A hedge fund reports 12% annualized returns with a 10% standard deviation. The Sharpe ratio measures return per unit of risk: average return minus the risk-free return, divided by the standard deviation of returns. With a 2% risk-free rate, this fund scores (12% - 2%) / 10% = 1.0, which looks strong. Then you check skewness: -1.8. Excess kurtosis: 6.2, which means a kurtosis of 9.2. These numbers tell you two things about the fund's returns. Large negative deviations dominate the asymmetry.
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.
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. Listing standard deviation as the second moment is incorrect; 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.
- Treating every positive autocorrelation as smoothing. Positive autocorrelation can come from genuine trending, barriers to trade, or appraisal-based prices. When autocorrelations are positive at every lag, the direct implication concerns horizons: multiperiod volatility exceeds single-period volatility times √T. Understated volatility is a separate conclusion.
Bottom line
- Four moments in order: mean, variance, skewness, kurtosis (standard deviation is the square root of variance, not a moment itself); zero skewness does not prove symmetry; leptokurtic = excess kurtosis > 0, mesokurtic = 0, platykurtic < 0; moments signal...
- Sample covariance = sum of deviation cross-products / (n - 1); correlation = covariance / (SD_A x SD_B), always between -1 and +1; +1 gives no diversification, -1 allows a zero-risk hedge
- Log returns and the lognormal distribution fix simple returns' -100% floor: log returns can be normal over any horizon, simple returns cannot
- Beta = correlation x SD_asset / SD_market: an average response, a regression slope, and additive across a portfolio; market beta is 1, risk-free beta is 0
Exam shortcut
When a question mentions a hedge fund or illiquid asset and asks about autocorrelation, first identify the cause. If the scenario establishes that the valuation process smooths returns, for example through infrequent appraisals or stale marks that lag true values, then reported volatility is understated. A pure reporting delay that only shifts observations later does not establish this on its own.
The full lesson (about 8,737 words, 58 min read) adds 2 worked examples, all 8 common mistakes, a self-check, free in the app.
Learning objectives
- defining alts
- blurred lines
- risk return characteristics
- goals
- history us
- history asia
- 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
- binomial trees
- single-factor default-free bond models
- reduced-form modeling credit risk
- single-factor equity pricing models
- single factor models
- hypothesis testing
- sampling problems
- forwards vs futures
- forward foundations
- forwards on rates
- carry forwards
- managing long short
- option exposures
- rate options
- option pricing
- risk measures
- var
- benchmarking
- ratio measures
- risk adjusted
- estimating alpha
- pricing data
- appraisals smoothing
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