Quantitative finance is built around one central challenge: how to measure and control risk. Whether you’re evaluating a portfolio’s downside exposure or comparing risk-adjusted performance, understanding common risk metrics is crucial for both interviews and real-world trading.
🧠 1. What is Value at Risk (VaR)?
Definition:
Value at Risk estimates the maximum loss over a given time horizon at a specified confidence level.
Mathematically:
For a portfolio return $R_p$,
\(P(R_p < -VaR_\alpha) = \alpha\)
Example: A 1-day 95% VaR of $1 million means there’s a 5% chance the portfolio will lose more than $1M in a day.
Methods to estimate:
- Parametric (Variance–Covariance): assumes normality.
- Historical Simulation: uses empirical distribution of past returns.
- Monte Carlo Simulation: models return distribution via repeated sampling.
Limitations:
VaR ignores tail losses beyond the cutoff and is not subadditive (violates diversification principle).
📉 2. What is Conditional Value at Risk (CVaR)?
Definition:
Also called Expected Shortfall (ES), CVaR measures the average loss beyond the VaR threshold:
\(CVaR_\alpha = E[R_p | R_p < -VaR_\alpha]\)
Interpretation:
While VaR tells you the loss threshold, CVaR tells you the expected loss when things go wrong.
Advantages:
- Coherent risk measure (satisfies subadditivity).
- Better captures tail risk — especially important for non-normal returns.
Common in:
Stress testing, portfolio optimization under tail constraints.
📊 3. What is Maximum Drawdown?
Definition:
The largest peak-to-trough decline in portfolio value over time:
\(MDD = \max_{t} \left( \frac{P_{peak} - P_{trough}}{P_{peak}} \right)\)
Intuition:
Represents worst-case cumulative loss an investor would have experienced.
Usage:
- Measures path-dependent risk, not just endpoint losses.
- Common in hedge fund reporting and strategy backtests.
Drawdown-based metrics:
- Calmar Ratio: $\text{Return} / \text{Max Drawdown}$
- Pain Ratio: $\text{Average Return} / \text{Average Drawdown}$
⚖️ 4. What is the Sharpe Ratio?
Definition:
Measures risk-adjusted return using total volatility as the risk proxy:
\(Sharpe = \frac{E[R_p - R_f]}{\sigma_p}\)
Interpretation:
How much excess return you earn per unit of total risk.
Limitations:
- Penalizes upside volatility as well as downside.
- Sensitive to non-normal returns and autocorrelation.
Interview Tip:
If asked how to improve Sharpe, mention position sizing, diversification, and turnover control — not just maximizing raw return.
📈 5. What is the Sortino Ratio?
Definition:
A refinement of Sharpe that penalizes only downside volatility:
\(Sortino = \frac{E[R_p - R_f]}{\sigma_d}, \quad \sigma_d = \sqrt{E[(\min(R_p - R_f, 0))^2]}\)
Intuition:
Rewards asymmetric return profiles (strategies with limited downside but high upside).
Used in:
Performance evaluation for options, hedge funds, or asymmetric payoff strategies.
🔍 6. What is the Kelly Criterion?
Definition:
Maximizes the expected logarithm of wealth (geometric growth rate):
\(f^* = \frac{p(b+1) - 1}{b}\)
where:
- $p$: probability of win
- $b$: odds (profit per unit bet)
Continuous version:
\(f^* = \frac{\mu - r_f}{\sigma^2}\)
Intuition:
Determines the optimal fraction of capital to invest in a risky asset to maximize long-run growth.
Pros:
- Log-optimal, maximizes compounded returns.
Cons: - Highly sensitive to estimation error, can lead to extreme leverage if $\mu/\sigma$ is misestimated.
🧩 7. How do VaR, CVaR, and Drawdown differ conceptually?
| Measure | Focus | Type | Pros | Cons |
|---|---|---|---|---|
| VaR | Threshold loss | Quantile-based | Simple, widely used | Ignores tail losses |
| CVaR | Average tail loss | Tail-based | Coherent, tail-sensitive | Harder to estimate |
| Drawdown | Peak-to-trough | Path-based | Intuitive, dynamic | Not distributional |
📏 8. How are these measures applied in practice?
- Risk management: daily VaR/CVaR reporting for trading desks.
- Portfolio construction: minimize CVaR or drawdown instead of variance.
- Hedge fund analytics: Sortino and Calmar ratios for performance review.
- Trading strategy sizing: Kelly or fractional Kelly for leverage control.
Quant Tip: Many modern strategies use CVaR optimization as a robust alternative to variance-based mean–variance optimization.
🧮 9. What are coherent risk measures?
A risk measure $\rho(X)$ is coherent if it satisfies:
- Monotonicity: If $X_1 \le X_2$, then $\rho(X_1) \ge \rho(X_2)$
- Translation Invariance: $\rho(X + a) = \rho(X) - a$
- Positive Homogeneity: $\rho(\lambda X) = \lambda \rho(X)$
- Subadditivity: $\rho(X + Y) \le \rho(X) + \rho(Y)$
VaR fails subadditivity; CVaR satisfies all four — hence it’s coherent.
🚀 10. How to combine return and risk metrics effectively?
For interviews, emphasize that risk and return are inseparable:
- Sharpe / Sortino: Risk-adjusted reward.
- VaR / CVaR: Tail exposure.
- Drawdown: Path-dependent resilience.
- Kelly: Optimal growth under repeated bets.
In portfolio design, practitioners often balance these through multi-objective optimization, e.g. maximizing Sharpe while constraining CVaR and drawdown.