| Active vs Passive: Part II ... a continuation of Part I |
We're talking about the validity of arguments like: "The S&P beat 75 percent of all actively managed Funds".
>And you don't like that argument, right?
Okay, let's be clear.
If you wanted to invest in an "active" Fund and you made your choice by putting all the Fund names in a box and picking one at random, then
that argument is relevant.
>Because there'd be a 75% probability of picking a lousy fund?
Exactly!
But suppose that, in seeking an active Fund, you were interested in where all the money is going ... so you ask:
"All those active investors ... where are they putting their money?"
>So we're back to "asset-weighted", right?
Exactly!
So if you wanted to compare returns, you might want to weight the BIG Funds more heavily than the little Funds and ...
>Yeah, I get it, but what else is there?
Didn't you listen to my Bill Gates story. He puts all his money in ...
>Yeah, yeah ... in Part I. So?
So, there are many ways to generate a comparison, "equal-weighted", "asset-weighted" ... well, let us count the ways:
| Counting the Ways |
The average of a bunch of numbers, x1, x2, ... xn is (x1 + x2 +...+ xn)/n.
We'll use the notation:
(1/n)Σxj = (x1 + x2 +...+ xn)/n
That's "equal-weighted".
If we assign weights to these numbers, like w1, w2, ... wn, then the
"weighted" average is: (w1x1 + w2x2 +...+ wnxn) /
(w1 + w2 +...+ wn)
We'll use the notation:
Σwjxj / Σwj
>And if all the weights are the same ... then?
Then the second, "weighted" formula, it becomes the first formula.
In fact, if all the weights equal 1, then (w1 + w2 +...+ wn) = n.
Okay, if there are N Mutual Funds with returns r1, r2, ... rN, then the Average Fund Return is:
[AFR] (1/N)Σrj
If we weight the Fund returns according to the number of dollars invested in each, say d1, d2, ... dN,
then we can
calculate a Weighted Fund Return:
[WFR] Σdjrj / Σdj
This average would be something like the return on the average invested dollar, for those dollars invested in the set of Mutual Funds.
>Set of Funds?
Yes, whatever set you like: active, passive, large cap growth, balanced ... whatever.
>What about the investors?
Patience! I'm getting to that!
Suppose there are K Mutual Fund investors, with returns R1, R2, ... RK.
The Average Investor Return is then:
[AIR] (1/K)ΣRj
If we weight the investor returns according to how much money each invests, say the dollar amounts Dj,
then we'd have the Weighted Investor Return:
[WIR] ΣDjRj / ΣDj
This average would be something like the return on the average investor dollar.
That's enough of "the ways'. The question is, which should we use to make the "active" vs "passive" comparison?
>I give up!
Well, let's think about it. If we ...
>Wait! Which one of those ways involves: "The S&P beat 75 percent of all actively managed Funds".
That'd be comparing the S&P with something like AFR ... but with Very Special Weights, namely:
[VSW] (1/N)Σwj
where wj = 1 if the Fund return rj was less than the S&P return and wj = 0 if the Fund beat the S&P.
Then, if 75% of the Funds had returns less than the S&P, all those w's would be 1 so we'd just get their sum which'd be
>That'd be 0.75N, so dividing by N would give 0.75, or 75%, right?
Exactly!
>Is that really an "average"?
Sure. It's the ordinary, garden-variety average of those Very Special Weights.
>So which one would you use?
I don't invest in Mutual Funds, so ...
>Okay! So which one would you suggest that I use?
In order to answer that, you have to answer my question, namely:
Which of the following is most important to you?
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>Is there some neat, mathematical relationship between all those returns, like AFR and AIR and ...
Yes, I think it's E = m c2.
>Very funny. Anyway, I like #3, because I'm an average investor and I don't have that much money and ...
So you don't want investor dollars to count, where the high rollers like Bill Gates ... where they get more weight, right?
Playing with Formulas ... and other details 
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Okay, we've got K Investors who invest Dj dollars and get returns Rj
... with j going from 1 to K.
We also have N Mutual Funds with dj dollars invested in the jth Fund which gives a return of rj
... j goes from 1 to N.
The total number of dollars invested in this particular set of Mutual Funds can be written as either ΣDj or Σdj.
[1] ΣDj = Σdj
But the collection of Investor returns (assuming buy-and-hold investing) isn't different from the collection of Fund returns.
In fact, if we look carefully at the K investors, some (say n1) will be investing in Fund #1
(getting that Fund's return: r1),
some (say n2) will be investing in Fund #2
(getting that Fund's return: r2), etc.
The average return of Investors, namely (1/K)ΣRj, is the same as
(1/K)Σnjrj
[2] ΣRj = Σnjrj
Now it ain't easy to calculate AIR = (1/K)ΣRj, the Average Investor Return.
However, since K = Σnj, we now see that we can also write:
[2A] AIR = (1/K)Σnjrj = Σnjrj / Σnj ... where it may happen that n123 = 0 if no Investors invest in Fund #123
>I haven't the faintest idea what you're ...
Patience.
Consider the ratio of the Average Investor Return (which ain't an easy number to get)
to the Average Fund Return (which is much easier to get):
[3] AIR / AFR = (N/K) Σnjrj / Σrj
Do you recognize the ratio of sums?
>You kidding?
It involves the weighted average of the numbers nj, the weights being the returns associated with each Fund.
>Huh?
It doesn't matter. I just thought it was interesting.
Just remember: nj investors are investing in Fund #j and get a return of rj.
To make it easier, we rewrite [3] like so:
[r] AIR / AFR = [ Σ(nj/K)rj ] / [ (1/N)Σrj ]
Now (nj/K) is the fraction of the K investors who invest in Fund #j.
The denominator is just the Average Fund Return.
>zzzZZZ
You see? If a greater fraction of Investors happen to choose the Funds with the larger returns, the Average Investor Return will be greater than the
Average Fund Return. Who knows, maybe Investors do tend to choose the Funds which have greater returns. Then again, maybe lots of Investors
choose those lousy Funds and ...
>zzzZZZ
Anyway, you may want to play with a spreadsheet which looks like this:
Click on the picture to play with the spreadsheet
It's a game. You press F9, you get a hundred Funds and associated Returns (selected at random from a normal distribution) and a bunch of Investors investing in them (these numbers are also selected randomly, each time you press F9) .
It's easy to see that it's quite possible that (for example) the average "active" Fund did poorly with respect to some market benchmark, yet the average Investor did better, so ...
>zzzZZZ
My sentiments exactly. However, it'd be neat to get data on the returns of the average Investor and ...
>Huh? Can you get that data?
Only if I interviewed every "active" and "passive" Investor.
Remember: it's possible to invest in a Fund yet have returns that are quite different from that Fund.
See, for example, your Mutual Fund and You.
You see? I don't think the average Investor would stick $120K into a Fund at t = 0, but rather stick $12K per year for ten years.
That DCA-type of investing would be more "normal", I'd think.
For example, if we assume that Sam stuck $12K per year into the S&P500, from 1996 to 2004, he'd get an
annualized return of 3.2%, whereas the S&P500 had an annualized return of 7.5%.
On the other hand, for the ten year period from 1972, Sam (with his DCA investing, as above) would have got a return of 3.9% while the S&P return was just 1.7%. (I'm talkin' annualized returns here, with Sam's return generated via the XIRR function, in Excel.)
>Yeah, okay ... so what's the answer? Who's best?
Huh? You asking me ?
| Investor's DCA Returns vs S&P Returns vs Fund Returns vs ... what? |
Well, since we've introduced the topic, let's consider whether the "typical" DCA Investor would beat the S&P500 and ...
>Assuming he's investing in the S&P500, right?
Yes. We'll look at how the S&P500 performed over ten years, starting in 1955, then 1956, then 1957, etc.
(ending in 1994) and how
Sam's portfolio would do, had he invested $1K per month for ten years, starting Jan1, 1956, or Jan 1, 1957 ... and so on..
The two sets of 10-year, annualized returns look like this:
The bottom charts show the S&P500 Index, for comparison, so we ...
>So we can see when DCA does better.
Yes, although I'm not so interested in that.
What's interesting is that the "typical" Investor, like our Sam, would "do-the-DCA", so using Mutual Funds returns to compare "Active" vs "Passive"
doesn't really capture the returns of the typical Investor. Don't you agree?
>Indubitably!
That makes two of us 
>But, ten years ain't much ... is it?
Okay, let's consider twenty years, from Jan 1, 1984 to Jan 1, 2004.
Here's the scoop, where
(to approximate what a 'typical" Investor might do)
we consider monthly DCA investments in the S&P500:
Well, for this time period, John Bogle, considering the 200 largest Equity Funds, finds the average return was 9.9%. However, I'm not sure whether he took the average annual return of all 200 funds, or asset-weighted them or used annualized or ... whatever. >But I'd say that "passive" beat "active", right?
>And the DCA return is even worser!
>But it's smaller than the S&P annualized, right?
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>So did Bogle do "asset-weighted" or not?
You asking me ? Actually, it looks like he counted the dollars in and out of the 200 largest Equity Funds and found that the return was 3.3% less than the buy-and-hold return so he subtracting that 3.3% from the above 9.9% and says the average Investor got 6.6%. Anyway, I looked up the Fidelity Magellan Fund and found data from 1987 and discovered that, during the past 18 years or so (including reinvested dividends!), the "average" annual return was 11.5% for that huge ($54B) actively managed Magellan Fund and, during that time period, the "average" annual return of the S&P500 was 10.0% and their total growths over that period were about 700%% and 300% respectively and ... >And Magellan invests in large cap growth, like the S&P500?
>Apples and oranges! Apples and oranges!!
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>I think you should start interviewing all those active and passive Investors.
Yeah, very funny.
However, if Sally asked me for a recommendation I'd say "passive".
Why? Because it'd perform better?
No, because Sally will never complain about giving her bad advice.
She sees her fund go down, because "the market" is down.
T'ain't my fault, eh?
| a Sharpe Argument ... in favour of "passive" |
I had almost forgotten William Sharpe's argument (not unusual, for me!) ... until reminded by mickeyd on NFB.
We'll use the magic equations we got above ... with somewhat different interpretations. They look like this:
[WPR] Σdjrj / Σdj
the dollar-Weighted Return of all Passive Investors,
where rj is the return of market security j and dj is how much passive Investors invest in that security.
Note that j runs through ALL securities in "the market".
[WAR] ΣDjrj / ΣDj
the dollar-Weighted Return of all Active Investors
where rj is (again) the return of the jth market security and Dj is how much active Investors invest in that security.
Note that D123 = 0 if no active Investor invests in security #123!
Note that the dollar-weighted Market Return will include ALL Investors, both active and passive.
Let's write this like so:
[WMR] ΣAjrj / ΣAj
the dollar-Weighted Return of the Market.
where rj is the return of the jth market security and Aj is the total dollar amount invested in that security.
Note: Aj includes both active and passive investments in the jth security!
Let's repeat that:
Aj includes both active and passive investments in the jth security!
That means that Aj = dj + Dj
... the total dollar amount invested in the jth security, by active and passive investors
>I haven't the faintest idea of what you're ...
Patience. Wait for the final result.
We can then write the Market return as:
[!!] WMR = ΣAjrj / ΣAj
= Σ(dj + Dj)rj / ΣAj
= PΣdjrj / Σdj
+ QΣDjrj / ΣDj
= P*WPR + Q*WAR
where P = Σdj / ΣAj
and Q = ΣDj / ΣAj
give the percentages of the total dollar amounts invested in the jth market security, by active and passive Investors, respectively.
>Uh ... isn't that backwards? I mean ...
Sorry 'bout that. You're right. P = Σdj / ΣAj
is the fraction invested by passive Investors and Q the fraction invested by active Investors.
>So P + Q = 1, right?
Yes.
Aah, but now consider the "passive" Investors.
They invest in "the market", allocating their portfolios according to the market weights.
As Sharpe puts it:
if security X represents 3 per cent of the value of the securities in the market,
a passive investor's portfolio will have 3 per cent of its value invested in X.
The (dollar-weighted) return of "passive" Investors will then equal the (dollar-weighted) Market Return :
WPR = WMR.
Then, from [!!], we see that :
WMR = P*WPR + Q*WAR
= P*WMR + Q*WAR
Then (1 - P)WMR = Q*WAR
and since, as you've pointed out, P + Q = 1,
then voila! ... WMR = WAR.
>Then active Investors get the market return ... same as passive Investors?
Yes!
>Even after commissions and fees and management expense ratios and ?
Aye, there's the rub. The above stuff assumes NO expenses for either active or passive.
Now add in the extra fees paid by the active Investors (especially the VERY active)
and, as Sharpe points out, their after-expenses returns will not ... cannot equal those obtained by passive Investors.
That's Sharpe's Arithmetic of Active Management.
>Then the debate is over, right?
Hardly. Sharpe is talking about ALL investors, not just those investing in actively or passively managed Mutual Funds.
So he's including investing klutzes, like me, who's a
lousy investor ... unlike professional Fund managers
He's also giving a HUGE weight to Bill Gates who has $10N invested.
He's also limiting the discussion to the active dollar invested vs the passive dollar invested.
He doesn't consider the average return of passive and active Fund Investors.
For example, for "active" Fund Investors, their average return is:
[2A]: Σnjrj / Σnj
... where nj "active" Investors invest in the jth Fund which has a return of rj
>Huh?
Haven't I said that before? It's what I think Fund Investors would really like to know:
If the average return of passive Fund Investors is X%
and the average return of active Fund Investors is Y%,
then is X > Y or Y > X ?
>So interview all Investors!
I'm workin' on it ...
