Valuation and Risk Models: FRM® Part I
It is 30% of the 2026 Part I exam. 130 learning outcomes across 16 learning modules.
Measures of Financial Risk
- Lay out the efficient frontier and the mean-variance framework behind it.
- Contrast the shape of the normal distribution with the way returns on risky financial assets, equities for example, are typically distributed.
- Set out what the VaR risk measure is, the assumptions it makes about return distributions and holding periods, and where VaR falls short.
- Calculate and interpret ES, and set it side by side with VaR to show how the two measures are alike and where they differ.
- State each property that a coherent risk measure must have and describe what each one means.
- Demonstrate that VaR violates the requirements of a coherent risk measure.
Calculating and Applying VaR
- Give examples of linear portfolios and of non-linear portfolios, and explain what separates them.
- Walk through how the historical simulation approach is used to compute both VaR and ES.
- Calculate VaR for non-linear derivatives by means of the delta-normal approach, after first explaining how that approach operates.
- Describe the VaR of linear derivatives and work it out.
- Point out where the delta-normal method breaks down.
- Explain how Monte Carlo simulation is used to compute VaR and ES, and pinpoint where it is strong and where it is weak.
- Discuss what a breakdown in correlations means for a VaR or ES analysis.
- Set VaR against worst-case scenario analysis, describing the latter first.
Measuring and Monitoring Volatility
- Explain the ways in which the distributions of asset returns typically depart from normality.
- Give the reasons why return distributions have fat tails and outline what follows from them.
- Distinguish conditional distributions from unconditional ones and outline what regime switching is.
- Compare the different methods for estimating conditional volatility, noting how they are alike and how they differ.
- Apply the exponentially weighted moving average (EWMA) method to volatility estimation, and outline the other ways in which older return observations can be given weights.
- Estimate volatility by applying a GARCH (1,1) model.
- Apply methods for estimating volatility or VaR over long horizons, and describe how mean reversion works under a GARCH (1,1) model.
- Assess implied volatility as a forecasting tool for future volatility, together with its weaknesses.
- Work through an example of how a correlation estimate is updated.
External and Internal Credit Ratings
- Describe the scales used by external rating agencies, how ratings are assigned, and how a rating connects to the chance of default.
- Distinguish default probabilities that are conditional from those that are unconditional, defining each.
- Define the hazard rate, then apply it to find the unconditional probability that a credit asset defaults.
- Define the recovery rate and, with it, compute the loan's expected loss.
- Contrast ratings built on a point-in-time basis with those built through the cycle, and explain each.
- Outline the options that can be used in place of the credit ratings issued by rating agencies.
- Weigh internal ratings against external ones, noting the strengths of each.
- Describe a rating transition matrix, calculate with it, interpret it, and explain what it is used for.
- Outline how credit default swap spreads, bond prices and stock prices tend to move when a credit rating is upgraded or downgraded.
- Explain where credit ratings have failed in the past and what difficulties arise when they are relied on for investment decisions.
Country Risk: Determinants, Measures, and Implications
- Relate a nation's risk exposure to its rates of economic growth, its political risk, its legal risk and the structure of its economy.
- Assess composite measures that combine several components of country risk into a single figure.
- Contrast defaults by sovereigns on debt in foreign currency with defaults on debt in local currency, and explain what typically causes sovereigns to default.
- Outline what happens as a result of a sovereign default.
- Identify the influences on how likely a sovereign is to default, then explain how rating agencies gauge that risk and assess their approach.
- Outline the features of sovereign credit default swaps (CDS) and of sovereign credit spreads, and contrast sovereign spreads with credit ratings as risk indicators.
Measuring Credit Risk
- Explain how economic capital is derived and how it differs from regulatory capital.
- Describe how strongly loan defaults in a bank's loan portfolio tend to depend on one another, and explain what this means for the portfolio's default rate.
- Define expected loss (EL) and calculate it.
- Define unexpected loss (UL) and explain what it represents.
- Using a binomial distribution for credit losses, estimate their mean and their standard deviation.
- Outline the Gaussian copula model and the uses to which it is put.
- Estimate a bank's default rate and its credit risk capital with the Vasicek model, and explain how that model is set up.
- Explain how CreditMetrics is applied to estimate economic capital, outlining how the model is built.
- Use Euler's theorem to find how much a given loan adds to the overall risk of a portfolio, and describe how it works.
- Explain why credit risk capital is harder to calculate for derivatives than for loans.
- Outline the difficulties involved in quantifying credit risk.
Operational Risk
- Outline the types of operational risk, with an account of how each type can arise.
- Contrast three ways of arriving at regulatory capital for operational risk: the advanced measurement approach, the standardized approach and the basic indicator approach.
- Explain why the Basel Committee introduced the standardized measurement approach and describe what it involves.
- Explain how Monte Carlo simulation builds a loss distribution out of a suitable loss frequency distribution and loss severity distribution.
- Outline the data problems commonly seen that make estimates of loss frequency and severity distributions inaccurate or biased.
- Outline using scenario analysis as a substitute when there is little data to work with.
- Outline how causal relationships are identified, and the roles that education, Risk and Control Self-Assessment (RCSA) and Key Risk Indicators (KRIs) play in understanding and managing operational risks.
- Outline how operational risk capital is allocated across business units.
- Explain how operational risk can be measured with the power law.
- Explain the link between insuring against operational risk and the moral hazard and adverse selection problems that insurance companies face.
Stress Testing
- Outline why stress testing is used as a risk management tool.
- Describe where stress testing sits alongside other risk measures, especially when it is done enterprise-wide.
- Compare how stressed VaR and stressed ES are determined with the way traditional VaR and ES are determined, describing the advantages and disadvantages of the stressed measures.
- Explain what must be weighed, and what proves difficult, when scenarios for stress testing are developed and stress testing models are built.
- Give an example of regulatory stress testing and describe what reverse stress testing is.
- Set out what the internal audit function, senior management and the board of directors each answer for when stress testing is governed.
- Outline how validation, independent review, and policies and procedures each feature in stress testing governance.
- Outline the Basel stress testing principles that apply to banks when they put stress testing into practice.
Pricing Conventions, Discounting, and Arbitrage
- Explain what a discount factor is and use a discount function to calculate present and future values.
- Explain the law of one price by way of an arbitrage argument, define it, and show how it is put to work in bond pricing.
- Spot where a fixed-income security with certain cash flows offers an arbitrage opportunity.
- Name the parts of a US Treasury coupon bond and contrast it with Treasury STRIPS, taking in how P-STRIPS and C-STRIPS differ.
- Use several fixed-income securities to build a replicating portfolio that reproduces the cash flows of a given fixed-income security.
- Distinguish clean bond pricing from dirty bond pricing and explain what accrued interest implies for bond prices.
- Summarise how interest on a fixed-income security is computed under the common day-count conventions.
Interest Rates
- Calculate how different compounding frequencies change a bond's value and interpret the result.
- Explain what a spot rate is and derive discount factors from spot rates.
- Calculate forward rates from spot rates and interpret what a forward rate tells you.
- Explain what a par rate is and outline how a bond's par rate is found.
- Interpret how spot, forward and par rates relate to one another.
- Assess how a bond's price is affected when its time to maturity changes.
- Explain what it means for a rate curve to flatten or steepen, and outline a trade that would express a view that it will do either.
- Describe a swap transaction and how par rates come to be defined in the swap market.
Bond Yields and Return Calculations
- Distinguish gross realized returns from net realized returns and calculate a bond's realized return over a holding period, reinvestments included.
- Define a bond's spread, interpret it, and explain its derivation from the bond price together with a term structure of rates.
- Explain what a bond's yield to maturity (YTM) is, interpret it, and use it in pricing the bond.
- Explain the way a bond's YTM is calculated when its structure and price are known.
- Determine the price of an annuity as well as that of a perpetuity.
- Explain how spot rates and YTM are related.
- Explain what the coupon effect is and how coupon rate, YTM and bond prices relate to one another.
- Break a bond position or portfolio's profit and loss (P&L) down into separate drivers, namely carry roll-down, rate change and spread change effects, and explain each.
- Calculate carry roll-down under the interest rate assumptions most commonly made for it, and outline what those assumptions are.
Applying Duration, Convexity, and DV01
- Outline a one-factor interest rate model and give examples of factors that are commonly used in it.
- Given the move in rates and the price change that results, define a fixed-income security's DV01 and compute it.
- Given the DV01 of each bond, calculate how much face amount is required to hedge a position that is sensitive to interest rates.
- Explain what effective duration is, interpret it, and calculate it for a fixed-income security from a given change in rates and the price change that results.
- Compare DV01 and effective duration as gauges of price sensitivity, noting how they are alike and how they differ.
- Explain what convexity is, interpret it, and calculate it for a fixed-income security from a given change in rates and the price change that results.
- Compute DV01, duration and convexity at the level of a fixed-income portfolio.
- Explain how effective duration and convexity are used to hedge a position.
- Construct a barbell that has the same cost and duration as a particular bullet investment, then explain what each of the bullet and barbell structures does well and badly.
Modeling Non-Parallel Term Structure Shifts and Hedging
- Identify which factors dominate term structure movements, and describe principal components analysis.
- Define key rate 01 (KR01) and outline what key rate shift analysis involves.
- For a set of key rates, work out a portfolio's KR01s.
- Calculate the hedging instrument positions that offset the key rate risks held in a portfolio.
- Estimate portfolio volatility by applying key rate analysis together with principal components analysis.
- Compare forward bucket 01s with KR01s, after defining forward bucket 01 and outlining an interest rate bucketing approach.
- Convert a KR01 or a forward bucket 01 into the duration measure that corresponds to it.
Binomial Trees
- Value American and European options, both calls and puts, with binomial models of one and of two steps.
- Outline how the binomial model captures volatility.
- Outline how a binomial model's computed value converges as further time steps are included.
- Explain what the delta of a stock option is and calculate it.
- Show how the binomial model is modified to price options on futures, currencies, stock indices and stocks that pay dividends.
The Black-Scholes-Merton Model
- Explain why stock prices are lognormal, how rates of return are distributed, and how expected return is computed.
- Compute the historical volatility and the realized return of a stock.
- Set out what the Black-Scholes-Merton option pricing model assumes.
- Value a European option on a stock with no dividends through the Black-Scholes-Merton model.
- Define implied volatilities and outline how the Black-Scholes-Merton model is used to back them out of options prices in the market.
- Explain the effect dividends have on whether American calls and puts should be exercised early.
- Use the Black-Scholes-Merton model to value a European option on a futures contract, a foreign currency or a stock that pays dividends.
- Outline what warrants are, calculate a warrant's value, and work out the dilution cost it imposes on existing shareholders.
Option Sensitivity Measures: The “Greeks”
- Assess the risks that naked and covered option positions carry, and describe them.
- Outline the advantages and disadvantages of a stop-loss hedging strategy, and explain why using it can result in naked and covered option positions.
- Work out an option's delta.
- Explain the dynamic nature of delta hedging an option position.
- Describe vega, gamma, theta and rho as they apply to option positions, and compute an option's gamma and vega.
- Explain how a delta-neutral and gamma-neutral position is put in place and then kept in place.
- Outline how delta, theta, gamma and vega are related to one another.
- Work out a portfolio's delta, gamma and vega.
- Outline how portfolio insurance is implemented and compare the strategy with delta hedging.
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