Financial Risk Management

Introduction, Risk Governance and Stock-Market Risk

The weights wᵢ represent capital proportions; covariance determines whether risks amplify or offset one another.

L is loss, defined as minus profit and loss; it may be negative when there is a gain. The confidence level α specifies the cumulative probability used to locate the quantile.

ES quantifies the average severity of losses in the tail; for discrete distributions a generalized tail mean definition is necessary.

A first-order one-period exposure formula: LGD is a loss fraction, and EAD is the exposure amount at default.

Capital is set aside for losses above average expected losses up to the stated tail quantile.

Expected return is linear in portfolio weights; risk is not.

Σ is the covariance matrix of individual security returns; w is the vector of portfolio weights.

For a one-period linear portfolio with normally distributed return: V₀ is value, μp return mean, σp return standard deviation, zα the standard-normal α quantile.

Beta measures portfolio sensitivity to a representative market index.

The minus sign denotes selling the index future to offset the positive market beta of long equities.

μ is instantaneous expected proportional drift; σ is volatility; W is a standard Brownian motion.

The drift correction −σ²/2 appears when simulating continuously compounded stock returns.

Expected return in excess of the risk-free rate per unit of total portfolio volatility.

Excess expected return per unit of systematic market exposure.

Foreign Exchange Risk and Corporate Hedging

Check the direction of the exchange-rate quotation before translating; inverse quotes require division.

The approximation is exact when q is home currency per foreign currency and the foreign-currency cash amount X stays fixed.

With an exchange rate $q$ quoted as home currency per unit of foreign currency and continuously compounded funding rates $r_h,r_f$, covered interest parity gives

The formula assumes matching maturities and conventions and frictionless access to the relevant funding and forward markets. Actual quotations may reflect a cross-currency basis and transaction costs.

Commodity Price Risk and Minimum-Variance Hedging

Sᵀ is the spot price at maturity; F₀,T is the delivery price contracted today.

Applies with continuous compounding, freely tradable underlying and no distributions or storage effects.

r financing rate, u storage cost rate, y convenience yield, T time to delivery.

Minimizes Var(ΔS + NΔF) for a linear hedge. Here ΔS is the monetary change of the entire exposure and ΔF the monetary change of one futures contract; using prices per unit instead requires adjustment for exposure quantity and contract size.

The regression slope of exposed value changes on futures changes gives hedge size with the opposite sign.

ρSF is the correlation of hedged exposure and futures value changes; R² measures the fraction of variance eliminated.

Interest Rate Risk, Duration and Banking-Book Exposure

F face value, y effective annual yield to maturity, T years to maturity.

Each payment time is weighted by its share of the bond price.

Discrete annually compounded convention; for other compounding conventions adjust the denominator accordingly.

C is price convexity, defined with a consistent yield convention; Δy is a decimal yield change.

PVBP (also DV01) is approximately the price loss from a one-basis-point yield rise for a positive-duration bond.

Under a normal-yield-shock and first-order duration model, use σ measured in basis points.

A and L denote present values of banking-book assets and liabilities; D terms their relevant effective duration.

Option-Related Risks and Greeks

Subtract the premium from the payoff to compute buyer profit; the seller receives the opposite payoff plus premium.

Assumes K₁ < K₂; the net premium is the lower-strike call cost less higher-strike call proceeds.

A long call and long put with a common strike generate a V-shaped terminal profit profile.

This is a local Taylor approximation. Here theta measures sensitivity to elapsed calendar time, volatility and rates are expressed as decimal inputs, and monetary exposures require position quantities. Higher-order and cross-effects may matter for large shocks.

Delta gives local exposure to underlying price; gamma is the change of that exposure with underlying price.

Sensitivity definitions rely on the pricing model and exact time/rate units.

Value at Risk Models and Market-Risk Measurement

α confidence level and h holding period specify the loss distribution being quantified.

With positive loss defined as negative P&L; sign convention for the mean must be checked.

Each Li is the loss after repricing the current holdings under a historical risk-factor scenario.

Generate M risk-factor paths or shocks and reprice the current portfolio in each scenario.

a contains portfolio P&L sensitivities to risk-factor shocks; Σ is their covariance matrix. This form assumes a zero P&L mean; otherwise subtract the expected P&L.

A complementary tail-risk statistic, particularly relevant when extremely bad outcomes are far beyond the VaR threshold.

The integral form of Expected Shortfall includes exactly the worst fraction of outcomes and remains valid for distributions with probability masses:

Here $u$ ranges over confidence levels, and the denominator is the tail probability. A simple conditional average above VaR can omit part of the boundary probability mass in a discrete distribution.

For a normal profit-and-loss variable with mean ${\mu }_{\Pi }$ and standard deviation ${\sigma }_{\Pi }$, the normal loss measures are

$z_{\alpha }$ is the standard-normal quantile and $\varphi$ its density. The normal-tail assumption is substantive: neither expression establishes that realised losses will be normally distributed.

The univariate EWMA forecast and a GARCH(1,1) specification are respectively

${ϵ}_t$ is the return innovation. For EWMA, $0<\lambda <1$ controls memory; in a conventional stationary GARCH model, $\omega >0$, $a,b\ge 0$ and $a+b<1$. These describe conditional variances, not a guarantee of normally distributed shocks.

Credit Risk, Ratings, Regulatory Capital and Portfolio Models

Separate probability of default from loss severity and the amount exposed at default.

The 12.5 factor equals the inverse of an 8% minimum total capital ratio; actual IRB computations include regulatory correlations, maturity adjustments and other adjustments.

Risk-neutral default probabilities are inferred from market prices, recovery assumptions and discount factors.

A price-implied risk-neutral default probability is generally not identical to the historical or physical probability of default.

xᵢ are measurable credit-quality variables; βᵢ are weights estimated or chosen by a credit modeller.

The logistic link maps a linear score to a probability between zero and one.

For a receiver operating characteristic (ROC) curve with default as the positive class, AUC is the area under that curve. This Gini expression must not be applied to the unnormalised area under a cumulative accuracy profile (CAP).

An institution needs credit portfolio scenarios to model dependence and extreme co-defaults.

A conditional annual default probability follows from cumulative default probabilities $F_t$:

The denominator is the probability of surviving to the start of year $t$. Cumulative and conditional probabilities should not be added or compared as if they represented the same horizon and population.

Credit Default Swaps and Asset-Backed Securities

A spread of 90 bps equals 0.90% annually; the actual timing and accrual depend on the contract.

Simplified cash settlement on a full notional N, ignoring accrued amounts and contractual nuances.

A is attachment point, D detachment point and Lpool total pool loss expressed in matching units.

Earnings at Risk and Cash Flow at Risk

C is constant annual cash coupon and r the discount yield. A change in r changes price even when C stays constant.

The expression gives the coupon rate. Multiply by notional and the accrual fraction to obtain a cash coupon. Frequent resets reduce rate-driven price exposure but make future coupon cash flows variable.

Liquidity Risk and Basel Liquidity Requirements

A standardized short-term liquidity stress measure; the general regulatory minimum is 100%.

Measures the extent to which asset and off-balance-sheet financing needs are supported by sufficiently stable funding; general minimum 100%.

For an LCR calculation with the standard 75% inflow cap, qualifying stressed outflows $O$ and inflows $I$ give

$N$ is the regulatory 30-day net cash outflow. Eligibility rules, haircuts, caps and permitted exceptions determine which amounts enter the calculation; this identity does not replace those rules. Both the liquidity buffer and the cash-flow amounts are measured in the same currency.