 # What is Modern Portfolio Theory?

The Modern Portfolio Theory (MPT) of Harry Markowitz (1952) introduced the analysis of portfolios of investments by considering the expected return and risk of individual assets and, crucially, their interrelationship as measured by correlation. Prior to this investors would examine investments individually, build up portfolios of favored stocks, and not consider how they related to each other. In MPT diversification plays an important role.

Example

Should you put all your money in a stock that has low risk but also low expected return, or one with high expected return but which is far riskier? Or perhaps divide your money between the two. Modern Portfolio Theory addresses this question and provides a framework for quantifying and understanding risk and return.

In MPT the return on individual assets are represented by normal distributions with certain mean and standard deviation over a specified period. So one asset might have an annualized expected return of 5% and an annualized standard deviation (volatility) of 15%. Another might have an expected return of —2% and a volatility of 10%. Before Markowitz, one would only have invested in the first stock, or perhaps sold the second stock short. Markowitz showed how it might be possible to better both of these simplistic portfolios by taking into account the correlation between the returns on these stocks.

In the MPT world of N assets there are 2N + N(N — 1)/2 parameters: expected return, one per stock; standard deviation, one per stock; correlations, between any two stocks (choose two from N without replacement, order unimportant). To Markowitz all investments and all portfolios should be compared and contrasted via a plot of expected return versus risk, as measured by standard deviation. If we write xa to represent the expected return from investment or portfolio A (and similarly for B, C, etc.) and ob for its standard deviation then investment/portfolio A is at least as good as B if The mathematics of risk and return is very simple. Consider a portfolio, n,of N assets, with W, being the fraction of wealth invested in the ith asset. The expected return is then and the standard deviation of the return, the risk, is where pij is the correlation between the ith and jth investments, with pii = 1.

Markowitz showed how to optimize a portfolio by finding the W's giving the portfolio the greatest expected return for a prescribed level of risk. The curve in the risk-return space with the largest expected return for each level of risk is called the efficient frontier.

According to the theory, no one should hold portfolios that are not on the efficient frontier. Furthermore, if you introduce a risk-free investment into the universe of assets, the efficient frontier becomes the tangential line shown in Figure 2.3. This line is called the Capital Market Line and the portfolio at the point at which it is tangential is called the Market Portfolio. Now, again according to the theory, no one ought to hold any portfolio of assets other than the risk-free investment and the Market Portfolio. Figure 2.3: Reward versus risk, a selection of risky assets and the efficient frontier (bold green).

Harry Markowitz, together with Merton Miller and William Sharpe, was awarded the Nobel Prize for Economic Science in 1990.

Ingersoll, JE Jr 1987 Theory of Financial Decision Making.Rowman& Littlefield

Markowitz, HM 1952 Portfolio selection. Journal of Finance 7 (1) 77-91

# What is the Capital Asset Pricing Model?

The Capital Asset Pricing Model (CAPM) relates the returns on individual assets or entire portfolios to the return on the market as a whole. It introduces the concepts of specific risk and systematic risk. Specific risk is unique to an individual asset, systematic risk is that associated with the market. In CAPM investors are compensated for taking systematic risk but not for taking specific risk. This is because specific risk can be diversified away by holding many different assets.

Example

A stock has an expected return of 15% and a volatility of 20%. But how much of that risk and return are related to the market as a whole? The less that can be attributed to the behaviour of the market, the better will that stock be for diversification purposes.

CAPM simultaneously simplified Markowitz's Modern Portfolio Theory (MPT), made it more practical and introduced the idea of specific and systematic risk. Whereas MPT has arbitrary correlation between all investments, CAPM, in its basic form, only links investments via the market as a whole. CAPM is an example of an equilibrium model, as opposed to a no-arbitrage model such as Black-Scholes.

The mathematics of CAPM is very simple. We relate the random return on the ith investment, Ri, to the random return on the market as a whole (or some representative index), Rm by The ei is random with zero mean and standard deviation ei, and uncorrelated with the market return Rm and the other €j. There are three parameters associated with each asset, ai, f\$i and ei. In this representation we can see that the return on an asset can be decomposed into three parts: a constant drift; a random part common with the index; a random part uncorrelated with the index, q. The random part q is unique to the ith asset. Notice how all the assets are related to the index but are otherwise completely uncorrelated.

Let us denote the expected return on the index by xm and its standard deviation by om. The expected return on the ith asset is then and the standard deviation If we have a portfolio of such assets then the return is given

by From this it follows that Writing we have Similarly the risk in n is measured by Note that if the weights are all about the same, N , then the final terms inside the square root are also 0(N-1). Thus this expression is, to leading order as N -—oo,

Observe that the contribution from the uncorrelated es to the portfolio vanishes as we increase the number of assets in the portfolio; this is the risk associated with the diversifiable risk. The remaining risk, which is correlated with the index, is the undiversifiable systematic risk.

Multi-index versions of CAPM can be constructed. Each index being representative of some important financial or economic variable.

The parameters alpha and beta are also commonly referred to in the hedge-fund world. Performance reports for trading strategies will often quote the alpha and beta of the strategy. A good strategy will have a high, positive alpha with a beta close to zero. With beta being small you would expect performance to be unrelated to the market as a whole and with large, positive alpha you would expect good returns whichever way the market was moving. Small beta also means that a strategy should be a valuable addition to a portfolio because of its beneficial diversification. Sharpe shared the 1990 Nobel Prize in Economics with Harry Markowitz and Merton Miller. 