Modern portfolio theory

Modern portfolio theory (MPT) is a framework for constructing and analyzing investment portfolios based on the relationship between risk and expected return. It was developed by Harry Markowitz and formalized in his 1952 paper "Portfolio Selection".

MPT emphasizes that investors should evaluate the risk and return of a portfolio as a whole rather than considering the risk and return of individual securities independently.

A simplified representation is:

$Portfolio Selection \to Expected Return + Portfolio Risk$

The theory shows that combining assets with imperfectly correlated returns can potentially reduce portfolio risk without proportionally reducing expected return.

Portfolio return

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

For a portfolio containing $n$ assets:

$E (R_{p}) = \sum_{i = 1}^{n} w_{i} E (R_{i})$

where:

$E (R_{p})$ = expected portfolio return;
$w_{i}$ = portfolio weight of asset $i$ ;
$E (R_{i})$ = expected return of asset $i$ ;
$n$ = number of assets.

The portfolio weights generally satisfy:

$\sum_{i = 1}^{n} w_{i} = 1$

For a fully invested portfolio without leverage, each weight is typically non-negative and the weights sum to 1.

Portfolio variance

Portfolio risk in MPT is commonly measured using the variance or standard deviation of portfolio returns.

For a portfolio containing multiple assets, portfolio variance is:

$σ_{p}^{2} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i} w_{j} Cov (R_{i}, R_{j})$

where:

$σ_{p}^{2}$ = portfolio variance;
$w_{i}$ and $w_{j}$ = portfolio weights;
$Cov (R_{i}, R_{j})$ = covariance between the returns of assets $i$ and $j$ .

Portfolio standard deviation is:

$σ_{p} = \sqrt{σ_{p}^{2}}$

Unlike the expected return calculation, portfolio risk depends not only on the individual risks of the assets but also on their relationships with one another.

Covariance and correlation

The covariance between two assets measures the extent to which their returns vary together.

Covariance can be related to correlation through:

$Cov (R_{i}, R_{j}) = ρ_{i j} σ_{i} σ_{j}$

where:

$ρ_{i j}$ = correlation coefficient between assets $i$ and $j$ ;
$σ_{i}$ = standard deviation of asset $i$ ;
$σ_{j}$ = standard deviation of asset $j$ .

Substituting this relationship into the portfolio variance equation gives:

$σ_{p}^{2} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i} w_{j} ρ_{i j} σ_{i} σ_{j}$

This relationship is central to the diversification principle of MPT.

Two-asset portfolio

For a portfolio containing two assets, portfolio variance can be expressed as:

$σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ_{12} σ_{1} σ_{2}$

where:

$w_{1}$ and $w_{2}$ = portfolio weights;
$σ_{1}$ and $σ_{2}$ = standard deviations of the two assets;
$ρ_{12}$ = correlation between their returns.

If the assets are not perfectly positively correlated, diversification can reduce portfolio volatility relative to a portfolio consisting entirely of one of the assets.

Diversification

Diversification is a central principle of modern portfolio theory.

A simplified relationship is:

$Imperfect Correlation \to Diversification \to Potentially Lower Portfolio Risk$

When assets have correlations below +1, changes in the value of one asset can be partially offset by changes in other assets.

If two assets have a correlation of -1, it may be possible under certain assumptions to construct a combination with zero variance.

If two assets have a correlation of +1, diversification does not reduce portfolio risk through correlation effects.

Efficient frontier

The Efficient frontier is the set of portfolios that offer the highest expected return for a given level of portfolio risk, or the lowest risk for a given expected return.

A simplified representation is:

$Efficient Frontier = {Portfolios with Maximum Expected Return for Each Risk Level}$

Portfolios below the efficient frontier are generally considered inefficient because another portfolio may offer a higher expected return for the same level of risk or lower risk for the same expected return.

Minimum-variance portfolio

The Minimum variance portfolio is the portfolio on the feasible set with the lowest possible variance under the specified investment constraints.

A simplified optimization problem is:

$min_{𝐰} σ_{p}^{2}$

subject to:

$\sum_{i = 1}^{n} w_{i} = 1$

and any additional portfolio constraints.

The minimum-variance portfolio does not necessarily provide the highest expected return.

Optimal portfolio

The preferred portfolio depends on an investor's risk tolerance and objectives.

A simplified relationship is:

$Optimal Portfolio = f (Expected Return, Risk Tolerance, Constraints)$

An investor with a greater tolerance for risk may choose a portfolio with higher expected return and higher volatility, while a more risk-averse investor may prefer a portfolio with lower expected volatility.

Risk-free asset and capital allocation

When a risk-free asset is introduced into the portfolio framework, investors can combine the risk-free asset with a portfolio of risky assets.

A simplified portfolio return is:

$E (R_{p}) = w_{f} R_{f} + (1 - w_{f}) E (R_{m})$

where:

$w_{f}$ = weight invested in the risk-free asset;
$R_{f}$ = risk-free rate;
$E (R_{m})$ = expected return of the selected risky portfolio.

Depending on the investor's preferences, the investor can allocate more or less capital to the risky portfolio.

Capital market line

The Capital market line (CML) represents the risk-return combinations available from combining a risk-free asset with the market portfolio under the assumptions of the Capital asset pricing model.

Its equation is:

$E (R_{p}) = R_{f} + \frac{E (R_{m}) - R_{f}}{σ_{m}} σ_{p}$

where:

$E (R_{p})$ = expected portfolio return;
$R_{f}$ = risk-free rate;
$E (R_{m})$ = expected return of the market portfolio;
$σ_{m}$ = standard deviation of the market portfolio;
$σ_{p}$ = standard deviation of the portfolio.

The slope of the CML is the Sharpe ratio of the market portfolio:

$CML Slope = \frac{E (R_{m}) - R_{f}}{σ_{m}}$

MPT and the Sharpe ratio

The Sharpe ratio measures excess return relative to total portfolio volatility.

A simplified expression is:

$Sharpe Ratio = \frac{E (R_{p}) - R_{f}}{σ_{p}}$

MPT and the Sharpe ratio are therefore closely related when evaluating portfolios on a risk-adjusted basis.

A higher Sharpe ratio indicates greater expected excess return per unit of total risk, subject to the assumptions and estimates used.

Portfolio optimization

Portfolio optimization involves selecting portfolio weights to achieve a specified investment objective.

A simplified mean-variance optimization problem can be expressed as:

$max_{𝐰} [E (R_{p}) - λ σ_{p}^{2}]$

where:

$𝐰$ = vector of portfolio weights;
$λ$ = parameter representing the investor's aversion to variance;
$E (R_{p})$ = expected portfolio return;
$σ_{p}^{2}$ = portfolio variance.

Alternatively, an investor may minimize portfolio variance subject to a required expected return:

$min_{𝐰} σ_{p}^{2}$

subject to:

$E (R_{p}) = R^{*}$

and:

$\sum_{i = 1}^{n} w_{i} = 1$

where $R^{*}$ represents the target expected return.

Assumptions

The traditional MPT framework relies on several simplifying assumptions, including:

investors consider expected return and risk when evaluating portfolios;
portfolio risk can be represented by variance or standard deviation;
investors have information about expected returns, variances, and covariances;
investment decisions can be evaluated over a specified horizon;
asset returns can be represented using statistical distributions or historical estimates;
investors can diversify across available assets.

Actual financial markets may differ substantially from these assumptions.

Limitations

MPT has several limitations.

Expected returns, volatility, and correlations must generally be estimated, and these estimates can be uncertain.

Correlations between assets can also change over time, particularly during periods of financial stress.

Other limitations include:

reliance on historical or estimated parameters;
sensitivity to estimation errors;
use of variance as a measure of risk;
difficulty in forecasting expected returns;
changing correlations;
transaction costs and taxes;
liquidity constraints;
leverage restrictions;
non-normal return distributions;
behavioral factors that may not be captured by the model.

Consequently, an optimized portfolio based on historical inputs may not remain optimal when market conditions change.

MPT and asset allocation

MPT provides a framework for allocating capital across different assets based on their expected returns, volatility, and correlations.

A simplified allocation process is:

$Expected Returns + Risk Estimates + Correlation Estimates \to Portfolio Optimization \to Asset Allocation$

The resulting allocation depends on the investor's objectives and the constraints imposed on the portfolio.

Relationship with other financial theories

MPT forms part of a broader set of theories concerning asset pricing and portfolio management.

It is closely related to:

The Capital asset pricing model extends portfolio theory by relating expected asset returns to systematic risk.

A simplified CAPM relationship is:

$E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]$

where $β_{i}$ measures the sensitivity of asset $i$ to market returns.

Modern portfolio theory

Contents

Portfolio return

Portfolio variance

Covariance and correlation

Two-asset portfolio

Diversification

Efficient frontier

Minimum-variance portfolio

Optimal portfolio

Risk-free asset and capital allocation

Capital market line

MPT and the Sharpe ratio

Portfolio optimization

Assumptions

Limitations

MPT and asset allocation

Relationship with other financial theories

See also