FRM Part I · FRM Exam Part I

The Building Blocks of Risk Management: formula sheet

Full chapter guide

Key formulas

Risk management process: Identify → Measure → Treat (avoid, reduce, transfer, accept) → Monitor and report → repeat; A continuous loop. Monitoring feeds back into identification.
Market risk: Loss from changes in prices or rates (equity, interest rate, FX, commodity); Arises from positions you hold, even if no counterparty fails.
Credit risk: Loss from a counterparty failing to pay in full and on time; Includes default, downgrade and settlement risk.
Liquidity risk: Market liquidity: cannot sell near fair price. Funding liquidity: cannot raise cash to meet obligations; Two distinct forms. Name the one in the question.
Operational risk (Basel): Loss from inadequate or failed internal processes, people, systems, or external events; The Basel definition includes legal risk and excludes strategic and reputational risk.
Risk treatment options: Avoid, reduce (mitigate), transfer, accept (retain: deliberately bear the risk, for example with capital or reserves held against it); Hedging and insurance are transfer. Diversification is reduction.
Hierarchy of risk terms: Risk appetite and risk tolerance both sit below risk capacity; limits are set within tolerance; Appetite is the chosen target level. Tolerance is the maximum acceptable deviation around it. Capacity is the maximum the firm can bear. Limits are the day-to-day controls, set within tolerance. This is a conceptual ordering, not a strict inequality or a calculation.
Limit utilization: Utilization = Current exposure ÷ Limit; Above 100% means a breach. Many firms also set early-warning triggers below 100%, such as 80%.
Risk-based capital headroom: Headroom = Available capital − Required (risk-based) capital; A simple way to compare appetite and capacity. Negative headroom means capacity is exceeded.
Expected loss for a single exposure (credit): EL = PD × LGD × EAD; PD is probability of default over the horizon, LGD is loss given default as a fraction, EAD is exposure at default. LGD = 1 − recovery rate.
Expected loss from a loss distribution: EL = Σ (loss i × probability i); This is the probability-weighted average of possible losses.
Unexpected loss as standard deviation: UL = σ(loss); Common definition. Always check how the question defines UL.
Unexpected loss for one exposure with fixed LGD and EAD: UL = EAD × LGD × √(PD × (1 − PD)); Holds when LGD is a known constant and default is Bernoulli. If LGD is random, UL is larger.
Economic capital: Economic capital = Loss quantile (at confidence level) − EL; Also written as the unexpected loss at the tail. Some questions use the quantile itself as capital, so read the wording.
Normal approximation for the tail loss: Loss quantile ≈ EL + z × σ; Gives capital ≈ z × σ. Example z values: 1.645 at 95%, 2.326 at 99%, 3.090 at 99.9%. Use only if losses are treated as normal.
Portfolio EL: EL(portfolio) = Σ EL(i); EL is additive. UL is not additive unless correlations are perfect.
Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; σp is the standard deviation of portfolio returns (total risk). Use for stand-alone portfolios.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; Uses beta (systematic risk). Use for a portfolio that is part of a diversified whole. Be careful if beta is negative or near zero.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Positive alpha means the portfolio beat the CAPM required return.
Information ratio: IR = (Rp − Rb) ÷ tracking error; Tracking error is the standard deviation of (Rp − Rb). Rb is the benchmark return.
RAROC: RAROC = (Revenues − Costs − Expected loss) ÷ Economic capital; Some versions add income earned on the capital. Compare with the hurdle rate (cost of equity).
CAPM required return: E(R) = Rf + β × (E(Rm) − Rf); Used to build Jensen's alpha and to judge hurdle rates.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2; Weights sum to 1. Take the square root for σp. Lower ρ means lower portfolio risk.
Perfect correlation case: If ρ = 1: σp = w1σ1 + w2σ2; No diversification benefit. For ρ < 1, σp is below this weighted average.
Equally weighted portfolio of n assets (same σ, same pairwise ρ): σp² = σ²/n + ((n − 1)/n)·ρ·σ²; As n grows, variance approaches ρσ². That floor is systematic risk.
Total risk decomposition: Total variance = Systematic variance + Idiosyncratic variance; Diversification removes only the idiosyncratic part.
Protective put payoff at expiry: Value = ST + max(K − ST, 0); Floor at K. The cost is the premium, which lowers net gains.
Net result of a hedged position: Net = Gain or loss on exposure + Gain or loss on hedge; Residual is the unhedged part, often basis risk.
Leverage ratio (simple form): Leverage = Total assets ÷ Equity; Higher leverage magnifies both gains and losses on equity. Used to compare banks and hedge funds.
Balance sheet identity: Assets = Liabilities + Equity; A loss on assets falls first on equity. If losses exceed equity, the institution is insolvent.
Insurance combined ratio: Combined ratio = Loss ratio + Expense ratio; Below 100% means an underwriting profit before investment income. Above 100% means an underwriting loss.
Net interest margin: NIM = (Interest income − Interest expense) ÷ Average earning assets; Measures the bank's spread from maturity and credit intermediation.

Quick revision

Main risk types: market, credit, liquidity, operational, plus others such as legal and reputational risk.
Risk management process: identify, measure, manage or mitigate, monitor and report.
Risk appetite is the amount and type of risk a firm is willing to accept; limits turn it into daily rules.
Expected loss = PD × LGD × EAD, where PD is probability of default, LGD is loss given default and EAD is exposure at default.
Expected loss is a cost of doing business and is covered by pricing and provisions.
Unexpected loss is variation of losses around the expected level and is covered by capital.
Sharpe ratio = (Rp − Rf) ÷ σp, using total risk.
Higher expected return generally requires accepting more risk, but the link is not guaranteed in any single period.
Diversification reduces idiosyncratic risk but not systematic risk.
Hedging cuts exposure to a risk but can also give up gains and add basis risk.
Insurance transfers loss for a premium; watch for moral hazard and adverse selection.
Banks transform maturity and size and manage credit and liquidity risk; insurers pool and transfer risk.

Common mistakes

Calling any loss from a falling price market risk, even when the cause is a default. Fix: Look for the root cause. If a counterparty failed to pay or was downgraded, it is credit risk.
Mixing up market liquidity risk and funding liquidity risk. Fix: Market liquidity is about selling an asset. Funding liquidity is about raising cash. Check which side of the balance sheet the problem sits on.
Treating risk appetite and risk tolerance as the same thing. Fix: Appetite is the desired overall level of risk. Tolerance is the acceptable deviation or maximum in a specific area. Limits are the operational controls.
Confusing risk appetite with risk capacity. Fix: Capacity is the most the firm can bear. Appetite is what it chooses to take and must be below capacity.
Using LGD as the recovery rate Fix: Convert first: LGD = 1 − recovery rate. A 40% recovery means 60% LGD.
Treating reserves as protection against unexpected loss Fix: Reserves cover EL. Economic capital covers losses above EL up to the confidence level.
Forgetting to subtract the risk-free rate in Sharpe or Treynor. Fix: Always write (Rp − Rf) first. If Rf is given, it is meant to be used.
Using beta in the Sharpe ratio or standard deviation in the Treynor ratio. Fix: Remember: Sharpe uses Standard deviation (S for S); Treynor uses beta (systematic risk).
Saying diversification can eliminate all risk. Fix: It removes only idiosyncratic risk. Systematic risk remains.
Treating hedging and insurance as the same thing. Fix: A hedge offsets gains and losses, so upside is lost. Insurance pays a premium and keeps the upside. Options behave like insurance, forwards like hedges.

Exam tips

Questions give a short scenario and ask for the primary risk. Find the root cause before reading the options.
Watch for options that are real risks but not the main one. Several may be true; choose the best fit.
Learn the Basel operational risk definition, including that it includes legal risk and excludes strategic and reputational risk.
Know the four treatment options and match hedging, insurance and diversification to them correctly.
Expect interaction cases, such as a credit event causing a liquidity squeeze. Pick the risk that started the chain.
Remember the conceptual ordering: appetite and tolerance sit below capacity, and limits are set within tolerance. Also know who owns each. Do not treat it as a strict inequality to memorise.
Expect questions on what a good risk appetite statement contains: strategy link, qualitative and quantitative parts, and communication.
For breach questions, choose answers with defined escalation and documented exceptions.