Korea Financial Risk Manager cheat sheet

Risk is the volatility of unexpected outcomes; it is distinct from uncertainty, which cannot be quantified with a probability distribution.

Expected shortfall (ES) is the probability-weighted average loss beyond the VaR threshold and is subadditive; VaR is not subadditive in general.

Coherent risk measures satisfy monotonicity, translation invariance, positive homogeneity, and subadditivity; ES is coherent, VaR fails subadditivity.

Enterprise risk management aggregates market, credit, operational, and liquidity risks under a single capital and governance framework rather than managing silos independently.

Systematic risk is undiversifiable and priced via beta; idiosyncratic risk is diversifiable and earns no risk premium in equilibrium.

The Sharpe ratio divides excess return by total volatility; the Treynor ratio divides excess return by beta and is appropriate for well-diversified portfolios.

Jensen's alpha is the difference between realized portfolio return and the CAPM-predicted return given beta and the market risk premium.

Basel III Pillar 1 covers minimum capital, Pillar 2 supervisory review, and Pillar 3 market discipline through disclosure requirements.

Risk appetite is the aggregate level of risk the firm accepts; risk tolerance is the acceptable variation around that target.

Variance of a two-asset portfolio is w1^2sigma1^2 + w2^2sigma2^2 + 2w1w2rhosigma1sigma2; covariance = rhosigma1*sigma2.

Covariance is unbounded and scale-dependent; correlation is bounded in [-1,1] and is the standardized covariance.

OLS estimators are BLUE under Gauss-Markov assumptions: linearity, exogeneity, homoskedasticity, no autocorrelation, and no perfect collinearity.

Heteroskedasticity leaves OLS coefficients unbiased but invalidates standard errors; use White or robust standard errors.

R-squared always rises when regressors are added; adjusted R-squared penalizes added regressors by degrees of freedom.

Omitted variable bias arises when a relevant regressor is excluded and it is correlated with included regressors.

A unit root implies non-stationarity; differencing once often restores stationarity, and regressions on I(1) variables risk spurious correlation.

Maximum likelihood estimates parameters maximizing the likelihood function; the log-likelihood is typically maximized numerically via Newton-Raphson.

The central limit theorem states that sample means approach normality as sample size grows, regardless of the underlying distribution.

A forward contract is a customized OTC agreement to trade at a set price on a future date; a futures contract is standardized, exchange-traded, and marked to market daily.

The cost-of-carry model prices a forward as F = S*e^((r - q + c)*T), where q is the dividend or convenience yield and c is storage cost.

A long call pays max(S_T - K, 0); a long put pays max(K - S_T, 0). The premium is paid upfront.

Put-call parity for European options on non-dividend stocks is C - P = S - Ke^(-rT).

An interest rate swap exchanges fixed for floating payments; the swap's value equals the difference between the two legs' present values.

A currency swap exchanges principal and interest in two currencies; the principal amount is exchanged at inception and at maturity.

A repo is a collateralized short-term borrowing in which the seller transfers securities and agrees to repurchase them at a slightly higher price.

Duration measures a bond's price sensitivity to yield; convexity captures the curvature and improves the estimate for large yield changes.

A credit default swap transfers default risk from the protection buyer to the protection seller in exchange for periodic premiums.

The binomial option pricing model prices a derivative by risk-neutral discounting: value = e^(-rT)(p*u + (1-p)d), with p = (e^(rT) - d)/(u - d).

Black-Scholes-Merton assumes lognormal returns, constant volatility, no arbitrage, continuous trading, and frictionless markets.

The Black-Scholes call formula is C = SN(d1) - Ke^(-r*T)*N(d2); the put follows from put-call parity.

Delta is the first derivative of option value with respect to the underlying price; gamma is the second derivative.

Vega measures sensitivity to volatility; theta measures sensitivity to the passage of time; rho measures sensitivity to interest rates.

The delta-normal approach approximates portfolio VaR as zsigma_pV, assuming linear exposures and normal returns.

Historical simulation reprices the portfolio under past scenarios; it captures fat tails and non-linearity but relies on the historical window.

Monte Carlo simulation generates random paths for risk factors and reprices the portfolio; it is flexible but computationally intensive.

The GARCH(1,1) model expresses conditional variance as omega + alphaepsilon^2 + betasigma^2, capturing volatility clustering.

VaR at the 99% confidence level over a one-day horizon is the loss that will not be exceeded with 99% probability.

The Basel internal models approach multiplies the greater of the previous day's VaR and the 60-day average VaR by a multiplier of at least 3, using a 10-day holding period and 99% confidence.

Stressed VaR uses a 12-month stress period of significant losses and is added to the standard VaR charge.

Backtesting compares daily P&L against VaR; the Basel traffic light system uses 250 trading days and escalates the multiplier with the number of exceptions.

The square-root-of-time rule scales VaR by the square root of the horizon, assuming i.i.d. returns.

Incremental VaR measures the change in portfolio VaR from adding or removing a position; component VaR sums to total VaR.

Duration-based VaR approximates interest rate exposure as modified duration times the yield change times the portfolio value.

Stress testing evaluates losses under extreme but plausible scenarios that VaR's historical window may not capture.

Expected loss equals PD times LGD times EAD; unexpected loss is the volatility of loss around the expected value and drives economic capital.

The Merton structural model treats equity as a call option on firm assets with strike equal to debt; default occurs when asset value falls below the default barrier.

In the KMV approach, distance to default equals (ln(V/D) + (mu - 0.5*sigma^2)T)/(sigmasqrt(T)).

Under Basel IRB, risk-weighted assets depend on PD, LGD, EAD, and maturity; the correlation formula is a function of PD.

Credit ratings map to default probabilities; migration risk is the risk that a rating downgrade increases credit spreads.

Credit value at risk measures the loss distribution of a credit portfolio over a horizon, incorporating default and migration.

The credit spread compensates investors for expected loss, liquidity, and risk premium; it widens as credit quality deteriorates.

Recovery rate and LGD sum to one; senior secured debt typically has higher recovery rates than subordinated unsecured debt.

Wrong-way risk occurs when exposure to a counterparty increases as the counterparty's credit quality deteriorates.