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The rule: the compact statistics toolkit

The Series 66 tests a small set of risk/return statistics. Know what each measures and how to compute or read it.

• Standard deviation (σ) measures total volatility — how widely returns disperse around their average. Higher σ = wider, riskier range. Between two investments with the same expected return, the one with the lower σ is preferred.
• Beta (β) measures systematic risk — sensitivity to the overall market (market β = 1.0). β = 1.3 moves ~30% more than the market; β = 0.7 moves ~30% less.
• Alpha measures return above or below what beta predicts via CAPM. Positive alpha = the manager added value beyond simply taking market risk.
• The Sharpe ratio measures return per unit of total risk — excess return over the risk-free rate divided by σ. Higher is better.
• R-squared measures how much of a portfolio's movement is explained by the benchmark (how meaningful its beta is).
• Duration measures a bond's price sensitivity to interest-rate changes (Chapter 12 expands this).

The rule: the formulas, stated plainly

• CAPM (required/expected return): r = Rf + β × (Rm − Rf), where Rf is the risk-free rate and (Rm − Rf) is the market risk premium.
• Alpha: actual return − CAPM-expected return.
• Sharpe ratio: (Rp − Rf) ÷ σ, where Rp is the portfolio return.
• Real (inflation-adjusted) return: approximately nominal return − inflation rate.

Worked example — CAPM required return

A stock has β = 1.2. The risk-free rate is 3% and the expected market return is 9%.

Market risk premium = Rm − Rf = 9% − 3% = 6%. Required return = Rf + β × (Rm − Rf) = 3% + 1.2 × 6% = 3% + 7.2% = 10.2%.

Interpretation: given its market sensitivity, this stock should return 10.2% to compensate for its risk.

Worked example — alpha

Suppose the stock above actually returned 12% over the year.

Alpha = actual − CAPM-expected = 12% − 10.2% = +1.8%.

A positive alpha of 1.8% means the investment beat what its beta alone would justify — evidence (over enough time) of manager skill or mispricing. Had it returned 8%, alpha = 8% − 10.2% = −2.2% (underperformance).

Worked example — the Sharpe ratio, comparing two funds

Risk-free rate = 3%.

• Fund A: return 11%, σ = 16%. Sharpe = (11 − 3) ÷ 16 = 8 ÷ 16 = 0.50.
• Fund B: return 9%, σ = 8%. Sharpe = (9 − 3) ÷ 8 = 6 ÷ 8 = 0.75.

Fund A has the higher raw return, but Fund B earns more return per unit of risk (0.75 > 0.50), so Fund B is superior on a risk-adjusted basis. The Sharpe ratio is the exam's favorite way to show that the highest-return fund is not always the best fund.

Worked example — reading beta both ways

A portfolio has β = 1.3. If the market rises 10%, expect roughly +13%. If the market falls 10%, expect roughly −13%. Beta cuts both ways — the aggressive portfolio that outperforms in rallies also underperforms in declines.

Worked example — the real rate of return

A bond yields a nominal 7% while inflation runs 3%. Real return ≈ 7% − 3% = 4%. If inflation jumped to 8%, the real return would be 7% − 8% = −1% — the investor is losing purchasing power despite a positive nominal yield. This is why "safe" long-term bonds still carry inflation (purchasing-power) risk.

The rule: the Treynor ratio and choosing the right denominator

The Sharpe ratio divides excess return by total risk (σ) and is the right tool for a stand-alone portfolio or a client's whole wealth. The Treynor ratio divides the same excess return by beta — return per unit of systematic risk — and is the right tool when the portfolio is one part of a larger, already-diversified holding (where unsystematic risk is assumed to be diversified away). Same numerator (Rp − Rf); the denominator changes with the question being asked:

• Sharpe = (Rp − Rf) ÷ σ — reward per unit of total volatility.
• Treynor = (Rp − Rf) ÷ β — reward per unit of market risk.

Worked example — Sharpe vs. Treynor can disagree

Risk-free = 2%. A fund returns 10%, with σ = 16% and β = 0.8.

• Sharpe = (10 − 2) ÷ 16 = 8 ÷ 16 = 0.50.
• Treynor = (10 − 2) ÷ 0.8 = 8 ÷ 0.8 = 10.0.

A second fund returns 11%, σ = 22%, β = 1.4.

• Sharpe = (11 − 2) ÷ 22 = 9 ÷ 22 = 0.41.
• Treynor = (11 − 2) ÷ 1.4 = 9 ÷ 1.4 = 6.43.

By both measures the first fund is superior here — but the exam sometimes crafts numbers where a high-beta fund wins on Sharpe yet loses on Treynor, testing whether you picked the denominator that matches the client's situation (whole-wealth → Sharpe; sleeve of a diversified whole → Treynor).

The rule: expected return from probabilities, and reading a distribution

Sometimes the exam gives scenarios with probabilities rather than a single number. The expected return is the probability-weighted average of the outcomes:

Expected return = Σ (probability × outcome).

And standard deviation describes how spread out those outcomes are. For a roughly normal distribution, about 68% of outcomes fall within ±1σ of the mean, about 95% within ±2σ, and about 99.7% within ±3σ — the "68-95-99.7" rule. This is why a fund with a 10% mean return and a 15% σ can plausibly return anywhere from −5% to +25% in a typical year (mean ±1σ).

Worked example — expected return under three scenarios

An analyst estimates a stock will return +20% with 30% probability, +8% with 50% probability, and −10% with 20% probability.

Expected return = (0.30 × 20%) + (0.50 × 8%) + (0.20 × −10%) = 6.0% + 4.0% + (−2.0%) = 8.0%.

The single-number expected return is 8%, even though no individual scenario equals 8% — a common exam framing that rewards careful weighting.

The rule: covariance, correlation, and R-squared — how they relate

Covariance measures whether two assets move together (positive) or oppositely (negative), but its raw size is hard to interpret. Correlation standardizes covariance onto the −1.0 to +1.0 scale, making it comparable across pairs. R-squared (the square of correlation with a benchmark) tells you how much of a portfolio's movement the benchmark explains — a high R-squared (near 100) means beta and alpha are meaningful; a low R-squared means beta is noise and alpha should be viewed skeptically.

Key figures — risk & performance - σ = total risk; β = systematic risk; alpha = actual − CAPM; Sharpe = (Rp − Rf) ÷ σ. - CAPM: r = Rf + β(Rm − Rf). - Market β = 1.0; risk-free asset β = 0. - Higher Sharpe = better risk-adjusted return. - Real return ≈ nominal − inflation.

Common traps

• Highest return ≠ best fund. Use the Sharpe ratio to reward risk-adjusted performance.
• Beta and standard deviation are not the same. σ = total risk; β = market-related risk only.
• CAPM order of operations. Compute the risk premium (Rm − Rf) first, multiply by β, then add Rf. A common wrong answer forgets to add Rf back.
• Positive alpha means outperformance of the CAPM benchmark, not merely a positive return.
• A high nominal yield can be a negative real yield when inflation exceeds it.

Check yourself

1. CAPM: Rf = 2%, β = 0.8, Rm = 10%. Required return? Answer: 2 + 0.8×(10−2) = 2 + 6.4 = 8.4%.
2. Fund returns 10%, σ = 20%, Rf = 4%. Sharpe? Answer: (10−4)/20 = 6/20 = 0.30.
3. A stock's CAPM-expected return is 9% and it returned 6%. Alpha? Answer: −3%.
4. Market +8%, portfolio β = 1.25. Expected move? Answer: ~+10%.
5. Nominal 5%, inflation 6%. Real return? Answer: ~−1%.