While browsing around on the web today I encountered an article from early 2009 that argued that most/many claims that are made in favor of fundamentally weighted indices are flawed. Before continuing reading this post, I would suggest reading the article. It’s just three pages in a large font, and I don’t want to repeat the complete argument :).

Intuitively the argument for fundamentally weighted indices is compelling. If the market is not efficient then stocks that are overvalued have a bigger market cap than they should have and stocks that are undervalued have a smaller market cap than they should have. If you now buy this market you must by definition underperform compared to the case where every security is fairly valued, right? Or perhaps wrong? Can you argue with the example below?

Consider a two-company world. Company A has a fair value of $10 billion with a market value of $9 billion, and Company B has a fair value of $5 billion with a market value of $6 billion. If we have a $150,000 market-capweighted portfolio, it will have $90,000 in Company A, the undervalued company, and $60,000 in Company B, the overvalued company. It will not have most of its money in companies that are above fair value—it will have most of its money in the company that is below fair value.

He is absolutely right that you have more money in the undervalued company in this example than in the overvalued company! But what is important to note that the undervalued company is less undervalued than the overvalued company is overvalued. The result is that the total market value of both companies remains equal and, as a result, both market efficiencies offset each other. If you would buy the complete market it would cost you $15 billion in both cases and in both cases you get company A and company B. Doesn’t matter if you overpay $1 billion for company B and underpay $1 billion for company A. The net result is exactly the same!

So when we add as a constraint that **if the market as a whole is fairly valued** the conclusion that the author draws is inescapable: **there is no hidden performance drag by buying a market cap weighted portfolio**. If you don’t overpay for the market as a whole it doesn’t matter that for some individual securities you overpay and underpay for others.

But this is of course only true if the market as a whole is correctly valued. Whether or not this is true depends on how market inefficiencies manifest themselves. Let’s take a look at a market with two companies, but to make things even simpler both companies have a $10 billion fair value. Let’s say that on a single day there is a 50/50 probability that a company drifts 1% away from fair value, either up or down. After one day the market is expected to look as following:

Fair value | Move | Market cap | |
---|---|---|---|

Company A | $10 billion | +1% | $10.1 billion |

Company B | $10 billion | -1% | $9.9 billion |

Total | $20 billion | $20 billion |

So it seems like everything is still well for the argument that market inefficiencies – represented by random movements away from fair value – don’t make a market cap weighted portfolio inferior compared to buying the stocks at intrinsic value. If you would buy the whole market you still get the same two companies at the same price!

While this model appears to be reasonable there is one major flaw, and that is the fact that these returns introduce a bias when we start compounding across multiple periods. A plus 1% return followed by a minus 1% returns doesn’t net out to zero. If we would alternate a long series of +1% returns and -1% returns the market cap of both companies would converge to zero. That wouldn’t make sense! To remove the bias from the expected returns we need to combine random movements of plus 1% with minus 0.9901%:

- $10.0 billion * 1.01 = $10.1 billion
- $10.1 billion * 0.990099 = $10.0 billion

What we are basically saying that market prices are log-normally distributed around fair value. The log-normal distribution is asymmetrical, as it should be because asset prices can’t go below zero. I think that by now you can guess what the impact will be if we redo our table:

Fair value | Move | Market cap | |
---|---|---|---|

Company A | $10 billion | +1.00% | $10.1 billion |

Company B | $10 billion | -0.9901% | $9.90099 billion |

Total | $20 billion | $20.00099 billion |

Because of the small difference in the up- and down move we lose the result that the market as a whole remains fairly valued, and as a result, a **market-cap weighted approach will be at a disadvantage** compared to a fundamentally weighted approach. Buying these two companies at fair value is better than buying them at their combined market cap.

Is this a reasonable result? Small-caps stocks are known to outperform large-cap stocks. The traditional academic view is that the difference in performance is the result of differences in risk between small and large stocks, and to some extent that is probably true. But a part of the performance differential could also be caused by the fact that undervalued companies have a smaller market cap than overvalued companies when they have an identical intrinsic value.

So, to summarize: I think that the intuitive idea that a fundamentally weighted index should outperform a market cap weighted index is actually correct. Whether or not it works in practice depends on how inefficient the market is compared to the added trading costs that will be incurred by a fundamental weighting.

xertyYour article link isn’t public, maybe you can update that?

Alpha VulturePost authorThink the new link should work for you.

DavidRYou’re missing something here. You’ve built your model so that the market weighted index will generally be overvalued? Come on. Your result simply means your model is incorrect.

Alpha VulturePost authorIn hindsight, I think I should have made a different argument since it isn’t exactly compelling and necessary.

But I’m not sure if the idea is indeed not correct. If you start with an efficient market and add some noise, wouldn’t it make sense that valuations on average increase since there is an asymmetry between the possible size of upside and downside errors? You can have a stock that is 200% overvalued compared to fair value, but you can’t have a stock that is 200% undervalued since prices can’t drop below zero.

pietjeI might be at the risk of sounding stupid but I really don’t understand anything about this blog post. First of all, the example given is so bad that I can only assume it’s a joke (it probably is?) Sure, if the $9 billion company is undervalued and the $5 billion company is overvalued you’d do great by buying a market cap weighted index! Why are you including this example it in your post? Why does the relative valuation even matter in that case?

And why does that lead towards the statement that “if the market as a whole is fairly valued there is no hidden performance drag by buying a market cap weighted portfolio.” ? And what does the first part of that sentence even mean? Is the total market cap of all companies equal to the sum of all intrinsic values? Is the average company trading at 1x fair value? Is the median trading at 1x fair value?

And how is this even relevant? I think that this: “but a part of the performance differential could also be caused by the fact that undervalued companies have a smaller market cap than overvalued companies when they have an identical intrinsic value” is far more important. If you buy SPY you own more AMZN and FB than WMT and PM. The individual analysis of these securities is far more interesting than a pseudo EMH-analysis of their respective stock movements.

Again, I could be missing the point completely.

Alpha VulturePost authorIf we start with the an efficient market as a base case we can buy a market cap weighted index and everything is well. But if the market is not efficient buying a market cap weighted index could lead to underperformance, and one of the claims of proponents in favor of alternative weighting mechanisms is that a market cap weighted index always has a hidden performance drag because you buy more overvalued companies and less undervalued companies.

What that example shows is that this isn’t a mathematical truth. You can, in fact, buy more of the undervalued company and less of the overvalued company. And if the market as a whole is correctly valued (meaning the sum of all market caps is equal to the sum of all intrinsic values) you also don’t underperform compared to the efficient market case when you buy a market cap weighted index.

Of course, this doesn’t mean that a superior alternative is not possible when you introduce market inefficiencies. If you only buy the undervalued companies you obviously do better than buying everything. But it is about the degree of how bad a market cap weighted index is, and as long as the market as a whole is correctly valued you are not losing performance. There is not necessarily a hidden performance drag.

The big assumption here is that the market as a whole is correctly valued, and in the second part of my post I tried to show that if you introduce random errors in market prices you probably lose the property that the market as a whole is still correctly valued. But that is perhaps a bit sketchy, and certainly unnecessary since an equal weighted portfolio would outperform in the first market model where we have a symmetric 1% up and 1% down move. But you need to introduce some conditions for this to be true in the generic case. You certainly could easily create a market where an equal weighted index (or some other fundamentally weighted index) underperforms a market cap weighted index.

pietjeIf the market cap of the index is equal to the fair value of the index then sure, you can buy a market-cap weighted fund. But this specific definition of ‘fair value’ is so arbitrary that any conclusion you derive from it is useless. If UberPinterest is trading at $100b but worth $10b why should the rest of the market ‘make up’ for this mispricing? If the only other stock in this fictional universe is Berkshire, trading at $100b but worth $120b you’d do horribly buying a cap-weighted fund.

If your goal was to prove that cap-weighted indices do not _always_ underperform then sure, you made your point but it was quite obvious beforehand that you could construct a scenario in which cap-weighted indices do fine. (i.e. a random scenario in which all large caps are undervalued).

Again, I don’t see the point of this all. If the market is efficient then you can do whatever you want. And if the market is not efficient then you’re better off exploiting the inefficiencies instead of buying the index. Didn’t we know that already?

Alpha VulturePost authorSure, but it is interesting to think about how the market could be/should be/is structured and what the possible implications are for various strategies.

DavidRYou should look up how options are priced using a binomial model and the underlying math. That will answer your questions. If my memory is correct, and it’s been a while, you need to set a condition such that the forward price of each security is equal to the current price accruing at the risk-free rate of interest. I believe that is done by setting a drift component that offsets the tendency for prices to increase more than they decrease through a number of steps as you have shown in a lognormal world.

DavidRWhoops. I meant that the expected forward price of each stock should equal the current price accruing at the risk-free rate. I don’t think that your example meets that necessary condition.

Alpha VulturePost authorI guess that makes sense 🙂

DThere are a lot of thing on the original post don’t quite make sense to me. Asness, btw, has some of the best commentary on this subject.

That said, this reply on binomial option models is even more baffling. In a simple binomial model (I’m using CRR, with no dividends to get to the core of the matter), you classically use risk-free rate assumptions *purely* because you can hedge out stock price variations at each and every step via buying m Stock and selling (aka borrowing) B bonds, or doing the opposite trade and shorting m Stock and investing B in Bonds.

You can use a binomial model with an expected arithmetic rate of return for the stock that is different than the risk free rate (note: since vol is an input, you automatically posit a geometric rate for the stock given an arithmetic rate and a vol, in a binomial world), and engage in the above mentioned dynamic hedging at each node, and your options will have the same value. Assuming the stock grows at the risk free rate just makes the calculations simpler.

What on earth this has to do with the original post, I have no idea.

Alpha VulturePost authorI wanted to go for a lognormal model to model inefficiencies, but if that introduces an unbounded drift to the upside it’s probably not a good model. But neither is some complex model based on option theory: unlikely that inefficiencies are willing to adhere to complex mathematics just because we would like it to. They are called inefficiencies for a reason.

So what I was trying to do was just a dead-end…

DavidR“You can use a binomial model with an expected arithmetic rate of return for the stock that is different than the risk free rate (note: since vol is an input, you automatically posit a geometric rate for the stock given an arithmetic rate and a vol, in a binomial world), and engage in the above mentioned dynamic hedging at each node, and your options will have the same value. Assuming the stock grows at the risk free rate just makes the calculations simpler.”

How can this possibly be true? If it were, why wouldn’t everyone just set the rate at zero since that must be the simplest way to do the calculation?

Counterexample to your claim:

Let’s consider the cost of at the money options one year forward.

With r=0, the value of the call equals the value of the put. The synthetic forward created by buying a call and selling a put demonstrates this.

Now let’s consider what happens when r>0%. The value of the call and put are equal when the strike price is above the current stock price since the forward price of the stock is now above its current price.

If the value of the call and put with r>0% are equal at a strike above the current price, they will not be equal when the strike is at the current price.

Therefore the prices of the calls and puts with the strike price set at the current stock price must be different than they were when r=0.

The rate which you choose for r in your option model matters and impacts the value of the options.

DIt sounds like you need to go back and read the original CRR paper here. I didn’t say you can set the risk free rate at whatever you want. I did say that you can set the expected rate of return of the assets (in this case a stock) at essentially whatever you want, and engage in dynamic hedging at each step. This yields the correct option value.

The rest of your response is a non-sequitor, that at best confuses the expected risk free rate of return and the asset’s expected rate of return, which for simplicity is set = to the former. The point is you can set the asset’s expected rate of return to whatever you want; put differently you don’t need to worry about the asset’s expected rate of return. Why? Dynamic hedging via buying (selling) m Stock and borrowing (investing) B bonds.

It’s all in the original CRR paper. It is also the core idea of dynamic replication that option theory is primarily built on.

neel prinskyAside from a purely theoretical perspective, I have to agree with the sentiment expressed by the other comments.

Obviously fair value is always going to be the most useful weighting. But how on earth do you accurately determine fair value? If you could you would be the richest person in the world?

Alpha VulturePost authorWell, you don’t necessarily need to be able to determine what intrinsic value is in order to create a strategy that can profit. If the market is not efficient buying things that go down and selling things that go up should work if you assume that mean reversion occurs.