Risk Management: formula sheet

Key formulas

Total risk

Total risk = Systematic risk + Unsystematic risk

In portfolio terms, variance splits into market-driven and specific parts. Diversification removes only the unsystematic part.

Risk management cycle

Identify → Measure → Respond → Implement → Monitor and report

Use this order as your answer skeleton for any process question.

Risk responses

Avoid | Reduce | Transfer | Accept

Hedging with derivatives is usually reduction or transfer, not avoidance.

Financial risk classes

Market (interest rate, FX, equity price, commodity) | Credit | Liquidity | Operational

Name the class first, then the sub-type.

Beta (systematic risk measure)

β = Cov(Rs, Rm) ÷ Var(Rm)

Beta measures only systematic risk. A security with β > 1 moves more than the market.

Parametric VaR (single asset or portfolio)

VaR = Z × σ × V

σ is the standard deviation of returns for the holding period. V is the portfolio value. Z is the standard normal value for the confidence level.

Scaling to t days

σ(t days) = σ(1 day) × √t

Valid when daily returns are independent with the same σ. Square root of time rule, not t.

Common Z values (one-tailed)

90% → 1.28; 95% → 1.645; 99% → 2.33

Use the Z value given in the question. VaR is a one-tailed measure.

Two-asset portfolio standard deviation

σp = √(wA²σA² + wB²σB² + 2 wA wB ρAB σA σB)

Gives diversification benefit when ρ < 1. Covariance = ρ × σA × σB.

Portfolio VaR with diversification

Portfolio VaR = Z × σp × V

Portfolio VaR is at most the sum of individual VaRs when ρ ≤ 1 and positions are long.

Historical simulation cut-off

VaR = loss at the (1 − confidence level) percentile of ranked outcomes

With 100 observations at 95%, the 5th worst outcome is a common reading. Follow the convention in the question.

Expected shortfall

ES = average of losses beyond the VaR cut-off

ES is always at least as large as VaR at the same confidence level.

Expected loss

EL = PD × LGD × EAD

Use PD for the same period as the exposure. EL is a statistical average, not the worst case.

Loss given default

LGD = 1 − Recovery rate

Recovery rate is the share of exposure recovered after default.

Credit spread (approximate)

Credit spread ≈ Yield on risky bond − Yield on risk-free bond

Compensation demanded for default risk. A rough approximation is spread ≈ PD × LGD.

Annual CDS premium

Premium = CDS spread × Notional

Spread is quoted in basis points. 100 bps = 1%.

CDS payout on credit event

Payout = Notional × (1 − Recovery rate)

Applies to cash settlement. With physical settlement the seller pays par and receives the bond.

Net protection value for the buyer

Net gain = Payout − Premiums paid

Premiums stop at default. Count only premiums actually paid.

Probability of survival

Survival probability over n years = (1 − PD)ⁿ

Valid only if the same annual PD is assumed each year and years are independent.

FRA settlement (buyer of FRA, i.e. borrower)

Settlement = [(R_ref − R_fra) × N × (days ÷ 360)] ÷ [1 + R_ref × (days ÷ 360)]

Positive means the FRA buyer receives. Negative means the buyer pays. Use the day-count given in the question (360 or 365).

Implied forward rate

(1 + r_long × t_long) = (1 + r_short × t_short) × (1 + f × t_forward)

Use simple interest for money-market periods under one year, with t in years.

Net swap payment

Net payment by fixed payer = (Fixed rate − Floating rate) × Notional × period

Only the net amount is exchanged. If negative, the fixed payer receives.

Cap payoff per period

Payoff = max(0, R_ref − Strike) × N × (days ÷ 360)

Net cost to borrower = interest paid + cap premium − payoff.

Floor payoff per period

Payoff = max(0, Strike − R_ref) × N × (days ÷ 360)

Used by lenders and investors to protect a minimum return.

Collar for a borrower

Buy cap at higher strike, sell floor at lower strike. Net premium = Cap premium − Floor premium

Effective rate stays between floor strike and cap strike, ignoring premium.

Interest rate futures price

Price = 100 − implied rate (%)

A borrower fears rising rates, so sells futures. A lender buys futures. Tick value = Contract size × 0.01% × period.

Interest rate parity (forward rate)

F = S × (1 + i_INR × n) ÷ (1 + i_foreign × n)

S and F are rupees per unit of foreign currency. n is in years. Use simple interest for periods up to one year unless told otherwise. For a quote in the other direction, invert the ratio.

Annualised forward premium / (discount)

(F − S) ÷ S × (12 ÷ months) × 100

Positive means the foreign currency is at a premium. Use the same side of the quote (bid or ask) for F and S.

Bid-ask side rule

Bank buys foreign currency at the bid; bank sells at the ask

You, the customer, get the opposite. An exporter sells USD to the bank at the bid. An importer buys USD from the bank at the ask.

Money market hedge for a payable

Deposit today (foreign) = Payable ÷ (1 + deposit rate × n); rupees needed today = Deposit × spot ask; rupee cost at due date = rupees today × (1 + INR borrowing rate × n)

If the firm holds surplus rupees, use the INR deposit rate as the opportunity cost instead of the borrowing rate.

Money market hedge for a receivable

Borrow today (foreign) = Receivable ÷ (1 + foreign borrowing rate × n); rupees today = Borrowing × spot bid; rupees at due date = rupees today × (1 + INR deposit rate × n)

The foreign receipt repays the foreign loan. Invest the rupees at the rupee deposit rate, or use the borrowing rate saved if the firm has rupee debt.

Option payoff for an exporter (put)

Net rupees per unit = max(strike, spot at expiry) − premium

The put is exercised only if spot is below strike. Add interest on the premium only if the question asks.

Option payoff for an importer (call)

Net rupees per unit = min(strike, spot at expiry) + premium

The call is exercised only if spot is above strike.

Number of futures contracts

Contracts = Exposure in foreign currency ÷ lot size

Use the lot size given in the question. Round to a whole number and show any unhedged balance.

Delta

Delta = Change in option price ÷ Change in price of underlying

Call delta lies between 0 and 1. Put delta lies between −1 and 0. Under Black-Scholes, call delta = N(d1) and put delta = N(d1) − 1, for a non-dividend-paying share.

Delta-neutral hedge

Number of options = Number of shares held ÷ Delta of option

Shares held long are hedged by writing calls (or buying puts). Neutral means shares × 1 + options × delta = 0, with sign for buy or sell.

Gamma

Gamma = Change in delta ÷ Change in price of underlying

Approximate new delta = old delta + gamma × price change. Gamma is highest for at-the-money options.

Vega, Theta, Rho

Vega = ΔPrice ÷ ΔVolatility (per 1%); Theta = ΔPrice ÷ Δtime (per day); Rho = ΔPrice ÷ ΔRate (per 1%)

Use the unit given in the question. Do not mix per-day and per-year theta.

Futures hedge ratio (beta method)

Number of contracts = (Portfolio value ÷ Value of one futures contract) × Portfolio beta

Value of one contract = futures price × lot size. Sell contracts to hedge a long portfolio.

Minimum variance hedge ratio

h = ρ × (σS ÷ σF); Number of contracts = h × Exposure ÷ Value of one contract

ρ is the correlation between spot and futures price changes. σS and σF are the standard deviations of those changes.

Changing portfolio beta

Number of contracts = (Target beta − Current beta) × Portfolio value ÷ Value of one contract

A negative answer means sell futures. A positive answer means buy futures.

Quick revision

• Risk process: identify, measure, treat, monitor.
• Main types: market, credit, liquidity, operational, and legal or regulatory risk.
• VaR estimates the loss that should not be exceeded at a stated confidence level over a stated period.
• VaR says nothing about how large the loss can be beyond the confidence level.
• Credit derivatives transfer default risk without selling the underlying loan or bond.
• Interest rate risk tools: FRAs, futures, swaps, caps, floors and collars.
• A swap exchanges cash flows, usually fixed for floating, on a notional that is not itself exchanged.
• Forex hedges: forward, money market, futures and options. Compare the final rupee amounts.
• A forward fixes the rate but removes any gain from a favourable move; an option keeps the gain but costs a premium.
• Delta is the change in option price for a small change in the underlying price.
• Gamma is the change in delta for a change in the underlying price.
• Vega measures sensitivity to volatility, theta to time decay, and rho to the interest rate.

Common mistakes

• Treating 'market risk' in portfolio theory and in the financial-risk classification as the same thing. Fix: In portfolio questions, market risk means systematic risk. In the risk classification, it means price-movement risk. Read the context before answering.
• Saying diversification removes all risk. Fix: State that it removes only unsystematic risk. Systematic risk remains.
• Scaling the daily σ by t instead of √t. Fix: Variance grows with time, so σ grows with √t. For 10 days multiply by √10 ≈ 3.162.
• Adding individual VaRs to get portfolio VaR. Fix: Use σp with the correlation. Simple addition is correct only when ρ = +1.
• Using the recovery rate as LGD in the expected loss formula. Fix: Always write LGD = 1 − recovery before multiplying.
• Treating credit risk and counterparty risk as the same thing. Fix: Say credit risk is the wider term covering loans and bonds. Counterparty risk is default by the other side of a contract, usually a derivative, where the exposure depends on market value.
• Taking the wrong sign on the FRA settlement. Fix: Remember that the FRA buyer gains when the reference rate is above the FRA rate. Write who receives before computing.
• Not discounting the FRA settlement. Fix: Divide by 1 + reference rate × days ÷ 360 when the question says settlement is at the start of the period.
• Using the wrong side of the bid-ask quote Fix: Write 'bank buys at bid, sells at ask' at the top. Exporter sells USD to the bank at the bid. Importer buys USD at the ask.
• Comparing rupee amounts at different dates Fix: Carry the rupee amount to the due date using the rupee interest rate, then compare with the forward result.

Exam tips

• In case-scenario MCQs, classify by the cause of the loss, not by where it appears.
• For descriptive answers, use the cycle (identify, measure, respond, implement, monitor) as headings, then add one example from the case.
• Always add the response and the tool after classification. Marks are often split between classifying and treating.
• In portfolio questions, state clearly that beta measures only systematic risk.
• Write one line on residual risk after hedging. It shows judgment and is often missed.
• Write the interpretation sentence every time. Many marks are given for stating VaR with confidence level and period.
• Show σp working in full for two-asset questions. Part marks are awarded for the variance step even if the final figure is off.
• If the question gives Z or √t values, use them exactly as given, even if your calculator differs slightly.