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# The Portfolio Separation Theorem

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This is about a theorem in finance. I will put it below the fold.

For many years, I have tried to come up with a way to explain the portfolio separation theorem that does not involve matrix calculus. I think have one now.

The significance of the portfolio separation theorem is that it demonstrates why every investor ought to hold shares in what is called "the market portfolio," rather than hold stocks that are peculiar to that investor. It is the theorem that leads to the Capital Asset Pricing Model, the idea of passive investing, indexing, and measuring the risk of an individual stock by its correlation with the market portfolio, or beta.

A portfolio of stocks is a set of percentage weights on stocks. The weight on each stock has to fall between 0 and 100 percent, inclusive. Because there are many stocks, there are many possible portfolios.

The portfolio separation theorem says that the number of portfolios that are needed to produce an optimum allocation is equal to the number of characteristics that investors care about. In particular, if investors only care about expected return (mean) and safety (minimizing variance), then every investor's portfolio can be computed as a mix of only two dominant portfolios.

Here is an argument for this theorem. Suppose that we start with two portfolios. Portfolio 1 has an expected return of 5 units and a safety of 5 units. Portfolio 2 has an expected return of 7 units and a safety of 2 units. Next, suppose that we introduce a new portfolio, 3. There are three possibilities:

a) portfolio 3 is dominated by each of the other portfolios, or by some combination of those portfolios. For example, if portfolio 3 has a return of 6 units and safety of 3 units, it can be dominated by a portfolio that is 50 percent of portfolio 1 and 50 percent of portfolio 2. That 50-50 combinatino would have a return of 7 units and safety of 3.5 units. So we throw out portfolio 3.

b) portfolio 3, either by itself or in combination with portfolio 2, dominates portfolio 1. For example, if portfolio 3 has a return of 4 units and safety of 8 units, then a 50%-50% portfolio of 3 and 2 would have a return of 5.5 units and safety of 5 units, which dominates portfolio 1. In that case, throw out portfolio 1 and replace it with portfolio 3.

c) portfolio 3, either by itself or in combination with portfolio 1, dominates portfolio 2. For example, if portfolio 3 has 8 units of return and 1 unit of risk, then an 20% -80% combination of portfolio 1 and portfolio 3 would have 7.4 units of return (20 percent of 5 plus 80 percent of 8) and 1.8 units of risk (20 percent of 5 plus 80 percent of 1). In this case, throw out portfolio 2 and replace it with portfolio 3.

When we are finished with portfolio 3, we can move on to another portfolio. We can keep trying this until you have tried all possible portfolios. The point is that there will always be just two portfolios in the dominant combination. The fact that we always end up with two portfolios when investors care about two characteristics is the portfolio separation theorem. Every investor, regardless of how he or she weighs those two characteristics, should choose some combination of the two dominant portfolios.

Next, assume that there is a risk-free asset. That is, there is an asset such that it has the maximum safety possible, a safety of infinity as it were. In that case, one of the two dominant portfolios has to be a portfolio consisting of nothing but the risk-free asset. The other portfolio will consist of risky stocks, and we can call it the market portfolio. Every investor will choose a combination of the market portfolio and the riskless asset, with relatively risk-averse investors putting more of their wealth into the riskless asset and withlatively risk-tolerant investors putting more of their wealth into the market portfolio.

An implication that Fischer Black emphasized is that no one is rewarded for taking unsystematic risk. If you choose a portfolio other than the market portfolio, then there exists some combination of the riskless asset and the market portfolio that could earn a higher return at the same risk as the portfolio you chose.

Kevin Dick writes:

What has always bothered me about the PST is that it kind of assumes it's own invalidity. From a technical standpoint, it uses exogenous prices when they should be computed endogenously given that you want to make statements about globally optimal behavior in the market.

Intuitively, you need prices to do the portfolio computation. But to have prices, you need some people willing to trade in individual assets. But if everyone accepts the PST, nobody will make such trades. So there will not be any prices. So nobody will be able to compute what the market portfolio should look like.

Of course, graduate school was almost 20 years ago so maybe somebody has made some progress on breaking this impasse.

Brendan writes:

Professor,

I recently read "Fischer Black and the Revolutionary Idea of Finance" from yours and Tyler Cowen's endorsement. As a CFA candidate, it was the perfect combination of substance and style for me. I find econ much more interesting than finance, but this book made finance far less dry and background on the econ revolution of the 70's was great too.

Can you think of anything similar to Perry Mehrling's book? Thanks a bunch.

writes:

Kevin,
I don't think of the portfolio separation theorem as taking prices as given. Properly done, I think it solves for the right prices for assets. My argument does not allow one to see this solution.

Brendan,
There is nothing else in that class. Justin Fox has a nice book on the history of finance theory.

Kevin Driscoll writes:

I'm no economist, but I am a physicist and it sounds like what you are trying to show is that the dimension of the portfolio space is equal to the number of interests that an investor cares about.

In your example, 2 interests means that the dimension of the portfolio space is 2 and so should be spanned by 2 basis vectors. These basis vectors are complete.

So, it sounds like what you really need is to define some "inner-product" function in portfolio space. I think your informal argument does a pretty good job of showing that your 2 portfolios span the set and I think you introduce the riskless asset to show that they are linearly independent but I think you might need a better argument as to how some non-riskless portfolio is linearly independent from another market portfolio. I'm not at all sure that an inner-product exists in these kinds of applications, but that would convince me.

Kevin Dick writes:

Arnold, I'm not sure that's right. Admittedly, it's been a long time, but IIRC an investor's choice of assets to hold is based on (1) his risk tolerance, (2) the volatility of the market portfolio, and (3) the borrowing/lending rate. The first is obviously endogenous. The third can pretty obviously be computed as a general equilibrium from (1).

However, the volatility of the market portfolio is dependent on expectations of future price fluctuations of the market portfolio. But the price of the market portfolio is a function of the prices of its individual components. So the volatility of the market portfolio is a function of the expected price fluctuations of individual assets.

Somebody has to trade in those individual components to set those prices and develop those expectations, right? I don't recall Tobin, or anyone else, explaining how this happens.

writes:

@Arnold

When you split it into just risk and return,
what shape does it assume the risks take?

Would it be better to split into kurtosis risk, skew risk, expected value of risk, and return? Or am I misunderstanding?

writes:

Kevin Driscoll,
I think that the vector space intuition is probably right. But if you are really interested, you can google "portfolio separation theorem Robert Merton" to find some material (much of it gated) on the topic. To the extent that Merton differs from me, you can be quite certain that I am wrong. But I have my intuition from the lecture notes that Merton handed out in the late 1970's.

Kevin Dick,
same thing--consult Merton to get a reliable answer.

Doc,
I think that if there is more than one risk measurement, then that adds another factor, so that we could have another portfolio.

fundamentalist writes:

I don’t want to advertise my ignorance about finance too much, but it frustrates me for a couple of reasons. 1) It is wedded to statistical probability exclusively and a limited set of distribution frequencies which rarely match reality and 2) it is divorced from micro economics, just as mainstream macro is divorced from it.

Finance seems to be built on the mainstream macro econ assumption that asset prices are random events, like a roulette wheel. Some have higher standard deviation and others lower standard deviations, so you should mix them with say four parts low SD and one part high SD. That way, the low SD portfolio will dampen the waves of the high SD while the high SD will occasionally wound the low SD portfolio, but not kill it, while occasionally benefiting the low SD.

And that works well most of the time. The only time it sucks is during major market crashes like 2008, but then the low SD portfolio allows us to say “It could have been worse” if we weren’t diversified. And that seems to make people feel better. And all of that is in line with mainstream macro theory that such crashes are unpredictable random events.

But many finance managers don’t believe it completely, because when the stock market is high they start shifting the mix toward low SD, and after a major crash they overweight high SD equities.

But what if Hayek was right and crashes aren’t random events? It should be possible to time the market to avoid major crashes. And I think it is possible with Hayek’s detailed version of the Austrian Business Cycle. Then probability becomes less important and micro economics far more important in choosing investments.

Kevin Dick writes:

Ahh, I assumed you were talking about Tobin's original result.

I found this by Merton:

http://mit.dspace.org/bitstream/handle/1721.1/46832/analyticderivati00mert.pdf

It's a much more rigorous treatment. However, he still appears to take the mean, variance, and covariance matrix of the component assets as exogenous.

I paged through the entire paper and didn't find anything that looked like general equilibrium equations solving for the prices of the component assets. But he's quite a bit more adept at mathematical symbology than I so I could have missed it.

Now, I suppose you could simply assume that there were a small proportion of "active" traders setting the market prices. But this smells like the old partial versus general equilibrium issue. They could be vastly different.

writes:

This sounds like a fine introduction to the topic, as long as you assume that risks are linear, when they aren't. But cows aren't spherical, either.

(If the only two portfolios are stocks and bonds, the safest investment is not 100% bonds, but about 90% bonds, since bonds do indeed have volatility, and it tends to be the inverse of stocks'.)

writes:

Kevin,
Think of the securities as real investment projects, like fruit trees. The variance-covariance matrix tells you how the various fruit trees behave in terms of average yield, variance, and covariance.

The optimization problem solves for the relative shares of those trees in the market portfolio. If you take the trees as given (impossible to plant new ones), then the derivation solves for the prices of the trees. If instead you take the costs of planting a tree as given, then the costs determine the prices, and the derivation solves for the number of each type of tree to plant.

Walt French writes:

Arnold, I wonder: I've seen both Sharpe and Markowitz discuss what I'll call “non-Euclidean CAPM” where they show consequences of altering one of the postulates that they seemingly find necessary to reach the simple results you present.

Fr'instance, even in the 2-D mean/variance space, the inability of investors to short-sell can make the portfolios on the “market line” between cash and the market, sub-optimal for all investors. Ditto, different borrowing and lending rates, as for instance I face if I try to buy on margin. Consensus views are not expressed in prices.

Neither of these distinguished scholars is hostile to CAPM nor are they just trying to show how clever they are. It is left as an exercise for the presentees to figure out how to separate the baby and the bathwater, but in my circles, there is no question which is which.

I'd love to see CAPM mathematically with one or two fewer assumptions underneath it, but I suspect that these counters limn the challenge.

Abhay Abhyankar writes:

Ref: Kevin Driscoll and Inner products etc.

Hansen and Richard (1987) uses the idea of inner products and Hilbert space etc to generate an elegant version of classical MV theory and portfolio seperation without the algebra in the standard textbook treatment. This approach of Hansen and Richard is nicely summarized by John Cochrane in a draft Chapter to his text book "Asset Pricing" available on his web page titled:A Mean-Variance Benchmark for Intertemporal Portfolio Theory.

Tom Mullaney writes:

Good piece. I hope that you will follow-up with a discussion of whether or not the market portfolio includes bonds and what the "bliss point" might be if bonds are included and what it might be if they are not.

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