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.