Descriptive Statistics

Descriptive Statistics

Statistics are the language the Series 66 uses to talk about risk and return. Mean and median describe what an investment typically returned; standard deviation and beta describe how wildly it bounced around. Three risk-adjusted ratios — Sharpe, Treynor, and Jensen's alpha — combine them. The trap the exam loves: knowing which risk measure each ratio uses, and when each is appropriate.

Why statistics matter on the Series 66

Descriptive statistics give advisers the vocabulary to discuss return (how well an investment performed) and risk (how reliable that performance was). The Series 66 doesn't ask you to compute long formulas by hand — it asks you to recognize which statistic answers a particular question, and to spot the common pitfalls that come from using the wrong one.

Two themes thread through the whole chapter. First, no single number tells the whole story — mean can hide outliers; standard deviation can't distinguish market risk from company-specific risk. Second, the right comparison metric depends on context — Sharpe vs. Treynor isn't a preference, it's a function of what the portfolio represents to the investor.

Mean, median, and mode

Three ways to describe the "typical" value in a dataset. They produce different answers when the data is uneven, which is most of the time in finance.

• Mean — sum of all values divided by the count. The most familiar measure; the most distorted by outliers. One spectacular year can pull a fund's average return well above what an investor typically saw.
• Median — the middle value when data is sorted. Half the observations are above it, half below. Less sensitive to extreme values, so often the more honest measure when outliers are present.
• Mode — the most frequently occurring value. Useful for categorical or repeated data, less commonly the right tool for continuous return data.

When a Series 66 question describes a return series with one or two unusually large or small values, the median is usually the answer the exam wants — and the trap is picking mean because it sounds more rigorous.

Geometric vs. arithmetic mean — for returns over time

When you're averaging multi-period returns, the arithmetic mean overstates what an investor actually experienced. The geometric mean is the honest annualized rate that would have produced the same ending value.

Why the gap matters. A fund returns +50% then −50% in two years. Arithmetic mean = 0%. Geometric mean = −13.4%. A $1,000 investment becomes $1,500, then drops to $750 — a real loss, masked by an arithmetic average that looks like a wash. For any multi-period return comparison on the Series 66, geometric mean is the honest answer. The bigger the volatility, the larger the gap.

Range and standard deviation

The two most common measures of how spread out a dataset is.

• Range — highest value minus lowest. Simplest possible measure of spread. Tells you the worst and best observed outcomes, but ignores everything in between.
• Standard deviation (σ) — average distance of each data point from the mean. The single most important measure of total risk for an investment, and the basis for the Sharpe ratio.

Key relationships to lock in for the exam:

• Higher standard deviation ⇒ more volatile ⇒ higher total risk.
• Standard deviation captures both systematic and unsystematic risk (this distinguishes it from beta, which captures only systematic risk).
• In a normal distribution: about 68% of returns fall within ±1σ of the mean, ~95% within ±2σ, ~99.7% within ±3σ.

Variance — the building block

Variance and standard deviation measure the same thing, just on different scales. Standard deviation is what gets reported because it's in the same units as the original data; variance is its mathematical building block.

If a fund's variance is 0.0196 (or 1.96 in percentage-squared units), its standard deviation is √0.0196 = 0.14 or 14%. The squaring step makes variance always positive but harder to interpret in plain English ("14% standard deviation" is meaningful; "196 squared-percentage-points of variance" is not).

Variance shows up in portfolio risk math because variances combine more predictably than standard deviations across multiple assets — but for any single-asset reasoning, standard deviation is the working number.

The normal distribution & the 68-95-99.7 rule

If returns are normally distributed (the simplifying assumption the Series 66 uses), the bell curve gives you a quick way to translate standard deviation into "how often" probabilities.

Standard deviation in practice — two funds, same return

Standard deviation only gets useful when you have something to compare it to. Consider two equity funds advertised at the same headline return:

Same expected return, dramatically different risk. Fund B is the clearly preferable choice for a risk-averse investor. This is precisely why headline return alone is meaningless — and why the Series 66 leans on risk-adjusted ratios like the Sharpe ratio to do the comparison properly.

Beta — measuring systematic risk

Beta measures a security's sensitivity to overall market movements. It tells you only one thing: how much a security tends to move when the market moves. It does not capture company-specific (unsystematic) risk.

A portfolio with beta 1.3 is expected to fall 13% when the market falls 10% — and rise 13% when the market rises 10%. Beta amplifies movement in both directions; it does not predict direction.

What beta does NOT measure: company-specific risk — a CEO scandal, product recall, accounting fraud. That's unsystematic risk, captured in standard deviation but not in beta. A single stock with very high beta might still be diversifiable away if held alongside enough other names; that's why beta gets paired with R-squared (Section 5) before drawing strong conclusions.

Systematic vs. unsystematic risk

Alpha & CAPM — what is the portfolio "supposed to" earn?

Alpha measures whether a manager added value beyond what was expected given the portfolio's risk. To compute alpha, you first need the expected return, which is where the Capital Asset Pricing Model (CAPM) comes in.

Plain English: the expected return on a portfolio equals the risk-free rate plus the portfolio's beta times the market risk premium. CAPM compensates investors for two things — time (the risk-free rate) and market risk (beta times the equity premium).

Once you have the expected return, alpha is the gap between what the portfolio actually delivered and what CAPM said it should have:

• Positive alpha — the manager beat the risk-adjusted benchmark. Skill (or luck).
• Negative alpha — the manager underperformed what their beta exposure alone would have produced.
• Alpha near zero — the manager delivered exactly market-equivalent risk-adjusted returns.

Worked example. A portfolio returned 14% during a year when the risk-free rate was 4%, the market returned 10%, and the portfolio's beta was 1.5. CAPM expected return = 4% + 1.5(10% − 4%) = 13%. Alpha = 14% − 13% = +1%. The manager added 100 basis points of value above the CAPM benchmark.

The three risk-adjusted ratios

The Series 66 tests three risk-adjusted performance measures. Each divides "excess return over the risk-free rate" by a different measure of risk. The choice of risk measure is the whole point.

• Sharpe ratio = (R_p − R_f) ÷ σ_p   — divides excess return by total risk (standard deviation). Use when the portfolio represents the investor's entire holdings.
• Treynor ratio = (R_p − R_f) ÷ β_p   — divides excess return by systematic risk only (beta). Use when the portfolio is one slice of a larger, already-diversified holding.
• Jensen's alpha = R_p − [R_f + β_p(R_m − R_f)]   — the dollar (or percentage-point) excess over CAPM. Use to measure pure manager skill.

The single most important question the exam asks: which risk measure does each ratio use? Sharpe → total risk (σ). Treynor → systematic risk only (β). Jensen's alpha → systematic risk via the CAPM expected-return calculation. Memorize this mapping cold.

Sharpe vs. Treynor vs. Jensen's alpha — when to use each

Information ratio — a brief mention

The information ratio shows up on some Series 66 forms as the "fourth ratio." It measures an active manager's excess return relative to a benchmark, divided by the consistency of that excess return:

Where tracking error is the standard deviation of (R_p − R_benchmark). A high information ratio means the manager not only beat the benchmark but did so consistently — a small information ratio with a high excess return suggests the outperformance came from one or two lucky bets rather than a repeatable process. Compared to Jensen's alpha (which measures the same excess but uses CAPM as the benchmark), the information ratio uses an actual investable benchmark.

Correlation — how two assets move together

Correlation measures how two investments move relative to each other, on a strict scale from −1.0 to +1.0:

• +1.0 — perfect positive correlation. The two assets move in the same direction by the same proportional amount. No diversification benefit.
• 0 — no linear relationship. Movements are unrelated.
• −1.0 — perfect negative correlation. Move in exactly opposite directions. In theory, a portfolio of two perfectly negatively correlated assets can be made riskless.

The key insight the exam tests: you do not need negative correlation to benefit from diversification. Any correlation below +1.0 provides some risk-reduction benefit. A portfolio of two assets with correlation +0.5 has lower volatility than the simple weighted average of the two assets' individual volatilities. As correlation falls, the benefit grows.

Correlation values and diversification benefit

The trap to watch for: the exam will say something like "this portfolio's two holdings have a correlation of +0.8" and ask whether diversification is helping. The answer is yes — even at +0.8, portfolio volatility is below the weighted average of the individual volatilities. Only perfect +1.0 correlation gives zero benefit.

Covariance — correlation's unscaled cousin

Covariance measures the same thing correlation does — whether two assets move together — but on an unscaled basis. The sign matches correlation: positive means they tend to move together, negative means they tend to move opposite. But the magnitude of covariance is meaningless on its own because it depends on the units (and standard deviations) of the underlying variables.

That's the formula tying them together: correlation is covariance normalized by the product of the two standard deviations, which strips out the unit dependence and produces a clean −1 to +1 scale. Covariance is the building block; correlation is the version you can interpret directly. The Series 66 rarely asks for covariance calculations, but it does ask you to recognize the relationship.

R-squared — is beta even meaningful?

R-squared (coefficient of determination) measures what percentage of a portfolio's movements can be explained by movements in its benchmark index. It ranges from 0 to 100 (or 0 to 1.0).

• R² = 100 — every movement is explained by the benchmark. The fund tracks the index perfectly (an index fund).
• R² ≥ 85 — beta is a meaningful measure of risk for this fund; the benchmark is appropriate.
• R² < 70 — beta is not a reliable risk measure for this fund. The benchmark doesn't capture what's driving its returns. Use standard deviation (and the Sharpe ratio) instead.

The deeper point. Beta only describes risk you can attribute to the benchmark. If a fund's R-squared is 45 against the S&P 500, less than half its variability comes from S&P movements — the rest comes from something else entirely (sector bets, security selection, alternative strategies). Reporting that fund's "beta to the S&P" is technically possible but practically misleading. R-squared is the gatekeeper for whether the beta number deserves any weight.

Diversification benefit as correlation falls