A longstanding debate in the financial services industry concerns whether investors are better off in the long run with actively versus passively managed investment accounts. Active management is simply an attempt to “beat” the market as measured by a particular benchmark or index such as the S&P 500. One of the more famous active fund managers is Legg Mason’s Bill Miller, who until 2006 had managed to outperform the S&P 500 for 15 years in a row. In 2006, Mr. Miller significantly underperformed the S&P 500. Going forward, no one (including Mr. Miller) really has any way of predicting with any degree of certainty whether Mr. Miller will be able to revert back to systematically beating the market over time.
Notwithstanding Mr. Miller’s impressive historical performance record, it is important to note that typically only 1020% of actively managed funds outperform the S&P 500 in any given year. Furthermore, the funds belonging to this elite group tend not to consistently replicate this performance in subsequent years; if anything, “winners” in any one period tend on average to subsequently be “losers”. Interestingly, to the extent that there is any persistence, it tends to be among funds which underperform the S&P 500 (e.g., see “Performance Persistence”).
I have worked out a simple numerical example (shown in the Addendum below) which shows that on average, active portfolio management can be expected to result in significantly worse investment performance than a passive (indexed) strategy, based upon the 1020% odds mentioned above. In order to make money (on an aftertransaction cost, riskadjusted basis) with an active management strategy, over time one has to be significantly better than average in order to have any hope of outperforming an indexed strategy. The numerical example shown below implies that an investor would need to pick winners nearly 50% of the time in order to make active portfolio management worthwhile. Furthermore, I have implicitly assumed that each period represents a completely independent lottery; thus the analysis does not consider the possibility of persistence in one’s investment performance. In order to model persistency, one would need to make the probability of beating the market in any given year (notated below as “p”) a function of previous p’s; I will leave this to the reader as an exercise.
Addendum
Suppose there are four time periods. Let u represent an “up” move where you beat the market,^{1} and d represent a “down” move where you underperform the market. Thus u > 1, and d < 1.^{2} Also suppose that you want to invest $1 at t=0. The following table lists all possible portfolio values across all dates and states for four periods:
t=0 
t=1 
t=2 
t=3 
t=4 




u^{4} 



u^{3} 



u^{2} 

u^{3}d 

u 

u^{2}d 

1 

ud 

u^{2}d^{2} 

d 

ud^{2} 



d^{2} 

ud^{3} 



d^{3} 





d^{4} 
The probability of an up move is p, whereas the probability of a down move is (1p). After 1 period, there are two possible outcomes; you either have an up move with probability p or a down move with probability p. After two periods, there are three possible outcomes; you either have two consecutive up moves with probability p^{2}, two consecutive down moves with probability (1p)^{2}, or two moves involving up and down moves with probability 2p(1p), etc. Taking this out to four periods yields the following set of probability distributions for 1, 2, 3, and 4 periods:
t=0 
t=1 
t=2 
t=3 
t=4 




P^{4} 



p^{3} 



p^{2} 

4p^{3}(1p) 

p 

3p^{2}(1p) 

1 

2p(1p) 

6p^{2}(1p) 

1p 

3p(1p)^{2} 



(1p)^{2} 

4p(1p)^{3} 



(1p)^{3} 





(1p)^{4} 
Next, suppose that the odds of beating the market in any one year (p) is 20%, and that u = 1.1. Thus d = 1/1.1 = .91. The following table lists the portfolio values in each possible state occurring 1, 2, 3, and 4 periods from now combined with associated probabilities based upon the previous table:
t=0 
t=1 
t=2 
t=3 
t=4 




$1.46 



$1.33 
0.160% 


$1.21 
0.80% 
$1.21 

$1.10 
4.0% 
$1.10 
2.560% 
$1.00 
20% 
$1.00 
9.60% 
$1.00 

$0.91 
32.0% 
$0.91 
15.360% 

80% 
$0.83 
38.40% 
$0.83 


64.0% 
$0.83 
40.960% 



51.20% 
$0.75 




40.960% 
Note that the probabilities in each column sum to 100%. You can calculate the expected value of your actively managed portfolio relative to an indexed portfolio 1, 2, 3, and 4 periods from now by calculating the state contingent portfolio values by their probabilities; thus the expected values at each of these dates are:
Expected (relative) portfolio values 

t=1 
$0.9473 
t=2 
$0.8973 
t=3 
$0.8885 
t=4 
$0.8332 
In other words, by following an active management strategy, on average you can expect to be significantly worse off than you would be if you follow a passive strategy. Furthermore, your underperformance grows worse over time (note that the indexed strategy by definition produces an expected value of $1.00 at each of these future dates).
Here is a copy of the spreadsheet that I used in order to perform these calculations. The reader can perform further sensitivity testing by changing the values of p, u, and d.
Endnotes
[1] By beating the market, I mean that one earns excess returns on an aftertransaction cost, riskadjusted basis by following an active management strategy rather than a passive (indexed) strategy. For example, suppose you invest in an actively managed technology fund which has transactions costs of 200 basis points per year and a portfolio beta of 2. If such a portfolio earns 12% when the market returns 8%, then this is not beating the market; e.g., if the riskless rate of interest is 5%, then the after transactions cost, risk adjusted return on this portfolio is 12% (gross portfolio return) – 2% (transactions costs) – 6% (risk premium = b(E(r_{m})r_{f}) = 2(85)) = 4%, whereas the passive (indexed) strategy returns ~ 5% on a risk adjusted, after transactions cost basis.
[2] For simplicity, assume that ud = 1; i.e., if you outperform (underperform) the market during the course of the next time period and underperform (outperform) the market during the subsequent time period, this means that your performance over two time periods is in line with the market.