Chartered Alternative Investment Analyst (CAIA) Quantitative Methods, Statistics, and Risk Foundations Syllabus

How can return smoothing (stale pricing) distort reported alternative-asset statistics?

Infrequent or appraisal-based valuations smooth returns, artificially lowering reported volatility and correlations and inducing positive autocorrelation, making the asset look less risky and more diversifying than it is.

What is the formula for covariance between two assets' returns?

Cov(X,Y) = E[(X − E[X])(Y − E[Y])], estimated as Σ(Xi − X̄)(Yi − Ȳ)/(n − 1). It measures how two return series move together; units are squared returns.

How is the correlation coefficient defined in terms of covariance?

ρ(X,Y) = Cov(X,Y) / (σX · σY). It is a standardized, unitless measure of linear co-movement bounded between −1 and +1.

What is autocorrelation (serial correlation) of returns?

The correlation of a return series with its own lagged values. Positive first-order autocorrelation means high returns tend to follow high returns; it often signals illiquidity or return smoothing in alternatives.

What problem does positive autocorrelation cause for volatility estimates?

It biases standard deviation downward when annualizing, because the simple √-time scaling assumes independent returns; positive autocorrelation means true multi-period volatility is higher than naively scaled.

What does a correlation of zero between two assets tell you, and what is its limitation?

Zero correlation indicates no linear relationship. Limitation: it does not rule out nonlinear dependence, and correlations can rise toward 1 during crises (tail dependence).

How do you annualize a standard deviation computed from monthly returns?

Multiply the monthly standard deviation by √12 (assuming independent returns). For periodic data with p periods per year, multiply by √p.

What is Value at Risk (VaR)?

VaR is the maximum expected loss over a given horizon at a specified confidence level (e.g., 95%). A 95% one-month VaR of $1M means there is a 5% chance of losing more than $1M in a month.

What is a key shortcoming of VaR, and how does Conditional VaR (Expected Shortfall) address it?

VaR says nothing about the size of losses beyond the threshold and is not sub-additive. CVaR (Expected Shortfall) measures the average loss given that the loss exceeds VaR, capturing tail severity and being coherent.

What is semi-variance / downside deviation?

A risk measure that considers only returns below a target (or the mean), capturing downside dispersion while ignoring upside variability. Downside deviation is its square root and is used in the Sortino ratio.

What is maximum drawdown?

The largest peak-to-trough decline in cumulative value over a period, measuring the worst loss an investor would have suffered buying at a peak and selling at the subsequent trough.

What is the formula and meaning of the Sharpe ratio?

Sharpe = (Rp − Rf) / σp. It measures excess return per unit of total risk (standard deviation). Higher is better; it is the most common risk-adjusted performance ratio.

How does the Sortino ratio differ from the Sharpe ratio?

Sortino = (Rp − target) / downside deviation. It replaces total standard deviation with downside deviation, penalizing only harmful (below-target) volatility — better for asymmetric/non-normal returns.

What is the Treynor ratio and how does it differ from Sharpe?

Treynor = (Rp − Rf) / βp. It measures excess return per unit of systematic (market) risk (beta) rather than total risk, appropriate for well-diversified portfolios.

What is the information ratio?

Information ratio = (Rp − Rb) / tracking error, where Rb is the benchmark return and tracking error is the standard deviation of active (excess) returns. It measures active return per unit of active risk.

What is Jensen's alpha?

Jensen's alpha = Rp − [Rf + βp(Rm − Rf)]. It is the portfolio's return in excess of what CAPM predicts given its beta; positive alpha indicates outperformance not explained by market risk.

State the steps of a hypothesis test.

1) State null (H0) and alternative (H1) hypotheses. 2) Choose a significance level (α). 3) Select a test statistic and its distribution. 4) Compute the statistic. 5) Compare to critical value / p-value. 6) Reject or fail to reject H0.

Define Type I and Type II errors.

Type I error: rejecting a true null hypothesis (false positive), probability = α. Type II error: failing to reject a false null hypothesis (false negative), probability = β. Power = 1 − β.

What is a p-value?

The probability of obtaining a test statistic at least as extreme as observed, assuming the null hypothesis is true. If the p-value < α, reject the null hypothesis.

What is a t-statistic used for in testing a mean or regression coefficient?

t = (estimate − hypothesized value) / standard error. It tests whether a parameter (e.g., a mean return or regression coefficient/alpha) differs significantly from a hypothesized value (often zero).

What does the R-squared of a regression measure?

R² is the proportion of the variance in the dependent variable explained by the independent variable(s); it ranges from 0 to 1. Higher R² means the model explains more of the return variation.

In a single-factor regression Ri = α + β·Rm + ε, what do α, β, and ε represent?

α is the intercept (return independent of the factor, i.e., skill/alpha); β is the sensitivity of the asset's return to the factor (systematic exposure); ε is the residual (idiosyncratic, unexplained return).

What is multi-factor (factor) analysis used for in alternatives?

To decompose returns into exposures to multiple systematic risk factors (e.g., equity, rates, credit, size, value, momentum), identifying the sources of return and separating true alpha from disguised beta.

What is the difference between an empirical model and a theoretical model of returns?

A theoretical model (e.g., CAPM) is derived from economic assumptions and equilibrium reasoning. An empirical model (e.g., Fama-French) is built by observing data and identifying factors that statistically explain returns, without requiring a full theoretical derivation.

What distinguishes alpha from beta as drivers of return?

Beta is return earned by bearing systematic market/factor risk (compensated exposure). Alpha is excess return from skill, mispricing, or unique strategy, independent of systematic factors and not explained by beta exposures.

What is 'alternative beta' or 'exotic beta'?

Returns that appear to be skill-based alpha but are actually compensation for systematic exposure to less common risk factors (e.g., liquidity, volatility, carry). It is replicable and should not command alpha-level fees.

What is ex ante alpha versus ex post alpha?

Ex ante alpha is the expected abnormal return anticipated before the fact (a forecast of skill). Ex post alpha is the realized abnormal return measured after the fact from actual data.

What is tracking error and what does it indicate?

Tracking error is the standard deviation of the difference between a portfolio's returns and its benchmark's returns. It quantifies how closely a portfolio follows its benchmark; high tracking error means more active deviation.

What are key challenges in benchmarking alternative investments?

Lack of investable/representative indices, self-reporting and survivorship/backfill biases, illiquidity and stale pricing, heterogeneity of strategies, and difficulty finding a passive replicable benchmark — making performance attribution unreliable.

What are survivorship bias and backfill bias in alternative-investment indices?

Survivorship bias: failed funds drop out, inflating average reported returns. Backfill (instant-history) bias: a fund's prior good track record is added when it joins the database, also overstating performance.

What is the difference between an investable and a non-investable benchmark?

An investable benchmark can actually be held/replicated by investors (with realistic capacity and liquidity). A non-investable benchmark (common in hedge funds/PE) reflects an average of reported returns that cannot be directly purchased, limiting its usefulness for evaluation.