FINC3017 · Investments And Portfolio Management
Risk, Return & the Two-Asset Portfolio
Risk, Return & the Two-Asset Portfolio (Week 3) is the quantitative heart of early portfolio theory. You define the equity risk premium and the Sharpe ratio, learn how to annualise returns and risk (variance scales by n, volatility and the Sharpe ratio by √n), and combine two assets: the portfolio mean is the weighted average of the asset means, while the portfolio variance adds a covariance term that shrinks as correlation falls. That covariance term is the whole point — it is why diversification reduces risk for less-than-perfectly-correlated assets.
What this chapter covers
- 01Equity risk premium = E[r_stock] − rf
- 02Sharpe ratio = (r̄_i − rf)/σ_i
- 03Annualising: variance scales by n; volatility and Sharpe scale by √n
- 04Portfolio weights w_i = N_iP_i/ΣN_jP_j (sum to 1; negative = short)
- 05Two-asset mean: μ_p = w₁μ₁ + w₂μ₂
- 06Two-asset variance: σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ
- 07How the diversification benefit grows as correlation ρ falls
- 08The amplifying effect of short (negative) weights on both return and risk
Mean and standard deviation of a two-asset portfolio
- 2 marksExpected return: μ_p = w_Aμ_A + w_Bμ_B = 0.6 × 8% + 0.4 × 14% = 4.8% + 5.6% = 10.4%.
- 2 marksSet up the variance terms (in %²). Own-variance terms: w_A²σ_A² = 0.36 × 225 = 81 and w_B²σ_B² = 0.16 × 625 = 100.
- 2 marksCovariance term: 2w_Aw_Bσ_Aσ_Bρ = 2 × 0.6 × 0.4 × 15 × 25 × 0.3 = 0.48 × 375 × 0.3 = 54.
- 1 markSum and take the root: σ_p² = 81 + 100 + 54 = 235, so σ_p = √235 ≈ 15.33%.
Key terms
- Sharpe ratio
- Excess return per unit of total risk, (r̄_i − rf)/σ_i. It ranks investments by reward for risk taken and is the slope of the capital allocation line; higher is better. It scales by √n when you re-express it over a longer horizon.
- Equity risk premium
- The extra expected return investors demand for holding stocks over the risk-free asset, E[r_stock] − rf. It is the reward for bearing systematic equity risk and anchors the slope of the security market line later in the course.
- Correlation (ρ)
- A standardised covariance between −1 and +1 that measures how two assets move together. In a portfolio it drives the covariance term of variance; the lower ρ, the larger the diversification benefit, and at ρ = +1 there is no benefit at all.
- Diversification benefit
- The reduction in portfolio risk below the weighted average of the individual asset volatilities, arising because assets are not perfectly correlated. It is free risk reduction and is the founding motivation for holding portfolios rather than single stocks.
- Annualising by √n
- Under i.i.d. returns, variance grows linearly with the number of periods n, so volatility and the Sharpe ratio grow with √n. Converting a monthly Sharpe to annual multiplies by √12; this square-root rule is a frequent MCQ point.
Risk, Return & the Two-Asset Portfolio FAQ
Why does a portfolio's standard deviation fall below the weighted average of the two assets' volatilities?
Because the covariance term is less than the product of the volatilities whenever correlation is below +1. Variance is the weighted sum of the own-variances PLUS a cross term scaled by ρ, so any ρ < 1 makes the portfolio less risky than a naive average would suggest. Only at ρ = +1 do the volatilities add up linearly with no diversification gain.
How do I annualise a Sharpe ratio?
Multiply by the square root of the number of periods per year, because variance scales by n while volatility and the Sharpe ratio scale by √n. A monthly Sharpe of 0.2 annualises to 0.2 × √12 ≈ 0.69. This assumes returns are independent and identically distributed across periods.
What happens to portfolio risk if one weight is negative (a short position)?
A negative weight means you have shorted that asset and levered up the other, so the position weights now exceed 100% of your capital. This amplifies both expected return and volatility, and the covariance term can flip sign — shorting a positively correlated asset can actually hedge and lower portfolio risk. Watch the signs carefully when squaring negative weights.
Exam move
Memorise the two-asset formulas cold and practise plugging numbers under time pressure, including cases with a negative (short) weight, since those are a favourite exam pattern. Always keep the variance in one consistent unit and root only at the end, and be ready to explain in one sentence why lower correlation means more diversification.
