Markowitz portfolio theory, CAPM, Arbitrary Pricing Theory
Markowitz portfolio theory
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
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.
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)
- 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.