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Corporate Finance Unit 6: Risk, Return and Diversification

Investors demand higher returns for bearing more risk, but not all risk counts. This unit measures return and risk and shows how diversification removes firm-specific risk, leaving only market risk to be rewarded.

Covers: Holding period, arithmetic and geometric returns, Historical risk premiums, Expected return and standard deviation from scenarios, Portfolio return and variance with correlation, Systematic vs unsystematic risk, Diversification and the Sharpe ratio

Study guide

Returns have two parts: income (dividends or coupons) and capital gains. The holding period return is (dividend + ending price − beginning price) ÷ beginning price. Over several years, the arithmetic average is the simple mean of the yearly returns, while the geometric average, [(1 + r₁)(1 + r₂)...(1 + rₙ)]^(1/n) − 1, is the compound growth rate actually earned. The geometric average is lower whenever returns vary. Use the arithmetic mean to estimate a typical single year and the geometric mean to describe or forecast long-run compounded growth.

Capital market history shows a clear risk-return trade-off. Over the past century, U.S. small-company stocks have earned the highest average returns with the most volatility, followed by large-company stocks, long-term bonds and Treasury bills. The risk premium is the average return on a risky asset minus the risk-free rate; large stocks have historically earned a premium of roughly 8 to 9 percentage points over T-bills. Volatility is measured by variance and standard deviation. For historical data, sample variance divides the sum of squared deviations by n − 1; for normal returns, about 68% of years fall within one SD of the mean and 95% within two.

For forward-looking analysis, analysts describe possible states of the economy with probabilities. Expected return is E(R) = Σ pᵢRᵢ, and variance is σ² = Σ pᵢ(Rᵢ − E(R))². A portfolio's expected return is the weighted average of its assets' expected returns, but its risk is not. For two assets, σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂, where ρ is the correlation (and ρσ₁σ₂ is the covariance). When ρ is below +1, portfolio SD is less than the weighted average SD; with ρ = −1, a perfectly hedged, riskless portfolio is possible.

Total risk has two parts. Systematic (market) risk comes from economy-wide events such as recessions, inflation and interest rate changes, and it affects nearly all firms. Unsystematic (firm-specific) risk comes from events unique to one company, such as a strike or a failed product. In a diversified portfolio, firm-specific surprises cancel out, so unsystematic risk shrinks as more stocks are added; most of the benefit comes from the first 20 to 30 stocks. Systematic risk cannot be diversified away, and the average covariance among stocks sets a floor on portfolio risk.

Because investors can remove unsystematic risk at almost no cost, the market does not reward it. This is the systematic risk principle: the expected return on an asset depends only on its systematic risk. A single stock's total SD can therefore be a poor guide to its expected return. To compare portfolios on reward per unit of total risk, use the Sharpe ratio, (E(R) − rf) ÷ σ. Mixing a risky portfolio with a risk-free asset gives returns and risk that are linear in the weight, which leads directly to the CAPM in the next unit.

Key ideas

• Total return equals income plus capital gain, measured relative to the starting price.
• The geometric average return is at or below the arithmetic average and measures compound growth.
• Riskier asset classes have historically earned higher average returns.
• Portfolio expected return is a weighted average, but portfolio risk depends on correlations.
• Diversification removes unsystematic risk but cannot remove systematic risk.
• Only systematic risk earns a risk premium in competitive markets.
• The Sharpe ratio measures excess return per unit of total risk.

Formulas and rules

Common mistakes

• Averaging portfolio standard deviations as if they were expected returns, ignoring correlation.
• Forgetting to weight scenario returns by their probabilities.
• Dividing by n instead of n − 1 for sample variance of historical returns.
• Thinking enough diversification eliminates all risk, including market risk.
• Using the arithmetic mean to compound wealth over long horizons.

Before the test, make sure you can

• Calculate holding period returns and arithmetic and geometric averages.
• Calculate expected return, variance and SD from scenario probabilities.
• Calculate sample SD from historical returns.
• Calculate the expected return and SD of a two-asset portfolio given correlation.
• Explain why diversification reduces unsystematic but not systematic risk.
• Identify events as systematic or unsystematic risk.
• Calculate and compare Sharpe ratios.
• Explain why only systematic risk earns a risk premium.

Words to know

Holding period return

The total return over a period: income plus price change, divided by the starting price.

Arithmetic average return

The simple mean of a series of returns.

Geometric average return

The compound average return per period over several periods.

Risk premium

The excess return on a risky asset over the risk-free rate.

Variance

The average squared deviation of returns from their mean.

Standard deviation

The square root of variance; a measure of total volatility.

Expected return

The probability-weighted average of possible returns.

Covariance

A measure of how two returns move together; equals ρσ₁σ₂.

Correlation

A standardized measure of co-movement between −1 and +1.

Portfolio weight

The fraction of a portfolio's value invested in an asset.

Systematic risk

Risk from economy-wide factors that affects most assets; cannot be diversified away.

Unsystematic risk

Firm-specific risk that can be eliminated through diversification.

Diversification

Spreading investment across many assets to reduce unsystematic risk.

Systematic risk principle

Expected return depends only on an asset's systematic risk.

Sharpe ratio

Excess return divided by standard deviation; reward per unit of total risk.

Normal distribution rule

About 68% of outcomes lie within 1 SD of the mean and 95% within 2 SD.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.