Market Risk Management: formula sheet

Key formulas

Market risk definition

Market risk = potential loss from adverse movement in interest rates, exchange rates, equity prices or commodity prices

Write the source of loss as a price movement, not a default.

Four components

Market risk = Interest rate risk + Equity price risk + Foreign exchange risk + Commodity price risk

This is a classification, not an arithmetic sum. Use it as a memory list.

Net open foreign exchange position

Net open position = Foreign currency assets − Foreign currency liabilities (including off-balance sheet items)

A positive position loses when the foreign currency weakens; a negative position loses when it strengthens.

Bond price and rate link

Interest rates ↑ ⇒ price of fixed-rate bond ↓; interest rates ↓ ⇒ price ↑

Holds for a fixed-coupon bond, other things being equal.

Trading book definition

Trading book = positions held with trading intent + positions held to hedge trading book positions

Intent is the test. The instrument type alone does not decide the book.

Banking book definition

Banking book = all positions not in the trading book

It is a residual category. Loans, deposits and held-to-maturity investments usually sit here.

Capital linkage

Trading book → market risk capital; Banking book → credit risk capital (plus IRRBB supervision)

Counterparty credit risk on trading book derivatives is still capitalised separately.

Valuation linkage

Trading book → fair value (mark to market); Banking book → cost or amortised cost, mostly

Exact accounting follows the applicable standards and RBI norms.

Reclassification rule

Switching books after initial designation is allowed only in exceptional cases, with senior management approval and board-approved policy

Any capital benefit from a switch is not allowed to the bank. State this as the principle.

Parametric VaR (1 day)

VaR = Portfolio value × z × σ

σ is daily standard deviation of returns. Mean return is taken as zero unless the question gives it.

Common z-values (one-tailed, normal)

95% → 1.645; 99% → 2.33

Use the z-value the question supplies if it gives one.

Holding period scaling

VaR (N days) = VaR (1 day) × √N

Valid under the square-root-of-time rule: independent returns with constant volatility.

Two-asset portfolio standard deviation

σp = √(w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2)

Use when VaR is needed for a portfolio. Low correlation ρ gives diversification benefit.

Historical simulation percentile

Loss at (1 − confidence) percentile of ranked returns

With 100 observations at 95%, VaR is the 5th worst outcome; at 99% it is the 1st worst (or interpolate, as the question directs).

Undiversified vs diversified VaR

Diversified VaR ≤ Sum of individual VaRs

Equality holds only when correlation is +1.

Backtesting exception

Exception = day when actual loss > VaR

Expected exceptions = observations × (1 − confidence). Under the Basel framework the number of exceptions over 250 days drives the multiplier on capital.

Expected shortfall (discrete, equally likely outcomes)

ES = Average of all losses that are greater than or equal to the VaR cutoff, i.e. the worst (1 − c) share of outcomes

c is the confidence level. At 97.5%, ES uses the worst 2.5% of outcomes.

Relation of ES to VaR

ES ≥ VaR at the same confidence level

ES is the average of tail losses. Never present ES as smaller than VaR.

Number of tail observations

Tail observations = N × (1 − c)

With 200 observations at 95%, the tail has 10 observations.

Stress loss from a sensitivity shock

Loss ≈ Σ (Exposure × Shock)

For a bond portfolio: Loss ≈ Modified duration × Δy × Market value, for small yield changes.

Stress loss as share of capital

Stress loss ÷ Capital × 100

Used to judge whether capital is adequate after the shock.

Repricing gap

Gap = RSA − RSL

Calculate for each time bucket. Cumulative gap is the running total up to the bucket you are testing.

Change in NII

ΔNII ≈ Gap × Δi

Use the cumulative gap for the horizon (usually 1 year). Δi as a decimal, e.g. 0.5% = 0.005. Assumes a parallel shift and that all items reprice at the start of the bucket.

Gap ratio

RSA ÷ RSL

Below 1 means liability-sensitive, above 1 means asset-sensitive.

Macaulay duration

D = Σ [t × CF_t ÷ (1 + y)^t] ÷ Price

t is time in years, CF_t is the cash flow at time t, y is yield per period. The price is the sum of discounted cash flows.

Modified duration

MD = D ÷ (1 + y ÷ m)

m is compounding periods per year. For annual compounding, MD = D ÷ (1 + y).

Price change with duration and convexity

ΔP ÷ P ≈ −MD × Δy + ½ × C × (Δy)²

C is convexity. Drop the second term if the question gives no convexity. Use Δy in decimals.

Convexity (annual cash flows)

C = Σ [t × (t + 1) × CF_t ÷ (1 + y)^(t + 2)] ÷ Price

Use only if the question asks you to compute it. Otherwise the value is usually given.

Duration gap

DG = D_A − (L ÷ A) × D_L

A is the market value of assets, L is the market value of liabilities, D_A and D_L are the weighted durations.

Change in equity value

ΔE ≈ −DG × A × Δy ÷ (1 + y)

Divide ΔE by E to get the percentage change in equity. A positive DG with rising yield gives a loss.

Net open position (single currency)

Net position = (Spot assets + Forward purchases) − (Spot liabilities + Forward sales)

Positive = long, negative = short. Include all on- and off-balance sheet items in that currency.

Overall net open position

Overall NOP = larger of (Σ net long positions) and (Σ net short positions)

Sum long and short currencies separately. Take the larger total. Do not net longs against shorts.

Limit check

Utilisation % = Overall NOP ÷ NOPL × 100

If utilisation is above 100%, the bank breaches the limit and must square off the excess.

Gap in a maturity bucket

Gap = Forward purchases − Forward sales (for that bucket)

Positive gap = net purchase. Negative gap = net sale.

Aggregate gap

Aggregate gap = Σ |gap in each bucket|

Add absolute values, ignoring signs, then compare with the AGL.

Gain or loss on an open position

P&L = Net position (in foreign currency) × (New rate − Old rate)

Positive for a long position when the rate rises. Reverse the sign for a short position.

Total market risk capital (standardised, older method)

Capital charge = Specific risk charge + General market risk charge

The older method uses a maturity or duration method for interest rate positions and fixed percentage charges for specific and general market risk on equity positions. Add foreign exchange and commodity charges separately.

Risk-weighted assets equivalent

RWA for market risk = Capital charge × 12.5

Under Basel, the market risk capital charge is converted into RWA by multiplying by 12.5, the reciprocal of the 8% minimum. RBI uses the same 12.5 conversion for market risk. Separately, RBI's minimum total capital ratio is 9% of total RWA, excluding buffers such as the capital conservation buffer. This ratio is applied to total RWA. It does not change the 12.5 factor.

Revised standardised approach (FRTB)

Capital = Sensitivities-based method charge + Default risk charge + Residual risk add-on

The sensitivities-based method covers delta, vega and curvature risk across risk classes.

Expected Shortfall

ES = average loss in the worst 2.5% of outcomes (97.5% confidence)

ES captures the size of tail losses. VaR only gives the threshold.

Liquidity horizon scaling

ES = √[ ES_T(P)² + Σ (j ≥ 2) (ES_T(P, j) × √((LH_j − LH_(j−1)) ÷ T))² ], with base horizon T = 10 days

FRTB uses horizons (LH) of 10, 20, 40, 60 and 120 days by risk factor category. ES_T(P) is computed for the 10-day base horizon using all risk factors. This first term covers LH_1 = 10 days. The sum runs over j ≥ 2, where LH_(j−1) is the previous horizon. Each ES_T(P, j) is scaled by √((LH_j − LH_(j−1)) ÷ 10), and the terms are aggregated. ES_T(P, j) is computed using only risk factors whose horizon is at least LH_j. Less liquid factors get longer horizons, so they attract more capital.

FRTB IMA test

Desk approval = back-testing pass + P&L attribution pass

A desk that fails the P&L attribution test, or breaches back-testing thresholds, is ineligible for IMA and is charged under the standardised approach. Minor back-testing exceptions lead to a higher capital multiplier first.

Net Open Position (forex)

NOP = larger of (Σ net short positions, Σ net long positions) across currencies

A simplified shorthand. Use it to compare against the NOP limit approved by the board and RBI. In a question, follow the method given in the problem.

Stop loss trigger

Breach if Cumulative loss ≥ Stop loss limit

Loss is measured from the cost or the start-of-period value. The usual action is to close the position or seek approval to continue.

Limit utilisation

Utilisation % = Actual exposure ÷ Approved limit × 100

Above 100% means an excess that must be reported. Many banks also set early warning triggers below 100%.

Hedge ratio

Hedge ratio = Value of hedge instrument ÷ Value of exposure hedged

1 means a full hedge. Below 1 is a partial hedge. The question may ask for the unhedged amount.

Futures hedge, number of contracts

N = (Exposure value ÷ Value of one contract) × Hedge ratio

Round to a whole number of contracts. Use the hedge ratio given, and 1 if it is not given.

Forward cover result

Hedged rupee amount = Foreign currency amount × Forward rate

The forward rate is locked, so the rupee amount does not depend on the spot rate at maturity.

Quick revision

• Market risk is loss from changes in interest rates, exchange rates, equity prices and commodity prices.
• The trading book holds positions for short-term trading and is marked to market; the banking book holds positions for the longer term.
• VaR estimates the loss that should not be exceeded at a stated confidence level over a stated holding period.
• The three VaR methods are variance-covariance, historical simulation and Monte Carlo simulation.
• Under the usual square-root-of-time scaling, N-day VaR = 1-day VaR × √N; it assumes independent returns and is only an approximation.
• VaR does not tell you how large the loss is beyond the cutoff; expected shortfall averages the losses beyond it.
• Stress tests examine extreme but plausible scenarios and complement VaR, not replace it.
• A positive duration gap generally means the value of equity falls when interest rates rise; a negative gap means the opposite.
• Net open position in a currency is the difference between assets and liabilities in that currency, including off-balance-sheet items.
• Capital for market risk may be set by the standardised approach or by an approved internal model.
• Limits such as position, loss-stop and VaR limits turn risk appetite into daily controls.
• Hedging reduces risk but has a cost and can leave basis risk.

Common mistakes

• Calling a borrower's default caused by a rate rise purely market risk. Fix: Separate the two effects. The price loss on the bank's own positions is market risk. The borrower's failure to repay is credit risk.
• Saying market risk applies only to the trading book. Fix: State that interest rate and currency risk also arise in the banking book, for example through repricing mismatches.
• Classifying by instrument type, for example saying all government securities are banking book. Fix: The same security can sit in either book. Decide by intent and documented strategy.
• Saying the banking book carries no interest rate risk capital consideration. Fix: Banking book positions carry IRRBB, which is managed and supervised separately from trading book market risk capital.
• Using the wrong z-value, such as 1.645 for 99% confidence Fix: Write 95% → 1.645 and 99% → 2.33 at the top of your answer before computing.
• Multiplying by N instead of √N when scaling the holding period Fix: Volatility scales with the square root of time. Multiply 1-day VaR by √N.
• Stating that ES is the loss at the VaR point. Fix: VaR is the cutoff. ES is the average of losses beyond the cutoff, so it is equal to or larger.
• Averaging all losses instead of only the tail. Fix: Compute N × (1 − c) first and average only that many worst losses.
• Using the total gap instead of the cumulative gap up to the one-year horizon (or the wrong bucket). Fix: Add buckets only up to the horizon asked. Then multiply by the rate change.
• Getting the sign wrong in ΔNII or ΔE. Fix: Decide direction first from the sign rule. Check that your final number agrees with it.

Exam tips

• In MCQs, spot the word that signals the cause: price, rate or volatility points to market risk; default or downgrade points to credit risk; fraud, process or system points to operational risk.
• In case-based questions, name the exposed position and the direction of loss before naming the risk type.
• For difference questions, compare on cause, source of loss, and typical control, using clear sentences.
• Practise one quick currency position calculation. Check whether the bank is long or short before judging gain or loss.
• Mention both trading book and banking book when asked about the scope of market risk.
• Begin every classification answer with the word 'intent'. Examiners look for it.
• In a case scenario, underline phrases such as 'to sell shortly', 'market making' and 'hold to maturity'. They decide the book.
• Always pair the book with both valuation basis and capital charge. A half pair loses marks.