Markowitz portfolio theory, CAPM, Arbitrary Pricing Theory

# Markowitz portfolio theory

Simple return:

Continuous return:

Markowitz (1952) is generically known as the mean-variance framework. With assumptions:

- decision based on expected return and risk, as measured by the mean and variance of return
- maximize mean return, minimize the standard deviation of the return

- all investors have same time horizon
- all investors agree on the means, variances, and correlations of returns on various investments
- all information is freely and simultaneously available to all market participants

- assets are arbitrarily fungible (mutually interchangable)

Portfolio with assets, each with return at time , which the mean and variance are respectively. Weight of asset in the portfolio is , and . The return, mean, and variance of the portfolio is denoted without subscript.

The mean of the portfolio is the linear combination:

the variance will be expressed as the weight vector and the covariance matrix :

Note that if we define the correlation matrix with , and diagonal matrix of std dev to be on diagonal and zero off-diagonal. Then and .

Diversification: If , i.e. whenever , and with identical weight , then . But if the assets are correlated, this variance approaches the average covariance over all pairs of distinct asset as .

## Effect of correlation

Consider a simplified example of a portfolio with two assets, then

The above describes in parametric form of . We can optimize for minimum , for example, which the minimum is attained at

If we take the correlation between the two asset to some extreme values:

- , the asset are uncorrelated, then
- , then
- , then

## Efficient frontier

In case we have more assets can potentially included in the portfolio, there is certain combinations such that they maximize the return for the fixed . These set of portfolio is the efficient frontier.

The investor should have a utility function that depends on and he should choose the portfolio in the frontier that maximized the utility, i.e.:

If short sales are allowed, we require only but not . Risk free lending/borrowing is modeled by an asset with and (constant), which lending/borrowing is long/short the asset.

# CAPM

Markowitz theory needs parameters. Sharpe (1964) provided a simplified model:

where is the index for a single security, are the return of the market and the security respectively at time , is noise term — a random variable of mean zero. The parameters can be obtained by regression using EWMA of across history:

Also, we can have:

Thus we have a simplier way to derive the covariance matrix of these assets:

If we consider a portfolio with weights for the assets, due to the linearity of expectation, we have

so if we have even weights , then

This gives the non-market risk (aka unsystematic risk, unique risk, or residual risk) and is the market risk (aka undiversifiable risk) of equity . The coefficient is the measure of the contribution of equity to the risk of the portfolio.

# APT (Arbitrary Pricing Theory)

Axioms:

- Investors are risk-averse and seek to maximise their wealth
- There is a risk-free rate for lending and borrowing
- No market friction, no arbitrage opportunities

The model describes the return of asset at time in terms of multiple indices:

where is the value of index at , is the alpha of asset , and is the beta (sensitivity) of asset to the index . The indices used can be unanticipated chyanges in inflation, bond spreads, slope and level of interest rates, or industry production.

Considering a portfolio with weights,

where are weighted sum of respectively.