A hedge fund reports 12% annualized returns with a 10% standard deviation. Looks great on a Sharpe ratio basis. Then you check skewness: -1.8. Excess kurtosis: 6.2. That "10% risk" is hiding a left tail three times fatter than a normal distribution predicts. The standard deviation told you nothing about the losses that actually matter.
Every return analysis starts with a basic question: are you looking forward or backward? Ex ante distributions represent the probability-weighted set of possible future outcomes. You build these from models, assumptions, and forecasts. Ex post distributions are constructed from realized historical returns after the fact.
The distinction matters because ex ante reflects expectations and uncertainty. Ex post reflects what actually happened. A fund's historical return distribution is just one possible outcome of what the ex ante distribution predicted. They will almost never match.
KEY: Ex ante means "before the event" (forward-looking). Ex post means "after the event" (historical). The exam tests whether you can identify which one a scenario describes.
Common mistakes
- Confusing variance with standard deviation as the second moment. The four moments are mean, variance, skewness, and kurtosis. Standard deviation is the square root of variance, not the second moment itself. When a question lists "mean, standard deviation, skewness, kurtosis" as the four moments, that is the trap answer. The correct sequence uses variance.
- Forgetting to convert percentages to decimals in the correlation formula. If Fund A has a standard deviation of 8% and Fund B has 15%, and the covariance is 0.0048, you must use 0.08 and 0.15. Using 8 and 15 gives 0.0048 / 120 = 0.00004.
- Claiming autocorrelation overstates volatility. It is the opposite. Positive autocorrelation from return smoothing understates true volatility. The standard deviation of smoothed returns is artificially low. After adjusting for autocorrelation, estimated volatility increases. Any answer choice saying "overstated volatility" when autocorrelation is positive is wrong.
Bottom line
- Four moments in order: mean, variance, skewness, kurtosis: standard deviation is the square root of variance, not a moment itself
- Correlation = covariance / (SD_A x SD_B): always between -1 and +1; convert percentages to decimals before dividing
- Log returns and the lognormal distribution fix simple returns' -100% floor: log returns can be normal over any horizon, simple returns cannot
- Negative skewness means a longer left tail (larger losses); leptokurtic = excess kurtosis > 0 (fat tails), mesokurtic = 0, platykurtic < 0
Exam shortcut
When a question mentions a hedge fund or illiquid asset and asks about the impact of autocorrelation, the answer is always the same direction: volatility understated, Sharpe ratio inflated, beta understated. If the question asks what happens after adjusting, reverse each one: volatility increases, Sharpe ratio decreases, beta increases. Durbin-Watson cheat: anchor on 2. Below 1 means positive autocorrelation (the smoothed alt-investment default). Above 3 means negative autocorrelation (mean-reversion).
The full lesson (about 4,378 words, 29 min read) adds 2 worked examples, all 8 common mistakes, a self-check, free in the app.
Learning objectives
- defining alts
- blurred lines
- history us
- history asia
- risk return characteristics
- goals
- buy sell side
- service providers
- legal structures
- fund types
- fund features
- fund terms
- drawdown fees
- waterfall calcs
- hedge fund fees
- fees and behavior
- return math
- irr
- irr problems
- modified irr
- other measures
- j curve
- notional principal
- return distributions
- moments
- covariance correlation
- beta autocorrelation
- std dev variance
- normality testing
- market efficiency
- time value
- forward rates
- arbitrage
- binomial trees
- single factor models
- hypothesis testing
- sampling problems
- forwards vs futures
- forward foundations
- forwards on rates
- carry forwards
- managing long short
- option exposures
- rate options
- rate swaps
- option pricing
- risk measures
- var
- benchmarking
- ratio measures
- risk adjusted
- pricing data
- appraisals smoothing
- alpha beta overview
- estimating alpha
- return attribution
- statistical issues
Browse all free CAIA Level I lessons or jump into free CAIA Level I practice questions.