Separation Theorem

The CAPM makes the following assumptions:

• Investors evalute portfolios by looking at the variance and expected returns over a one-period horizon.
• Investors, when given a choice between two otherwise identical portfolios, will choose the one with the higher expected return.
• Investors are risk-averse.
• Individual assets are infinitely divisible.
• There is a riskfree rate at which an investor may either lend or borrow money.
• Taxes and transaction costs are negligible.

Investors are considered to be a homogeneous bunch; have the same expectations, the same one-period horizon, the same riskfree rate and that information is freely and instantly available to all investors. This is an extreme case, but it allows the focus to change from how an individual should invest to what would happen to security prices if everyone invested in a similar manner. The first feature of the assumptions we examine is often referred to as the separation theorem, which states that:

Theorem 1   The optimal combination of risky assets for an investor can be determined without any knowledge of the investor's preferences toward risk and return.

The proof of which is pretty trivial since each person faces the same linear efficient set, where the investor will borrow or lend according to his/hers indifference curves, but the risky portion of each investor's portfolio (which we will denote by T, for tangency portfolio) will be the same. The linearity of the efficient set is because of the riskfree lending and borrowing introduced.