Add a jumpy, volatile asset to your portfolio and your total risk can go down. Harry Markowitz won a Nobel Prize for showing why.
Learning objectives. After this week you can (1) compute a two-asset portfolio's expected return; (2) show how correlation lowers portfolio risk; and (3) explain the efficient frontier.
Remember this: returns average, risk does not. Below a correlation of +1, the mix is less risky than its parts.
1) Portfolio expected return: it averages
E(rp) = wA × E(rA) + wB × E(rB)
w is each holding's weight (Week 6) and E(r) its expected return. $60,000 of Stock A (you expect 10%) and $40,000 of Stock B (you expect 6%) in a $100,000 account: 0.60 × 10% + 0.40 × 6% = 8.4%.
2) Portfolio risk: it doesn't average
With σ for each asset's standard deviation (Week 4) and ρ (rho) for the correlation between them:
σp2 = wA2σA2 + wB2σB2 + 2 wAwB ρ σAσB
The last term is where correlation lives. At ρ = +1 nothing cancels, and portfolio risk is just the weighted average. Any ρ below +1 shrinks that term, so risk falls while expected return stays put. Same 60/40 portfolio, with σA = 20% and σB = 10%:
| Correlation ρ | σp2 | Portfolio σp |
|---|---|---|
| +1.0 (lockstep) | 144 + 16 + 96 = 256 | 16.0% |
| +0.2 (loosely linked) | 144 + 16 + 19.2 = 179.2 | 13.4% |
| −1.0 (perfect hedge) | 144 + 16 − 96 = 64 | 8.0% |
Expected return is 8.4% in every row. Only the correlation changed, and risk fell from 16% to 8%. That is the "free lunch" of diversification. Two stocks in the same industry often sit near +0.8; stocks and bond ETFs often sit near zero.
3) The efficient frontier
Plot every possible mix of your assets as a dot: risk across, expected return up. The top edge of the cloud is the efficient frontier, the mixes with the highest expected return for each level of risk. A mix below it is dominated: another mix gives more return for the same risk. The lowest-risk dot of all is the minimum-variance portfolio. A single stock usually sits well inside the cloud, because it carries firm-specific risk a mix would have cancelled. Markowitz's lesson: judge each holding by what it does to the portfolio's risk and return, not by its own risk alone.
Quick self-check
(a) Weights 50/50, expected returns 12% and 8%. Portfolio expected return? (b) σ of 20% and 10%, weights 60/40, ρ = +1. Portfolio σ? (Answers: 0.5 × 12 + 0.5 × 8 = 10%; the weighted average, 16%.)
Common mistakes
- Averaging risks. That only holds at ρ = +1.
- Fake diversification. Ten tech stocks are closer to one big tech bet.
- Judging a holding alone. What matters is what it does to the whole portfolio.
This week in the arena
Compute the weights of your two largest holdings from your Holdings table and Total value. Then ask how closely they move together: same industry, same customers, same reaction to interest rates? Write down the correlation you would honestly guess for the pair, and what the table above says that means for its risk.