Modern Portfolio Theory (MPT)

Modern portfolio theory (MPT) is a framework for building investment portfolios that treats risk and return as properties of the whole portfolio, not of each holding taken alone. Its central insight is that combining assets whose prices do not move in step reduces overall risk without necessarily reducing expected return. An investor should therefore choose holdings for what they contribute to the portfolio and, for any target level of return, hold the combination with the lowest risk.

The theory was set out by Harry Markowitz in his 1952 paper Portfolio Selection. He shared the 1990 Nobel Memorial Prize in Economic Sciences with Merton Miller and William Sharpe. MPT gave a mathematical basis to diversification and is the foundation of asset allocation, of index investing and of the capital asset pricing model.

The expected return of a portfolio is the weighted average of the expected returns of its assets:

Risk, measured by variance, is not a simple average. For two assets it is:

• $w_i$ is the share of the portfolio invested in asset $i$.
• $E(R_i)$ is the expected return of asset $i$.
• $\sigma_i$ is the standard deviation of asset $i$'s returns.
• $\rho_{12}$ is the correlation between the returns of the two assets.

The last term carries the theory. Unless the two assets are perfectly correlated, the portfolio's standard deviation is less than the weighted average of the individual figures.

Suppose 60 percent is invested in shares with an expected return of 8 percent and a standard deviation of 20 percent, and 40 percent in bonds with an expected return of 4 percent and a standard deviation of 10 percent. The correlation between them is 0.2. The expected return is $0.6 \times 8 + 0.4 \times 4 = 6.4$ percent. The variance is $0.36 \times 0.04 + 0.16 \times 0.01 + 2 \times 0.6 \times 0.4 \times 0.2 \times 0.20 \times 0.10 = 0.01792$, and its square root gives a standard deviation of 13.4 percent. The weighted average of the two standard deviations is 16 percent. Diversification has removed 2.6 percentage points of risk at no cost in expected return.

Repeating the calculation for every possible set of weights produces a region of achievable combinations of risk and return. Its upper-left edge is the efficient frontier: the portfolios that offer the highest expected return for each level of risk. A portfolio below the frontier is inefficient, since another exists with more return for the same risk or less risk for the same return. Which efficient portfolio an investor should hold depends on risk aversion.

Adding an asset with a certain return, lending or borrowing at the risk-free rate, changes the picture. The best combinations then lie on a straight line from the risk-free rate that just touches the frontier. The point of contact, the tangency portfolio, is the risky portfolio with the highest Sharpe ratio. Every investor, in this model, holds the same risky portfolio and adjusts risk only by mixing it with cash or borrowing. This result, known as the separation theorem, leads directly to the capital asset pricing model, in which the tangency portfolio is the whole market.

As more assets are added to a portfolio, the risks peculiar to individual companies offset one another and fade. This diversifiable part is unsystematic risk. What remains is the risk common to all assets, systematic risk, which diversification cannot remove. The theory implies that investors are rewarded only for bearing the second kind, since the first can be eliminated at no cost.

• Inputs. The method needs expected returns, variances and correlations for every asset. These must be estimated, and small changes in the estimates produce large changes in the recommended weights. The practical side of this problem is treated under portfolio optimization.
• Variance as risk. Variance penalises gains above the average as much as losses below it, and it describes risk fully only if returns are normally distributed. Actual returns have fat tails and are skewed.
• Unstable correlations. Correlations between risky assets tend to rise in a crisis, so diversification is weakest when it is most needed.
• Behaviour. Research in behavioral finance shows that investors do not choose as the theory assumes. They weigh losses more heavily than gains and treat separate pots of money differently.

Later approaches keep the framework and change parts of it. Some replace variance with downside measures. Others avoid return forecasts altogether and allocate by risk alone, as in risk parity and hierarchical risk parity. Despite its limits, the idea that an asset should be judged by its effect on the whole portfolio remains the starting point for professional investment management.

• The Nobel Prize: Prize in Economic Sciences 1990