# Modern Portfolio Theory

The modern portfolio theory (MPT) is a theoretical framework for constructing and analyzing a portfolio.

The ideas of diversification and risk–return trade-off were long known before the development of the Modern Portfolio Theory.

But these ideas leave several important questions unanswered. How should one measure the risk of an asset? What should be the quantitative trade-off between risk (properly measured) and expected return? One would think that risk would have something to do with the volatility of an asset’s returns, but this guess turns out to be only partly correct. When we mix assets into diversified portfolios, we need to consider the interplay among assets and the effect of diversification on the risk of the entire portfolio. Diversification means that many assets are held in the portfolio so that the exposure to any particular asset is limited. The effect of diversification on portfolio risk, the implications for the proper measurement of risk, and the risk–return relationship are the topics which have come to be known as modern portfolio theory.

The central idea of MPT is that financial assets should not be viewed in isolation, but each asset should be viewed as part of a portfolio and how in contributes to the portfolio risk and return.

## A Brief History

Investing is about making choices of how to allocate wealth among different assets. Before the development of the MPT, investors analyzed each investment option individually. If one believes a company is going to perform well, he buys the stock. The concepts of diversification (e.g. Don’t put all your eggs into one basket) and risk–return trade-off (e.g. “You have to take more risks if you want more reward”) existed, but were not formalized.

In 1952, Harry Markowitz delineated what is now known as the modern portfolio theory. The most significant insight is that assets should not be selected based on their individual characteristics, but on their contribution to the entire portfolio (in terms of expected return and risk). This way, one can construct a portfolio with the same expected return but less risk compared to a portfolio constructed the old way.

The development of this theory brought two of its pioneers, Harry Markowitz and William Sharpe, Nobel Prizes.

## How Does It Work

The MPT suggests that one can construct a portfolio of lower risk, while keeping the same return. How does it work exactly?

Under the MPT framework, investors care only about two things: expected return and risks, measured by variance. For a portfolio, the expected return of the portfolio is simply the weighted average of returns of each component. However, risk, measured by variances, is not a simple weighted average of component variances.

Because asset prices tend to move together, co-variances actually reduce the overall variances.

Asset prices have a tendency to move together. If you take into account these interactions, under certain assumptions, one can construct a portfolio of lower risk, while keeping the same return. People can do that because portfolio return is a simple weighted average of component returns. However,

It is important to keep in mind of the restrictions of this framework. Under the MPT, risk is measured by variances, extrapolated from historical data.

## Assumptions

Under this framework, investors care about two things in a portfolio: expected return and risk. Investors like expected return and dislike risk. This is not true for all investors. For example, socially responsible investing (SRI) is a strategy that seeks social or environmental benefits in addition to financial rewards. (WSJ: BlackRock Plans to Block Walmart, Dick’s From Some Funds Over Guns) Therefore, MPT is not suitable for social investors.

Risk and expected return. MPT assumes that investors are risk averse, meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns.

Risk is measured about standard deviation.

Modern financial theory rests on two assumptions: (1) securities markets are very competitive and efficient (that is, relevant information about the companies is quickly and universally distributed and absorbed); (2) these markets are dominated by rational, risk-averse investors, who seek to maximize satisfaction from returns on their investments.

The first assumption presumes a financial market populated by highly sophisticated, well-informed buyers and sellers. The second assumption describes investors who care about wealth and prefer more to less. In addition, the hypothetical investors of modern financial theory demand a premium in the form of higher expected returns for the risks they assume.

## Mean-Variance Analysis Variance Analysis

For individual investment options:

• The expected return, or the mean return, is denoted by $\mu_i$
• The variance, the measure of risk, is denoted by $Var[R_i] = E[(R_i-u_i)^2] = \sigma_i^2$
• The standard deviation is simply the square room of the variance.

For a portfolio:

• The expected return is the simple weighted average of individual returns $\mu_i$
• The variance, the measure of risk, is denoted by $Var[R_i] = E[(R_i-u_i)^2] = \sigma_i^2$
• The standard deviation is simply the square room of the variance.