11.1 Portfolio Return, Risk, and Diversification

Return, risk, and the diversification engine

A portfolio is a set of weighted exposures that interact, not a list of securities. Expected return is the easy part: multiply each asset's expected return by its weight and sum. With 60% at 8% and 40% at 4%, E(Rp) = 0.60(8%) + 0.40(4%) = 6.4%. Weights must sum to 1.0; a short position carries a negative weight.

Risk is the subtle part. A security's standard deviation measures stand-alone uncertainty, but a portfolio's standard deviation also depends on how holdings co-move. Covariance captures joint movement in squared units; correlation standardizes it to the range -1 to +1 via Corr(1,2) = Cov(1,2) / (sigma1 * sigma2).

The two-asset variance formula

This is the single most-tested formula in 11.1. For two risky assets:

Var(Rp) = w1^2 * sigma1^2 + w2^2 * sigma2^2 + 2 * w1 * w2 * Cov(1,2)

Because Cov(1,2) = Corr(1,2) * sigma1 * sigma2, the cross term shrinks as correlation falls. The diversification benefit is the gap between actual portfolio standard deviation and the naive weighted average of the two standard deviations.

Worked example: w1 = w2 = 0.5, sigma1 = 20%, sigma2 = 30%, correlation = 0.0. Var = 0.25(0.04) + 0.25(0.09) + 0 = 0.0325, so sigma_p = 18.03% - below the 25% weighted average and even below the less-risky asset alone. If correlation were +1.0 instead, sigma_p would equal exactly 0.5(20%) + 0.5(30%) = 25%, with no diversification benefit. If correlation were -1.0, a weight set exists that drives sigma_p to zero. This is why the curriculum stresses that correlation, not the number of holdings, is the lever.

Reading the numbers correctly

Covariance carries the same sign as correlation but its magnitude is hard to interpret because it is in squared, asset-specific units. Correlation rescales it to -1 to +1 so two pairs can be compared. A correlation of +0.3 between a stock and a bond is low, but it is not zero - during an inflation shock both can fall together as discount rates rise, so historical correlation is not a guarantee of future independence. The portfolio manager still has to identify the economic forces behind the statistic, not just trust the number.

Diversification does not erase risk; it changes how risk combines, and it works best when assets carry low or negative correlation and meaningful weights. A 2% sleeve of a diversifier cannot transform a portfolio on its own.

Systematic versus unsystematic risk

Unsystematic risk (also called firm-specific, idiosyncratic, or diversifiable risk) is tied to one company, industry, or narrow exposure: a product recall, lawsuit, fraud, or plant fire. Holding many imperfectly correlated issuers averages these shocks toward zero. Systematic risk (market or non-diversifiable risk) comes from economy-wide forces - interest rates, recession, inflation, credit cycles - and cannot be diversified away within risky assets.

The key CFA principle: investors are compensated only for systematic risk. Bearing avoidable concentration risk earns no expected reward, because a diversified investor would not pay for it. Total risk = systematic risk + unsystematic risk.

From the efficient frontier to the CML

The efficient frontier is the set of portfolios offering the highest expected return for each level of risk; a portfolio is dominated if another has equal return at lower risk or higher return at equal risk. Adding a risk-free asset turns the frontier into a straight line - the Capital Market Line (CML) - that runs from Rf through the tangency (market) portfolio:

E(Rp) = Rf + [(E(Rm) - Rf) / sigma_m] * sigma_p

The slope is the market Sharpe ratio. The CML uses total risk (sigma) because efficient portfolios hold no diversifiable risk.

Beta, CAPM, and the SML

For an individual security, only systematic risk matters, measured by beta: beta_i = Cov(i, m) / Var(m) = Corr(i,m) * sigma_i / sigma_m. The market's beta is 1.0; Rf has a beta of 0. The Capital Asset Pricing Model (CAPM) gives required return:

E(Ri) = Rf + beta_i * [E(Rm) - Rf]

Plotted, this is the Security Market Line (SML), with beta on the x-axis. A security above the SML is undervalued (offers more than its required return). The Sharpe ratio (Rp - Rf)/sigma_p and M-squared rank total-risk performance; Treynor ((Rp - Rf)/beta) and Jensen's alpha rank systematic-risk performance.

Distinguishing the CML from the SML

Candidates routinely confuse the two lines. The CML plots expected return against total risk (sigma) and applies only to efficient portfolios that combine the risk-free asset with the market portfolio. The SML plots expected return against beta and applies to any asset or portfolio, efficient or not, because beta isolates the priced systematic component. An inefficient single stock can sit on the SML (it is fairly priced for its beta) yet lie far below the CML (it carries diversifiable risk that earns nothing).

When a stem asks about an individual security's required return, reach for CAPM/SML and beta; when it asks about an efficient portfolio's total-risk trade-off, reach for the CML and sigma.

Common trap: computing portfolio risk as a simple weighted average of standard deviations. Whenever a stem gives two sigmas plus a correlation, the test is whether you apply the covariance term. A second trap is using sigma in CAPM - CAPM and the SML use beta, never total risk. A third trap mixes up the performance ratios: Sharpe and M-squared divide by total risk and so reward diversification, while Treynor and Jensen's alpha use beta and assume the portfolio is already well diversified. Choosing Treynor for an undiversified portfolio overstates its risk-adjusted quality because the leftover unsystematic risk is invisible to beta.