More on Threshold Distortion

I stand one of William Bernstein’s articles on end. He claims that actively managed mutual funds are losers. I use his own data to show you how to beat the S&P500 index by 2% to 3% using actively managed mutual funds. Price discipline is the key.

I selected one of William Bernstein’s articles as an example of the threshold distortion. William Bernstein represents himself as presenting the consensus of academia. I believe that his views represented the consensus of academia at one time, but that academia is moving away from his position.

The article that I selected is A Sucker's Bet dated April 1, 2001. It is posted at Financial-Planning.com. The relevant data are in Figure 1.
A Sucker's Bet

Mutual Fund Data

William Bernstein selected the top 30 mutual funds according to how well they performed in a specified five-year interval. He compared this to their performance over the remaining years through 1998.

This is a little awkward, because the time intervals before and after differ. The initial time interval was always five years. Subsequent intervals ranged from 23 years to 3 years in five-year increments.

He collected similar data for all funds. The funds examined were diversified mutual funds. The top funds were selected from this category. There is internal evidence (from the performance numbers) that the total number of mutual funds differed for each start year, as it should.

Measuring the Threshold Distortion

The annualized rate of return in the initial period r1 satisfies this equation:(Balance at year 5/starting balance) = (1 + r1)^5 since all initial periods last 5 years.

The annualized rate of return in the second period r2 satisfies the equation:(Balance in 1998/balance at year 5) = (1 + r2)^n where n is the number of years from year 5 until 1998.

The process of selecting the top 30 mutual funds is equivalent to setting a threshold that all 30 exceed, but which no other funds exceed. If no skill were involved, we would expect this group of funds to underperform until it gave back this entire amount.

Here is a sketch of the mathematical formalism.

If we have three totally random prices, each with an expectation of zero, and if we set a selection threshold based on the second price, the expected increases and decreases would be:
1) The rate of return in the first period would be determined from (the price after 5 years/the initial price). We would report the percentage increase. [Mathematically, we do this by taking logarithms. The logarithm of a ratio is the difference between the logarithm of the numerator and the logarithm of the denominator.]
2) The rate of return in the second period would be determined from (the final price/the price after 5 years). Again, we would report the percentage increase. [Mathematically, we would use logarithms to convert the division of numbers to a difference of their logarithms.]
3) In terms of percentages, the assumption of total randomness means that the expectation of the price of each fund (relative to the average) is zero. That is, the expectations in the three time intervals of start, year 5 and end satisfy:
E(relative price at the start) = E(relative price at year 5) = E(relative price at the end) = 0%.
4) The initial rate of return is determined by expressing the price at year 5 as a percentage increase from the initial price. [Since the selection process seeks the top 30 funds, this will normally be a positive percentage change.]
5) The subsequent rate of return is determined by expressing the final price as a percentage increase from the price at year 5. [Assuming total randomness, this would normally be a negative percentage change.]
6) When we introduce the threshold at year 5, the expectation of those prices that exceed the threshold is no longer zero.
7) The initial rate of return has the form (when using logarithms):
E(relative price at year 5)-E(relative price at the start) = E(relative price at year 5)-0 = E(relative price at year 5 as a result of having a threshold).
8) The subsequent rate of return is determined from:
E(relative price at the end)-E(relative price at year 5) = 0 – E(relative price at year 5) = -E(relative price at year 5 as a result of having a threshold).

Once again, since we are assuming total randomness: on average, in the second interval, we expect to give back everything that we gained by introducing a threshold at the end of the first interval. The intermediate price links the gains and losses in the two intervals.

The expected prices at the start and finish still equal to the average prices, independent of the threshold in the middle.

Mutual Fund Comparisons

Here are the returns of the top 30 mutual funds and of all funds in each period.

Starting from 1970:

Top 30:
1970-1974…...0.78%
1975-1998…..16.05%

All Funds:
1970-1974…...-6.12%
1975-1998…..16.38%

Starting from 1975:

Top 30:
1975-1979…..35.70%
1980-1998…..15.78%

All Funds:
1975-1979…...20.44%
1980-1998....15.28%

Starting from 1980:

Top 30:
1980-1984…..22.51%
1985-1998…..16.01%

All Funds:
1980-1984…...14.83%
1985-1998…...15.59%

Starting from 1985:

Top 30:
1985-1989…..22.08%
1990-1998…..16.24%

All Funds:
1985-1989…...16.40%
1990-1998…...15.28%

Starting from 1990:

Top 30:
1990-1994…..18.94%
1995-1998...21.28%

All Funds:
1990-1994…....9.39%
1995-1998…...24.60%

Annualized Returns

The article provides the annualized returns for each interval, but not for the entire period.

These are the returns for the top 30 funds category:
1970-1998: 13.16%
1975-1998: 19.85%
1980-1998: 17.78%
1985-1998: 18.45%
1990-1998: 18.94%

These are the returns for the all funds category:
1970-1998: 12.00%
1975-1998: 16.38%
1980-1998: 15.39%
1985-1998: 15.71%
1990-1998: 14.86%

These are the annualized returns for the top 30 fund groups minus the annualized returns of all funds:

Starting in 1970:
initial period: 6.90%
subsequent period: (0.33%)
total 1970-1998: 1.16%

Starting in 1975:
initial period: 15.26%
subsequent period: 0.50%
total 1975-1998: 3.47%

Starting in 1980:
initial period: 7.68%
subsequent period: 0.42%
total 1980-1998: 2.39%

Starting in 1985:
initial period: 5.68%
subsequent period: 0.96%
total 1985-1998: 2.74%

Starting in 1990:
initial period: 9.55%
subsequent period: (3.32%)
total 1990-1998: 4.08%

Hurdles to Overcome in the Second Interval

The performance during the first 5 years sets the threshold, which makes a hurdle to overcome in the subsequent interval. If performance were completely random, we would expect the second interval to return all of the increase in the first interval.

We convert the increase in the first five years to a ratio. Then we assume that the mutual funds give back everything in the second interval (in the years remaining until 1998).

Mathematically, we solve:
(1 + r12)^[years remaining] = [(1 + r1) for the top 30 funds / (1 + r1) for all of the funds)]^5

Here are the solutions for r12, the annualized amount that would be returned if there were no skill whatsoever:

Starting in 1970: 1.56%
Starting in 1975: 3.37%
Starting in 1980: 2.52%
Starting in 1985: 3.02%
Starting in 1990: 14.97%

Measurement of Skill

We can compare the advantage of the selected funds in the second interval to the hurdles created by using thresholds. We add the two percentages together:

Skill equals second (or subsequent) interval outperformance + hurdle

Starting in 1970: 1.23%
Starting in 1975: 3.87%
Starting in 1980: 2.94%
Starting in 1985: 3.98%
Starting in 1990: 11.65%

This is a tremendous amount of skill. These top 30 fund managers delivered 2.94% to 3.98%.

There can be some very bad intervals, as the 1970 sequence shows. We should not use the 1990 sequence. It is much too short to be useful.

Top-notch mutual fund managers can deliver 3% to 4%, annualized.

Why Too Many Economists are Broke

An individual who does not try to solve a problem is unlikely to do so. A casual observer of these data will gain very little, if anything.

Notice that the big payoff from skill came during the first five years. Here are the later payoffs, annualized:

Starting in 1970: (0.33%)
Starting in 1975: 0.50%
Starting in 1980: 0.42%
Starting in 1985: 0.96%
Starting in 1990: (3.32%)

With frustrating regularity, I hear people referring to these numbers as the payoff from skill. That is, we are told that all that we can get is 0.42% to 0.96% by selecting skilled managers. [I have excluded the 1970 and 1990 results for ease of discussion.]

My tremendous 3% to 4% advantage has fallen into oblivion, to be consumed by fees.

Let me be harsh. Let me be blunt. I am tired of such nonsense.

We lose our 3% to 4% skill advantage only if we abandon price discipline.

Has anybody considered waiting on a favorable price? Apparently, too many economists can’t imagine such a thing. They would have us pay full price at year 5 for the hottest fund of the last 5 years. Or, at least, that is how they conduct their studies. If that is the limit of your investment sophistication, perhaps you really should buy index funds on a mechanical basis. Their fees are almost always low. Buying them mechanically is far better than paying top dollar for today’s hottest hand.

Peter Bernstein in his book Against the Gods suggests buying mutual funds among today’s losers. Alternatively, he recommends waiting two years and, only then, buying today’s winners.

I disagree strongly with ever buying the losers. Mark Hulbert reports that some newsletter writers consistently do poorly. I have seen similar findings related to mutual funds. If you buy mutual funds from among today’s losers, you are in danger of finding a manager with exceptional skill at losing money.

Waiting two years is another matter. You are likely to get a better price after the delay. Academic researchers might like buying right away (or at a preprogrammed interval) because of the ease of use with computer models. Real investors should be looking at prices. It is worth knowing that it will take a couple of years before you get the right price.

What about the S&P500?

I don’t know what to do about the S&P500 data. The numbers [in William Bernstein’s article] didn’t seem right and my calculator [which uses Robert Shiller’s database] tells me that they are wrong.

Here are the total returns (annualized, nominal) of the S&P500 with all dividends reinvested and with zero expenses:

1970-1998:
William Bernstein: 13.32%
My calculator (Robert Shiller): 12.59%
Difference: 0.73%

1975-1998:
William Bernstein: 17.03%
My calculator (Robert Shiller): 15.68%
Difference: 1.35%

1980-1998:
William Bernstein: 17.63%
My calculator (Robert Shiller): 16.27%
Difference: 1.36%

1985-1998:William Bernstein: 18.80%
My calculator (Robert Shiller): 17.17%
Difference: 1.63%

1990-1998:
William Bernstein: 16.96%
My calculator (Robert Shiller): 16.59%
Difference: 0.37%

Here are the performance comparisons between the top 30 funds and William Bernstein’s S&P500 data and between the top 30 funds and my calculator’s S&P500 data (annualized, nominal, in order):

Total period:
1970-1998: (0.16%) and 0.57%
1975-1998: 2.82% and 4.17%
1980-1998: 0.15% and 1.51%
1985-1998: (0.35%) and 1.28%
1990-1998: 1.98% and 2.35%

Initial 5 years:
1970-1998 sequence: 3.13% and 1.39%
1975-1998 sequence: 20.94% and 22.10%
1980-1998 sequence: 7.75% and 8.55%
1985-1998 sequence: 1.67% and 3.96%
1990-1998 sequence: 10.25% and 9.43%

Final period:
1970-1998 sequence: (0.99%) and 0.37%
1975-1998 sequence: (1.89%) and (0.49%)
1980-1998 sequence: (2.75%) and (1.16%)
1985-1998 sequence: (1.57%) and (0.34%)
1990-1998 sequence: (10.90%) and (8.12%)

Previously, I wrote the equation:
Skill equals second (or subsequent) interval outperformance + hurdle
We have determined the skill level as being 3% to 4%. The outperformance term corresponded to the final period listed above.

Using Bernstein’s data and Shiller’s data in order, a skill level of 3% to 4% beats the S&P500 index by:
1970-1998 sequence: 2.01% to 3.01% and 3.37% to 4.37%
1975-1998 sequence: 1.11% to 2.11% and 2.51% to 3.51%
1980-1998 sequence: 0.25 % to 1.25% and 1.84% to 2.84%
1985-1998 sequence: 1.43% and 2.43% and 2.66% to 3.66%
1990-1998 sequence: (7.90%) to (6.90%) and (5.12%) to (4.12%)

The advantage available from skill ranges between 0.25% to 2.43% if William Bernstein’s S&P500 data are correct and between 1.84% to 3.66% if Robert Shiller’s S&P500 data are correct.

I have confidence in Professor Shiller’s data. My conclusion is that you can beat the S&P500 index by 2% to 3% or more, annualized (nominal), by taking advantage of demonstrated skill.

Detailed Procedure

I strongly prefer buying individual stocks for a variety of reasons, including taking advantage of price dips and gaining better control of tax consequences. Not everyone agrees. Here are procedures for those who prefer to buy mutual funds.

This is how you convert William Bernstein’s study into a 2% to 3% edge over the S&P500 index.

Price discipline is essential.

1) Identify the top diversified mutual funds over the past 5 years. You don’t need to identify 30, but you do want to identify more than one or two.
2) Eliminate funds that are no longer suitable. Typical reasons would include changes of management, style drift and/or increased fees.
3) Eliminate funds that you do not like for other reasons such as too much volatility or self-serving management. Retirees are likely to focus on dividends.
4) Follow 5 to 10 of them carefully.
5) Identify before and after prices of these superior funds. Find out how far your fund would have to fall to bring its total return down to the level of the S&P500 index.
6) Set aside a time interval between 2-5 years to purchase one or two of these mutual funds. Not all of them will return to favorable prices. Some of them will. Depending upon market conditions, you may be able to pick one up below the price that you have calculated (i.e., equal to what it would have been if it had matched the S&P500 index from the start).

Have fun.

John Walter Russell
July 4, 2005