QM 8 The Return and Risk of a Financial Portfolio

A portfolio return is a weighted average of the returns on the assets it holds, where each weight is the fraction of portfolio value invested in that asset. Figures like a portfolio expected return, its standard deviation, and the correlations among its holdings can be built from historical data, which look backward at realized outcomes, or estimated in a forward-looking way from economic and financial forecasts. Historical figures help investors set realistic expectations, yet they can mislead when the sample is short or when the future no longer resembles the past.

The expected return of a portfolio is the sum of each asset expected return multiplied by its weight.

Arithmetic and geometric averages

Over several periods the arithmetic average simply adds each period return and divides by the number of periods, ignoring compounding. The geometric average instead compounds the period returns and then takes the root, so it captures the growth actually realized. For a two-asset portfolio held at fixed weights of 50 percent each, where Asset A returns 10 percent then negative 10 percent and Asset B returns negative 5 percent then 20 percent, the period returns are 2.5 percent and 5.0 percent. The arithmetic average is 3.750 percent, while the geometric average is slightly lower at 3.742 percent, the usual gap between the two measures.

Portfolio drift

When weights are held fixed only on paper, the actual weights shift as some assets outperform others. This portfolio drift moves value toward the winners without any deliberate trade. Left unchecked it can cause style drift, where the portfolio wanders away from its intended strategy, so periodic rebalancing keeps the mix aligned with the original plan. In an equally weighted portfolio, drift tilts performance toward the high-return assets; in an unequally weighted portfolio, the largest allocations dominate the outcome.

Variance and standard deviation measure how far returns spread from their mean, and they are the standard gauges of portfolio risk. Portfolio variance depends not only on the individual asset variances and weights but also on how the assets move together, captured by their covariances.

Because standard deviation is the square root of variance and shares the units of the returns themselves, it is the more intuitive risk figure. Covariance can be awkward to interpret on its own, so it is often rewritten through the correlation coefficient, which rescales it to lie between negative one and positive one.

The two-asset case

For a portfolio of just two assets, the double summation expands into three terms: one variance term for each asset plus a single cross term that carries the covariance.

The sign of the cross term is what makes diversification work. When the correlation is positive the term adds to risk, but when it is low, zero, or negative, the term is small or subtracts, pulling portfolio risk below the weighted average of the parts. One caution: if a problem hands you the covariance directly, use it in place of the product of correlation and the two standard deviations; do not multiply the covariance again by the standard deviations.

Moving from two assets to many raises the workload sharply. The variances and covariances are collected in a variance-covariance matrix, a square grid holding each asset variance on the diagonal and the covariance of every pair off the diagonal. The matrix is symmetric, since the covariance of asset i with asset j equals the covariance of asset j with asset i.

The number of distinct covariances grows far faster than the number of assets. For N assets there are N variances and a much larger count of unique covariances.

With 7 indices that is (49 minus 7) divided by 2, or 21 unique covariances alongside 7 variances. For a 100-security portfolio the count jumps to 4,950 unique covariances, which is why large-scale optimization leans on matrix notation and computers. A closely related grid, the correlation matrix, rescales every covariance to the range from negative one to positive one, with ones on the diagonal, and it makes the diversification structure easy to read: assets with low or negative correlations are the ones that reduce overall risk.

As a portfolio adds assets, the influence of any single asset variance on total portfolio risk shrinks, and the average covariance among the holdings takes over as the main driver of risk. If every asset shares the same variance and the average correlation across pairs is a single number, portfolio variance simplifies to a clean two-part expression.

When N grows large the first term fades toward zero and the second term approaches the average covariance contribution, so portfolio variance settles near the product of the average correlation and the common variance.

How correlation shapes the benefit

The lower the average correlation, the greater the risk reduction from diversification, and a higher average correlation limits it. The table below tracks a two-asset portfolio, with Asset 1 at 7.0 percent return and 12.0 percent risk and Asset 2 at 15.0 percent return and 25.0 percent risk, as the correlation between them varies. Read across any row: the portfolio return is fixed by the weights, but the portfolio standard deviation falls as the correlation drops, and it falls most steeply when the correlation is negative one.

At a correlation of positive one the risk-return combinations trace a straight line; at negative one they form two straight lines that reach a minimum risk very close to zero.

The risk that diversification removes is the unsystematic risk, specific to individual assets, while the systematic risk inherent in the whole market cannot be diversified away. A common rule of thumb holds that holding roughly 20 to 30 assets captures most of the benefit on offer, especially when those assets have low correlations.

Portfolio optimization means finding the best mix of assets subject to constraints such as a required return or a maximum risk. The set of portfolios an investor can actually build from the available assets is the feasible set. Plotting many feasible portfolios in risk-return space fills a region, and its upper-left boundary is where the attractive portfolios live.

The Markowitz efficient frontier is that boundary: the set of portfolios that offer the highest expected return for each level of risk, or equivalently the lowest risk for each level of return. Portfolios above and to the left of the frontier cannot be constructed from the available assets, and portfolios below it are inefficient, because some frontier portfolio beats them at the same risk.

The minimum-variance portfolio

The leftmost point of the frontier is the global minimum-variance portfolio, the single combination with the lowest possible risk. For a two-asset portfolio, calculus gives closed-form weights that minimize variance, and these depend only on the two variances and their covariance, not on the returns.

The minimum-variance frontier adds a target-return constraint to the same problem, tracing the lowest risk available at each return level. Its upper segment, from the global minimum-variance portfolio and rising to the right, is the efficient frontier; the lower segment is inefficient and is ignored, because a higher-return portfolio always sits directly above it at the same risk.

Choosing among efficient portfolios depends on the investor. The theory here assumes investors are rational and risk averse, share a single-period horizon, pay no taxes or transaction costs, and can borrow and lend freely at a constant risk-free rate. Under these assumptions, a simple utility function ranks portfolios by trading expected return against risk.

Utility rises with expected return and falls with variance, and the penalty on risk grows with the coefficient A. A risk-averse investor has A greater than zero; a risk-neutral investor has A equal to zero and cares only about expected return; a risk-seeking investor has A below zero and actually gains utility from added risk. Risk tolerance is the mirror image of risk aversion, and it shifts with age, wealth, health, and the economic environment.

Indifference curves

An indifference curve joins the risk-return combinations that give the same utility. For a risk-averse investor these curves are convex and slope upward: as risk rises, ever larger increases in return are needed to hold utility steady. A steeper curve signals greater risk aversion. The investor maximizes utility at the point where the highest reachable indifference curve just touches the available opportunity set.

The capital allocation line

Introducing a risk-free asset expands the choices beyond the curved frontier of risky assets. Mixing the risk-free asset into a chosen risky portfolio traces a straight line in risk-return space, because a risk-free asset carries no standard deviation of its own and no covariance with the risky holding. That straight line is the capital allocation line.

The slope of this line is the Sharpe ratio, the excess return earned per unit of risk. A steeper line means a higher Sharpe ratio and a better risk-adjusted deal. A risk-averse investor moves left along the line, holding more of the risk-free asset; a risk-tolerant investor moves right, holding more of the risky portfolio and possibly borrowing to lever it. Every combination along a single line shares the same Sharpe ratio, so a leveraged position in the risky portfolio has the same risk-adjusted quality as an unleveraged one.

The capital market line

The capital market line is the special capital allocation line drawn to the market portfolio, the value-weighted combination of all risky assets. Because the market portfolio has the highest Sharpe ratio of all portfolios, the capital market line is the best line available.

The capital asset pricing model, or CAPM, is a straightforward equilibrium model. Since the market portfolio sits on the frontier of efficient portfolios, an asset expected return is governed solely by its systematic risk, measured through beta. The model assumes that investors share the same views, hold mean-variance efficient portfolios, act as price takers, and trade in a market where supply and demand are balanced.

Systematic risk and beta

Beta measures how sensitive an asset return is to movements in the market, and it is the only risk that earns a reward, because the asset-specific portion can be diversified away.

An asset with a beta above one is expected to swing more than the market and to earn a higher expected return; a beta below one means smaller swings and a lower expected return. Only systematic risk moves expected returns, because non-systematic risk disappears in a diversified portfolio.

The CAPM equation

The CAPM ties an asset expected return to the risk-free rate plus a reward for its systematic risk, where the reward is beta multiplied by the market risk premium.

The market risk premium is the added return that choosing the risky market portfolio, rather than the risk-free asset, is expected to earn, though it is never guaranteed. Beta scales that premium to each asset. In practice both beta and the market expected return must be estimated from data, since neither can be observed directly; beta is usually found by regressing an asset excess returns on the market excess returns over one to five years of weekly or monthly data.