Risk: the free lunch and its limits
Introduced| English | Meaning |
|---|---|
| expected return | the probability-weighted average of possible returns |
| volatility/ˌvɒləˈtɪlɪti/ | the standard deviation of returns - the spread around the mean |
| diversification/daɪˌvɜːsɪfɪˈkeɪʃn/ | Expanding into different products, markets or activities to spread risk or seek growth. |
| risk premium | the extra return demanded for carrying risk, over the risk-free rate |
| specific risk | risk tied to one firm or asset that diversification can remove |
| systematic risk | market-wide risk that diversification cannot remove |
The hurdle rate is earned, not given
- The 8% is what investors DEMAND for the risk they carry. The second site's scenarios: boom 30% (p = 0.3), base 12% (p = 0.5), bust -10% (p = 0.2).
- Expected return 预期收益率 = 0.3 x 30 + 0.5 x 12 + 0.2 x (-10) = 13.0% ✓; volatility 波动率 = 13.9% ✓.
Diversification, the free lunch
- A 60:40 stock-bond portfolio (sd 20% / 8%, rho 0.3) has sd 13.31% ✓ - below the 15.2% weighted average, because the covariance term shrinks with rho.
- Risk diversification cannot remove is systematic 系统性风险; risk it can is specific 非系统性风险. Only the systematic kind is priced.
Scenarios 30% (p 0.3), 12% (p 0.5), -10% (p 0.2). The expected return, in %?
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0.3(30) + 0.5(12) + 0.2(-10) = 13.0%.
The same distribution. The volatility, in % (one decimal)?
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Variance 0.0193 -> sd 13.89% - the spread around the 13% mean.
Risk-free 4%, expected 13%. The risk premium, in percentage points?
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13 - 4 = 9 points of demanded excess - the price of carrying the spread.
The correlation knife
- At rho = 1: sd 15.2 (no benefit). rho = 0: 12.65. rho = -0.2: 11.6. The benefit IS the distance below 1 - which is why "another bakery" (rho ~ 0.9) barely helps while the bond fund does.
- In 2008 correlations rose toward 1 and the lunch briefly vanished. The honest course says so.
What kind of risk is this?
Separate systematic from specific risk, and match each portfolio move to its effect on sd.
60% stock (sd 20), 40% bond (sd 8), rho 0.3. Portfolio sd, in % (one decimal)?
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sqrt(0.36 x 0.04 + 0.16 x 0.0064 + 2 x 0.6 x 0.4 x 0.3 x 0.2 x 0.08) = 13.31% - below the 15.2 weighted average.
The same portfolio at rho = 1. The sd, in % (one decimal)?
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0.6(20) + 0.4(8) = 15.2% - the weighted average, zero benefit. Diversification IS the distance below 1.
Why does the market NOT pay a premium for the site losing its chef?
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Only systematic risk survives the portfolio; the single-site founder FEELS it, the diversified investor does not price it.
Portfolio volatilities add: sd_p = 0.6 x 20 + 0.4 x 8 always.
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That is the rho = 1 case only. VARIANCES add, with the covariance term - the whole point of the formula.
Match each risk to its treatment.
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Name the kind, then the correlation - the two numbers that decide the portfolio.
The risk premium
- 13% expected against 4% risk-free is a 9-point risk premium 风险溢价. A diversified investor bears little of a single site's specific risk - and should not price it.
Three risk slips
- Past average as expectation without scenarios or a model.
- Adding volatilities - variances add, with the covariance term.
- Pricing specific risk - the market pays for systematic only.
Recap
- E[r] = sum p x r; sd = the spread. Portfolio sd falls with the covariance term.
- Only systematic risk earns a premium.
- Diversification is free, exactly as far as rho < 1.