The Upside Potential Ratio

By

Frank Sortino, Mike Wilkinson, and Hal Forsey

Introduction

Consider an investor who wants to retire at a specified age. In the first quarter of 2001 she visits a financial planner for assistance. He determines that she will have to earn 10% on her current 401K plan in order to achieve that goal. In the course of the planning session she expresses a desire to find managers who have demonstrated the potential to exceed 10% subject to an acceptable level of downside protection. How should the financial planner evaluate managers in a way that is consistent with this client’s wishes? Sortino et al [1999] proposed a new performance measure called the Upside Potential ratio (U-P ratio) designed for this purpose.

We begin with a quantitative analysis of some properties of the U-P ratio and conclude with an example of how this financial planner could use the U-P ratio for ranking possible investment opportunities in mutual funds. All data, graphs and statistics were generated using "Investment Tools", a proprietary product of LCG Associates, Atlanta, Georgia.

Properties of the Upside Potential Ratio

The following notation will be used:

f = probability density function of return

The formal definitions are:

The relationship of the upside probability and downside probability to the MAR is a well understood starting point.

. 1.

Pu is the probability of exceeding the MAR, and is the area under the curve above the MAR.

Pd is the downside probability. Pd + Pu must sum to 1.

The relationship of the upside potential (Ru), and the downside potential (Rd), to the Mar and X is not as obvious:

2.

Ru is the upside potential and measures how much the return might exceed the MAR, so it a measure of excess return. It is a probability weighted function of the returns above the MAR where the function is the simple differences between the MAR and all values above the MAR. This implies that the investor is risk neutral for returns above the MAR.

Ru is not the conditional mean of the excess return given the the return exceeds the MAR. It is however closely related. The conditional mean is Ru / Pu and is a misleading measure for upside potential when Pu is small. Also, it would not be consistent with Fishburn’s equation for downside risk (1977). For these reasons, we do not use the conditional mean.

Figure 1 presents a graph of the upside probability of the Dodge and Cox stock fund relative to a mid cap value index. The table below the graphs shows how upside probability decreases as the MAR increases. For an investor with an MAR of 0%, Dodge & Cox has the potential of exceeding his MAR by 23.7% and t-bills have the potential of exceeding his MAR by 6.9%.

For the investor in this study who must earn at least 10% in order to accomplish her goal, the Dodge and Cox stock fund has an upside potential of 15.2% while t-bills have no upside potential. Dodge and Cox has an upside potential of 25.2%, or 15.2% above the MAR of 10%. It is perhaps more meaningful to say that Dodge and Cox has 680 basis points (15.2% – 8.4%) more upside potential than the average Mid Cap Value fund.

Figure 1

Table 1 provides a simple example of how upside potential is calculated from a discrete

Distribution. This example is for illustrative purposes only. In Sortino and Forsey (1996) we

Explain why one should not do this, but instead, use the continuous distribution as shown in Figure 1 and equation 2.

Table 1

Both fund 1 and fund 2 have an average return over the past 10 years of 10.8% and they both have an upside probability of 70%. However, fund 2’s upside potential is 90 basis points higher than fund 1. Notice, all values below 10 are given a value of zero, but are counted when determining the probability of each value above the MAR, i.e., the sum of values above the MAR is divided by 10, not 7.

Of course, one should not make evaluations on the basis of returns alone. We now look at the risk of falling below the MAR. The relationship of upside variance and downside variance with respect to M and X is:

Therefore:

Vu + Vd – (X-M)² = Var = total variance 3.

Vd is the downside variance, a measure of the spread of the distribution below the MAR. Vd is the mean of the squared deviations below the MAR counting returns greater than the MAR as having 0 deviation. Vd is the measure of downside risk first proposed by Peter Fishburn (1977). The upside variance plus the downside variance minus (X-M)² = the total variance.

The upside potential ratio (U-P ratio) is:

Ru / (Vd)^1/2 4.

The numerator is linear which is appropriate for an investor that is not averse to making more money and enjoys the next dollar just as much as the last dollar. However, Vd is a quadratic function and is consistent with the notion of risk aversion for outcomes below the MAR. The farther below the MAR returns fall, the greater the aversion. Taking the square root of Vd converts the denominator to the same units as the numerator, allowing an intuitive interpretation of the ratio. Figure 2 provides an example of these calculations.

Figure 2

The upside potential for Fidelity Magellan (11.2) plus the downside potential (-4.4) = the mean minus the MAR (16.8 - 10) as shown in equation 2. Note that downside potential is linear and therefore a poor measure of risk since it implies risk neutral behavior.

The U-P ratio described in equation 4, shows the Magellan fund has 19% more upside potential than downside risk for this investor, while the Average Large Cap Core fund has18 % more downside risk for this investor (.82-1). This is an easily understood and intuitively appealing interpretation.

Application

In the first quarter of 2000, Sortino ranked 100 mutual funds for Pensions and Investments magazine (P&I). Let’s suppose the financial planner used this ranking to choose mutual funds for the equity portion of the asset allocation. For illustrative purposes, Table 2 shows only the funds ranked at the top and bottom in each style category.

Table 2

The difference between the top and bottom rankings in each style category are large. Dreyfus Basic S&P had 36% more upside potential than downside risk, while Fidelity Growth had 2% less upside potential than downside risk. How did these funds fair in the end of the year decline? Figure 3 indicates the top ranked U-P funds did better in all three style categories, and on average, did approximately 4 times better than the bottom ranked funds. In this instance, the U-P ratio was able to forecast bad as well as good performance. We believe this was one of the first ex ante research studies, i.e., the rankings were made before the fact and published in P&I.

We are not suggesting the U-P ratio be used for predicting short term performance. Only that it did provide some degree of protection in a subsequent market decline. Traditional measures did not.

Figure 3

The financial planner in this example assumed the ranking from a magazine article was appropriate for this client. However, the MAR in the P&I ranking was 8.6%, while the clients MAR is 10%. Figure 4 was generated by Plantinga et al (2001) and illustrates how the rankings for European funds changed dramatically as the MAR changed.

                                              Correlation coefficients as a function of monthly MAR’s.

Another advantage the U-P ratio has over the Sharpe ratio or Sortino ratio is that it can never be negative, which would imply a risk taking attitude. Plantinga et al provide an example of how the Sharpe ratio decreases with the level of put protection while the U-P ratio increases.

Conclusion

We have presented evidence to support the use of the Upside Potential ratio for an individual investor. It is an individualized performance measure and is only suitable for investors with the same MAR. We assumed the individual was a 401K plan participant. However, we believe the U-P ratio is just as appropriate for an individual pension fund. We chose the 401K example because of the view expressed by Sally Atwater [2001], that financial planners are more accepting of a tailor made analysis because their focus is on how to accomplish the individual investor’s goal. Pension funds, she claims, focus on who will manage the money. The result is that pension funds measure a manager’s ability to beat the market, while financial planners measure performance in terms of accomplishing the client’s financial goal.

If pension fund sponsors believe the goal is to beat the market, their consultant can do one ranking of managers and use it for all clients. A tailor made approach would show how all managers performed with respect to the MAR necessary to accomplish the individual client’s goal.

References

Atwater, S. 2001. "Managing Downside Risk in Financial Markets." Butterworth-Heinemann, Publisher, Ch 3

Plantinga, A., R. van der Meer, and F. Sortino. 2001. "The impact of downside risk on risk-adjusted performance of mutual funds in the Euronext markets." Social Sciences Research Network, www.sscn.com

Fishburn, P. C. 1977. "Mean-risk Analysis with Risk Associated with Below Target Returns, American Economic Review, March.

Sortino, F.A., and H. Forsey. (1996) "On the Use and Misuse of Downside Risk." Journal of Portfolio Management, Winter.

Sortino, F., R. van der Meer, and A. Plantinga. 1999. "The Dutch Triangle." Journal of Portfolio Management, Fall.

End Notes

 
The Minimal Acceptable Return

In order to evaluate these policy decisions, a comprehensive approach that identifies the impact on the various interest groups is required.  We believe an asset liability management (ALM) analysis is admirably suited to the task. It will identify those asset allocations across asset categories that best accommodate the various decisions dealing with the level and volatility of premium contributions, the indexing of future benefits and the funding level or solvency of the pension fund. 

The strategic mix of assets in the ALM study will have an expected return, which is a valuable estimate of the return that must be earned at minimum in order to accomplish the policy goals of the plan.  We will refer to this as the minimal acceptable return (MAR). See Exhibit 3.

                                                             Exhibit 3

The bootstrap procedure for generating this distribution was developed by Bradley Effron [1993].  The MAR is what links the decisions of top level management at the strategic level to the management decisions at the tactical level and the operational level.  It is also the MAR that serves as the point from which risk is measured for both performance measurement and asset allocation.  Thus, linking performance measurement with asset allocation. 

It is this crucial link that distinguishes the Dutch triangle from more traditional approaches for pension management.  This structure shaped the following policy statements:

1.       The goal is to fund the pension plan within the constraints identified in the ALM study.

2.       The investment objective is to maximize the expected return above the MAR, subject to the risk of falling below the MAR. 

Notice that the objective supports the goal, in that, if the objective is achieved, the goal will be accomplished.  The rate of return that separates success from failure to accomplish the goal is the MAR.  Only returns equal to or greater than the MAR assure success. The goal is not to make money.  Making money is how one accomplishes the goal.  The MAR identifies how much money is needed at minimum.

 Unless the MAR is established at the strategic level there is a danger that it will either be absent from the performance measurement and asset allocation decisions, or it will be misspecified.  Without a directive from above, those responsible for implementing policy usually look outside the organization for advice on what tools to use for performance measurement and asset allocation.  If the MAR decision gets pushed down to the operational level, the consultant and/or portfolio manager may select a substitute for the MAR that presents their results in the most favorable light, but has little or nothing to do with the return necessary to accomplish the stated goal of the pension plan.  Such a case is described in Sortino [1999]. 

 While this paper focuses on the corporate pension fund, some public officials are also planing to implement this concept.  Charles Valdez [1998, p 38], chairman of the investment committee of CALPERS said the $141 billion fund is considering a plan to "set a minimum acceptable return" to be used in assessing manager performance.  Valdez said "volatility (standard deviation) is the one risk I don't think is as important as many consider it."  Robert Boldt, senior investment officer for CALPERS called this a "major major" change in investment policy.

 The Tactical Level

At this level, management is concerned with implementation of policy. Actions are concerned with risk-return tradeoffs with respect to performance measurement and asset allocation.  The task is to determine which combination of active managers and passive indexes to hold in the portfolio.  It is the responsibility of the Chief Investment Officer for the pension fund to obtain the necessary tools for accomplishing this task in a manner that is consistent with established policy.  This process begins with performance measurement relative to the MAR. 

 Should the MAR for both equities and fixed income portfolios be the mean of the benchmark identified in the ALM study, or should the risk for equity managers be measured relative to the equity component, and the risk for bond managers measured relative to the fixed income component?  Exhibit 4 shows three distributions.  A is the distribution of the strategic benchmark, B is the distribution for a bond index, and C is the distribution for an equity index. 

 Suppose one decided to measure the performance of equity managers relative to the mean of C, and bond managers relative to the mean of B.  Now suppose a bond manager D invests only in government notes and earns a constant return over some interval represented by the spiked broken line.  The downside risk for D measured relative to the mean of B is zero and the return is greater than the index for bond managers.  It is also true that the standard deviation of returns for D is zero, and that government notes have no default risk.  All three measures confirm the riskless nature of the strategy pursued by manager D.  But what about the risk of not accomplishing the goal? Only by measuring risk relative to the return necessary to fund the plan within their cost constraints (the mean of A) would management be aware of the risk that was incurred.

 Ergo, the performance of a bond manager should be measured relative to a bond index and/or other bond portfolio managers, but the risk for both

index and manager should be measured relative to the MAR of the strategic benchmark.  Using the mean of the strategic benchmark as the MAR

for all managers keeps everyone focused on the return necessary to accomplish the goal of the pension plan and clearly identifies the returns that

will contribute to the risk of not achieving that goal. Furthermore, to have a different MAR for each asset category (stocks, bonds, real estate,

etc.) would imply multiple utility functions within one organization.  We assume all participants have the same utility function.  Of course, this does

not take into consideration covariance relationships.  However, that is best handled by the asset allocation model we describe later.  

                                                                 Exhibit 4

The Upside Potential Ratio

 One of the great pioneers in behavioral finance was the late Amos Tversky, professor of psychology at Stanford University.  Some of his empirical studies disputed the assumptions of modern portfolio theory (MPT) that investors are rational.  In a discussion of prospect theory, Tversky  [1995] called attention to the tendency of investors to make risk-averse choices in gains and risk-seeking choices in losses, resulting in suboptimal portfolios. The S shaped utility function of prospect theory indicates investors are very risk-averse for small losses but will take on investments with a small chance of very large losses. 

While Taversky's work describes how investors do behave, Peter Fishburn's normative utility function [1977] describes how investors should behave.  Fishburn assumes investors are risk averse below the benchmark MAR, and risk neutral above the MAR, i.e., they have an aversion to returns that fall below the MAR, and the farther they fall below the MAR the more they dislike them.  On the other hand the higher returns are above the MAR the more they like them.

Recent research in the behavioral finance area describes how investors want to behave.  In general,   investors do not seek the highest return for

a given level of risk, as portfolio theory assumes.  According to Statman and Shefrin [1998] investors seek upside potential with downside

protection. Olsen [1998] says, "investors desire consistency of return and therefore choose decision processes that preserve appropriate future

financial flexibility."  Rather than maximize the expected return, they want to maximize a "satisficing" strategy.  Sebastiaan de Groot [1998]

studied one hundred wealthy investors to determine if they made decisions in a manner consistent with expected utility theory or behavioralfinance theory.  He found that approximately half the questions were answered in a manner consistent with expected utility theory and the other

questions were answered in a manner consistent with behavioral finance.  But most of these investors said they wanted "wealth growth that is as

stable as possible where a trade-off between risk and return has been made [2]."

How investor's want to behave and how investors should behave can be accommodated in one statistic.  Maximizing the expected value above the MAR instead of maximizing the mean of the entire distribution, would imply a linear utility function above the MAR, and would also capture the notion of upside potential.   A performance measure that is consistent with this attitude is the upside potential ratio (U-P ratio) shown below.

 The numerator of the U-P ratio is the expected return above the MAR and can be thought of as the potential for success. This is the opposite of

shortfall risk.  The denominator is downside risk as calculated in Sortino and van der Meer [1991] and can be thought of as the risk of failure. 

 Sortino, Miller, and Messina [1997] claim that more stable estimates of risk are possible by employing style analysis.   William Sharpe [1992] developed a procedure for identifying a manager's style in terms of a set of passive indexes, which we refer to as the manager's "style benchmark."  If a manager's style can be identified in terms of a style benchmark of passive indexes, one can use twenty or more years of data on the style indexes instead of being limited to five years of data, or less, on the manager. Downside risk is then calculated from the distribution of returns of the style benchmark instead of the manager's return distribution.[3]  

 Exhibit 5 illustrates the type of investor who would use the U-P ratio to select a manager.  On the upside: Manager A invests in securities that are perceived as safe, but that guarantee the investor will not earn a high enough return to accomplish the stated goal.  Manager B provides the highest chance of success, and manager C provides the highest potential for success.  On the downside: Manager A has the lowest standard deviation, but the highest downside risk, and manager B has a lower downside risk than manager C. 

 How might these risk-return tradeoffs affect an investor's choice of manager?  An investor who wanted to maximize the probability of exceeding the MAR for a given level of downside risk would choose B.  An investor who wanted to maximize the potential for success for a given level of downside risk would choose C. Investors such as A confuse credit risk with investment risk and therefore, choose safety of principal and ignore the MAR.

                                                            Exhibit 5

 Linking performance to asset allocation: (see Addendum at bottom)

 The link between performance measurement and tactical asset allocation is the mean of the strategic benchmark, which is the MAR (see Exhibit 3). In the first stage of tactical asset allocation, an efficient frontier consisting solely of passive indexes is generated.  The benchmark establishes that segment of the efficient frontier that is most often relevant for implementing policy decisions (see Exhibit 6).  This segment lies between the efficient portfolio that has the highest return for the same risk as the strategic benchmark (vertical arrow), and the efficient portfolio that has the least risk for the same return as the strategic benchmark (horizontal arrow).  

For active managers to replace passive indexes they would have to lie beyond the passive efficient frontier, i.e. they would have to add value.  One procedure for accomplishing this is to calculate alphas for managers to see if they provide a higher return for the same level of risk.  This procedure results in a different mix of styles for each portfolio on the efficient frontier in the first stage of the optimization process.  We use a linear programming model to keep the style mix constant.  Each vertical line that extends above the efficient frontier in Exhibits 6 and 9 represents a combination of active managers and passive indexes that have the same style mix as the point on the original frontier, but have a higher return. 

                                                                                         Exhibit 6

Operational Level

 To make the tactical decisions operational, active and passive management firms must now be hired in accordance with the results gathered at the tactical level.  Funds are transferred to each manager and purchases of securities made.  Each manager should be informed as to the risk-return procedures that will be used to evaluate their management style and their future performance.  The active managers should understand that their goal is the same as the plan sponsor's, which is the same as the participants: to maximize consistency and magnitude of returns above the MAR.  They incur risk of failing to accomplish the client's goal if they fall below the MAR.

 Empirical Example

 We now demonstrate how this is applied in practice.  Let's assume the strategic benchmark mix is 75% in fixed income, 25% in large Dutch equities, with no foreign and no small cap exposure. Using MSCI style data from Independence International Associates converted to local currency, we used the style analysis procedure developed by William Sharpe [1992] to identify styles of six Dutch mutual funds as of the end of June 1998.  Because of van der Meer's current position with Fortis, and his previous association with Aegon, we decided to omit any funds from these corporations. The style analysis results for ABN Amro Netherlands Fund, Holland Fund, ING Bank Dutch Fund, AXA Aandelen Netherlands Fund, EOE Index Fund, Orange Fund are shown in the table in Exhibit 7.

                                                               Exhibit 7

 The R² for all but the Orange Fund are very high, indicating style analysis can be applied successfully  to many Dutch investment firms. Roger Otten at Maastricht University informed us that the low R² for Orange is due to the fact that the IIA small cap indexes are not small enough to capture the very small cap securities in the Orange Fund.  Something is missing from our specification of the returns generating process resulting in an unreliable estimate of risk.  This calls attention to an important consideration.  There is a tradeoff between including all the indexes necessary to explain the variance in returns for all funds and the requirement of independence between indexes.  At some point, adding indexes increases the R squared at the expense of increasing multicolinearity.   

           lace.  The difference is due primarily to measuring the manager's style risk over twenty-three years instead of simply using the manager's risk over the last five years.  So, while the style fit for the Orange fund is admittedly poor ( R² = 66% ), the style based Sharpe ratio captures the inherent risk of micro cap securities better than just using the fund's realized returns for the past five years.

 The third column measures performance relative to the MAR. The upside-potential ratio ranks the Orange fund twelfth and EOE jumps up to second place.  This performance measure indicates that those who are concerned with maximizing the potential for success relative to the risk of failure would rank the Orange Fund lower than any other measure.   For this reason, plus the low R squared, we will not include the Orange Fund in the asset allocation phase.

 Linking Performance To Asset Allocation

 The strategic benchmark in Exhibit 9 is very close to the passive efficient frontier because it is not globally diversified. In this example, we chose to use the mean of the benchmark as a random variable instead of an absolute.  In other words, the MAR becomes a moving target that rises and falls with the market. In this instance, the strategic benchmark mix had a realized return of  12.79% for the five years ending June of 1998.

                                            Exhibit 9

Mean-variance versus Downside

The passive mix that is closest to the benchmark is 0% in cash, 72% in bonds, 5% in large cap growth, 10% in large value, 4% in small growth, 6% in small value, and 1% each in Japan, the U.S., and the pacific basin without Japan.  Because the benchmark does not have any foreign or small cap it is below the efficient frontier.  Exhibit 9 indicates that an additional 100 basis points is possible by replacing the passive indexes with active managers EOE and AXA.  An investor who used standard deviation as a risk measure would over weight EOE with respect to AXA.  An investor who used downside risk (down arrow) would do just the opposite.

Summary:

 While there are some differences in the management of assets and liabilities in Europe versus other countries, there is one universal commonality, there is some rate of return that must be earned at minimum on the assets in order to meet the payments on the liabilities.  Establishing this minimal acceptable return (MAR) is one of the most important policy decisions of the plan sponsor.  Failure to achieve this MAR could have a dramatic impact on corporate earnings.  Therefore, it deserves the careful attention of those responsible for setting policy as well as those responsible for implementing it.  To implement this policy, the Chief Investment Officer of the pension fund and her staff  must select performance measurement and asset allocation tools designed to carry out the established goals and objectives.  If top level management fails to identify the MAR as a policy decision and establish goals and objectives incorporating the MAR, managers at the tactical level may well make decisions that have little or nothing to do with the goal of funding the plan within established constraints. 

For any performance measure to be oriented toward this goal, risk must be measured relative to the MAR that will achieve that goal.  Similarly, asset allocation should focus on those portfolios that provide the highest return for a given level of risk of falling below the MAR.    A decision framework is needed that ensures policy decisions will be implemented and made operational.  We believe the Dutch triangle is such a framework. 

Conclusions:

q       We have shown how performance measurement can be linked to asset allocation in a cohesive management framework that ensures policy decisions will be implemented and made operational.

q       We have shown that style analysis can be successfully applied to small European markets. 

q       We have shown that downside risk produces different results in performance measurement and asset allocation than standard deviation. 

q       We have introduced a new performance measure designed to identify asset managers with the highest upside potential relative to their downside risk.

The following are our suggestions to the questions implicit in the Dutch government's requirement for pension fund benchmarks:

q       The strategic benchmark should be constructed in an ALM framework.

q       The benchmark enters the decision framework at the policy level.

q       The benchmark affects all investment decisions because performance measurement and asset allocation are driven by risk and return measures calculated relative to the mean of the strategic benchmark (the MAR).

                                                               References

·       Burr, Barry. "The Plan Contribution Holiday is Over Folks." Pensions and Investments, November 2, 1998.

·       De Groot, J. Sebastiaan. "Behavioral Aspects of Decision Models in Asset Management." Labyrint Publication, The Netherlands, 1998

·       Effron, Bradley, and Robert J. Tibshirani, "An Introduction to the Bootstrap." Chapman and Hall.1993

·       Fishburn, Peter C. "Mean-Risk Analysis With Risk Associated With Below Target Returns." The American Economic Review, March 1977.

·       Griffin, M. “The Global Pension Time Bomb and its Capital Market Impact”, Goldman Sachs, Global Research. 1997

·       Markowitz. Harry M. "Portfolio Selection: Efficient Diversification of Investments." Blackwell Publishers, 1991 

·       Merton, R.C. and A. Perold. "The Theory of Risk Capital for Financial Institutions." The Journal of Performance Measurement, Spring 1998.

·       Olsen, Robert A. "Behavioural Finance and its Implications for Stock-Price Volatility." Financial Analysts Journal, March 1998.

·       Sharpe, William F. "Asset allocation: Management style and performance measurement." Journal of Portfolio Management, Winter 1992.

·       Sortino, Frank A. "The Price of Astuteness" Pensions & Investments" May 3, 1999.

·       Sortino, F.A., G. Miller and J.Messina  "Short Term Risk-adjusted Performance: A Style Based Analysis." Journal of Investing, Summer 1997.

·       Sortino, F. and R.A.H. van der Meer. “Downside Risk.” Journal of Portfolio Management, Summer 1991.

·       Statman, Meir. H Shefrin, "Behavioral Portfolio Theory", Unpublished, Leavey School of Business, Santa Clara University 1998

·       Stewart, Scott D. "Is Consistency of Performance a Good Measure of Manager Skill" Journal of Portfolio Management, Spring 1998.

·       Tversky, Amos. "The Psychology of Decision Making."  ICFA Continuing Education no 7, 1995

·       Valdez, Charles. "CalPERS' eyes on risk" Pensions & Investments, November 30, 1998.

·       Van der Meer, R.A.H. and M. Smink. “Applying Downside Risk to Asset-Liability Management: A Pension Fund Case Study.” Journal of Performance Measurement

                                                                    Endnotes