskless rate can be created using a combination of investments in these two assets. On other hand, a ortfolio that lies above RP can be created through borrowing at the riskless (using leverage) and investing the roceeds in ortfolio RP. Now that the riskless rate has been introduced, we can see that one can make a case for investing a low risk ortfolio and then using leverage to increase the risk and hoefully the exected return on the ortfolio. In the above figure, we have assumed that both the risk arity 5

6 ortfolio and the 60/40 ortfolio are on the original efficient frontier. We can see that ortfolio Z, which is a combination of the risk arity ortfolio and leverage, has the same risk as the 60/40 ortfolio but with a higher exected rate of return. This aears to resent a comelling reason for using a risk arity aroach to asset allocation. However, there needs to be a word of caution: if the risk arity ortfolio is far away from the efficient frontier, the leveraged aroach to risk arity asset allocation may lead to oor erformance. In other words, it is critical for the risk arity ortfolio to be close to the efficient frontier. In addition, leverage reresents a source of risk that many institutional investors may not wish to assume. The following table summarizes the results for the risk arity ortfolio and its leveraged version /40 Risk Parity Risk Parity (Unlevered) (Levered) Monthly Mean 0.59% 0.59% 0.73% Monthly Standard Deviation.95%.9%.95% Monthly Information Ratio Monthly Share Ratio It is imortant to note that to raise the volatility of the rate of return on the risk arity ortfolio to the same level as the volatility of the rate of return on the 60/40 ortfolio one needs to emloy 54% leverage. That is, for each $00 caital, one needs to borrow $54 and then to invest $54 in the risk arity ortfolio. This leverage figure is given by: Volatility Target Leverage = Volatility of Unlevered.95% =.9% This level of leverage may be too high for many institutional investors. However, in ractice it may not be necessary to use that much leverage to reach reasonable exected returns. Of course, given historical erformances of equity and bond indices, no amount of leverage was needed to achieve the same rate of return as the 60/40 ortfolio because the unlevered risk arity ortfolio already has the same average return as the 60/40 over eriod (both earned 0.59% er month). Other Related Aroaches The idea of leveraging u a relatively low volatility ortfolio to generate a given exected rate of return can be alied to other ortfolios as well. Risk arity is one aroach to creating a low volatility ortfolio. Any aroach that leads to a low volatility well-diversified ortfolio can be used to create higher exected returns using leverage. The key is for the low volatility ortfolio to have a Share ratio that is higher than the 60/40 or other high volatility ortfolios. If the Share ratio of the low volatility ortfolio is lower than the riskier ortfolio, then leverage will actually lead to a ortfolio that will be inferior to the riskier ortfolio. This is the key: for the risk arity to work it has to lead to a relatively high Share ratio and the investor should be able and willing to use some degree of leverage. One simle aroach to creating a low volatility ortfolio is to use an equally weighted ortfolio. This ortfolio is by definition rather well diversified and is likely to have relatively high allocations to less risky assets. The other aroach would be to use an otimization ackage to identify the minimum variance ortfolio. This ortfolio is created by finding the weights that 6

7 minimize the volatility of the rate of return on the ortfolio. M on the efficient frontier dislayed in the above grah is such a ortfolio. Finally, a volatility-weighted ortfolio can be used to create a low volatility ortfolio. In this aroach the weight of each asset class is given by: w i = N σ [ Ri ] σ R j j= This means the weight of each asset class is roortional to the inverse of its volatility. This aroach is in fact identical to the risk arity aroach when we have only two assets and it will be the same as risk arity in the more general case if correlations between asset returns are the same. We are going to use our numerical examle to demonstrate this aroach. MSCI World Barclays Caital Index Global Aggregate Monthly Standard Deviation 4.50%.6% Weights 6.46% 73.54% Here 6.46% = 73.54% = 4.50% %.6%.6% %.6% Since we have only two asset classes, it can be seen that the weights are same as in the risk arity ortfolio. Risk Parity and Alternative Investments To the degree that alternative investments tend to have low volatility and low correlations with other asset classes, the allocations to alternative investments will be relatively high in a risk arity ortfolio. However, many institutional investors may have a difficult time acceting relatively large allocations to alternative investments. Let us use a numerical examle to demonstrate this. We are going to consider three asset classes, Barclay Caital Global Bond Index, MSCI World Index, and HFR Hedge Fund Index. The following table dislays the statistics for these three asset classes as well as those of three different ortfolios. 7

8 990-0 HFRI Fund Weighted Comosite Index MSCI World Index Barclays Caital Global Aggregate 0/50/40 Volatility Weighted Risk Parity Monthly Mean 0.99% 0.60% 0.59% 0.63% 0.74% 0.73% Monthly Standard.03% 4.50%.6%.66%.7%.64% Deviation Monthly Share Ratio Weights in 0/50/40 Weights in Volatilityweighted Weights in Risk Parity 0% 50% 40% 37% 7% 46% 35% 4% 5% A few observations are in order. First, as exected, both the volatility-weighted and the risk arity ortfolios require significant allocations to hedge funds and bonds. Second, the volatility-weighted and the risk arity ortfolios are rather similar. Third, both the volatilityweighted and the risk arity ortfolios have much higher Share ratios than the 0/50/40 ortfolio. This means that if these two low volatility ortfolios are levered u to have the same volatility as the 0/50/40 ortfolio, they will have higher mean return than the 0/50/40 ortfolio. Conclusion In this note, we introduced the basic ideas behind the risk arity aroach to asset allocation and examined its economic foundation. It turns out that risk arity aroach is a viable aroach to asset allocation and is in fact suerior to ad hoc asset allocation models emloyed by the industry. In the absence of a full otimization aroach, risk arity aears to rovide a close aroximation to the original model of Harry Markowitz. The key in using this aroach is the willingness to use leverage and the ability to manage the risks osed by the use of leverage. While risk arity is a viable aroach to asset allocation, it does not reresent a trading strategy that can be emloyed by active managers. The reasons are that it does not require any estimate of exected return on an asset class (otentially a source of skill for active managers) and it always leads to ositive weights for asset classes (long/short strategies cannot be imlemented). It is a suitable model for institutional and high net worth investors who do not face significant constraints on their asset allocation olicies and are able to use leverage. Finally, investors who are able and willing to use derivatives, could use these instruments to lever u their risk arity ortfolios. 8