VOLUME 41 NUMBER 2 WINTER The Voices of Influence iijournals.com

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1 VOLUME 41 NUMBER WINTER 015 The Voices of Influence iijournals.com

2 On the Holy Grail of Upside Participation and Downside Protection EDWARD QIAN EDWARD QIAN is the chief investment officer in the Multi-Asset Group at PanAgora Asset Management in Boston, MA. One of the objectives of active management is to provide upside participation and downside protection. With the global financial crisis of 008 etched on the psyche of the investment community, this objective, especially the latter half of it, has taken on much more significance in recent years. To many investors and managers, this has become the new holy grail of investing. What else can account for the attraction and popularity of investment strategies with the characteristics of low risk and/or low beta, or strategies that come with drawdown controls? The former category includes, for eample, minimum variance or defensive equity strategies while the latter includes strategies with built-in tail-risk hedging based on portfolio insurances schemes, offered by either derivatives such as options and swaps, or stop-loss policies using trend signals. These two categories are not necessarily mutually eclusive. Whether or not these defensive strategies would provide investors protection in the net financial crisis or competitive returns over the long run is an open question. But how do we define and evaluate a defensive strategy in the first place? A naïve interpretation would mean a strategy should go up when the market is up but it will go down less when the market is down. But less than what? It cannot just be less than the market. Were that to be the case, one could simply divest some of one s portfolio into cash to achieve the goal. For eample, a simple static portfolio with 50% in the S&P 500 Inde and 50% in cash would certainly participate on the upside and lose less than the inde on the downside. We can stipulate that downside protection must mean, at a minimum, the degree with which the strategy goes down in a down market is less than the degree with which it goes up in an up market. As a first step to properly analyze and evaluate these strategies, we need a rigorous quantitative definition of upside participation and downside protection. Another important question involves the relationship between these defensive strategies and traditional active strategies that focus on alpha or ecess return relative to their benchmarks. For eample, do good defensive strategies imply positive alpha and vice versa? Do good defensive strategies necessarily outperform the overall market in the long run? These questions are especially relevant to institutional investors who are often required to evaluate their performance against traditional indices and benchmarks. In this article, we shall answer these questions. We first define the participation ratios of a strategy in both up and down markets, respectively, and then introduce the participation ratio difference (PRD) as a quantitative measure for the goodness of upside participation and downside protection. We show WINTER 015 THE JOURNAL OF PORTFOLIO MANAGEMENT

3 that the participation ratios and PRD can be derived analytically when the returns of a strategy and its underlying inde form a bivariate normal distribution. In fact, our eamples show the analytic results fit the sample results very well. One of the main results of the article is that PRD is approimately proportional to the alpha of the strategy relative to the inde based on a onefactor capital asset pricing model (CAPM) (Markowitz [1959], Sharpe [1964], Lintner [1965]). In other words, PRD and alpha are closely related. Therefore, traditional active strategies with positive information ratio (IR) can be epected to deliver positive PRD, even though their downside protection might be limited. On the other hand, their upside participation ratio can be epected to be higher than 1. Throughout the article, we shall use the S&P 500 Inde and the S&P 500 sectors to illustrate our results numerically. The empirical results show that most defensive sectors possess positive PRD, thus positive alpha, while many cyclical sectors have negative PRD, thus negative alpha. From this perspective, the PRD results of different sectors are closely related to the empirical rejection of the CAPM model and the low volatility anomaly. PARTICIPATION RATIOS We define participation ratio as the ratio of conditional mean of a strategy or investment versus conditional mean of the corresponding inde, conditioned on the sign of the inde return relative to cash. Suppose the inde return (in ecess of cash) is denoted by r and the strategy return (in ecess of cash) is denoted by r y. Then the upside participation ratio is E ( ) P + = (1) E ( ) The notation E( ) in the equation denotes epectation or average. Similarly, we define the downside participation as E ( ) P = () E ( ) This definition can be interpreted as ratio of conditional averages. An alternative definition would be average of conditional ratios, such as ry P + = E > 0 (3) r However, this definition suffers an infinity problem when r is near zero. 1 It is then necessary for us to modify the definition Equation (3) by requiring the magnitude of the inde return r to eceed a threshold. This alternative definition would then depend on the choice of the threshold and could be a potentially useful measure of participation ratio at the tails of the return distribution. While this is an area of further research, we employ the definitions in Equations (1) and () in the present article. The use of ecess return instead of actual return also warrants some eplanation. The nominal return on cash is almost always positive (most countries still have positive, albeit small, risk-free rates at the time of this writing). If we had defined the participation ratios in absolute return terms instead, then cash would have a positive upside participation ratio and a zero downside participation ratio. While cash is the ultimate defensive investment, its participation ratios should be noneistent or zero, because it really does not participate in any way. Defining participation ratios in ecess cash terms also makes their values intuitive for any strategy that incorporates the use of cash. For eample, an investment strategy of 50% in the S&P 500 and 50% in cash would naturally lead to a 0.5 upside participation ratio as well as a 0.5 downside participation ratio. By the same token, a leveraged portfolio that is long 150% in the S&P 500 and short 50% in cash would have 1.5 for both upside and downside participation ratios. With both participation ratios defined, we denote the participation ratio difference as ( ) ( ) ( ) ( ) E PRD r 0 E y r y 0 P+ P = (4) E 0 E 0 r When the upside participation ratio is greater than the downside participation ratio, PRD is positive. Naturally, all else equal, one would prefer a strategy with a positive PRD to one with a negative PRD. And a strategy with positive PRD would be one that provides upside participation and downside protection. As we shall see later in the article, the PRD has a direct relationship to the strategy s alpha. Even on the face of Equation (4), it can be roughly interpreted as an r ON THE HOLY GRAIL OF UPSIDE PARTICIPATION AND DOWNSIDE PROTECTION WINTER 015

4 incremental return of the strategy when the inde makes a round trip of a positive return followed by a negative return of the same magnitude. An additional benefit of the PRD measure is that it is a linear function of individual PRDs when a portfolio consists of individual investments. In other words, if a portfolio has weights (w 1, w,, w N ) in N investments, then the portfolio s participation ratios and PRD would be the weighted average of individual participation ratios and PRDs. One might also opt to use the ratio of P + over P to measure the tradeoff between upside and downside participation. There are several drawbacks with this ratio. First, there is no intuitive interpretation. Second, the two ratios are not necessarily of the same sign. One can envision an etreme (but very rare) case in which P is negative, that is, a strategy on average achieves positive returns when the overall market is down. In that case, a negative ratio is even harder to interpret. Lastly, a definition based on the ratio of participation ratios wouldn t be a linear function, which makes it hard to use in a portfolio setting. EXAMPLES OF PARTICIPATION RATIOS FROM THE S&P SECTORS The S&P 500 is often used as an indicator of the broad U.S. equity market. Other broad market indices such as the Russell 1000 inde closely mimic the S&P 500. Hence, participation ratios measures against the S&P 500 are reasonable proies for U.S. equity strategies. Here, as a numerical eample, we present the participation ratios and PRDs for two of the S&P 500 inde sectors: the consumer staples and the financial sector. Later in the article, we shall use all 10 sectors to illustrate the relationship between PRD and alpha. Ehibit 1 shows the conditional averages of the S&P 500 and the two sectors, based on monthly returns from October 1989 to April 014. During months when the inde generated positive returns, the S&P 500 gained an average of 3.11% per month. During months when the inde generated negative returns, the S&P 500 declined an average of 3.74% per month. By default, the inde itself has participation ratios of one and a PRD of zero. In the months when the S&P 500 inde s return was positive, the consumer staples sector s average return was.11%. This implies an upside participation ratio of On the other hand, when the S&P 500 s return was E XHIBIT 1 Conditional Means, Upside/Downside Participation Ratios, and PRDs of the S&P 500 Inde and Consumer Staple and Financial Sectors negative, the consumer staples sector s average return was 1.74%. This implies a downside participation ratio of Both ratios are lower than one mainly because the consumer staples sector is a low beta sector. But their difference or PRD is a positive 0.1. In contrast, the financial sector is a high beta sector. Its conditional averages are higher than (in magnitudes) those of the S&P 500. Both participation ratios are greater than 1, that is, on average it gains more than the inde but it also loses more than the inde. The upside participation ratio is 1.18, while the downside participation ratio is higher at 1.5. As a result, its PRD is Among the two sectors, based on these results, it is evident that the consumer staples sector is the one that provided upside participation and downside protection and the financial sector did not. ANALYTIC DERIVATION OF PARTICIPATION RATIOS Participation ratios obviously depend on the statistical properties of the underlying return distributions of both the inde and the strategy under consideration. We now derive analytical results for participation ratios for a bivariate normal distribution of r and r y. In the net section, we show these results match those obtained from the sample data very closely. The ecess return of the inde and the strategy follows a normal distribution with a mean vector and a covariance matri r μ ~ N, r y μ y σ ρσ σ ρσσ y σ y y (5) WINTER 015 THE JOURNAL OF PORTFOLIO MANAGEMENT

5 The parameters μ and μ y are unconditional means of the inde and the strategy; σ and σ y are their return volatilities; and ρ is the correlation between the two. Based on (5), we can derive four conditional means of r and r y, conditioned on the sign of the inde return r. The eact analytic formulas are given in an appendi. The results involve eponential functions as well as error functions. Fortunately, the results can be greatly simplified when the Sharpe ratio of the inde is small. We have S = μ σ (6) Sharpe ratios for many types of asset classes and investment strategies are usually low. This is especially true over the long run when calculated on a monthly basis. For an annualized Sharpe ratio of 0.5, the monthly value is only about When Sharpe ratios are small, the eponential term in Equation (A-1) is approimately one and the error function term is nearly zero. We have E( ( r > 0) σ μ q 1 (7) Hence, the conditional mean of positive inde returns consists of two terms divided by the probability q. The first volatility term is actually the mean of the half of the return distribution that is to the right of the mean. The second term adds to it half of the unconditional mean. The conditional mean of negative inde returns can be derived analytically and we give the result in an appendi. It can also be simplified to a linear form as follows E( ( r < 0) ( 1 q) σ μ 1 (8) Similarly, the first volatility term is the mean of the left half of the return distribution. In the appendi, we also list the analytical results of conditional means of r y. The upside conditional mean can be approimated as ρσ y μy E( ( y r > 0) q π 1 (9) Equation (9) is of the same form as Equation (7), ecept that the volatility term is modified by the correlation coefficient between the inde and the strategy. The downside conditional mean is approimately E( ( y r < 0) ρσy μy 1 q ( ) 1 (10) Dividing Equation (9) by (7), we obtain the upside participation ratio as P + ρσπ μ y y π S ρ σ μ =β y / π S π β + π Sy π 1 S ρ (11) The final epression in Equation (11) is a key result, which states that the upside participation ratio is directly proportional to the beta of the strategy, multiplied by a fraction derived from Sharpe ratios and the correlation coefficient. The beta of the strategy is given by β =ρσ y / σ. Suppose both Sharpe ratios and correlation are all positive, then higher strategy Sharpe ratios and low correlations will lead to higher upside participation. However, if the Sharpe ratio of the strategy is much less than the Sharpe ratio of the inde, or is even negative while the other two parameters are positive, then the upside participation ratio would be lower than the beta. Dividing Equation (10) by (8), we obtain the downside participation ratio as P 1 π Sy / ρ β π β 1 1 S π Sy + π S ρ (1) Similarly, the downside participation ratio is directly proportional to the beta of the strategy multiplied by a fraction derived from Sharpe ratios and the correlation coefficient. Suppose both Sharpe ratios and correlation are all positive, then the higher strategy Sharpe ratios and low correlation will lead to lower downside participation, which is desirable. However, if the Sharpe ratio of the strategy is negative while the other two ON THE HOLY GRAIL OF UPSIDE PARTICIPATION AND DOWNSIDE PROTECTION WINTER 015

6 parameters are positive, then the downside participation ratio would be higher (or worse) than the beta. From Equation (11) and (1), we obtain the participation ratio difference as 1+ π Sy / ρ 1 PRD = P+ P β β π 1+ S 1 π Sy / ρ π S This equation can be simplified algebraically to PRD π ( μy βμ ) πα π = 1 π S σ 1 S σ (13) Notice the term in the parenthesis of the numerator is CAPM alpha (in sample), that is, α μ βμ (14) y Therefore, Equation (13) establishes the fact that the participation ratio difference or PRD is proportional to alpha. Note that the scaling constants in the denominator depend only on the inde return not on the return statistics of a given strategy. It is now clear that at least under the normal return assumption, there is a one-to-one linear relationship between PRD and alpha. A positive/negative/zero PRD implies a positive/negative/zero alpha and vice versa. Of course, a single PRD could be from different sets of upside and downside participation ratios. It is epected that investors might have different preferences on the levels of participation ratios and in addition those preferences can be time varying. For eample, during and after the 008 global financial crisis, some investors preference might have shifted toward a lower downside participation ratio. During the height of the technology bubble, many investors preference moved toward a higher upside participation ratio. In other words, while PRD is important for alpha, investors preference might depend on individual upside and downside participations, a behavior that is consistent with the prospect theory (Kahneman and Tversky [1979]) where investors utility depends on profit and loss relative to initial wealth. Here the participation ratios are relative to a market inde. Regardless of an investor s preference for individual market participations, we have shown that the overall PRD is no more and no less important than alpha. If upside participation and downside protection is the new holy grail, apparently so is good old alpha. PARTICIPATION RATIOS AND PRDS OF S&P 500 SECTORS As an application of the analysis, we now present a full empirical analysis of the participation ratios and PRDs of the S&P 500 sectors. In the process, we eamine differences between values obtained from sample returns, values based on models of normal distributions, and values based on approimation to the models. We find ecellent accuracy for values based on the approimation. As a result, there is an almost perfect relationship between sectors PRDs and their alphas against the S&P 500. Ehibit shows monthly return statistics of the inde and the 10 sectors. The inde and all 10 sectors had positive average ecess returns (arithmetic), 3 with the energy (ENE), healthcare (HLH), and technology (TEC) sectors being the top three, and the materials (MAT), telecommunication (TEL), and utility (UTL) sectors being the bottom three. The volatility varies more widely across the 10 sectors than the average return. E XHIBIT Sample Return Statistics of the S&P Inde and the 10 S&P Sectors WINTER 015 THE JOURNAL OF PORTFOLIO MANAGEMENT

7 The financial (FIN) and technology (TEC) sectors have the highest return volatility at 6.4% and 7.30%, respectively, while the consumer staples (CSS) and utility (UTL) sectors have the lowest volatility at 3.85% and 4.33%, respectively. The Sharpe ratios on a monthly basis are all quite modest, but they also show some differences. Two low volatility, defensive sectors consumer staples and healthcare have the two highest Sharpe ratios. The two high volatility, cyclical sectors: financials and technology have relatively low Sharpe ratios. The lowest Sharpe ratio resides in the telecom sector. Both correlation and beta tend to be high for cyclical sectors and low for defensive sectors. The utility sector has the distinction of having the lowest correlation and beta to the inde. Lastly, we derive alpha according to Equation (14) using average returns and betas. In general, defensive sectors have positive alpha and cyclical sectors have negative alpha. There are a couple of eceptions. One is the energy sector with a large positive alpha of 0.36% per month, and the other is the telecom sector 4 with a negative alpha of 0.1% per month. We shall have more to say about the relationship between the alphas and PRDs, but now we turn our attention to the participation ratios of these sectors. First, we present in Ehibit 3, Panel A, the sample participation ratios and PRDs, based on conditional averages from actual monthly returns. In Ehibit 3, Panel B, we present the ratios based on normal distributions with the estimated parameters given in Ehibit and the full analytic results in the appendi. Ehibit 3, Panel C, shows the differences between the two, in other words, the errors of model-based values. We make several observations about these results. First, the modelbased errors (see Ehibit 3, Panel C) are, in general, very small. There are a couple of sectors (energy and financial) that have slightly larger errors in their upside and downside participation ratios. However, these errors offset each other, resulting in minimal errors in their PRDs. Therefore, even though sector return distributions may not be normal due to fat tails, for eample, PRDs derived under normality assumption match those sample values. One possible reason, supported by Panel C, is that non-normality affects both left and right tails of the distribution equally, and PRDs cancel those errors. Our second observation is about the participation ratios. As we proved in the previous section, participation ratio, both upside and downside, is greatly influenced by beta. This fact is confirmed in Panels A and B of Ehibit 3. For low beta sectors, or sectors with a beta less than one, their upside and downside participation ratios are also less than one. Conversely, high beta sectors, or sectors with a beta greater than one, all have both upside and downside participation ratios greater than one. Of course, alphas can move the participation ratios away from beta. But in the case of sector portfolios, alphas are not strong enough to move any participation ratio to the other side of beta, that is, increase an upside participation ratio of a low beta sector to above one or decrease a downside participation ratio of a high beta sector to below one. An ideal combination of upside and downside participation ratios is that in which the former is higher than one while the latter is lower than one. Not only does such a strategy lead to a positive PRD, E XHIBIT 3 A. Sample Participation Ratio and PRD of the 10 S&P Sectors B. Model Participation Ratio and PRD Based on Normal Return Distributions C. Model Participation and PRD Error vs. the Sample Estimates ON THE HOLY GRAIL OF UPSIDE PARTICIPATION AND DOWNSIDE PROTECTION WINTER 015

8 hence positive alpha, but it is also a strategy that goes up more when the market is up, while going down less when the market is down. Such a strategy would allow investors the psychological benefit of keeping up with the Joneses in good times and harboring some degree of Schadenfreude in bad times. From Ehibit 3, we see this doesn t occur naturally with sector portfolios. It is of course possible if we apply an appropriate amount of leverage to some of the low beta sectors. Ehibit 4 provides the participation ratios and PRDs using approimation results derived from Equations (11), (1), and (13). Compared with Ehibit 3, Panel B, the results are almost identical, so we have decided not to present the table of approimation errors. In all cases, the magnitude of errors is less than It thus appears that the approimation results of PRD work very well for all sectors. PRD, ALPHA, CAPM, LOW-VOLATILITY ANOMALY According to Equations (13) and (14), PRD and alpha share an approimate linear relationship. We have listed sector alpha in a one-factor CAPM model in Ehibit and sample PRD in Ehibit 3, Panel A. As it turns out, the correlation between the two is Ehibit 5 is a scatter plot between PRD and alpha for the 10 sectors. They nearly lie on a straight line, which fits the data with a R-squared of This reconfirms our analysis. E XHIBIT 4 Approimated Participation Ratio and PRD E XHIBIT 5 The Scatter Plot and the Fitted Line of PRD and Alpha for the 10 Sectors WINTER 015 THE JOURNAL OF PORTFOLIO MANAGEMENT

9 The eistence of both positive and negative PRDs and equivalently alphas is a refutation of the one-factor beta version of CAPM (Black et al. [197]). As we showed previously, positive PRDs tend to be associated with defensive sectors and negative PRDs tend to be associated with cyclical sectors. Thus defensive sectors that have low beta and low volatility have provided upside participation and downside protection, while cyclical sectors have not. From this perspective, it could be concluded that our results regarding PRD are another manifestation of the low beta/low volatility anomaly (Haugan and Heins [197]). To illustrate this point graphically, we plot in Ehibit 6 the sectors beta versus their PRDs. The correlation of the two is quite negative. The fitted line is downward sloping sectors with beta greater than one tend to have a negative PRD and sectors with beta less than one tend to have a positive PRD. The relationship between volatility and PRD (see Ehibit 7) is very similar. Low volatility sectors tend to have a positive PRD, and high volatility sectors tend to have a negative PRD. Both cases offer support of the low risk anomaly at the sector level. E XHIBIT 6 The Scatter Plot and the Fitted Line of Beta and PRD for the 10 Sectors E XHIBIT 7 The Scatter Plot and the Fitted Line of Volatility and PRD for the 10 Sectors ANOTHER APPLICATION: PARTICIPATION RATIOS OF STYLES In addition to sectors, investment styles such as growth, value, and size can also be studied from the perspective of participation ratios. In this section, we analyze the Russell indices for the U.S. equity market. We use the Russell 3000 as the market inde and measures upside/downside participation ratios and PRDs of the following si indices: the Russell 1000, 1000 growth, 1000 value, 000, 000 growth, and 000 value indices. Ehibit 8 displays the return statistics (on a monthly basis) of the indices based on monthly returns from January 1979 to April 014, and participation ratios and PRDs of ON THE HOLY GRAIL OF UPSIDE PARTICIPATION AND DOWNSIDE PROTECTION WINTER 015

10 E XHIBIT 8 Return Statistics and Participation Ratios of the Russell Indices the si style indices versus the Russell 3000 benchmark. First, we note the large-capitalization Russell 1000 inde shows little difference versus the Russell 3000 benchmark. Second, the small-cap Russell 000 inde has higher return and higher volatility than the Russell 1000 inde, but its Sharpe ratio is actually lower. Third, between growth and value, both value indices have higher returns and yet lower volatilities than their growth counterparts do. As a result, they ehibit much higher Sharpe ratios and positive alphas. On the other hand, both growth indices have negative alpha. Hence, with respect to styles of growth and value, there also eists a low beta/volatility anomaly, perhaps related to the low beta/volatility anomaly of sectors. Participation ratios of the si indices are consistent with these relationships between beta/volatility and alpha. The two growth indices and the small-cap inde have high beta and their participation ratios (both upside and downside) are greater than one. But their PRDs are all negative. The worst case is the Russell 000 growth inde. The Russell 1000 value inde s beta is low and both participation ratios are less than one, with a positive difference. Lastly, the Russell 000 value inde stands out as a shining eample of upside participation and downside protection. Its beta is close to one at 0.97, and more significantly, its upside participation is greater than one while its downside participation ratio is less than one. This is remarkable in two ways. First, as we noted previously, none of sectors possess such desirable combination of participation ratios. Second, as we shall see soon in the article, this type of participation ratio is typically associated with beta-neutral active equity management strategies with positive information ratio. It is therefore not a coincidence that almost all active equity strategies employ in some way both value and size factors because two of them together offer balanced beta eposures. PARTICIPATION RATIOS OF BETA NEUTRAL ACTIVE STRATEGIES Traditional benchmark-relative active strategies strive to outperform a benchmark in both up and down markets. Some fundamental managers might try to do so with beta timing or beta bias; most quantitative managers try to do so while maintaining beta neutrality. In quantitative terms, these quantitative strategies relative performance versus a benchmark is determined by the tracking error to the benchmark and the information ratio (IR), or the ratio of average ecess return over tracking error. While these strategies usually don t have upside participation and downside protection as an eplicit goal, it is interesting to analyze their participation ratios in the framework of the present article. Since these strategies beta is close to one, it is reasonable to epect that their participation ratios are also close to one. Deviations from one would depend on the information ratio and tracking error. If we denote the information ratio by IR and the tracking error by σ α, then the average alpha would be IR σ a. The mean of the active strategy is μ y = μ + IR σ y a Assuming the active risk and the benchmark are uncorrelated, then the active strategy is of beta one and the volatility of the strategy is σ = y σ + σa WINTER 015 THE JOURNAL OF PORTFOLIO MANAGEMENT

11 By Equation (11), we derive the upside participation ratio as P + + π σ a 1 + S IR σ + π 1+ 1 S = 1+ π α σ π σ IR σ a (15) This approimation is appropriate when the Sharpe ratio of the benchmark is small. Hence, the upside participation ratio is one plus a correction term that is proportional to IR and the ratio of active risk relative to benchmark volatility. For eample, for a strategy with an annualized IR of one (monthly IR of 0.9) and an annualized tracking error of 3%, and a benchmark annualized risk of 15%, P + is roughly The downside participation ratio is given by P π σ a 1 + S IR σ π 1 1 S = 1 π α σ π α IR σ a (16) So the participation ratio difference is approimately σ π π α a PRD= + IR = σ σ (17) With the same parameters, the downside participation ratio P is roughly 0.93 and PRD would be An active strategy with annual IR of one is not easily achievable in practice and its PRD is just Recall the PRD of the S&P 500 sectors from the previous section. Four of the ten sectors had a PRD close to 0.14 or higher: consumer staples, energy, healthcare, and utility. From this e post comparison, it does appear that a sector strategy that relies on low volatility/beta anomaly for providing upside participation and downside protection is more appealing than a traditional beta-neutral active strategy. Anecdotally, the advantage of minimum variance strategies over traditional beta-neutral active strategies became much stronger after the 008 financial crisis, in terms of both quantitative and psychological effects. DOWNSIDE PROTECTION WITHOUT SACRIFICING OVERALL RETURN One of the potential pitfalls of focusing too much on downside protection is low upside participation and, consequently, low overall returns over time. For eample, a strategy with a very low beta would definitely have low downside participation but could also miss too much of the upside. Without additional alpha, the overall return of such a strategy would lag the return of the inde. This begs the question: when is alpha or PRD high enough to propel a low beta strategy to outperform the inde? Historically, three low beta sectors (see Ehibit ), consumer staples, energy, and healthcare, outperformed the S&P inde in absolute terms. Two low beta sectors, telecom and utility, underperformed the inde. Note that the first three sectors had very high PRDs while the last two sectors had rather low PRDs. We now derive a PRD threshold which determines if a low beta strategy could match or outperform the inde. The threshold is determined by the condition μ y = μ. The average return of a given strategy also equates to σ μ y =βμ +α βμ + PRD π Equating this epression to μ gives the threshold PRD= π(1 β )S (18) This equation is rather intuitive. First, the higher the Sharpe ratio of the inde, the higher the PRD threshold has to be. Second, the lower the beta of the strategy, the higher the PRD threshold is. For eample, for a beta of 0.5 and an inde Sharpe ratio of 0.15, the PRD must be 0.19 or above for the strategy to match or outperform the inde. According to Equation (18), three low beta sectors (see Ehibit and Ehibit 3), consumer staples, energy, and healthcare, clear threshold hurdles while the other two low beta sectors, telecom and utility, do not. ON THE HOLY GRAIL OF UPSIDE PARTICIPATION AND DOWNSIDE PROTECTION WINTER 015

12 SUMMARY Upside participation and downside protection has been a qualitative objective for active management. In this article, we have developed a quantitative framework to define and evaluate investment strategies from this perspective in terms of upside and downside participation ratios. It is demonstrated that results obtained under normality assumptions match actual results well. More importantly, we have established clear and intuitive relationships between participation ratios and traditional portfolio statistics including such as beta, alpha, Sharpe ratio of benchmarks, and information ratio of active management. In this contet, it seems there is nothing new about the new holy grail of upside participation and downside protection. But this is a false impression. The framework offered in this article is based on average participation ratios over the entire return distribution. For investment strategies with etreme fat-tailed distributions or investors who focus on etreme losses or gains, different kinds of participation ratios, such as the one suggested in the article, might be needed and they warrant further research. Another area of research is portfolio construction that incorporates participation ratios. For eample, instead of return and risk, how do we construct a portfolio with targeted upside and down participation ratios? We shall eplore these topics in future research. A PPENDIX In this section, we provide the analytical results of conditional upside and downside means of two return variables that form a bivariate normal distribution given in Equation (5). For the inde, we simply integrate the probability density function (PDF) from zero to infinity to get E( ( r e erf( > 0) σ + μ + μ s s ) q π The function erf() is the error function defined as erf ( s) = e d π s 0 1 (A-1) And the parameter s is linearly proportional to the Sharpe ratio of the inde S s = μ σ = The parameter q istheprobability 1 q = P ob ( ( ) = [ ] The downside conditional mean of the inde is given by E( ( erf( r < 0) σ e + μ μ s) ) 1 q s 1 ) (A-) To derive the conditional mean of the active strategy, we first utilize the conditional epectation of r y given r and then integrate r over its designated interval. We arrive at and ρσ y E( ( r e erf( y > 0) + μ + μ s y y s ) q π ρσ y s y y E( ( r e erf( y < 0) + μ μ s) ) 1 ENDNOTES 1 q) 1 1 (A-3) (A-4) The author thanks Nick Alonso, Mark Barnes, Bryan Belton, and Kun Yang for helpful discussions on the topic. 1 Mathematically, the epectation integral becomes divergent at r = 0. Data are from Bloomberg. 3 Arithmetic averages are used to model return distributions. In addition, they are consistent with the analytic framework for participation ratios and CAPM model. However, if we use geometric average instead, the results do not change qualitatively at all. 4 The underperformance of the telecom sector in the tech bubble burst is largely responsible for the negative alpha of the entire period. It is fair to say during the tech bubble, the telecom sector became a cyclical sector due to its eposure to broadband networks. REFERENCES Black, F., M.C. Jensen, and M. Scholes, The Capital Asset Pricing Model: Some Empirical Tests. Studies in the Theory of Capital Markets, edited by M. Jensen. New York: Praeger, 197, pp WINTER 015 THE JOURNAL OF PORTFOLIO MANAGEMENT

13 Haugen, R.A., and A.J. Heins. On the Evidence Supporting the Eistence of Risk Premiums in the Capital Markets. Working paper, December 197. Kahneman, D., and A. Tversky. Prospect Theory: An Analysis of Decision under Risk. Econometrica, XVLII (1979) pp Lintner, J. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics (February, 1965) pp Markowitz, H.M. Portfolio Selection: Efficient Diversification of Investments. Cowles Foundation Monograph 16, Yale University Press, New Haven, CT, Sharpe, W.F. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, Vol. 19, No. 3 (1964) pp To order reprints of this article, please contact Dewey Palmieri at dpalmieri@iijournals.com or ON THE HOLY GRAIL OF UPSIDE PARTICIPATION AND DOWNSIDE PROTECTION WINTER 015

14 PanAgora Asset Management 470 Atlantic Avenue, 8th Floor Boston, MA 010 The opinions epressed in this article represent the current, good faith views of the author(s) at the time of publication, are provided for limited purposes, are not definitive investment advice, and should not be relied on as such. The information presented in this article has been developed internally and/or obtained from sources believed to be reliable; however, PanAgora does not guarantee the accuracy, adequacy or completeness of such information. Predictions, opinions, and other information contained in this article are subject to change continually and without notice of any kind and may no longer be true after the date indicated. The views epressed represent the current, good faith views of the author(s) at the time of publication. Any forward-looking statements speak only as of the date they are made, and PanAgora assumes no duty to and does not undertake to update forward-looking statements. Forward-looking statements are subject to numerous assumptions, risks and uncertainties, which change over time. Actual results could differ materially from those anticipated in forward-looking statements. PanAgora is eempt from the requirement to hold an Australian financial services license under the Corporations Act 001 in respect of the financial services. PanAgora is regulated by the SEC under US laws, which differ from Australian laws. DISCLOSURES The opinions epressed in this article represent the current, good faith views of the author(s) at the time of publication, are provided for limited purposes, are not definitive investment advice, and should not be relied on as such. The information presented in this article has been developed internally and/or obtained from sources believed to be reliable; however, PanAgora does not guarantee the accuracy, adequacy or completeness of such information. Predictions, opinions, and other information contained in this article are subject to change continually and without notice of any kind and may no longer be true after the date indicated. Past performance is not a guarantee of future results. As with any investment there is a potential for profit as well as the possibility of loss. This material is directed eclusively at investment professionals. Any investments to which this material relates are available only to or will be engaged in only with investment professionals. Predictions, opinions, and other information contained in this presentation are subject to change continually and without notice of any kind and may no longer be true after the date indicated. The views epressed represent the current, good faith views of the author(s) at the time of publication. Any forward-looking statements speak only as of the date they are made, and PanAgora assumes no duty to and does not undertake to update forwardlooking statements. Forward-looking statements are subject to numerous assumptions, risks and uncertainties, which change over time. Actual results could differ materially from those anticipated in forward-looking statements. INDEX DESCRIPTIONS The Russell 1000 Inde measures the performance of the 1,000 largest securities in the Russell 3000 Inde. The Russell 1000 Growth Inde measures the performance of the Russell 1000 securities with higher price to book ratio and higher forecasted growth values. The Russell 000 Inde is an unmanaged list of common stocks that is frequently used as a general performance measure of U.S. stocks of small to midsize companies. The smallest,000 securities in the Russell 3000 Inde are included in this inde. The Russell 000 Growth Inde includes the Russell 000 securities that have higher price to value ratios and higher forecasted growth values. The Russell 000 Value Inde includes the Russell 000 securities with lower price to book ratios and lower forecasted growth values. The Russell 3000 Inde lists the 3,000 largest U.S. securities, as determined by total market capitalization. The S&P 500 Inde is an unmanaged list of common stocks that is frequently used as a general measure of U.S. stock market performance.