Measuring and Managing Market Risk: formula sheet

Key formulas

Interpreting VaR

Probability(loss ≥ VaR) = 1 − confidence level, over the stated horizon

A 95% confidence VaR is the same as a 5% VaR. The loss is a minimum in the tail, not a maximum.

Parametric VaR (normal returns, return form)

VaR = [−(μ − z × σ)] = z × σ − μ, where z is the critical value

Common one-tailed z values: 1.645 for 5% and 2.33 for 1%. Multiply by portfolio value for a money figure.

Money VaR

VaR (₹) = VaR (%) × portfolio value

Use the starting portfolio value unless the vignette says otherwise.

Scaling the horizon

VaR over T periods ≈ VaR over 1 period × √T

Assumes returns are independent and identically distributed and the mean is close to zero. State this when you use it.

Converting periods for the mean and volatility

μ over T = μ × T; σ over T = σ × √T

Use this when the mean is not negligible. The mean scales with T, volatility with √T. With a large positive mean, VaR may not rise as the horizon lengthens.

Parametric VaR (return form)

VaR% = −[R̄ − z × σ] = z × σ − R̄

z is the one-tailed critical value: 1.645 for 5%, 2.33 for 1%. Use the same time unit for R̄ and σ.

Parametric VaR (currency)

VaR = (z × σ − R̄) × Portfolio value

Reported as a positive loss amount. If the mean is ignored, the VaR is z × σ × value.

Scaling volatility over time

σ(T days) = σ(1 day) × √T

Assumes independent returns. Scale the mean by T, not √T.

Annual to daily conversion

σ(daily) = σ(annual) ÷ √(number of trading days)

Use the number of trading days stated in the vignette, often 250 or 252.

Historical simulation percentile

VaR at α% = loss at the α percentile of ranked outcomes

With 500 observations, 5% VaR is around the 25th worst loss. When α% of the observations is not a whole number, the position depends on the convention the question states (round up, round down or interpolate).

Conditional VaR (expected shortfall)

CVaR = E[Loss | Loss ≥ VaR]

Average loss in the tail beyond the VaR cutoff. CVaR ≥ VaR at the same confidence level.

Incremental VaR

IVaR = VaR(portfolio with change) − VaR(portfolio without change)

Use for sizeable changes such as adding or removing a position. Can be negative for a hedge.

Marginal VaR (concept)

MVaR ≈ ΔVaR ÷ Δposition, for a very small change in position

A rate of change per unit of position value. A first-order approximation, not a full revaluation.

Component VaR

Component VaR(i) = position value(i) × MVaR(i); Σ component VaRs = portfolio VaR

MVaR is per unit of position value. Allocates total VaR across positions. The components add up exactly to the total. If MVaR is instead defined per unit of portfolio value, multiply by weight(i) rather than position value.

Relative VaR

Relative VaR = VaR of (portfolio return − benchmark return)

Measures risk of underperforming the benchmark. Absolute VaR ignores the benchmark.

Price change using duration and convexity

%ΔP ≈ −ModDur × ΔY + ½ × Convexity × (ΔY)²

ΔY is in decimal form. Use effective duration and convexity for bonds with embedded options.

Option price change (delta-gamma)

ΔC ≈ Delta × ΔS + ½ × Gamma × (ΔS)²

Delta-only is the first-order estimate. Adding gamma improves it for larger moves.

Option price change from vega

ΔC ≈ Vega × Δσ

Vega is usually quoted per 1 percentage point change in volatility. Check the units in the vignette.

Beta

β = Cov(Ri, Rm) ÷ Var(Rm)

Expected stock move ≈ β × market move, ignoring the stock-specific part.

Delta of a position

Position delta = number of options × Delta × contract multiplier

Long calls and short puts have positive delta. Long puts and short calls have negative delta.

RAROC

RAROC = (Revenue − Costs − Expected loss + Return on economic capital) ÷ Economic capital

Exact numerator varies by vignette. Always use what the vignette defines. Expected loss is deducted; unexpected loss is what capital covers.

Simple RAROC form

RAROC = Risk-adjusted expected return ÷ Economic capital

Compare with the hurdle rate. RAROC above hurdle means the unit adds value.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

Uses total risk. Best for a standalone portfolio, not for a small slice of a larger one.

Risk budget share

Share of risk = Unit risk contribution ÷ Total portfolio risk

Contributions use marginal or component risk, so they add up to the total. Standalone risks do not.

Capital allocation by ratio

Rank units by RAROC (or Sharpe) and shift capital to the highest ratio, subject to limits

Ranking assumes comparable risk measures and capital definitions.

Futures hedge ratio (minimum variance)

h = ρ × (σ_S ÷ σ_F)

Equal to the slope of spot changes regressed on futures changes. Number of contracts = h × (exposure value ÷ value of one futures contract).

Equity beta adjustment with futures

N = [(β_T − β_S) ÷ β_F] × (S ÷ f)

S is portfolio value, f is the futures contract value, β_T target beta, β_S current beta. Positive N means buy; negative means sell.

Bond duration adjustment with futures

N = (BPV_T − BPV_P) ÷ BPV_F

BPV_T is the target portfolio basis point value. BPV_P is the current portfolio basis point value. BPV_F is the basis point value of the futures contract (for a bond future, based on the cheapest-to-deliver bond, adjusted by the conversion factor). Positive N means buy futures to raise duration; negative means sell to lower it.

Delta hedge

Options needed = − (Position delta ÷ Option delta)

Hedge is only valid for small price moves. Gamma changes delta, so rebalance.

Portfolio risk of two assets

σ_p = √(w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂)

Lower correlation ρ lowers σ_p, which is why diversification reduces VaR.

Parametric VaR (normal)

VaR = (z × σ_p − μ_p) × portfolio value

Use z = 1.65 for 95% and 2.33 for 99% one-tailed. With μ_p near zero, cutting σ_p cuts VaR roughly in proportion.

Quick revision

• VaR is the minimum loss that would be expected to be exceeded with a given probability (for example 5%) over a set horizon, so it is the loss cutoff at the chosen tail probability.
• Parametric VaR under normality uses a z-score times standard deviation, adjusted for the expected return and horizon.
• Scaling VaR to a longer horizon with the square root of time assumes returns are independent and identically distributed.
• Historical simulation uses actual past changes and needs no distribution assumption, but depends on the chosen window being representative.
• Monte Carlo VaR is the most flexible, handles options and non-linear exposures, but is slow and depends on model assumptions.
• CVaR is the expected loss given that the loss is at or beyond the VaR cutoff, so it is at least as large as VaR.
• Marginal VaR is the change in portfolio VaR for a very small change in a position; incremental VaR is the change from adding or removing a whole position.
• VaR does not say how large losses can be beyond the cutoff, and it can understate tail risk when returns are fat-tailed.
• Stress tests and scenario analysis cover extreme events VaR may miss, and can be historical or hypothetical.
• Sensitivity measures such as duration, delta and beta give the change in value for a small change in one risk factor.
• Risk limits, such as VaR limits, position limits and stop-loss rules, tie risk taking to a budget and to allocated capital.
• Hedging reduces exposure but costs money and can create basis risk and leave residual risk.

Common mistakes

• Treating market risk as only equity price risk. Fix: List all four main factors: interest rate, equity, currency and commodity. Bond portfolios and foreign assets carry market risk too.
• Saying the risk manager's job is to eliminate risk. Fix: The goal is to keep risk consistent with tolerance and objectives. Taking risk is how returns are earned.
• Calling VaR the maximum loss. Fix: Say VaR is the minimum loss expected in the worst tail of outcomes. Losses beyond it can be much larger.
• Mixing up confidence level and tail probability. Fix: Tail probability = 1 − confidence. Use the tail for z values and for the "1 day in 20" reading.
• Using a two-tailed z-value such as 1.96 for a 5% VaR. Fix: VaR is one-tailed. Use 1.645 for 5% and 2.33 for 1%.
• Scaling the mean by √T along with the standard deviation. Fix: Scale σ by √T and the mean by T.
• Treating CVaR as the loss at a higher confidence level. Fix: CVaR is an average of all losses beyond the VaR cutoff, not a single percentile.
• Using marginal VaR for a large position change. Fix: Marginal VaR is for tiny changes. For sizeable changes, recompute VaR and use incremental VaR.
• Using ΔY in percent instead of decimal in the duration-convexity formula. Fix: Convert to decimal first: 1% = 0.01. The convexity term uses (0.01)², which is 0.0001.
• Forgetting the ½ in the convexity or gamma term. Fix: Always write ½ × Convexity × (ΔY)² and ½ × Gamma × (ΔS)².

Exam tips

• Read the vignette for who is acting. Many questions test roles, not calculations.
• Memorize the four risk factors and match each exposure to one or more of them, including currency risk on foreign assets.
• Look for words like tolerance, limit and independent. They signal the governance point being tested.
• Prefer answers that align risk with objectives over answers that remove all risk.
• There is no penalty for wrong answers, so answer every question.
• Read the vignette for the tail probability and horizon before anything else. Many wrong answers come from using 95% as if it were the tail.
• Expect interpretation questions with three similar-sounding options. Eliminate any option that says "maximum" or "will not exceed".
• For a longer horizon, scale volatility by √T and, with a zero or negligible mean, check the answer rises. If the mean is large and positive, VaR may not rise. State the assumption of independent returns if the question asks why scaling may fail.