Skip to content

CFA Level I · CFA Level I Exam

Statistical Characteristics of Asset Returns: CFA Level I Guide

Statistical characteristics of asset returns are the numbers that describe a return series: its centre (mean, median, mode), spread (variance, standard deviation), shape (skewness, kurtosis) and how it moves with other assets (covariance, correlation). You solve questions by picking the right measure, applying its formula, and interpreting the result.

What this chapter covers

This chapter teaches you how to summarise a set of returns with a few numbers. You start with the centre of the data, then its spread, then its shape, then how to read it in charts and quantiles, and finally how two return series move together.

The tools here are the base for the rest of Quantitative Methods. Probability, sampling, hypothesis testing and regression all assume you can read a mean, a standard deviation and a correlation. Without them, those later chapters feel like memorised formulas.

The chapter also feeds other topics. Portfolio Construction uses variance, covariance and correlation to explain diversification. Equities and Fixed Income use returns and risk measures. Derivatives and Risk Management uses percentiles and tail behaviour. Fat tails and negative skew also help you judge why a normal-distribution assumption can understate risk.

Quantitative Methods carries a meaningful share of the 180 questions, and this chapter supplies ideas that reappear in Portfolio Construction and other topics. The questions are mostly short calculations and interpretation items, which suit the three-option format. If you can compute a geometric mean, a sample standard deviation or a correlation quickly on your BA II Plus or HP 12C, you gain marks at about 90 seconds per question. You can also eliminate wrong options by checking direction: for example, whether the mean is above the median in a positively skewed set. There is no minimum score per topic, but steady marks here protect your total against the 1600 scale-score pass mark.

Statistical Characteristics of Asset Returns: topics in the order to study them

  1. 1Measures of Central TendencyEvery later measure is built around the mean, so you learn arithmetic, geometric, harmonic and weighted means first.
  2. 2Measures of Dispersion and RiskVariance and standard deviation measure spread around the mean you just learned, and they are needed for skewness and correlation.
  3. 3Skewness and Kurtosis of ReturnsShape measures use deviations and standard deviation, and they show where mean, median and mode sit relative to each other.
  4. 4Quantiles, Percentiles and Data VisualizationOnce you know centre, spread and shape, quartiles, percentiles and charts give you ways to see and describe the same data.
  5. 5Correlation and Covariance of ReturnsThis topic extends spread to two variables, so it comes last and links directly to diversification and regression.

How to prepare Statistical Characteristics of Asset Returns

Aim for a clear idea of what each measure tells you, then speed on the calculations. Most marks are lost on choosing the wrong measure or the wrong divisor, not on arithmetic.

  1. Read each topic once for meaning. For every measure, write one sentence on what it tells you and when it is the right choice.
  2. Learn the formulas in plain form, including the difference between population (divide by N) and sample (divide by n − 1) versions.
  3. Practise the calculator workflow. On the BA II Plus, use the 2nd DATA statistics worksheet (enter data, then read the mean and standard deviation, noting which of Sx and σx is sample or population). On the HP 12C, use Σ+ and the mean and standard deviation functions.
  4. Do small numerical sets by hand until you can check a calculator answer for reasonableness, for example that a geometric mean sits below the arithmetic mean when returns vary.
  5. Practise interpretation questions: positive or negative skew, excess kurtosis, and what a correlation of a given sign or size implies. Use direction checks to remove two options.
  6. Finish with mixed timed sets of three-option questions, about 90 seconds each, and log every error by type: wrong formula, wrong divisor or wrong interpretation.

Common mistakes in Statistical Characteristics of Asset Returns

  • Using the arithmetic mean to describe multi-period compound growth.

    Fix: If the question asks for the compound rate over several periods, use the geometric mean of (1 + R) and subtract 1.

  • Dividing by N instead of n − 1, or the reverse.

    Fix: Check the wording first. A sample of returns uses n − 1. A complete population uses N. Check which standard deviation your calculator is showing.

  • Mixing up skew direction with where the long tail sits.

    Fix: Name skew by the long tail. A long right tail is positive skew, and the mean is pulled above the median.

  • Treating a high kurtosis as high spread.

    Fix: Standard deviation measures overall spread. Kurtosis measures tail weight relative to a normal distribution. Compare using excess kurtosis.

  • Reading correlation as causation or as the strength of a non-linear link.

    Fix: Remember that correlation measures only linear association. Interpret sign and size, and avoid claiming one variable drives the other.

  • Treating covariance as comparable across data sets.

    Fix: Use covariance for its sign and for portfolio variance formulas. Use correlation when you need to compare the strength of relationships.

Last-day revision: Statistical Characteristics of Asset Returns

  • Arithmetic mean = Σx ÷ n. Geometric mean of returns = [Π(1 + R)]^(1/n) − 1.
  • For returns that vary, the geometric mean is lower than the arithmetic mean. They are equal only when all returns are identical.
  • Harmonic mean = n ÷ Σ(1/x), used for averaging prices paid with equal money amounts per period.
  • Population variance divides by N. Sample variance divides by n − 1.
  • Standard deviation = √variance and is in the same units as the returns.
  • Coefficient of variation = standard deviation ÷ mean, a measure of risk per unit of return. The mean must be nonzero (and positive for the usual interpretation).
  • For a typical unimodal distribution, positive skew gives mean > median > mode and negative skew gives mean < median < mode.
  • Excess kurtosis = kurtosis − 3. A positive value means fatter tails than a normal distribution (leptokurtic).
  • Quartiles split ordered data into four parts, and the interquartile range is Q3 − Q1.
  • Covariance = Σ(X − mean X)(Y − mean Y) ÷ (n − 1) for a sample. Its size depends on units.
  • Correlation = Cov(X, Y) ÷ (σX × σY) and always lies from −1 to +1.
  • Correlation shows linear association only. It does not prove causation, and a zero correlation can hide a non-linear link.

Statistical Characteristics of Asset Returns practice questions

Statistical Characteristics of Asset Returns in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Statistical Characteristics of Asset Returns: frequently asked questions

What topics are in the statistical characteristics of asset returns chapter?

The chapter covers measures of central tendency, measures of dispersion and risk, skewness and kurtosis, quantiles and data visualization, and correlation and covariance. Together they describe the centre, spread, shape and co-movement of returns.

Do I need a calculator for this chapter on the CFA Level I exam?

Yes, it helps. You may use the TI BA II Plus or HP 12C, and both can compute a mean and standard deviation from a data set. Practise entering data and reading the correct sample or population result, because that is where errors happen.

What is the difference between covariance and correlation?

Covariance shows whether two return series move in the same direction, but its size depends on the units. Correlation divides covariance by the product of the two standard deviations, so it always falls between −1 and +1 and is easier to compare.

How do I remember the effect of skewness on the mean and median?

Follow the long tail. In positive skew, the tail is on the right and the mean is above the median. In negative skew, the tail is on the left and the mean is below the median.