## The Cost of Assuming Symmetry in a Skewed World

Frank Sortino & Kal Salama

In an Uncertain World

If you believe the uncertainty associated with investing in the indexes shown in Figure 1 is bell shaped, and it is not, all your metrics for performance measurement and asset allocation will be wrong. Well, the law of large numbers may be right in the long run, but we all live and operate in the often skewed world of the here and now.  This paper offers a tool for recognizing those times when there is more downside risk than upside potential and vise versa and how to manage it better by formally recognizing a different shape to uncertainty that is better suited to the task at hand.

That “better shape” is the brain child of two professors at Cambridge University, Aitchinson and Brown, who developed a form of the lognormal distribution that can use negative as well as positive returns to skew the distribution either left or right. Fortunately, you don’t need a PhD in mathematics to calculate the shapes shown in Figure 1.  All you need is the Forsey-Sortino model; a simplified version of software that Professor Emeritus in Mathematics, Hal Forsey and Professor Emeritus in Finance Frank Sortino developed at the Pension Research Institute (PRI) at San Francisco State University.  And all you need to know as inputs to the model is what you already know: the average return and standard deviation of each index, plus one more return that we call: the Desired Target Return®, or DTR®.  Given the better shape, the model can then calculate better statistics.  Better, in that one could have beat the market more often than not over the past 26 years.  Of course, the more people who use this model, the less inefficiencies one would expect to find in the stock market.  History indicates that will not be a problem.

Figure 1

The picture on the left is the positively skewed large core equity index (LC) and the picture on the right is the negatively skewed small cap growth index (SG). If you are forced to viewing both of these indexes as symmetric, bell shaped distributions, you may well conclude that you should just buy the market index because in the long term, the market is bell shaped, and everyone knows, you can’t beat the market, right?  Maybe not.  The key forecasting element in Figure 1 is the Upside Potential ratio (UP ratio)[1], which was first mentioned in an article published with my Dutch friends’ way back in 1991.  PRI has repeatedly recommended replacing the Sortino ratio with the UP ratio, but to no avail.  We will conduct three tests to indicate the superiority of the UP ratio.

How it works:

Imagine two investment committees with two different investment objectives. The first investment committee is a foundation that needs to earn 6% annually to meet their scheduled philanthropic payouts. The second investment committee has a defined benefit plan and their Desired Target Return (DTR®) is the actuarial rate of return of 8.5%.  Both committees use the Forsey-Sortino model (F-S model) shown in Figure 1 that can calculate the UP ratio on data provided by their consultant.  Beginning in 1990 they both use the average return and standard deviation for the last three years as input to the F-S model.  Figure 2 shows how the foundation would key in the historic data and DTR of 6%.

Figure 2

The output from the F-S model is shown in the right hand side of Figure 2.  The UP ratio of 3.59 means this large value index they own has 3.59 times more upside potential than downside risk. The committee intuitively understands that upside potential is good and downside risk is bad.[2] They don’t know how these terms are calculated, nor do they care.  They decide to see if some combination of the nine style indexes they own could beat the market the following year by using the UP ratio.  The pension committee acted the same as the foundation, only they used the actuarial return of 8.5% as their DTR. Based on this simple framework, both committees performed three tests:

First Test: Can an Asset Allocation based on the UP ratio perform better than the market mix of 60% S&P 500 and 40% Barclays A+ Aggregate bond index? Any negative UP ratios will be allocated to the bond index.

Second test: Holding the 60/40 asset allocation constant, can the UP ratio weights select style indexes that beat the 60/40 market mix?

Third test: Can the UP ratio select equity style Indexes, remaining 100% in equities, that beat the S&P 500?

Figure 3 presents the results they would have seen on the first test year after year.

Figure 3

In 15 of the 26 years the UP ratio 6% strategy (UPR) beat the market mix. In one year it beat the market mix by less than 100 basis points so they called that a tie.  The market mix beat the UPR 6% strategy 10 times; so, using the UP ratio won  50% more often than the 60/40 market mix.  That is astounding. The pension committee, using a DTR of 8.5%, found the UP strategy beat the market mix as often as it lost but earned a higher cumulative return.  Importantly, this shows the sensitivity of the UP ratio to the return the user needs to earn in order to accomplish their investment objective.  When the committees held the asset allocation mix constant to test the ability of the UP ratio to add value, by identifying sectors of the market that would perform better than the S&P 500, they got the second test results shown in Figure 4.

Figure 4

Once again both UP ratio strategies beat the market mix even when the asset allocations were held to a 60/40 mix. So, it wasn’t just the asset allocation decision that was adding value. It was something buried in the calculus used to generate the picture of uncertainty shown in the F-S model output.

Third test: Can the UP ratio weighted equity style Indexes beat the S&P 500?

The all equity 6% UP ratio strategy averaged 9.63% versus 7.64% for the S&P 500 over 26 years. The all equity 8.5% UP ratio averaged 9.10% over 26 years.

The S&P 500 was only able to beat the UP ratio strategies 6 out of the 26 years and two of those times it was by less than 100 basis pts.

Figure 5

Figure 5 shows using the UP ratio weights to invest in the indexes while remaining 100% in equity performed better than the S&P 500 most of the time over the past 26 years, in spite of the fact that the three year interval following the 2008 market collapse resulted in negative UP ratios for all indexes. To ensure we were measuring 100% equity in both strategies, we assumed all funds were invested in the S&P 500 for these three years.

Figure 6

The tests above were performed with no constraints on the asset allocations. The result shown in Figure 6 identifies years when excessive allocations (100% Stock or 100% Bond) were made that could be considered outside of acceptable investment policy guidelines due to lack of diversification.  The F-S model was developed as a tool to be used by financial professionals to make better portfolio decisions. The authors caution that the model should not be the decision maker.

We now return to Figure 1, repeated below, for a more detailed explanation of the otherwise unavailable information disclosed in the F-S model output.

Figure 1

On the left is the forecast for the Surz large centrix index (LC) which had the highest UP ratio of the nine Surz Style Pure indexes®.  LC is all the stocks that are neither growth nor value.  This is different than all other core style indexes which are a mish-mosh of everything (For more details see www.ppca-inc.com.).  The UP ratio of 1.66 indicates It has 66% more upside potential than downside risk.  Because the mean is positive it is anchored at -25% causing the shape of uncertainty to be positively skewed, not symmetric and not a normal or bell shape.

The small cap growth index shown on the right has a negative mean of -3.01% so we anchored it at +25% causing it to be negatively skewed with below target risk of 23.3% as opposed to 4% for the LC index. This obviously had the worst UP ratio.  While there is a 41% chance returns could be above 6%, the UP ratio of .12 indicates this index has 88% more downside risk than upside potential.  If one doesn’t believe the inputs, you can play,” what if” games by changing them.  This is unique information one should want to know.  We don’t believe any other model has this information with only these 3 inputs.  How did these projections do so far?  As of June 1st the LC index was up 8.4% and SG was up 2%.

Figure 7

Traditional mean-standard deviation risk measures would make the all equity UP ratio strategies appear to be the most risky over the last 26 years; highest return and highest risk. Nobody would choose the S&P 500 risk-return trade-off and everybody would choose the 8.5% UPR, really? If someone measured risk relative to their DTR and most of the volatility was above the DTR?  Risk for the committee members in this study is not a bumpy ride, it is that they don’t accomplish their investment objective, causing a failure to accomplish their goal of a specified pay out.  For them, beating the market is not the goal and is not the investment objective.  For them, the traditional framework of performance measurement is misleading.  For them, performance should be measured relative to their DTR, as shown in the Journal of Performance Measurement, Summer 2016.

Summary & Conclusions:

To prove to yourself that the Forsey-Sortino model offers you valuable information not available anywhere else, download the model as shown below and follow these steps:

1. Enter the means and standard deviations shown at the bottom of Figure 1. Then see if the upside potential ratios give you indications of which styles to emphasize and which to deemphasize. Does this provide a unique insight?
2. Enter the year end means and standard deviations of the three best performing and worst performing assets in your portfolio and see if the information in the F-S model would have been useful, given current results.

Kal Salama, CFA

Chief Investment Officer

(415)464-9144

To download the software: Control click on the highlighted link, install the VBruntime.exe file to ensure your computer can run software written in Visual Basic. Then install and run the F-S model.

This dropbox link may ask you to sign in but at the bottom you can skip this step. This zip file contains two files: Forsey-Sortino.exe and a Microsoft VBRuntime.exe.  If you get an error message with F-S Sortino.exe you must install the enclosed VB runtime.exe.

[1] Sortino, F., van der Meer, R.A.H and Plantinga, A (1999) The Dutch Triangle, Journal of Portfolio Management, Fall

[2] Researchers at Groningen University in The Netherlands found that people they interviewed intuitively understood these terms without having to explain exactly what upside potential or downside risk were or how they were calculated.

DTR® Calculator

Dr. Hal Forsey, and I, are pleased to offer another model we developed that will calculate the return a Defined Contribution participant needs to earn in order to achieve a desired pay-out.  We call this the Desired Target Return® or DTR®.[1]  Using the Department of Labor actuarial tables for male or female the model calculates the DTR with only 6 inputs.

Tutorial:

If it does not open, download either the 32 bit (i586) or 64 bit .exe file and install.

Double Click again on the.jar icon.

You will see the following screen:

Click on the “Calculate DTR” tab.  The DTR of 7.9% will appear as shown below.  Our research indicates that 60% equity or more will be required to earn around 8%.  That’s right, this does not imply a precision to 1/10th of 1%.  This is only a guide post to get you headed in the general direction.  In our research we used 4%, 6%, 8% 10% and 12% as guideposts and we found these break points to be sufficient for portfolio construction purposes.

Suppose you are not a male.  Click the New tab and then click on the F button. Now click “Calculate” and you will see that the Department of Labor actuarial tables estimate that you will probably need to earn 9% compounded until you retire because women are expected to live longer than men.

In order for a woman to reduce her risk to that of a man of 30 she could increase her contributions.

Go up to “Current Annual Contributions”, click on the “slider” to the right of 3500 and move it right to 4550 and you will see the DTR change to 8%.

But what if you are 50 years old, not 30 years old?

Click the “NEW” tab and change the age to 50 then click, “Calculate” to see that the DTR is now 20.4%

How much longer would you have to wait to be able to retire without taking more risk?

Move the slider to the right of “Projected Retirement Age” until the DTR is 8%.  No way are you going to be allowed to work until you are 80 years old.  So start adding to your retirement assets (move slider to the right), or start an IRA.

Now Comes the Hard Part:

You will need to find someone who can construct a portfolio for you that might compound at your final DTR between now and your eventual retirement. We wish you lots of luck…

Because, all of the traditional portfolio construction frameworks available to investors ignore any future payout that will be needed and they ignore the direction to the financial destination.

The most popular framework for investing is an asset management framework, e.g., the Capital Asset Pricing Model (CAPM).  The basic assumptions of this framework are that future payouts are irrelevant, just make as much money as you can, given your tolerance for risk and hope for the best.  Proponents of this framework almost always favor a heavy allocation to equities.  The other popular framework views future payouts as liabilities and therefore uses liability driven investment strategies (LDI). This framework is almost always invested heavily in fixed income securities no matter what your contribution schedule is or where interest rates are.

To illustrate the importance of the DTR:

Assume you want to send a valuable package to New York. Asset Management advocates don’t care where you are sending it from or where you want it to go.  They assume you want to send it as far as possible as fast as they can.  LDI proponents put all packages on a truck no matter how far it needs to go or when it needs to get there.  Neither one bothers to determine the direction to send it, due to their myopic concern about which vehicle to choose.

PMPT assumes you are trying to transport your portfolio from where it is to where it needs to be at some specific time in the future in order for you to use its contents for a specified cash outflow.  Well, we built an optimizer that measured risk and reward relative to a DTR and gave that software away in a book by Dr. Steve Satchell at Cambridge University and myself.  The title is, “Managing Downside Risk in Financial Markets, Amazon.com”

Oh, I almost forgot.  There is one other framework that claims: all they need to know is your age In order to determine what vehicle to put your valuable package on.  It is called: “Target Date Funds” Target Date Funds also neglect to determine what direction to take.  So, they know when it has to get there, they just don’t know where or what the target is.  Low tech snake oil.

[1] Desired Target Return and DTR are registered Trademarks of Dr. Frank A. Sortino

## A New Beginning

The Upside Potential Ratio 2017 update with this link to:

The Forsey-Sortino Model-1 software

This dropbox link may ask you to sign in but at the bottom you can skip this step. This zip file contains two files: Forsey-Sortino.exe and a Microsoft VBRuntime.exe.  If you get an error message with F-S Sortino.exe you must install the enclosed VB runtime.exe.

Frank Sortino and Hal Forsey

The concept developed in this paper was first presented in Pensions and Investments Magazine on 9/26/99, page 22. A paper incorporating this ratio was first published in the Journal of Portfolio Management, Fall 1999 issue. The authors, Frank Sortino, Hal Forsey, Auke Plantinga and Robert van der Meer were all professors from SFSU and Groningen University, who, along with Professor Joseph Messina[1] produced numerous studies over the past 16 years showing superior results of the Upside potential ratio relative to the Sortino ratio, Sharpe ratio and information ratio.  This working paper will update this body of research in preparation for this first release of the Forsey-Sortino Model announced on Linkedin.

Post Modern Portfolio Theory basics

If you want to know how well your manager is doing with respect to accomplishing your goals, then both the return and the risk associated with achieving your goal must be incorporated in the performance statistic. Instead of searching for the manager who had the highest average return over some period of time, those investing for some payout in the future should discount that future payout to present value.  That discount rate separates the good outcomes that achieve the goal from the bad outcomes that identify failure.  In our early work we called this the MAR and later changed it to DTR® for Desired Target Return®. As support for our framework for managing portfolios we initially relied on the emerging field of behavioral finance and the esoteric area of utility theory.

Two of the great pioneers in behavioral finance were the late Amos Tversky, professor of psychology at Stanford University and Nobel Prize Laureate Daniel Kahneman. While Tversky and Kahneman’s work described how investors do behave, Peter Fishburn’s normative utility function [1977] described how investors should behave. Rational investors should be risk averse below the benchmark DTR, and risk neutral above the DTR, i.e., they should have an aversion to returns that fall below the DTR, and the farther they fall below the DTR the more they should dislike them. On the other hand the higher returns are above the DTR the more they should like them. Fishburn showed how this utility function was consistent with expected utility theory.

More recently we found support in an article by Dr. Robert Merton in the July-August, 2014 issue of the Harvard Business Review, who said; The seeds of an investment crisis have been sown. The only way to avoid a catastrophe is for plan participants, professionals, and regulators to shift the mind-set and metrics from asset value to income. Merton echoes our view that everyone is focusing on the wrong goal andif the goal is income for life after age 65, the relevant risk is retirement income uncertainty, not portfolio value.”  Merton goes on to make many cogent observations. What we would like to focus on is this quote: Clearly, the risk and return variables that now drive investment decisions are not being measured in units that correspond to savers’ retirement goals and their likelihood of meeting them. Thus, it cannot be said that savers’ funds are being well managed.” In the spirit of that comment we offer the following:

You can’t manage what you can’t describe so we begin with two different descriptions.

Figure 1

Modern Portfolio Theory (MPT) says that all the statistical characteristics of the portfolio on the left which you are trying to manage can be described by two numbers, the mean (average Return) and the standard deviation of those returns. The framework we developed at The Pension Research Institute, that came to be called Post Modern Portfolio Theory (PMPT), says you need to bring up that asymmetric blue distribution on the right by its bootstraps and then fit to it that black curve you can barely see, called the three parameter lognormal.  Then calculate the mean, standard deviation, upside potential, downside risk and upside potential ratio.

Now let’s examine the three most popular metrics for decision making:

Figure 2

In Figure 1 all three ratios show the manager’s average return (the mean) in the numerator. The Sharpe ratio compares the mean to a risk-free rate of return.  The information ratio compares the mean to an index return.  The Sharpe and Information ratios do not include a reference to the return (DTR) the investor has to earn in order to accomplish a desired goal.  The Sharpe and Information ratios measure risk in term of standard deviation.  These metrics do not meet the demands of Merton or Fishburn nor the findings of behavioral finance. The Sortino ratio does present a risk-return relationship relative to the DTR.  Now let’s look at the Upside Potential equation:

Figure 3

In Figure 4 we convert the Upside Potential Ratio into a picture that can be more easily understood.

Figure 4

The numerator is more than just the simple differences of two returns as shown in all three of the ratios in Figure 1. Also, It is more than just the probability of exceeding the DTR (shown in blue to the right of the DTR). The upside potential calculation incorporates a magnitude factor that no other ratio has, shown by the green ark that exceeds the upside probability. In the denominator the difference between R and DTR are squared (-4% becomes -16%).  The Upside Potential Ratio presents a very different picture than any of the 3 ratios shown in Figure 1 and it is consistent with the Fishburn utility function.  Unlike any other ratio anywhere, it shows the portfolio has 25% more upside potential than downside risk, which we have shown has more predictive power than the ratios in Figure 1.

An example of the shortcomings of standard deviation and correlations:

Figure 5

Manager G has much less volatility than manager F and because it has a smaller standard deviation G is less risky than F according to the Sharpe and Information ratios. Also, if we could find a manager with returns portrayed in time series 3 that was perfectly negatively correlated with manager G we could create the MPT version of the Risk-free asset shown in the Sharpe ratio in Figure 2. Many consultants and government agencies might well recommend putting all 401 (k) accounts in the concocted risk-free asset.  But if almost all 401(k) participants need to earn more than 4% to be able to retire with dignity, almost all participants would be guaranteed failure to accomplish their goal.  The Upside potential ratio would not be fooled.

Figure 6 presents a simplified calculation of the upside potential ratio which should never be used in practice. We show this flawed methodology in an attempt to help you understand the concept of upside potential and downside risk and how they differ from the mean and standard deviation.

Figure 6

Looking at Figure 6 All MPT believers would say the 4% mean and 7% sigma of G versus the 8.5% and 13.8% numbers of F mean you could only choose between them based on risk-tolerance. Some PMPT believers might say, well, same logic applies using downside risk if you don’t know the upside potential ratio, which is 3 times higher for F. The Sortino ratio gets the sign right but if the DTR was 4% and you earn 4% then the numerator would be zero. But what do you do when both F & G have more downside risk than upside potential as shown in Figure 6?

Let’s say your total portfolio is 75% equity and 25% fixed income and so far this year it is up10%. The market is at an all time high of 2400 on the S&P 500 and the investment committee is considering whether or not to take some profits in equity and increase fixed income.  The DTR given by your actuary is 8% and the expected return is 8% with a standard deviation of 12%.  What does that tell you to aid you in this decision?  Not much.  Suppose you put this information into the Forsey-Sortino Mode as shown in Figure 7.

Figure 7

You now see some additional information. The Upside Potential ratio of .64 shows 36% more downside risk than upside potential for the total portfolio.  Might this influence your decision regarding profit taking?  Would you like to be able to change the mean, standard deviation or DTR to see what the impact was? The link at the top will take you to dropbox where you can download it

We will not receive any remuneration. Our only motivation is to provide a tool that might improve the decision making process in portfolio management.

Original References

• 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
• 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

[1] see our first book, “Managing Downside Risk in Financial Markets” p 74, Amazon.com

## The Facts about Who Developed PMPT & The Sortino Ratio

PRI-Rom Contract

The term, “Post-modern portfolio theory (PMPT)” was first used by Brian Rom to describe the underlying theory of an asset allocation model developed by The Pension Research Institute (PRI) called Primix. Mr. Rom wrote a very favorable article about Primix in Pensions and Investments magazine, May 2, 1988, page 34. The article noted that Premix “exactly calculates the risk of falling below the target return “ and later states: “users should request a white paper titled Risk and Relevance by the Institutes Director, Frank Sortino.” A few weeks later Mr. Rom signed an agreement with PRI to market PRI’s software. The agreement required Rom’s firm “include the following words on the opening screen: “This product is an extension of a model developed by the Pension Research Institute;” furthermore, the agreement limited Mr. Rom to “designing all Non-Technical Features” , such as the user interface.[i] PRI was responsible for all technical features.[ii] Therefore, all of the body of knowledge constituting PMPT (including all equations) prior to 1995, when Rom terminated the agreement, were developed by PRI. The source code for all Technical Features in the software including the Sortino ratio was written by Dr. Hal Forsey, professor of mathematics at S.F.S. U. A copy of the PRI Agreement was hand delivered to the San Francisco office of Wikipedia 9/8/2015.

The supporting evidence can be viewed by clicking on the PRI-Rom.pdf  contract at the top.

[i] Defined in the PRI agreement with Brian Rom as:”All those attributes of the Products that relate to on-screen menus and user interaction.”

[ii] Defined in the PRI Agreement as: “The algorithms, mathematical and statistical operations and other functions required to compute the results.”

## Portfolio Navigation For DB Plans

By

Frank Sortino & Hal Forsey[1]

`Would you tell me, please, which way I ought to go from here?’                           `That depends a good deal on where you want to get to,’ said the Cat.                                                                                                             Lewis Carroll

Introduction:

Some people give advice on how to manage pension fund assets. Others focus on the liabilities. We focus on the link between the two; The Desired Target Return®. If you don’t know your DTR®, you don’t know how to get from where you are to where you need to be.  At any point in time the DTR is the return that must be earned on assets in order to return to a fully funded status within a designated time horizon.[i]  This can be viewed as a navigation problem.  Let’s assume the relevant actuarial statistics are as follows:

 Table 1 00000000 PV of future benefits 60 Assets 60 Discount Rate 4.00% Yearly Benefit 2.832 DTR 4.00% PV of PV of Year Assets total benefits annual benefits 0 60 59.99769493 2.832 1 59.568 59.45232273 2.723076923 2 59.11872 58.88513564 2.618343195 3 58.6514688 58.29526106 2.517637688 4 58.1655276 57.68179151 2.420805469 5 57.6601487 57.04378317 2.327697566

When setting up the plan the actuary assumed a discount rate of 4% on projected liabilities and a yearly benefit (\$283.2M discounted to present value at the discount rate) in order to solve for the contribution schedule on this \$6 billion dollar plan (add 8 zeros).  The plan is assumed to be fully funded at the top of the market in 2007, after the year 2000 high-tech sell off and the 9/11 attack on the twin towers.  Table 1 indicates a DTR of 4% would maintain the fully funded status for the next five years without any further contributions.

PMPT Portfolio Construction:

We recently asked a large institutional investment advisory firm to provide us with five asset allocations for portfolios with absolute returns of 4%, 6%, 8%, 10%, and 12%.[ii]  Each portfolio was constructed as described in a recent book.[iii]  Figure 1 shows the portfolio we constructed for their 8% absolute return strategy, given their manager data base.  We did not want to know the names of their active managers so they used a number for each category (e.g., LG – 20 is the 20th large growth manager).

Figure 1 (Absolute 8% return portfolio)

This portfolio may well have had only an 8% average using historic returns for the past few years but our methodology indicates the mean of their asset allocation using passive indexes is 9.5%.  When we include their active managers when they add value and ETFs when they don’t, the mean is 12.5%.  The DTR-alpha indicates their active managers could add 300 bp in any given year.  The Upside potential ratio indicates the actual portfolio with active managers has 30% more upside potential than downside risk (1.3) while a totally passive portfolio would have 20% more downside risk than upside potential (.8).  We believe this is useful information not contained in the Sortino ratio or Sharpe ratio.  We assume this is the portfolio the plan has had for some time and they simply rebalance each year to maintain this asset mix.

The Portfolio Navigation strategy begins with a DTR 4 portfolio because that is the internal return that discounts the liabilities to the present value of the assets. However, this portfolio does have the potential to earn 590 basis points more than the DTR in any given year.[iv]

Figure 2 traces the results for the 8% absolute strategy (AR-8) and the DTR-Portfolio Navigation strategy (black dashed line) beginning at the top of the market in May, 2007 when the S&P 500 was at 1425 on its way to a 50% decline by February of 2009.  We chose 8% as the bogey in Figure 2 because it is a popular return that actuaries assume will be earned on assets.  The assumed rate of return on assets does not figure in determining the present value of the liabilities but under FAS 87 it can affect what the firm declares as profits.

Figure 2 – Fully Funded

Actively Managed (DTR) vs an 8% Absolute return

As shown in Figure 2, whether one used Portfolio Navigation, an 8% absolute return strategy or simply bought the S&P ETF (SPY), the plan would have returned to a fully funded status in four years as indicated by the red dotted line that declines as benefits are paid each year. However, all happy endings are not the same.  The absolute return strategy was down 32% when the market finally bottomed while the DTR strategy was only down 11.7%.  Also, the DTR strategy is more than a billion dollars better off after four years.  How was this achieved?

In Figure 2 the DTR was calculated at the end of every quarter and when the asset value reached \$5.77B the DTR increased to 6%. At that point the allocation to equity was increased from 30%, as given in the 4% portfolio, to 45% in the 6% portfolio.  Importantly, this change was not due to an economic forecast or some market timing strategy.  Based on the funding status, it was the return calculated to return to a fully funded status within a 5 year interval.  Five years is not a magic number that must be used in portfolio navigation.  We did so, so as not to be accused of making propitious changes with 20/20 hindsight.  We chose it before conducting the study and purposefully did not try other intervals.  In practice we would present the plan sponsor with other options.

Less than a year later the pension assets were down another \$470 M and the DTR was calculated as 8% as shown in Figure 2. Again, in less than a year, the DTR was calculated to be back to 4%.  Would a pension fund have been willing to make such changes, buying as the market crashed and selling as the market recovered.  Well, that’s better than doing the opposite.  Of course, the pension committee would have to make that decision each time the DTR changed and they would want some evidence to support the decision.  This study is intended to provide that evidence.

The Underfunded Case

What if the pension plan was underfunded at the beginning of this study? Figure 3 illustrates the results if the plan had a shortfall in May of 2007.  All the statistics in Table 1 were kept constant except the pension assets were assumed to be \$5.37 B.  In this example the plan would return to fully funded status if it earned 6% for the next 5 years.  This time we compare Portfolio Navigation (black dashed line) to absolute return strategies of 4%, 6%, 8%, 10% and 12%.  All absolute strategies are rebalanced annually.  The DTR- Portfolio Navigation strategy begins with a DTR of 6% and changes three times over the next four years.

After four years none of the strategies has succeeded in returning to a fully funded status. Portfolio Navigation does not guarantee success anymore than navigation aids for air travel guaranty one will reach their destination.  In both cases it is a matter of improving your odds of success.  Navigation has steadily improved over the decades to become a reliable science.[v]  We are a long way from that level of reliability. What if the market never recovered?  What if the airplane crashes?  Risk is situation specific.  The way one should measure the risk of flying is not appropriate for measuring the risk of investing in financial markets. We have set forward in books and papers what we believe to be an improvement in the way to calculate risk and reward when investing.  This paper describes a way to use these improvements as tools to navigate from where a pension plan’s assets are to where they need to be over a prescribed time horizon.

[1] Frank Sortino is Professor Emeritus in Finance and Hal Forsey is Professor Emeritus in Mathematics from San Francisco State University, California.

Figure 3 – Under Funded

(DTR® vs Absolute Return Strategies)

The reason the DTR strategy stays at 8% for the last two years is that time is running out in the initial 5 year horizon. Of course the pension committee could decide to reset the time horizon to 3 years or more in May of 2011.  The operative questions are: as fiduciaries, do they know how much risk they will be taking to get from where they are (\$5.58 B) to where they want to be (\$5.7 B) in just one more year?  Is that too much risk? Should they start making contributions and/or increase the time horizon?  These are questions that have an answer, if you have the right tools.

Throughout this study we have assumed the goal was to fund the plan within the cost constraints provided by the actuary. The investment objective then became, maximize the potential to exceed the DTR relative to the risk of falling below the DTR.  What if the goal was to maximize the excess return of the assets over the current costs in order to increase earnings?  Even so, there is still some return that must be earned in order to accomplish that goal and the investment objective remains the same.  It is simply another application of Portfolio Navigation for another Desired Target Return®.

Conclusions:

• Constructing a portfolio around the assumed actuarial return on assets ignores the link between pension assets and pension liabilities.       Therefore, constructing a portfolio around that link, the DTR®, makes more sense. Likewise, the mean and standard deviation and beta are also unrelated to the link between assets and liabilities.
• Absolute return strategies come in many forms. Some attempt to minimize the risk of falling below a specific return that may or may not be related to the funding ratio and totally ignore the upside potential. We have written our opinions about this subject on our blog, pmpt.me (see, “Absolute Nonsense”).
• Portfolio navigation outperformed a wide range of absolute return strategies with less downside risk during one of the worst stock market periods in our history.
• It also outperformed all absolute return strategies when the plan was underfunded and began with a 6% DTR or 8% DTR (available on our website download folder).
• Maintaining the same asset mix by rebalancing may have been appropriate in bygone years, but not in today’s volatile markets. In the last twelve years we have witnessed three major disruptions in the financial markets and we are still suffering from the last one. We believe it is time to find a new way to reach your destination. This study offers a way to accomplish this with less risk in a manner that is understandable.

[i] This could be calculated by any actuarial firm as the internal rate of return that discounts the projected liability stream to present value.  All we are using in this study is the statistics shown in Table 1.

[ii] These are the asset allocations provided:

[iii] “The Sortino Framework for Constructing Portfolios,” Elsevier Publishing 2010 (see Amazon.com).

## Defining Risk Video

Many people are talking about risk in a way that has nothing to do with what is at stake.  The myRA  bond introduced by President Obama at his state of the union address is the latest example.  This 3 minute video is my explanation of what is at stake. The 2nd video is a 4 minute video on Portfolio Navigation.  The 3rd video is an interview with the former CEO of The City and County of San Francisco Retirement System and an interview with professor Sam Savage of Stanford University.  The full 30 minute video that covers the research findings of the past 30 years is available at the KMVT link below the interview video.