Nov 04

2008 FRM Cram Session

by David Harper, CFA, FRM, CIPM

FRM |

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Yesterday I published to the member page the 2008 FRM cram session (in two parts. total running time: 2+ hours). The cram session is not good for learning anything new; it moves way too quickly to serve as an introduction to anything. I assume you’ve seen the material before. Rather, the cram is only meant to be a rapid-fire review of selected, key ideas in this year’s curriculum. I reviewed the AIMs (learning outcomes) and the fourteen episodes from this year, and then I extracted the key ideas that I thought were most thematic or relevant to the exam. The ideas that deserve to be stored in short-term memory.

Here is a rough outline of the cram session contents:

I. Quantitative Methods

  • Theoretical advantage but practical disadvantage of the normal distribution. We use normal distribution (e.g., delta normal VaR) but observed asset returns are not normal. Asset returns are skewed (skew > 0), fat-tailed (kurtosis>3) and unstable (time-varying).
  • Parametric volatility: STDEV (i.e., moving average), exponentially weighed moving average (EWMA), and GARCH(1,1). GARCH(1,1) and EWMA are both conditional and they assign exponentially declining weights (better than moving average because more recent data gets greater weight). But GARCH(1,1) adds a term for mean reversion while EWMA does not. GARCH(1,1) models unconditional fat-tails and mean-reversion toward a long-run average variance.
  • Nonparametric volatility: Historical simulation (sort actual returns from highest to lowest), multivariate density estimation (MDE) is like ARCH(m) but weights based on kernel function (current versus historical state); Hybrid (sort returns like historical simulation, but gives greater weight to more recent like EWMA)
  • Value at risk (VaR). Relative VaR is special case for short intervals (daily) where we assume mean = 0. (Note Culp's LVaR uses full absolute version). Individual VaR is not coherent (not sub-additive).
  • The square root rule requires i.i.d.
  • Taylor series approximation: option delta is analogous to bond duration. Option gamma is analogous to bond convexity (but they are not exactly the same because bond metrics include price whereas option Greeks are pure derivatives)
  • Geometric Brownian motion (a Weiner process). GBM is a lognormal diffusion process.
  • Extreme value theory (EVT): Block maxima (GEV) and POTs (GPD). Generally, POT is more modern.
  • Distributions: discrete versus continuous; density (PMF, PDF) vs cumulative. Distributions: marginal (unconditional), conditional, and joint
  • Variance, covariance, correlation. Mean, sample mean. Variance, sample variance. Population vs. sample.
  • Normal distribution (one-tailed): 1.645 normal deviate (critical z) @ 95%; 2.33 @ 99%
  • Important distributions: student's t; chi-square; F distribution
  • Hypothesis & inference: Confidence interval. Type I/Type II. p value
  • Type I versus Type II error
  • Linear regression: Definition. Standard error of parameters. Significance test
  • Linear regression: RSS + ESS = TSS
  • Multivariate linear regression: Joint hypothesis test, ANOVA, F vs. R^2, adjusted R^2

II. Market Risk

  • All hedges involve some basis risk. Tradeoff between basis risk versus liquidity; i.e., exchange-traded instruments are low basis, high-liquidity but OTC contracts are low basis, low liquidity.
  • Must know optimal hedge ratio. Beta = optimal hedge ratio (and closely related to marginal VaR).
  • Cost of Carry Model. Forward/Future = F[financing cost, storage cost, income/dividend, and convenience yield]
  • Interest rate futures. Day count convention. Cheapest to deliver (CTD). Convexity adjustment.
  • Swaps. Features (e.g., interest rate does not swap principal). Comparative advantage.
  • Swap valuation: as two bonds, the floating rate bond only requires one cash flow (a floating rate bond is worth exactly par at coupon payment date).
  • Swap valuation: be familiar with all four illustrations (interest rate swaps as bonds, FRA. currency swaps as bonds, FRA).
  • Stock options. Put-call parity (relates to lower bound and Black-Scholes) is essential.
  • Stock options valuation. Binomial (lattice) versus Black-Scholes (closed).  Black-Scholes employs Geometric Brownian Motion (GBM), a lognormal diffusion process.
  • Greeks. All are first derivatives (delta, vega, theta, rho) except Gamma (which is a 2nd derivative; rate of change of delta).
  • Greeks. Work the gamma-neutral position exercise.
  • Culp Commodity Forwards: Equilibrium formula
  • Culp Commodity Forwards: Stack-and-roll hedge. This stack hedge was employed by Metallgesellschaft.
  • Fixed Income: Compound frequency and conversion (e.g., semi-annual to continuous). Please practice several conversions: discrete to continuous, vice-versa
  • Fixed Income: Forward rate implied by two spot rates (bootstrap the implied forward curve). Definitely extra forward from spots.
  • Fixed Income: discount factor, spot rate, forward rate, and yield (YTM). YTM versus forward/spot curves
  • Accrued interest. Clean/Full price.
  • FI Sensitivities: DV01 & Duration (DV01 = Price * Duration/10,000; or, Duration = DV01*10,000/Price)
    FI Sensitivities: Macaulay versus Modified duration. Macaulay duration is "time," measured in years (10 year zero = 10 Macaulay). Modified duration is sensitivity (best measure): % of price change given 1% change in yield. Modified = Macaulay/[1+(y/k)]. Convexity.
  • Key rate shift.
  • Jorion Value at Risk (VaR). Methods
  • Jorion Value at Risk (VaR). Mapping
  • Jorion Value at Risk (VaR). Stress Testing
  • Foreign Exchange. On-balance sheet versus off-balance sheet hedge.
  • Stulz. CFaR. VaR impact of SMALL and LARGE project.

III. Credit Risk

  • Saunders. Individual Loan. Gross promised return. Expected return. Linear discriminant (Altman's Z)
  • Credit Ratings. Through-the-cycle versus At-the-point-in-time (compare to through-the-cycle in Subprime securitization).
  • Credit Ratings are procyclical
  • Implied probability of default (PD) given term structure (corporate & riskless yield). Cumulative default probability.
  • De Servigny Ch 3. Equity as a call option (i.e., use the Black-Scholes to solve for equity value). The Merton model for PD. The KMV approach is Merton except PD is not NORMSINV().
  • Loss Given Default (LGD). Primary determinants of LGD are seniority and collateral. Beta distribution is used because it is flexible and LGD is hard to parameterize.
  • Ong. EL = AE * EDF * LGD (= EAD * PD * LGD)
    Ong. Loans (more complex, asymmetric, less liquid) than bonds
  • Ong. Adjusted exposure = Outstanding (OS) + Unused Commitment (COM) * Usage Given Default (UGD). UGD is a "credit option."
  • Ong. Unexpected loss (UL) = One standard deviation
  • Ong. Portfolio EL & Portfolio UL
  • Ong. Risk contribution is the first derivative concept
  • Counterparty. Expected & Worst Exposure. Current, Potential Future, and Expected Exposure.
    Counterparty. Lending versus Counterparty risk.
  • Securitization. Features.
  • Securitization. Internal & External enhancements; Liquidity support; Net Excess Spread
  • Subprime Securitization. Key frictions.
  • Subprime Securitization. Ratings are procyclical.
  • Meissner. Synthetic Structures. Credit linked note (CLN) versus Credit default swap (CDS); i.e., funded versus unfunded.
  • Meissner. Know which instruments hedge against the three primary credit risks: default, deterioration, and market (interest rate) risk.
  • Credit Portfolio Models. Just know key dimensions; e.g., which incorporate deterioration, default, or interest rate risk.
  • Hull Credit Derivatives. CDS Valuation. PV(payment leg) = PV(payoff leg)

IV. Operational Risk

  • Allen OpRisk. Definition of OpRisk excludes strategic and reputational risk. Top down versus bottom up.
  • DB LDA @ work. Event type/business line matrix.
  • DB LDA @ work. Frequency distribution and severity distribution (more important)
  • DB LDA @ work. Piece-wise distribution: empirical (10K-1M, internal), empirical (1M -50M, internal + external), parametric (>50M)
  • Operational VaR. DB LDA is a bottom-up “numerical” approach to generating/simulating the loss distribution. Boecker gives an “analytical” approximation (i.e., can use a formula). Part of the simplification is only the mean of the frequency distribution is needed.
  • ERM: regulatory versus economic capital
  • Stulz Ch 2. CAPM. WACC. Risk management cannot create value.
    Stulz Ch 3. Due to "imperfections," risk management can create value (financial distress, tax)
  • Crouhy. RAROC and ARAROC.
  • Gallati Cases. Metallgesellshaft (hedging long-term forwards with short-term futures. Market shifted to contango. Accounting problem: futures M2M but not forwards. No fraud here), LTCM (model risk, funding liquidity risk, correlations spiked during crisis, leverage concentrated market risk), Barings (lack of supervision), Sumitomo (market manipulation [trying to corner the market], lack of supervision)
  • Liquidity. Funding versus market liquidity risk.
  • Liquidity. Factors that impact liquidation cost (time horizon, asset type, fungibility, market microstructure [call/continuous, centralized/decentralized]
  • Liquidity. Liquidity-adjusted VaR: add one-half spread.
  • Basel II: Know the weaknesses of original accord Basel. Capital ratio (regulatory capital/RWA >= 8%). Tier I/II/III capital. Credit risk standardized and IRB. Credit risk mitigants. Market risk IMA: VaR requirements and backtest (green/yellow/red). Operational risk.
  • Basel II: Note “advanced/internal” credit risk and operational risk require one-year horizon with 99.9% confidence but market risk is ten-day horizon with 99% confidence.

V. Investment

  • Traditional: CML (vs. standard deviation), SML (vs. beta), CAPM, beta = covariance [instrument, market]/variance[market]
  • RAPM: Treynor (excess return/beta; diversified portfolio), Sharpe (excess return/volatility; when diversified portfolio is not well diversified), Jensen's (actual versus CAPM-implied performance), Tracking error (volatility of active/residual return), Information ratio (residual return/tracking error), Sortino
  • Grinold. Jensen's alpha, t-stat for Jensen's alpha
  • Grinold. Information ratio = portfolio alpha/residual risk
    Jorion. Individual VaR, Diversified VaR, Marginal VaR (first derivative: change in portfolio VaR given change in position), Component VaR (only coherent VaR. Sums to Portfolio VaR), Incremental VaR
  • Jorion. Risk Budgeting
  • Stulz. Hedge funds.
  • Andrew Lo. Replication.
    Andrew Lo. Quants.
  • Jaeger. Hedge Fund Strategies

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