Statistics Distributions And Portfolio Metrics

40 essential concepts in portfolio statistics and risk metrics

What You'll Learn

Master portfolio statistics, risk metrics, and distribution analysis with 40 comprehensive flashcards. Learn Sharpe Ratio, kurtosis, skewness, correlation, diversification, and portfolio variance calculations for quantitative finance.

Key Topics

  • Sharpe Ratio calculation and annualization formulas
  • Distribution analysis: kurtosis, skewness, and normality tests
  • Portfolio variance and correlation mathematics
  • Diversification principles and risk metrics

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How to study this deck

Start with a quick skim of the questions, then launch study mode to flip cards until you can answer each prompt without hesitation. Revisit tricky cards using shuffle or reverse order, and schedule a follow-up review within 48 hours to reinforce retention.

Preview: Statistics Distributions And Portfolio Metrics

Question

What is the formula for the Sharpe Ratio?

Answer

Sharpe Ratio = (Mean Return - Risk-Free Rate) / Standard Deviation of Returns. When risk-free rate ≈ 0: Sharpe Ratio = Mean Return / Standard Deviation

Question

How do you annualize a daily Sharpe Ratio?

Answer

Annual Sharpe Ratio = Daily Sharpe Ratio × √252. Note: 252 is the number of trading days in a year, and √252 ≈ 15.87

Question

How do you annualize daily volatility (standard deviation)?

Answer

Annual Volatility = Daily Volatility × √252. Volatility scales with the square root of time because variance is additive.

Question

How do you annualize daily returns?

Answer

Annual Return = (1 + Daily Return)^252 - 1. Returns compound, so you CANNOT just multiply by 252. Example: 0.08% daily = (1.0008)^252 - 1 ≈ 21.9% annually

Question

Why do we multiply by √252 for volatility but compound returns differently?

Answer

Volatility (standard deviation) scales with √time because variance is additive over time (σ²_annual = 252 × σ²_daily). Returns compound multiplicatively because each day's return builds on the previous balance.

Question

What does a Sharpe Ratio of 1.5 indicate?

Answer

Good risk-adjusted performance. General guidelines: <0 = losing money, 0-1 = mediocre, 1-2 = good, >2 = excellent, >3 = exceptional (rare)

Question

What is kurtosis and what does high kurtosis mean?

Answer

Kurtosis measures the 'fatness' of distribution tails. High kurtosis (>3) = fat tails = extreme events happen MORE FREQUENTLY than a normal distribution predicts. This applies to both positive and negative extremes.

Question

What is skewness and what does negative skew mean?

Answer

Skewness measures asymmetry of a distribution. Negative skew = left tail is longer = occasional large losses outweigh gains = when you lose, losses are bigger than typical gains. The distribution leans right with a long left tail.

Question

What's the difference between kurtosis and skewness?

Answer

Kurtosis = HOW OFTEN extremes occur (frequency). Skewness = WHICH DIRECTION extremes lean (asymmetry). High kurtosis = more extremes (both ways). Negative skew = losses are bigger than gains when they occur.

Question

What does the Jarque-Bera test do?

Answer

Tests whether returns follow a normal distribution by examining both skewness and kurtosis. Rejection (high test statistic) means returns are NOT normally distributed. This does NOT mean returns are untrustworthy—just non-normal.

Question

If a strategy fails the Jarque-Bera test, what does that mean?

Answer

Returns don't follow a normal distribution. This means: (1) Be cautious using normal-distribution-based tools, (2) Standard deviation may not fully capture risk, (3) Need additional risk metrics. It does NOT mean the returns are fake or the strategy is bad.

Question

Why is standard deviation limited for non-normal distributions?

Answer

Standard deviation only measures average dispersion from the mean and treats all deviations equally (upside and downside). For non-normal distributions (fat tails, skew), it misses tail risk and asymmetric risk. It underestimates the probability and magnitude of extreme losses.

Question

What is the limitation of the Sharpe Ratio?

Answer

Assumes returns are normally distributed and treats upside volatility the same as downside volatility. Two strategies with identical Sharpe Ratios can have very different risk profiles if one has skew or kurtosis. It doesn't capture tail risk or drawdown risk.

Question

What is the formula for portfolio variance with two assets?

Answer

σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂, where w = weights, σ = standard deviations, ρ = correlation. The last term is the covariance term.

Question

What happens to portfolio volatility when correlation = 1.0?

Answer

Portfolio volatility = weighted average of individual volatilities. There is ZERO diversification benefit. Formula: σ_p = w₁σ₁ + w₂σ₂. Assets move perfectly together.

Question

What happens to portfolio volatility when correlation = 0?

Answer

Portfolio gets diversification benefits. The covariance term in the variance formula equals zero, so σ²_p = w₁²σ₁² + w₂²σ₂². Portfolio volatility will be lower than the weighted average of individual volatilities.

Question

What happens to portfolio volatility when correlation = -1.0?

Answer

Maximum diversification benefit. With proper weights, portfolio volatility can theoretically be reduced to zero. Assets move in perfect opposite directions, creating a perfect hedge.

Question

Does adding more stocks to a portfolio always reduce volatility?

Answer

NO. Adding stocks reduces volatility ONLY when correlation < 1.0. If correlation = 1.0 (perfect positive correlation), adding more stocks provides zero diversification benefit. Portfolio volatility would just be the weighted average.

Question

What is the relationship between correlation and covariance?

Answer

Covariance = Correlation × σ₁ × σ₂. They move together: low correlation means low covariance (in magnitude), high correlation means high covariance. Correlation is just standardized covariance (always between -1 and +1).

Question

If you lower correlation while keeping everything else constant, what happens to portfolio volatility?

Answer

Portfolio volatility ALWAYS decreases. Lower correlation reduces the covariance term (2w₁w₂ρσ₁σ₂) in the portfolio variance formula, which lowers total variance and thus volatility.

Question

What is the principle of diminishing marginal returns in diversification?

Answer

The first few assets added to a portfolio provide massive diversification benefits. Each additional asset still helps, but less and less. Example: going from 1 to 5 stocks is huge; going from 50 to 55 stocks barely matters. Most benefits captured with 20-30 stocks.

Question

Why is Option B (US stocks, international stocks, bonds, real estate, commodities) better diversification than Option A (10 US tech stocks)?

Answer

Different asset classes have LOW CORRELATION with each other (maybe 0.2-0.5). They respond to different economic factors. Tech stocks have HIGH CORRELATION with each other (0.7-0.9) because they share similar risk factors. Lower correlation = better diversification.

Question

What is tail risk?

Answer

The risk of extreme negative outcomes (the 'tails' of the distribution). Fat tails (high kurtosis) mean tail risk is higher than normal distribution would predict. Standard deviation underestimates tail risk for non-normal distributions.

Question

If two portfolios have identical mean returns and standard deviations, are they equally attractive?

Answer

NOT necessarily. They have identical Sharpe Ratios, but could have very different risk profiles. One might have negative skew (occasional large losses) or high kurtosis (frequent extremes). Investors care about the full distribution shape, not just mean and standard deviation.

Question

What's the formula for daily return?

Answer

Daily Return = (P_t / P_{t-1}) - 1, where P_t is today's price and P_{t-1} is yesterday's price. Can also be written as: (P_t - P_{t-1}) / P_{t-1}

Question

What's the formula for cumulative return?

Answer

Cumulative Return = (P_t / P_0) - 1, where P_t is the final price and P_0 is the initial price. This is the total return over the entire period.

Question

What does 'risk-adjusted return' mean?

Answer

Return per unit of risk taken. It's not just about how much you made, but how much risk you took to make it. Sharpe Ratio is the most common measure: return / volatility. Higher Sharpe = better risk-adjusted performance.

Question

In the portfolio variance formula, which term can be negative?

Answer

The covariance term: 2w₁w₂ρσ₁σ₂. It's negative when correlation (ρ) is negative, meaning assets move in opposite directions. This REDUCES total portfolio variance. The individual variance terms (w²σ²) are always positive.

Question

Can portfolio volatility be lower than ALL individual asset volatilities?

Answer

YES, when assets have low or negative correlation. The diversification benefit from the negative covariance term can reduce portfolio volatility below even the least volatile individual asset. Example: two 20% volatility assets with ρ=-0.5 can create a portfolio <20% volatility.

Question

What does it mean when assets are 'uncorrelated' (ρ ≈ 0)?

Answer

Assets move independently—no linear relationship. When one goes up, the other might go up, down, or stay flat with no pattern. This provides good diversification because losses in one don't predict losses in the other.

Question

Why do we use 252 days for annualization, not 365?

Answer

252 is the approximate number of TRADING days in a year (markets closed weekends and holidays). Financial calculations use trading days because returns only happen when markets are open. Using 365 would underestimate annualized figures.