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FRM Part I Updated
What are spectral risk measures, and how do they generalize expected shortfall through risk aversion weighting?
Spectral risk measures generalize expected shortfall by applying a risk-aversion-weighted function across quantiles of the loss distribution. While ES weights all tail losses equally, spectral measures assign increasing weight to more extreme losses, making them consistent with expected utility for risk-averse agents.
Why does the collateral rate specified in a CSA determine the discount curve, and how does this affect derivative valuations?
The collateral rate determines the derivative discount curve because collateral earns interest, and this interest rate defines the economic cost of carrying the position. Different CSA terms (cash vs. bonds vs. uncollateralized) lead to different discount curves and therefore different valuations.
How does a digital (binary) option pay a fixed amount, and why does the discontinuous payoff create hedging challenges?
Digital options pay a fixed amount if the underlying finishes beyond the strike, and zero otherwise. The discontinuous payoff creates extreme delta and gamma near the strike at expiration, forcing dealers to use spread replication and position limits for risk management.
What is the Anderson-Darling test, and why is it preferred over Kolmogorov-Smirnov for testing normality in risk management applications?
The Anderson-Darling test is a goodness-of-fit test that places extra weight on the tails of the distribution, making it superior to the Kolmogorov-Smirnov test for detecting fat tails in financial return data. This matters because risk managers care most about tail behavior.
How do you read a QQ plot to assess whether financial return data follows a normal distribution?
A QQ plot compares observed data quantiles against theoretical normal quantiles. If points fall on the diagonal, the data is normal. Fat tails appear as S-shaped deviations, indicating that normal-distribution VaR underestimates tail risk.
How does a longevity swap work and why do pension funds use them?
A longevity swap transfers the risk that a group of people live longer than expected from a pension fund to a counterparty. The pension fund pays a fixed leg based on expected mortality and receives a floating leg based on actual mortality.
How do you use the chi-squared test to test a hypothesis about variance?
The chi-squared test for variance is a natural extension of hypothesis testing to the second moment of a distribution. While z-tests and t-tests focus on the mean, the chi-squared test focuses on dispersion — critically important in risk management.
How do jump-diffusion models improve on geometric Brownian motion for risk modeling?
Jump-diffusion models add sudden, discrete price changes to the continuous GBM framework. This produces fat tails and negative skewness, matching real market behavior far better than Normal-distribution models.
What exactly is a variance swap and how does the payoff work relative to implied volatility?
A variance swap is an OTC derivative where one party pays a fixed rate (the 'strike variance') and receives the realized variance of an underlying asset over the contract period. It gives pure exposure to volatility without delta-hedging complications.
How do you bootstrap zero-coupon rates from coupon bond prices step by step?
Bootstrapping extracts zero-coupon spot rates from coupon bond prices by solving sequentially from the shortest maturity to the longest. Each step uses previously derived rates as inputs, building the complete term structure one maturity at a time.
What is entropic VaR, and how does it provide a tighter bound on tail risk than traditional VaR?
Entropic VaR is derived from the Chernoff bound and uses the moment-generating function to provide a tighter tail risk bound than VaR. It sits in the ordering VaR <= EVaR <= ES, offering a coherent risk measure connected to information theory and relative entropy.
How does the funding spread adjustment work for uncollateralized derivatives, and why is FVA controversial?
FVA captures the cost of funding uncollateralized derivative positions at rates above the risk-free rate. It equals the dealer's funding spread multiplied by expected exposure over the trade's life. The controversy centers on whether FVA double-counts with DVA.
What is the Calmar ratio, and how does it compare to the Sharpe ratio for evaluating hedge fund performance?
The Calmar ratio relates annualized return to maximum drawdown, providing a risk-adjusted performance measure that is more relevant than the Sharpe ratio for hedge fund strategies where investors care about worst-case loss paths rather than volatility.
How is maximum drawdown calculated, and why do risk managers use it alongside VaR?
Maximum drawdown measures the largest cumulative loss from a peak to a subsequent trough before a new peak is established. Unlike VaR which measures single-period risk, MDD captures the worst path-dependent experience and is widely used in hedge fund due diligence.
How do embedded call and put options affect bond valuation and risk?
Bonds with embedded options have cash flows that depend on the path of interest rates. A callable bond equals a straight bond minus the call option value, while a putable bond equals a straight bond plus the put option value. OAS strips out the option effect to isolate credit spread.
What is the Jarque-Bera test and how do you use it to check if financial returns are normal?
The Jarque-Bera (JB) test is a joint test of whether a sample's skewness and excess kurtosis are consistent with a normal distribution. It is one of the most commonly referenced normality tests in FRM.
How does excess kurtosis (fat tails) affect VaR calculations, and what can you do about it?
Fat tails mean extreme returns occur far more often than Normal predicts, causing standard VaR to underestimate risk by 3-700x depending on the event severity. Fixes include Cornish-Fisher expansion, Student-t distributions, and Extreme Value Theory.
How do principal-protected notes work, and what are the hidden risks most investors miss?
Principal-protected notes (PPNs) are structured products that promise return of principal at maturity while offering upside participation. They combine a zero-coupon bond with a call option, but carry hidden risks including issuer credit risk and opportunity cost.
What are the steps in the risk management process cycle, and how do they connect?
The risk management process is a continuous five-stage cycle: identification, assessment and measurement, mitigation and control, monitoring and reporting, and governance review. Each stage feeds into the next, with a critical feedback loop from governance back to identification that keeps the framework current.
What are the main types of exotic options, and how do they differ from vanilla options in risk characteristics?
Exotic options include barrier options (activated or deactivated at trigger prices), Asian options (payoff based on average price), and lookback options (payoff based on the extreme price observed). Each has different premium levels, path dependencies, and hedging complexities compared to vanilla options.
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