CHOICE AND CHANCE
A Statistical Approach To Mutual Investment Fund Selection
Lawrence Christensen
Adjunct Instructor of Mathematics
Monroe Community College
December 13, 2007
Table of Contents
Introduction 4
You Will Be Wrong 5
The Basics 5
The Sample 6
Statistical Concepts 7
The Annual Rate Of Return Confidence Interval 10
100 Sample Funds At The 90% Confidence Level 12
The Best and Worst Funds For A Mildly Risk-Adverse Investor 15
The Other Sample Funds 16
Risk Tolerance 17
Even-Odds Gambler 19
The Slightly Conservative Gambler 20
The Moderately Conservative Investor 22
The Conservative Investor 24
The Highly Conservative Investor 26
Funds Selection Sensitivity To Risk Level 27
Optimizing The Strategy 28
Hypothetical Frontier To Annual Return vs. WSD 31
The Final Fund Filters 34
In Conclusion 35
Bibliography 37
Appendix A: 100 Sample Investment Funds 39
Appendix B: Not All Percentages Are Created Equal 41
List of Figures
Figure 1: Percent Returns for All Sample Funds and Years 9
Figure 2: Scatter Diagram of 100 Funds; 90% Confidence Level, α = .1 14
Figure 3: Scatter Diagram of 100 Funds; 75% Confidence Level, α = .25 21
Figure 4: Scatter Diagram of 100 Funds, α = .05 23
Figure 5: Scatter Diagram of 100 Funds, α = .01 25
Figure 6: Upper Boundary of Return 33
List of Tables
Table 1: Top Ten Fund Picks at a Confidence Level of 90% 12
Table 2: Point Pairs For Three Funds With Identical Lf Values 90% 15
Table 3: Ten Lowest Minimum Returns at a Confidence Level of 90% 16
Table 4: Risk Level and Game Odds 18
Table 5: Top Ten Fund Picks at a Confidence Level of 50% 19
Table 6: Bottom Ten Funds at a Confidence Level of 50% 19
Table 7: Top Ten Fund Picks at a Confidence Level of 75% 21
Table 8: Bottom Ten Funds at a Confidence Level of 75% 22
Table 9 : Top Ten Fund Picks at a Confidence Level of 95% 23
Table 10: Bottom Ten Funs at a Confidence Level of 95% 24
Table 11: Top Ten Fund Picks at a Confidence Level of 99% 25
Table 12: Bottom Ten Funds at a Confidence Level of 99% 26
Table 13: Top Ten Fund Picks at a Confidence Level of 99.9% 26
Table 14: Bottom Ten Funds at a Confidence Level of 99.9% 27
Table 15: Ranking in Top Ten for Four “Robust” Funds 28
Table 16: Selected Funds With The Least Sensitivity To Risk Level 29
Table 17: Selected Funds Using The Top Pick For Each Risk Level 29
Table 18: Selected Funds With Near “Ideal” Scatter Diagram Pairs 30
Table 19: Upper Boundary Parameters 32
Table 20: Upper Boundary Values For Annual Return 34
Introduction:
If you are invested in the American economy through mutual funds, you are not alone. According to the Investment Company Institute and Department of Commerce, in 2007 51 million households in America were invested in 8015 investment funds worth a total of 12.4 trillion dollars. Again according the Institute, about 2/3 of us invest through a professional investment advisor, leaving the decision making process to the paid consultant. Conversely a third of us, or about 17 million American families, make our own decisions on our investment funds for better or worse. If you are oriented toward making your own investment fund decisions, then please pay attention to this paper as it is directed at those of us that do. We will explore a mathematical process that can give an investor as good a chance of selecting the best mutual funds as the professionals.
In this paper there is no endorsement of any particular portfolio of investment funds, but rather a description of a process available to any investor that can help winnow down his or her options in an orderly and systematic fashion using statistical mathematical techniques using easily available public information. The investment funds discussed below represent a sample of 100 investment funds of all 8000 available funds. These funds will be used to illustrate the logic and procedures, not to make recommendations for or against any specific investment fund.
You Will Be Wrong:
One thing that life should teach us all is that… “you will be wrong some percentage of the time in everything you do, and that includes selecting investments”. Conversely, your will be right a percentage exactly equal to one minus the percentage that you are wrong. So our goal should be to minimize the percent of wrong decisions… not to eliminate error completely, an impossibility, or accept simply flipping coins every time we look to add an investment fund to a portfolio. There must be a statistical basis for managing fund selection using readily available data on fund returns that would allow us to avoid negative return on our investments most of the time and give us some level of confidence in a positive outcome.
The fundamental question we will address in this paper is: “Over future years for a specified investor risk level, what is the likelihood a specific fund of interest may produce a negative return (loss) versus the likelihood that it will produce a positive return (gain)”.
The Basics:
Almost all investment publications and websites will provide their top ten picks and bottom ten dogs for the past week, month or year and these selections will be based on average annual returns over some period of time. We have all read the disclaimer in prospectuses and other investment resources that states: “…past performance is not an indicator of future market performance…”, but guess what, it’s the history of returns that the professionals will use in making and publishing their choices. Many considerations may be present in deciding on an investment such as our age, preferences for a particular market sector, style, tax efficiency, management personnel or fees to mention just a few, but sooner or later we look to past performance as a major decision factor. The performance statistic we will focus on will be percent annual rate of return after adjustment for fees. There is an inherent bias in percent return that causes negative returns to have a greater affect on overall performance that positive returns. This bias is discussed in Appendix B.
So if we agree that past performance is the best quantifiable guide to fund selection, then what is wrong with simply ranking the performances over some time period and selecting the top funds for our portfolio? Well nothing if you’re a real gambler. As we will demonstrate later in this article, if your approach to investing is to assume you only need to be right half the time, i.e. a 50%-50% chance of being right or wrong on each decision, then flipping a coin is an acceptable strategy for selecting a portfolio. If you are not so inclined, then we need to examine ways to understand and perhaps mitigate risk and parse funds that lessen our risk of making bad choices, and also provide a way to decide between funds that may have equally good or bad historical track records.
The Sample:
Taken from the total 8000 US funds, the 100 samples discussed in this paper are for the most part the recommendations of investment professionals as “good investments” with a few extra funds sprinkled in that where selected from newspaper and magazine articles highlighting so-called “hot” funds of the day. In all, 100 investment funds were examined over as many years as the fund existed or had reported results. The statistic we focused on was the annual rates of return as reported by the fund managers in their annual reports and prospectuses and augmented by Morning Star performance summaries as reported through the on line financial website Yahoo-Finance.
To obtain an unbiased sample for each fund, several years of performance data were necessary that would bridge both good and bad periods in stock market performance so that we would not be overly optimistic or overly pessimistic about the average performance and variability of a particular fund. Some funds had as little as four years of data but most had reported returns up to 21 years of data between the years 1986 and 2006.
The sample of 100 funds used here included funds from all investment styles from low risk, stable value bond funds to high risk, foreign emerging market equity funds. Most of the 100 funds are widely traded on the three major exchanges and are well known with identifiable ticker symbols. A few are specific industrial investment plans unique to a particular industry and not publicly traded. In a best-case scenario, the sample represents the “best picks” by the investment experts. In a worst-case scenario the sample is representative of the full range of the 8000 investment vehicles and the experts are no better at picking the best funds than we are!
To reiterate, the author makes no claims about the selected funds other than they had the endorsement of investment professionals and that they are a typical of the range of investment styles available and that they represent a reasonable random sample of the overall 8000 funds. The author concedes that picking a different 100 samples would produce somewhat difference results but we can be reasonably comfortable that those results would be similar and that the approach we discuss here would be equally valid.
A complete list of the 100 funds of the sample which shows the fund profile or style can be found at the end of this paper in the Appendix. The author has chosen to create a three-letter code ID for each fund in the sample to preserve the integrity of the analysis and prevent the paper from turning into another stock and bond pick-de-jour article. Rest assured that they are all real investment funds with real life returns data behind them. Upon request, the author will provide a cross reference between the fund codes and the actual ticker symbols.
Statistical Concepts:
To accomplish the task of statistically selecting funds we need to introduce a few statistical concepts:
· Statistical calculation of mean and standard deviation
· Application of the central limit theorem
· Classical hypothesis testing
· Use of the Student t-distribution for finite sample sizes
We must first ask the obvious question concerning the average or mean rate of return µf. The mean is calculated by summing the individual annual rates of return Xy,f over Nf years and dividing the total by Nf.
µf = (1/ Nf) x Σ Xy,f
Where the variable X(y,f) is the individual rates-of-return data for a particular year y and a particular fund f.*
The next question we ask about our data is how do individual return results for a particular fund vary about the average or mean long-term annual rate of return for that fund. That is, what is the standard deviation of the individual annual returns, σf. This is calculated as by summing the differences between the individual values and the mean, squaring the result, summing over all the years and dividing by the degrees of freedom, Nf -1 and taking the square-root of the result:
σf = √1/(Nf –1) x √ Σ(Xy,f - µf)2
Figure 1 is a graphic presentation of all the rates-of-return in the 100 sample funds for all available years that demonstrates the frequency distribution of annual returns. There is a bias and a suggestion of a skewness toward higher returns… lucky for us! In fact any fund in any year picked at random from the sample of 100 funds has an 82% chance of yielding a positive annual rate of return. Conversely, there is an 18% chance that any fund-year might result in a loss. The average return for the 1494 sample fund-years was 13.4% with a standard deviation of 20.7%. We may assert that with 90% confidence that the true mean for all sample fund-years lies between 12.5% and 14.3%. It is interesting to note that the distribution is not exactly Normal or classic bell-shape, but even so could be closely approximated by the Normal distribution. Later when we calculate confidence intervals the assumption of near Normality will work in our favor.
Figure 1: Percent Returns for All Sample Funds and Years
For every fund we have two measures, the mean annual rate of return µf and standard deviation σf of annual rate of return. One tells us the fund long-term performance and the other the fund volatility or variability, and as you might expect these can be mathematically combined to quantize risk. In statistics the central limit theorem states that the standard deviation of the mean µf decreases by the square-root of the sample size √ Nf, so the more years of data we have for a particular fund the smaller the standard deviation of the mean σµ,f.
There also exists a concept in statistics whereby one can test a hypothesis that a long-term mean will be within a lower or upper bounding values at a predetermined confidence level, or odds in favor of making a correct decision. This interval is referred to as the “confidence interval of the mean” and brackets the region where the true mean can be found. The confidence interval is based on an assumed probability (or risk) of being wrong in a conclusion and includes a measure for the sample size and the standard deviation of the individual data values. The construct of the confidence interval allows us to test a “null” hypothesis, i.e. what we assume to be true cannot be disproved and is therefore can be accepted. Conversely, if we reject the null hypothesis we will accept an alternative hypothesis. In this paper we shall explore the hypothesis that over several future years the mean or average return for a particular fund will be non-zero and positive, i.e. an investment that will gain in value. We can state the null and alternative hypotheses mathematically as follows:
Ho (Null): “Fund f will result in a mean rate of return of greater than zero percent”
Ha (Alternative): “Fund f will not result in a mean rate of return greater than zero ( or restated, will result in a zero or negative return)”
The Annual Rate Of Return Confidence Interval:
Using the math above to compute the mean and the standard deviation for each of the sample funds we obtain the mean annual rate of return, µf, the standard deviation of the rate of return, σf and a sample size Nf , and set up a hypothesis Ho for each fund f that we can test at various risk levels. But before we can proceed we need to introduce two more statistical concepts to compute the desired confidence interval for each fund. The first concept is the additional variance associated with finite sample sizes. For our sample of 100 funds we have between 4 to 21 years of historical returns data for each fund. Because our samples within a fund are limited, the assumption that the distribution of individual returns data is Normally distributed must be modified to apply a related function, the Student t-distribution. Inherently what the Student t-distribution does is to increase variance for finite sample sizes that widens the confidence interval for a particular hypothesis test (i.e. less sure of where the mean truly lies).
The second concept is another assumption that must be made to form the proper confidence interval for our data. That is, are we placing all our risk of being wrong in concluding a zero or negative return below a specific threshold or are we spreading the risk into a lower threshold AND a higher threshold. Known as the one-tail test or two-tail hypothesis respectively, we will assume that a return can never be too high and that we are interested in only the lowest possible limit rate of return only, i.e. all the risk of an error α, is therefore in a one-tail test. Stated mathematically, the lower confidence limit rate of annual return Lf is:
Lf = µf – σf x t(α, Nf-1) x √1/ Nf
If Lf is non-zero and positive (i.e. the interval does not include zero), we will accept the null hypothesis and conclude that the fund will continue to produce positive gains in the future. If Lf is zero or negative (i.e. the interval does include zero), we will reject the null hypothesis and accept the alternative hypothesis and conclude that the fund will not produce positive gains in the future.
This may be way more than what anyone wants to know about the statistical concepts, but suffice it to say that what we have done is to combine these several factors into an estimate of a lower rate of return value for each fund that uses all the information available about the fund’s historical returns and we will use this information in more than one way to help us with our investment decisions.
If we select one fund at random from our 100 samples we can illustrate the lower return confidence interval calculation. For instance fund FEF had 19 years of data available, had an average return µFEF over those N FEF = 19 years of a very respectable 15.2% and a standard deviation of returns σFEF of 14.1%. If we use a confidence level of 90%, or a risk value α of 0.1, there is 90% chance of being correct in our hypothesis test of the mean return and a 10% chance of being wrong in our hypothesis test or decision. We can estimate a lower performance value of LFEF:
LFEF = 15.2 – 14.1 x t(0.1, 18) x √1/19 = 15.2 – 14.1 x 1.33 x √1/19
LFEF = 10.9%
In other words, we can state with a 90% confidence that FEF over the long term mean fund performance should be AT or ABOVE a return of 10.9% per year and therefore will satisfy our hypothesis criteria of a positive return.
Of course if the fund closes it’s door next week we will be back to square one, and you will be looking at another fund to replace it… there is more homework to be done than just statistical calculations. In every future year we will add and delete funds and add or delete data to our analysis. We therefore will need to revisit our results periodically no matter how comfortable we are with this year’s results.
100 Sample Funds At The 90% Confidence Level:
Continuing the above calculations for the other 99 funds results in a table of minimum expected returns Lf over several years for all 100 sample funds. These minimum return values can be used to test the hypothesis Ho and we would find that at the 90% confidence level, five funds would fail the hypothesis that they would produce positive investment results and 95 would pass the hypothesis test and would be expected to produce positive investment results. This should be reassuring to us investors in general, but of limited help in selecting our best investments.
However, we can also use the Lf minimum rates of return data in a slightly different way. We can use the lower limit of our hypothesis test statistic Lf to rank order the funds from highest to lowest minimum rates of return and select the top few funds for inclusion in our preferred portfolio. Using this approach and using the rank order at the 90% confidence level, the top 10 fund picks will be as shown in Table 1. In doing so we must recognize that we will be wrong in our selection about 1 time in 10… one of our selected funds will prove to be a dog!
Table 1: Top Ten Fund Picks at a confidence level of 90%
Fund Fund Style Lf
1. BMC Mid Cap Stocks 21.6%
2. CSR Realty 12.9%
3. LLP Mid & Large Cap Stocks 12.2%
4. SSE Small Corp Stocks 11.9%
5. FMF Common Stocks 11.4%
6. TAV Undervalued Stocks 11.3%
7. PSC Small Corp Stocks 10.9%
8. FSF Financial Services 10.9%
9. FEF Restructured Corp Stocks 10.9%
10. RIT Large Corp Stocks/Bonds 10.6%
But immediately we need to make some additional observations. The top picks 7 through 9 PSC, FSF and FEF all have substantial histories and all have identical minimum lower limits for their confidence interval 10.9%. You could invest in all three or you might look for another view of the same data that might help parse these funds further.
We can obtain some additional insights into the relative comparison of one fund and another by examining mean fund returns and standard deviation of the mean returns together as data pairs (σμf, µf). To do this we need to recognize that if the mean return µf is subtracted from the confidence interval minimum return Lf the result is a weighted standard deviation of the mean σwf.
σwf = Lf - µf
Note that the weighted standard deviation σwf incorporates into a single statistic the variability of the fund returns, the number of years of data, the risk level for incorrect decisions and the additional variance associated with finite sample sizes. The weighted standard deviation can be graphically depicted along with the mean annual return in a scatter diagram that depicts each fund as key data pairs (σwf, µf). In Figure 2 is plotted the average annual return against the weighted standard deviation of the mean for all 100 funds at the 90% confidence level.
Notice that the units for both axes are the same and that there is a diagonal line running from lower-left to upper-right that represents the set of points for which the annual return is exactly equal to the weighted standard deviation and also incidentally, exactly equal to a Lf of 0% minimum return. Notice also that there are exactly 5 data points below this zero return line, the same funds that failed the hypothesis test. The zero-line separates the funds that are expected to produce at least some positive return (those to the left of the line) and those that could possibly result in zero or a negative return (those on or to the right of the line). Points equidistant from the line have the identical expected minimum rates of return, but are now separated in terms of annual rate of return and volatility. Of the three funds highlighted above, fund FEF has the lowest volatility for the same minimum rate of return σwFEF = 4.3%, versus 4.7% for fund PSC and 7.1% for fund FSF. The same expected minimum rate of return and with the lowest volatility… hmm… sounds like a winner, at least for the moment.
Figure 2: Scatter Diagram of 100 Funds; 90% Confidence Level, α = .1
Another observation from Figure 2 is that the highest minimum return for a given volatility (furthest from the diagonal line) is the BMC fund mentioned earlier. Note that it sits alone and distinctly different from the majority of the funds, a clear signal that one should take a closer look. As it turns out, BMC closed in 2002 and owes its high standing to only 4 years of data mostly preceding the tech stock meltdown of 2001. Because it is so different from the bulk of the data and an obvious outlier, we can and should discard this fund from inclusion in further analysis.
One can only image that the investors who saw BMC as a great rising star in 1998 were sharing the kennel with the dogs by 2002. The message here… “even the most rigorous mathematical analysis requires a few common sense judgment calls along the way!”
The Best and Worst Funds For A Mildly Risk-Adverse Investor:
At the 90% confidence level, i.e. one chance in ten of being wrong, we can separate the optimal funds available in the sample of 100 funds from all other investment possibilities. These will be the upper-most points furthest from the 0% return line. Conversely, the least desirable funds will be those at or below the 0% line. The reason for this differentiation is the logic that for a given risk level the point representing the highest average annual return would be the optimal choice among funds of equal volatility. Imagine a vertical line directed straight up from the weighted standard deviation axis that represents a line of constant volatility. The fund at the highest point along this line would be the optimal pick.
At the 90% confidence level, the funds PSC, FSF and FEF all have a minimum rate of return of 10.9% and as we pointed out earlier, cannot be further differentiated unless we examine the scatter diagram data point pairs (σwf, µf) in more detail. For instance funds PSC and FEF have almost identical point pair values and we might conclude that in terms of expected return they are interchangeable. Fund FSF however, has a noticeably different point pair values with a higher volatility and higher mean rate of return, but the same minimum rate of return value Lf = 10.9%. If we were willing to accept greater volatility in exchange for a higher potential return, we might select fund FSF over either PSC or FEF. We will explore this more in a later section of this paper.
Table 2: Point Pairs For Three Funds With Identical Lf Values;
90% Confidence Level
Fund σwf µf
PSC 4.7% 15.6%
FEF 4.3% 15.2%
FSF 7.1% 18.0%
The top ten fund picks at the 90% confidence level have already been presented in Table 1 above. The lowest ten funds at the 90% confidence level for a positive return long-term are shown in Table 3. It should not be any surprise that for a mildly risk-adverse investor, i.e. a confidence level of 90%, the preponderance of desirable funds would favor real estate and corporate stock funds and avoid riskier overseas investments and small capitalization stocks, BMC (Mid Cap Stocks) versus FEA (Asia Stocks).
Table 3: Ten Lowest Minimum Returns at a Confidence Level of 90%
Fund Fund Style Lf
1. FEA Asia Stocks -4.6%
2. KSF Eastman Kodak Stock -4.3%
3. BUS Small Corp Stocks -2.6%
4. ACE Int’l Emerging Stocks -4.8%
5. TMC Mid Cap Stocks -1.1%
6. TMG Mid Size Corp Stocks 0.3%
7. SEM Int’l Emerging Markets 1.2%
8. MPT Int’l Asia Stocks 1.3%
9. FAK Korea Stocks 1.3%
10. POE OTC Hi Tech Stocks 1.3%
The Other Sample Funds:
All the other funds in our 100 samples at the 90% confidence level fall somewhere in between the extremes of Tables 1 and 3. Many funds could arguably be every bit as good (or bad) a choice as those listed, but we should always be looking toward the upper limit of the mean annual return, µf and the lower limit of weighted standard deviation, σwf pairs for the best selection for a portfolio.
Fund Data Pairs: (σwf , µf ) = (σf x t(α, Nf -1) x √1/ Nf, µf)
In Figure 2 we can imagine an upper boundary for the paired-data scatter diagram that suggests a number of things. First there may exist a minimum-annual-return maxima such that there exists a fund or set of funds that would provide the highest expected return for the least risk i.e. those pairs farthest from the zero-line. Figure 2 suggest that these might be funds where σwf is in the area of 10% to 12% with a minimum annual return of approximately 12% to 13% per year. Lower volatility such as money market funds and very high volatility risky high tech markets would be expected to result in a long-term lower minimum rates of return for their respective volatility, i.e. closer to the zero-line. Furthermore, there appears to exist a upper frontier to the minimum rate of return data that bounds all the 100 samples. If this frontier exists, it represents an upper bound to what we can ever expect in the long term for a minimum annual return for any given level of volatility.
If we had the capability to add all 8000 available funds to our data base, we would come to a different set of best and worst funds because we would have more to pick from. But the basic process would remain the same. The author believes that the 100 samples provide an adequate insight into the investment selection process and can be generalized to the full range of possible investments. With the proper resources these procedures could be extended to sort all 8000 funds.
Risk Tolerance:
Financial advisors and investment firms are quick to point out to an investor that a key decision factor in how they put a portfolio together is the individual’s tolerance to risk. Age, wealth and personality all play a role in this assessment. Attempts at quantifying an inherently qualitative scale have been attempted and usually take the form of an assessment or survey of an individual’s “ability to sleep at night” (not counting trips to the bathroom). This is not a scale with which most analytical people can get comfortable.
But there may be a better way to quantize the risk scale if we can relate investment decisions to probabilities of winning or losing familiar games and at the same time quantify the specific risk level that determine specific fund choices. Table 4 offers some game odds that might be illustrative of the meaning of alternative values of α as we apply it to fund selection risk.
Table 4: Risk Level and Game Odds
Conf. Invest.
Level α Risk Odds Equivalent Game Odds
50% 50% 1:1 Getting one tail in one flip of a fair coin
75% 25% 3:1 Getting 2 tails in two flips of a fair coin
90% 10% 9:1 ~ Rolling “snake eyes” in one roll of a pair of
dice
95% 5% 19:1 ~ Being dealt two-pairs in a five card poker
hand
99% 1% 99:1 ~ Obtaining eleven or more tails in thirteen flips
of a fair coin
99.9% 0.1% 999:1 ~ Being dealt a full-house in a five card poker
hand
Since we need to make decisions about what risk levels would be appropriate, we must first understand our own tolerance to risk. To personalize and make sense of our own risk tolerance we could use the Table 4 odds and ask ourselves the following question: “If you had to pay out half of your savings account every time your opponent succeeded in winning a game of table 5, or you doubled your savings account every time your opponent lost, which game would you feel comfortable playing?”. 5% or 1% or even 0.1% might be an appropriate risk levels if you need to be fairly certain that you will not lose the game more than a few times in many repeats of the same gamble… we already know where our true “gambler” will be in his or her risk tolerance!
The odds we are using here assume a fair game, so this may be a good place to ask yourself another question: “Can the financial investments ever be a completely fair game?”. The answer is probably not, but it sure beats the state lottery as a retirement plan.
Even-Odds Gambler:
For an investor that completely tolerates risk, the opening scenario discussed in the “Basics” i.e. someone that is willing to flip a coin to make a financial decision, can be modeled by selecting an α equal to .5. At this confidence level, volatility ceases to be a decision factor and the investor will make choices purely on the basis of average annual return. At this risk level and an annual return for 10 years of 31.8%, it is hard to argue that the fund FAK shouldn’t be first on the risk-tolerant investor’s list. Table 5 shows the top ten picks for a high risk portfolio for our even-odds investor and Table 6 shows the bottom ten funds for the same confidence level of 50%. With minimum returns of between 18 and 32% per year, this would be a portfolio that would turn heads, at least until one realizes that at this confidence level, five of the ten picks could turn out to be bad choices and actually could lose money.
Table 5: Top Ten Fund Picks at a confidence level of 50%
Fund Fund Style Lf
1. FAK Korea Stocks 31.8%
2. SSR Small Co. Value Stocks 29.4%
3. FMF Common Stocks 23.7%
4. MND Int’l Dev Country Stocks 21.5%
5. FST Technology Securities 21.2%
6. TMG Mid Corp Stocks 20.3%
7. SLA Int’l Latin Am Stocks 20.0%
8. HWM Mid Size Value Stocks 19.8%
9. RMC Emerging Growth Stocks 19.1%
10. FSH Healthcare Stocks 18.8%
Table 6: Bottom Ten Funds at a Confidence Level of 50%
Fund Fund Style Lf
1. FPF Cash 2.7%
2. EAR Ultra-short Bonds 2.9%
3. KSF Eastman Kodak Stock 3.0%
4. TIA US & Corp Bonds 6.1%
5. SBI Corp Hi Yield Bonds 7.0%
6. THY Corp Hi Yield Bonds 7.3%
7. LBA US & Corp Bonds 7.9%
8. BUS Small Corp Stocks 8.5%
9. MNA Fixed Income Securities 8.6%
10. FIF Stable Value Bonds 8.6%
But while fund FAK works for our high risk taker, it would not likely be on the list for anyone with a lower risk tolerance. Which begs the question, how do you account for individual investor’s risk tolerance if we are not all totally risk takers? The answer lies in the confidence level and risk factor α.
The Slightly Conservative Gambler:
As we recall, the 50% confidence level represents a risk factor α of .5 which means the high risk taker is willing to make decisions by the flip of a coin. We also suggested that a modestly conservative investor might be better characterized by a 90% confidence level or α = .1. Where does that leave an investor willing to take some risk but doesn’t accept using a coin to pick an investment fund?
We can repeat our analysis using the same data we used for the prior 50% and 90% confidence level scenarios but changing the α risk factor to .25, equivalent to being wrong 1 time in 4 decisions. Figure 3 shows the 100 sample data pairs (σwf , µf ) for the 75% confidence level scenario. Notice that only one fund has a minimum rate of return that is negative and therefore would fail the hypothesis test. Table 7 shows the top ten picks for the conservative gambler and table 8 the bottom worst ten picks for the conservative gambler at confidence level of 75%.
Figure 3: Scatter Diagram of 100 Funds; 75% Confidence Level, α = .25
Table 7: Top Ten Fund Picks at a confidence level of 75%
Fund Fund Style Lf
1. FMF Common Stocks 17.4%
2. SSR Small Corp Value Stocks 17.3%
3. FAK Korea Stocks 16.3%
4. MND Int’l Developed Stocks 15.7%
5. CSR Real Estate Trusts 15.5%
6. HWM Mid-sized Value Stocks 15.1%
7. SSE Small Corp Stocks 14.6%
8. FSH Health Care Stocks 14.6%
9. FST Technology Securities 14.5%
10. LLP Mid & Large Cap Stocks 14.3%
Table 8: Bottom Ten Funds at a Confidence Level of 75%
Fund Fund Style Lf
KSF Eastman Kodak Stock -0.8%
EAR Ultra-short Bonds 2.2%
FPF Cash 2.3%
BUS Small Corp Stocks 3.0%
FEA Asia Stocks 3.2%
TMC Mid Cap Stocks 4.3%
ACE Int’l Emerging Stocks 4.9%
SBI Corp Hi Yield Bonds 4.9%
TIA US & Corp Bonds 5.0%
THY Hi Yield Corp Bonds 5.1%
The Moderately Conservative Investor:
A moderately conservative investor might choose a game whereby if his or her opponent is dealt two pairs in a game of five-card poker, corresponding to an α of 5% (equivalent to losing 1 game in 20) the player loses money and conversely win 19 games out of 20 and wins money each time. Depending on how much is risked in the one losing hand, not a bad game plan.
Figure 4 shows the scatter diagram for (σwf , µf ) pairs at 95% confidence level or α = 5%. At this risk level the 85 out of the 100 funds would be expected to have a positive rate of return and 15 of the 100 funds may result in a negative returns. Tables 9 and 10 provide the top best and bottom worst picks at the 95% confidence level.
Figure 4: Scatter Diagram of 100 Funds, 95% Confidence Level, α = .05
Table 9 : Top Ten Fund Picks at a confidence level of 95%
Fund Fund Style Lf
CSR Real Estate Trusts 11.2%
LLP Mid & Large Cap Stocks 10.8%
SSE Small Corp Stocks 10.2%
TAV Undervalued Equities 9.9%
FEF Restructured Corp Stocks 9.6%
PSC Small Corp Stocks 9.5%
RIT Large Corp Stocks/Bonds 8.9%
PEI Established Corp Stocks 8.8%
FSF Financial Services 8.7%
SSS Multi-sized Corp Stocks 8.7%
Table 10: Bottom Ten Funds at a Confidence Level of 95%
Fund Fund Style Lf
ACE Int’l Emerging Stocks -11.5%
FEA Asia Stocks -9.7%
FAK Korea Stocks -8.7%
SSR Small Corp Value Stocks -7.7%
BUS Small Corp Stocks -6.7%
KSF Eastman Kodak Stock -6.6%
TMG Mid-sized Corp Stocks -6.1%
TMC Mid Cap Stocks -4.6%
MPT Int’l Asia Stocks -2.9%
POE OTC Hi Tech Stocks -2.5%
The Conservative Investor:
A conservative investor might apply a risk factor α of 1% and would result in an alternative set of ten top and ten bottom funds. To illustrate, Figure 5 is a similar plot of average annual returns versus weighted standard deviation similar to shown in Figure 2 but for a confidence level of 99%, i.e. one chance in 100 of being wrong. Notice that the annual rates of return remain the same but the weighted standard deviation for each fund is now further to the right. More points are also below the threshold of 0% return and fewer remain above so as a result of our hypothesis test, the number of possible investments has been reduced from 100 for the gambler to 57 for the conservative investor. Again, ignore the spurious point represented by fund BMC. Table 11 consists of the top ten best funds for a conservative investor and Table 12 presents the bottom ten worst funds.
Figure 5: Scatter Diagram of 100 Funds, 99% Confidence Level, α = .01
Table 11: Top Ten Fund Picks at a confidence level of 99%
Fund Fund Style Lf
1. LLP Mid & Large Cap Stocks 8.0%
2. CSR Real Estate Trusts 7.8%
3. FIF Stable Value Bonds 7.5%
4. FEF Restructured Corp Stocks 6.9%
5. TAV Undervalued Equities 6.8%
6. SSE Small Corp Stocks 6.7%
7. PSC Small Corp Stocks 6.6%
8. PEI Established Corp Stocks 6.5%
9. SSS Multi-sized Corp Stocks 6.2%
10. FPU Balanced Stocks/Bonds 6.0%
Table 12: Bottom Ten Funds at a Confidence Level of 99%
Fund Fund Style Lf
1. SSR Small Corp Value Stocks -42.2%
2. FAK Korea Stocks -30.5%
3. ACE Int’l Emerging Mkt Stocks -27.6%
4. FEA Long Term Growth Stocks -21.0%
5. TMG Mid Corp Stocks -20.4%
6. BUS Small Corp Stocks -16.9%
7. TMC Mid Cap Stocks -12.6%
8. MPT Int’l Asia Stocks -12.0%
9. KSF Eastman Kodak Stock -11.1%
10. POE OTC Hi Tech Stocks -10.3%
The Highly Conservative Investor:
If we further increase the confidence level to 99.9%, almost certain of a positive minimum return in the long run, the number of available investments again drops, this time to only 20 possible funds. Table 13 presents the top ten best funds for this highly conservative, risk-adverse investor and Table 14 presents the bottom ten worst funds. In these two tables there is a not-so-surprising clear bias toward bond funds such as FIF and away from emerging market funds such as SSR.
Table 13: Top Ten Fund Picks at a confidence level of 99.9%
Fund Fund Style Lf
1. FIF Stable Value Bonds 7.0%
2. LLP Mid & Large Cap Stocks 4.5%
3. PR1 Balanced Stocks/Bonds 3.9%
4. PEI Established Corp Stocks 3.9%
5. FPU Balanced Stocks/Bonds 3.7%
6. PTR Bonds 3.7%
7. FEF Restructured Corp Stocks 3.5%
8. LBA US & Corp Bonds 3.4%
9. CSR Real Estate Trusts 3.3%
10. PNE Natural Resource Stocks 3.1%
Table 14: Bottom Ten Funds at a Confidence Level of 99.9%
Fund Fund Style Lf
1. SSR Small Corp Value Stocks -131.6%
2. FAK Korea Stock -63.1%
3. ACE Int’l Emerging Stocks -55.2%
4. TMG Mid Cap Stocks -41.7%
5. FEA Asia Stocks -37.9%
6. BUS Small Corp Stocks -36.1%
7. TMC Mid Cap Corp Stocks -25.0%
8. EEA US & Global Securities -25.0%
9. MPT Int’l Asia Stocks -24.8%
10. MLG Global Stocks -22.1%
Funds Selection Sensitivity To Risk Level:
In each case by changing the risk level for the investor it was possible to sort and rank the 100 sample funds to arrive at a list of possible investments with a known probability of a positive annual return that best fits our risk tolerance. Each risk level produces a different set of top best and bottom worst funds that can change position between a low confidence level, high risk scenario to a high confidence, low risk scenario.
The relative rankings and the changes in top and bottom sets should not be terribly surprising as we change risk tolerance and actually might follow our instinctive assumptions about bonds versus stock funds and emerging markets versus established markets funds. For example FAK which specializes in investment in Korean company stocks is the top pick of all funds for the gambler at a 50% risk level and second from the bottom of the ten worst picks for the extremely conservative investor at a risk level of 0.1%. Conversely, PTR and LBA bond funds do not even appear on the radar until we reach the 99.9% confidence level for the extremely conservative investor.
There are other funds we might pay particular attention to such as funds CSR, LLP, FEF or SSE. For these funds there are ample annual return data and a surprisingly robust staying power over a range of risk levels. Although there respective positions in the top ten continually change they are consistently close to the frontier of returns for a given volatility. Table 15 provides the four investment fund rank positions as a function of confidence (or risk) level.
Table 15: Ranking in Top Ten for Four “Robust” Funds
Confidence Fund CSR Fund LLP Fund FEF Fund SSE
Level Top Ten Rank Top Ten Rank Top Ten Rank Top Ten Rank
50% --- --- --- ---
75% 5 10 --- 7
90% 1 2 7 3
95% 1 2 5 3
99% 2 1 4 6
99.9% 9 2 7 ---
Optimizing The Strategy:
Another constant reminder from the investment professionals is diversification. By this they mean that you should spread your risk over several investments and not place your bets on a single fund, fund style or worst yet, a single stock (think ENRON). Commonly you will encounter recommendations that you split your investments among 5 to 10 funds. Assuming that this is sound advice, and it most certainly is, we should seek strategies that spread our investments over a broad range of types of investment that vary risk exposure from low expected return to high expected return with a minimum exposure to volatility.
Since we have defined risk both in terms of the fraction of incorrect decisions we will make (α-risk) and in terms of fund variability (σwf), there are several ways to spread risk in selecting funds. For instance we could use an approach that selects funds on there relative insensitivity to risk level, i.e. pick those funds that are least sensitive to risk level change but are consistently in the top funds of α risk several scenarios. We would select the funds from the previous section discussion and our list would look as follows in table 16.
Table 16: Selected Funds With The Least Sensitivity To Risk Level
Fund Fund Style (σwf , µf)
CSR Real Estate Trusts (7.0%, 18.2%)
LLP Mid & Large Cap Stocks (5.8%, 16.6%)
FEF Restructured Corp Stocks (5.6%, 15.2%)
SSE Small Corp Stocks (7.3%, 17.5%)
This strategy unfortunately results in a fairly narrow range of volatility, σwf 5.6% to 7.3%, also limits our expected minimum return µf to a range of 15.2% to 18.2%, not bad but maybe we could do better.
Another strategy might be to use the α-risk tables and selecting one or more funds from each α-risk category top ten funds. For instance we might select the top in each risk category as shown in table 17:
Table 17: Selected Funds Using The Top Pick For Each Risk Level
α-risk Fund Fund Style (σwf , µf)
50% FAK Korea Stocks (31.8%, -6.1%)
25% FMF Common Stocks (16.3%, 23.7%)
5% CSR Real Estate Trusts (7.0%, 18.2%)
1% LLP Mid & Large Cap Stocks (5.8%, 16.6%)
0.1% FIF Stable Value Bonds (0.8%, 8.6%)
This strategy results in a wide range of volatility, σwf ranging from 0.8% to 31.8% with expected minimum returns µf of between –6.1% to 23.7%. This strategy has opened us up to the possibility of negative returns but also holds the possibility of greater minimum returns. Both Table 16 and Table 17 lists could be expanded, but they would not necessarily provide a spread in risk as defined by σwf nor would they ensure that we were in the sweet spot of investment, i.e. maximum return for minimum volatility.
Yet a third strategy should be considered that spreads risk over several selected levels of the weighted standard deviation and at the same time maximizes the minimum expected return. If we make a rational choice for our personal α-risk tolerance and then use the scatter diagrams for (σwf , µf) pairs and some imagination for the “frontier” curve of funds which maximizes the minimum returns, we can select from the 100 sample funds just those that are at or near a selected volatilities σwf and also at or near the upper frontier of return for that particular volatility, i.e. those pairs that have near “ideal” values. At the even-odds risk level of 50% there is no “frontier” to be estimated, but for any other risk level one can imagine that there exists a theoretical optimal upper bounding curve. For the sake of discussion, let us assume an α-risk of 5%, and arbitrarily select 5% increments for σwf. We can then estimate the frontier minimum return values µfmax and select actual funds with data pairs the same or close to the ideal (σwf , µfmax). These selections are shown in Table 18.
Table 18: Selected Funds With Near “Ideal” Scatter Diagram Pairs
Frontier Pair Closest Pair Fund Fund Style
(σwf , µfmax) (σwf , µf)
(0%, 7%) (.8%, 8.6%) FIF Stable Value Bonds
(5%, 15%) (5.6%, 15.2%) FEF Restructured Corp Stocks
(10%, 18%) (9.3%, 18%) FSF Financial Services Stocks
(15%, 23%) (15.1%, 21.5%) MND Int’l Developing Stocks
This approach results in reasonable range of volatility σwf 0.8% to 15.1%, and a reasonable minimum return µf of between 8.6% and 21.5% while avoiding funds with negative minimum returns. This strategy would seem to combine the best of each approach and would result in as optimum fund selection as can be reasonably expected. The only true improvement would be to expand the sample size to all available funds so that the true “frontier” curve could be estimated over the full range of fund volatilities.
Hypothetical Frontier To Annual Return vs. Weighted Standard Deviation:
As discussed earlier, one can imagine a curve that represents the theoretical upper limit, or frontier to annual return for an incremental change to the weighted standard deviation. If such a function exists, it would provide a reference against which all funds could be compared and provide a limit to our long-term expectations with regard to annual returns as a function of confidence level and fund variability.
Unlike earlier discussions in this paper which are based on solid statistical and mathematical approaches, this section is devoted to some conjecture about how optimal funds might best perform in the real world. From the sample of 100 funds I have estimated the frontier of return versus variability by visualizing an envelope surrounding the (σwf , µf ) data points. This visualization assumes that the upper boundary will increase with fund variability to some upper level and not decrease after some high level of variability is reached, i.e. we will reach an asymptotic value and not turn down again. The upper boundary could then be approximated by an exponential function whose parameters are an exponent (a) and multiplier (b) operating on the weighted standard deviation statistic (σwf). This is a conjecture that a mathematical equation for the upper boundary might be of the form:
Upper Boundary = β(σwf) = b x σwf a
This type of equation produces a fast rising return values that tapers off and eventually reaches an asymptotic annual return value for very high weighted standard deviation values. The parameters themselves vary with confidence level α so the complete equation for the upper boundaries would be as follows:
β(σwf, α) = b(α) x σwf a(α)
Using the individual (σwf , µf ) data pairs from the sample 100 funds I have made an educated guess of the data pair values for the limiting upper boundary for return versus weighted standard deviation. From the upper boundary data pairs it was possible to estimate the functional parameters using a regression technique that best fit the upper boundary data pairs. Table 19 provides the best estimates for values for the frontier equation parameters.
Table 19: Upper Boundary Equation Parameters
Confidence Level α b(α) a(α)
75% .25 15.0 0.27
90% .10 10.2 0.33
95% .05 9.6 0.33
99% .01 6.4 0.42
99.9% .001 4.6 0.47
Each parameter set produces a unique upper boundary curve that purports to represent the best possible annual return for a particular fund weighted standard deviation at a particular level of confidence. Figure 6 shows the upper boundary curves for the parameters and confidence levels (α – risk) of table 19. All the upper boundary curves will eventually intersect the straight line that represents the points at which the annual return equals the weighted standard deviation. When the upper boundary curve is below the line we have reached the point at which even the best performing fund will fail the hypothesis test for a positive outcome and face us with a possibility of a long-term negative return.
Figure 6: Upper Boundary of Return
As can be observed from Table 19, the multiplier decreases with higher confidence levels and the exponent increases. The curves generated by these parameter values suggest that as we become more conservative, i.e. higher confidence level, we will expect lower fund variability and the accompanying lower potential annual return. Table 20 provides the estimated intersection points of the upper boundary curves. Note that the upper boundary rate of return is exactly equal to the weighted standard deviation (representing the lower limit of the confidence interval for annual return). This is the point where we fail the hypothesis test.
Table 20: Upper Boundary Values For Annual Return
Confidence Level σwf µf
75% 41% 41%
90% 33% 33%
95% 29% 29%
99% 24% 24%
99.9% 17% 17%
Table 20 provides an interesting perspective on fund investing. If we are willing to risk our capital with an expectation that we will make a bad decision on 1 out of 4 choices (i.e. we invest in a losing fund 25% of the time) and we invest in only the highest return funds for the highest variability then we might expect to see an annual return of 41%. At the other end of the scale if we risk only making 1 wrong decision in 1000 choices (i.e. we almost always invest in funds with a positive return) and we invest in only the highest return funds with a moderate amount of variability then we might expect to see an annual return of 17%. At 41% per year return we could double our money in slightly over 2 years, but you must be able to tolerate large swings in year-to-year performance. At 17% it would take just under 5 years to achieve the same goal with less year-to-year variability. The message?… A conservative approach combined with a statistical basis for fund selection should yield solid investment gains within 5 years. If you wish to do it in less time, stick with the state lotto.
The Final Fund Filters:
Having isolated those funds that have the best net returns for a particular level of risk, the final numerical issue to be considered is the cost to invest in the fund in the first place and the annual management fees thereafter. Many of the 100 sample funds are no-load (i.e. no fees charges for opening the fund account) but not all. Since load funds can charge 4, 5 or even 6% to open the account, this is a serious hit to your initial investment nest egg. Later, management fees of 1%, 2% or even higher can be a serious drag on performance. Although they are included in the net performance data, the annual fees may be higher for one fund of otherwise equivalent funds. Another road block to picking the best can be if the fund is closed to new investors. If you have isolated your pick as the best investment and find that it is indeed closed, investigate if there are exceptions available.
Once you have sorted funds that meet you risk and return objectives take a hard look at the load and annual fees. Use this final filter to help sort between statistical close calls and make your final fund selections. In the 100 sample funds used in this paper the no-load funds actually out performed the load funds with an average overall return of 14.4% for the no-load funds versus 11.9% for the load funds. In addition they showed less variability.
In Conclusion:
If we could analyze all 8000 funds in the same manner as the 100 samples, the 16 million of us who invest in mutual funds and rely on our own council could increase our confidence that we understand our risk and have made choices that best fit our risk tolerance. Given enough time and resources to assemble all the available data we could in fact rank-order all investment funds for specific levels of risk tolerance and provide a rational basis for our investment decisions.
That not withstanding, from our sample of 100 investment funds we may conclude the following:
· Average annual return alone is not a sufficient statistic to make rational decisions about fund investments.
· Standard deviation of fund returns, while it is a measure of a particular fund’s volatility, is not sufficient in and of itself to characterize probable long-term results.
· We can employ the statistical concepts of classical hypothesis testing in making rational investment decisions.
· For a particular risk tolerance level, hypothesis testing confidence intervals can be used to reduce the number of investment funds to be considered.
· Scatter diagrams of mean return and weighted standard deviation of the mean is a rational way to compare funds and parse funds with similar minimum annual returns.
· Some funds will be preferable to high risk takers and inappropriate for low risk takers and visa versa.
· Some funds are more robust to changes in risk and remain on the preferred list for low and high risk takers.
· There exists a frontier of returns for given levels of volatility that represents the best possible set of investment funds long-term.
· The results of any analysis are time dependent and are subject to change in the future.
· Any mathematical analysis must be tempered with an additional level of “old fashion common sense”.
What we hope to have accomplished here is to have solidified a few of our investment instincts and to put a mathematical framework around them. Use of standard deviation as a measure of an investment volatility is not a new concept but the author believes that the use of confidence intervals of the mean and return versus weighted standard deviation scatter diagrams, which are not commonly found in the open investment literature, may be an appropriate and novel approach to filtering investment funds. The author invites your comments.
Bibliography
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Modern Elementary Statistics
John E. Fruend
Prentice-Hall 1967
Introduction to the Theory of Statistics
Alexander M. Mood and Franklin A. Grabill
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Trends In Mutual Fund Investing October 2007
Investment Company Institute
Washington DC
(US Department of Commerce statistics)
The Individual Investor’s Guide to The Top Mutual Funds
26th Edition 2007
American Association of Individual Investors
Chicago, IL
Appendix A: 100 Sample Investment Funds
Fund ID Code* Fund Profile
1 EAA Balanced US & Foreign Stock Funds
2 PTR Bonds
3 MTH Bonds-Corp
4 SBI Bonds-Corporate Hi Yield
5 FIF Bonds-Stable Value
6 PR1 Bonds-Stocks Balanced
7 LBA Bonds-US & Corp
8 FPF Cash
9 FMF Common Stocks
10 ABC Corporate Bond
11 FAH Corporate High Yield (Bond)
12 RMC Emerging Growth Stocks/Bonds
13 ACF Emerging Small & Mid Stocks
14 FSF Financial Services
15 MNA Fixed Income Securities
16 MLG Global Stock
17 MMI Growth & Current Income
18 FSH Health Care Company Stocks
19 THS Health Related Common Stocks
20 PHS Health Stocks
21 THY High Yield Corporate Bonds
22 VKH Income/Appreciation
23 ANP Industrial Stocks World Wide
24 ANE International Equity
25 OMS Large Cap Energy/Indust/services
26 EVT Large Cap Finacial/Services
27 LMV Large Cap Stocks
28 FCA Large Cap/Growth/Value
29 CLF Large Company Stocks
30 RIT Large Company Stocks/Bonds
31 MVA Large Undervalued Stocks
32 MIG Long Term Growth
33 MEG Long Term Growth
34 FAG Long Term Growth
35 FAE Long Term Growth
36 APG Long Term Growth
37 ABG Long Term Growth
38 VAW Mid & Large Cap Stocks
39 LLP Mid & Large Cap Stocks
40 ARA MidCap Growth/Value
41 DNL MidCap Growth
42 POE OTC Emerging Hi Tech Stocks
43 CSR Real Estate Trusts
44 AQF Small Company Growth
45 SSR Small Company Stocks Value
46 FEC Stock-European Countries
47 MND Stock-Inernational, Developed Countries
48 ACE Stock-International Emerging Markets
49 KSF Stock-Kodak
50 HWM Stock-Mid Sized Value
51 FNO Stock-Nordic Countries
52 JNF Stocks
53 BMC Stocks
54 AGF Stocks Broadcasting&Electronics
55 AEP Stocks Europe & Pacific Rim
56 VKE Stocks High Risk
57 TLC Stocks Large Cap Companies
58 TMC Stocks Mid Cap Companies
59 FEA Stocks-Asia
60 PBC Stocks-Blue Chip
61 TH2 Stocks-Bonds Balanced
62 FPU Stocks-Bonds Balanced
63 PEI Stocks-Established Corp
64 MPT Stocks-Intl Asia
65 SMI Stocks-Intl Corp
66 NUS Stocks-Intl Developed & Developing Countries
67 MSI Stocks-Intl Developed Countries Large Corp
68 TDM Stocks-Intl Emerging Markets
69 SEM Stocks-Intl Emerging Markets
70 SLA Stocks-Intl Latin America
71 ACI Stocks-Intl Small&Mid Corp
72 ARI Stocks-Intl World Wide
73 FAK Stocks-Korea
74 PUI Stocks-Large Corp
75 FGI Stocks-Large Corp
76 AFG Stocks-Large Corp Growth
77 TMG Stocks-Mid Corp
78 PVI Stocks-Mid Corp
79 PGF Stocks-Mid Corp
80 SSS Stocks-Multisize Corp
81 PVO Stocks-Multisize Corp
82 MTV Stocks-Multisize Corp
83 MIR Stocks-Multisize Corp
84 PST Stocks-Multisize Corp High Tech
85 PNE Stocks-Natural Resources
86 PR3 Stocks-S&P500 Plus
87 SSE Stocks-Small Corp
88 RSS Stocks-Small Corp
89 PSC Stocks-Small Corp
90 BUS Stocks-Small Corp
91 APO Stocks-Small Corp
92 SPS Stocks-SP500
93 FEF Stocks-Undervalued/Restructured Corp
94 ATF Technology
95 FST Technology Securities
96 EAR Ultrashort Bonds
97 TAV Undervalued Equities
98 TIA US & Corporate Bonds
99 EEA US & Global Securities
100 OMK US Common Stocks
*Note: This is a substitute fund code not the actual fund ticker symbol.
Appendix B: Not All Percentages Are Created Equal
There is a caveat to using annual percent return as a measure of fund performance. Negative returns are more influential to overall performance than positive percent returns. Return to Figure 1 and note that the distribution of annual returns for our sample of 100 funds can be both negative in any fund-year or positive in any fund-year. If we rank all funds for all years for which there was data, we have 1494 fund-year data points which range from the lowest annual return observed of –65% to the highest annual return observed of +126%. If we break the distribution into quartiles (blocks of 25% of the distribution (i.e. each block contains 374 data points and is equally probable to contain a fund-year data point when selected randomly) we have the following quartile break points for annual return:
Table B-1: Annual Return Quintiles
Quartiles Return
1st -65% to +2%
2nd +2% to +12%
3rd +12% to +24%
4th +24% to +126%
In table B-1 we observe that 25% of the time an annual return will be between: –65% and +2%, +2% to +12%, +12% to +24%, and +24% to +126%. Thus 75% of the time returns will be 2% or better but about 20% of the time returns will be negative.
We can use the quartile break points to understand how negative returns affect performance to a greater degree than positive returns of the same magnitude. For example a –20% fund decrease affects the portfolio value greater than a +20% increase. Consider an investment of $10,000 with a
–20% loss the first year and a +20% gain the next; we will have lost $400 after two years. As another example consider that in a disastrous year where we picked the worst performing fund out of the 100 funds which loses –65% in one year and consider what will happen to our $10,000 investment. Our initial investment at the end of that year would be worth only $3500 because of the –65% decrease in the fund value. To regain our original investment would require that we see two consecutive annual returns of +69%, not a likely outcome when 75% of time returns are below +24%. Even if on the following year we where lucky enough to select the best year for the best fund (i.e. +126%) we would only recover $7910 of our original investment after that first disastrous year and would still need another year of +26% to break even.
Another way to examine the affect of negative returns is to imagine a five year investment where on each year the return is the quartile point values: -65%, +2%, +12%, +24% and +126% (the exact order does not matter). In this scenario the overall return will be +12% but it will take four solid positive years to counteract the effect of the one very negative year. So percentages on the down side are more affecting of our fund performance then percentages on the up side and avoiding disastrous selections for a portfolio will be key to long term success.
Fortunately we will not be completely random in our selections and the range of returns will be much smaller than the extremes. The effect of negative returns out-weighing positive returns will not be so obvious. For instance a -10% decrease followed by a +10% increase will bring us back to within $100 of being equal to our original $10,000 investment. Because the negative versus positive effect is less problematic for typical returns which are going to be positive over 75% of the time, we can ignore the bias effect for purposes of establishing hypothesis test confidence intervals discussed in this paper. That is to say negative returns will be given the same “weight” as positive returns even though they do in fact have a slightly greater affect.
