FNCE30001 Investments
Risk and Return
Every investment decision trades off expected return against risk. Return is measured as a holding-period return over a realised period, or as an expected return across probability-weighted scenarios; risk is the variance / standard deviation of that return. The chapter then builds the machinery of diversification: covariance and correlation measure whether two assets move together, and any correlation below +1 is the engine that lets a two-asset portfolio's risk fall below the weighted average of its parts — while the portfolio's expected return stays a simple weighted average. As you add assets, total risk falls steeply then flattens to a systematic floor: the firm-specific (diversifiable) risk averages away, but the market risk that hits every asset together remains. The punchline carries straight into the CAPM: only systematic risk is rewarded, because the market refuses to pay a premium for risk you could have diversified away. The AU framing uses the ASX 200 as the market and Bank-Accepted Bills as the risk-free proxy.
What this chapter covers
- 014.1 Holding-period return, expected return, variance and SD
- 024.2 Probability-weighted scenario returns
- 03Covariance, correlation and the risk premium
- 044.3 The historical Australian equity premium
- 05The two-asset portfolio: return is a weighted average, risk is not
- 064.4 Systematic vs unsystematic risk and the diversification limit
- 07Why only systematic risk is priced
Worked example: two-asset portfolio risk and return
- +1Identify. Two-asset Markowitz — expected return is a weighted average, but variance carries a covariance term.
- +1Expected return: E[r_p] = 0.6(12) + 0.4(8) = 10.4%.
- +1Covariance: cov = ρσ_Dσ_E = 0.30(20)(14) = 84 (in %²).
- +1Variance: σ_p² = 0.6²(20²) + 0.4²(14²) + 2(0.6)(0.4)(84) = 144 + 31.36 + 40.32 = 215.68 (%²). Show all three terms.
- +1Standard deviation: σ_p = √215.68 = 14.69%.
- +1Interpret: the weighted-average SD would be 0.6(20) + 0.4(14) = 17.6%, but the actual σ_p is 14.69% — nearly 3 points lower, purely because ρ < 1. That gap is diversification.
Key terms
- Holding-period return (HPR)
- Total return over a period: (P₁ − P₀ + Income) / P₀ = P₁/P₀ − 1 — the price change plus income, divided by what you paid.
- Expected return and variance
- Across scenario states with probabilities p(s): E[r] = Σ p(s)·r(s) and σ² = Σ p(s)(r(s) − E[r])², with σ = √σ². Weight by the stated probabilities, not 1/n, unless the states are equally likely.
- Covariance and correlation
- Covariance σ_XY = Σ p(s)(r_X − E[r_X])(r_Y − E[r_Y]) measures whether two returns move together. Correlation ρ_XY = σ_XY / (σ_Xσ_Y) rescales it to [−1, +1]. Correlation below +1 is the engine of diversification.
- Two-asset portfolio variance
- σ_p² = w_D²σ_D² + w_E²σ_E² + 2w_Dw_Eρσ_Dσ_E. The cross term is why portfolio risk is not the weighted average of the SDs: for any ρ < 1, σ_p is strictly below the weighted average, and at ρ = −1 a zero-variance mix exists.
- Systematic vs unsystematic risk
- Unsystematic (firm-specific) risk — a recall, a scandal, a flooded mine — averages out across many holdings and vanishes as n grows. Systematic (market) risk — a rate decision, a recession — hits every asset together and cannot be diversified away. Only systematic risk is rewarded with a premium.
Risk and Return FAQ
Why isn't portfolio risk just the weighted average of the individual risks?
Because of the covariance term. Expected return IS a weighted average, but variance is w_D²σ_D² + w_E²σ_E² + 2w_Dw_Eρσ_Dσ_E. Only when ρ = +1 does that collapse to the weighted-average SD. For any ρ < 1 the portfolio SD is strictly below the weighted average, and it can even fall below the less-risky asset's SD. Always quote the variance formula with all three terms, then square-root — never weighted-average the SDs.
Do I weight scenario returns by probability or by 1/n?
By the stated probabilities (the population form): E[r] = Σ p(s)r(s) and σ² = Σ p(s)(r(s) − E[r])². Only divide by n if the states are explicitly equally likely. A historical sample is a different setting — there you use the sample SD with an n − 1 divisor.
What does correlation do to the diversification benefit?
The lower the correlation, the more the frontier of (σ, E[r]) combinations bows to the left — same returns, less risk. At ρ = +1 there is no benefit; at ρ = 0 the portfolio SD can fall below both assets; at ρ = −1 a zero-variance mix exists. Negative correlation is the most powerful diversifier, though it is rare in practice. You can read the covariance sign off a scenario table: same-direction deviations give positive covariance, opposite deviations give the strongest diversifier.
Why does the market only reward systematic risk?
Because firm-specific risk can be diversified away for free — bearing it earns no extra expected return, so a concentrated single-stock bet is uncompensated risk. As you add assets, portfolio variance decays toward a floor set by the average covariance (the systematic component), while the own-variance term shrinks like 1/n. The market pays a premium only for the floor it cannot remove. This is exactly why the priced risk in the CAPM is beta (covariance with the market), not total σ.
Exam move
Make the statistics mechanical, because these formulas underpin everything that follows (portfolio theory and the CAPM). For scenario questions, lay out a probability table and compute E[r], then deviations, then σ² — weighting by p(s), not 1/n. For two assets, always write σ_p² with all three terms visible before taking the square root; markers drop method marks for a bare SD. Internalise the two headline traps: portfolio σ equals the weighted-average SD only when ρ = 1, and the market pays no premium for diversifiable risk. Finish by connecting the chain — single-asset stats → two-asset covariance → the systematic floor → the CAPM — so the bridge to beta and required return is obvious.
