Financial Risk Manager (FRM) Market Risk Measurement and Management Syllabus

Write the general single-factor short-rate SDE and identify Model 1 (Vasicek with no drift / 'Model 1').

General: $$dr = \lambda(t)\,dt + \sigma\,dw$$ Model 1 sets a constant drift $\lambda$ and constant volatility $\sigma$: $dr = \lambda\,dt + \sigma\,dw$. Here $dw$ is a Brownian increment with $E[dw]=0$, $\mathrm{var}(dw)=dt$. Model 1 has normally distributed rates and no mean reversion.

State the Vasicek model SDE and its key property of mean reversion.

$$dr = k(\theta - r)\,dt + \sigma\,dw$$ where $k$ = speed of mean reversion, $\theta$ = long-run (central tendency) rate, $\sigma$ = volatility. Rates are normally distributed (can go negative), and the expected rate reverts toward $\theta$; the half-life of a shock is $\frac{\ln 2}{k}$.

What is the Cox-Ingersoll-Ross (CIR) model and how does its volatility differ from Vasicek?

$$dr = k(\theta - r)\,dt + \sigma\sqrt{r}\,dw$$ Volatility is proportional to $\sqrt{r}$, so it rises with the rate level and vanishes as $r\to 0$ — preventing negative rates (basis points volatility increases with rate). Vasicek instead has constant absolute volatility $\sigma$ that allows negative rates.

Distinguish equilibrium (e.g. Vasicek, CIR) from arbitrage-free (e.g. Ho-Lee, Hull-White) term-structure models.

Equilibrium models specify the short-rate process with constant parameters and derive the curve — they generally do NOT fit the current observed term structure exactly. Arbitrage-free models include time-dependent parameters (e.g. a drift function $\theta(t)$) calibrated to exactly reprice today's curve, ensuring no arbitrage against observed bond prices.

State the Ho-Lee model and explain how its drift achieves an arbitrage-free fit.

$$dr = \theta(t)\,dt + \sigma\,dw$$ The time-dependent drift $\theta(t)$ is chosen to match the current term structure exactly (no arbitrage). It has constant volatility and normally distributed rates (allows negative rates) but no mean reversion.

State the Hull-White (extended Vasicek) model and what each time-dependent term accomplishes.

$$dr = \left[\theta(t) - k\,r\right]dt + \sigma\,dw$$ It adds mean reversion (speed $k$) to Ho-Lee. $\theta(t)$ is calibrated to fit the current term structure (arbitrage-free), $k$ controls reversion, and constant $\sigma$ governs volatility. Rates remain normally distributed.

In 'the art of term structure modeling,' how is time-dependent volatility introduced and what is the Model 3 form?

Volatility is made a deterministic function of time, e.g. Model 3: $$dr = \lambda(t)\,dt + \sigma\,e^{-\alpha t}\,dw$$ The exponential decay $\sigma e^{-\alpha t}$ lets short-dated volatility be high and decline over time, helping fit the volatility (cap/swaption) term structure. Caution: it can imply unrealistic future declining volatility.

What is the trade-off / caveat of using time-dependent (deterministic) volatility functions in term-structure models?

They fit today's volatility term structure but assume future volatility is a fixed, known function of calendar time. This can imply that volatility in the distant future is predetermined and declining, which is economically unrealistic; mean reversion is often preferred to generate a declining term structure of volatility more naturally.

Define a mortgage pass-through security and the cash flows it distributes.

A pass-through is a security backed by a pool of mortgages; principal and interest payments from homeowners (less a servicing/guarantee fee) are 'passed through' pro rata to investors. Investors receive scheduled principal, scheduled interest, and prepayments. Agency pass-throughs (Ginnie/Fannie/Freddie) carry credit guarantees so the main risk is prepayment (interest-rate) risk.

What is the weighted average maturity (WAM) and weighted average coupon (WAC) of a mortgage pool?

WAC is the average note rate of the underlying mortgages weighted by outstanding principal balance; WAM is the average remaining months to maturity weighted by balance. They summarize the pool's coupon and maturity profile used in cash-flow and prepayment modeling.

What does prepayment risk consist of, and define extension risk vs contraction risk.

Prepayments are principal repaid ahead of schedule. Contraction risk: when rates FALL, prepayments accelerate (refinancing), shortening the security's life and forcing reinvestment at lower rates. Extension risk: when rates RISE, prepayments slow, lengthening the life just as higher reinvestment rates are available — hurting the holder. This creates negative convexity.

Define SMM (single monthly mortality) and CPR, and the relationship between them.

SMM is the fraction of the beginning-of-month balance (net of scheduled principal) prepaid in a month. CPR is the annualized conditional prepayment rate. They relate by: $$\mathrm{CPR} = 1 - (1 - \mathrm{SMM})^{12}$$ $$\mathrm{SMM} = 1 - (1-\mathrm{CPR})^{1/12}$$

Describe the PSA prepayment benchmark (100% PSA).

100% PSA assumes CPR rises linearly by 0.2% per month for the first 30 months, reaching 6% CPR, then stays constant at 6% CPR thereafter. For month $t \leq 30$: $\mathrm{CPR} = 6\% \times \frac{t}{30}$. A speed like 200% PSA doubles these CPRs (capping at 12%).

List the main drivers of mortgage prepayment in a prepayment model.

(1) Refinancing incentive — spread between the mortgage rate and current market rates (the dominant driver); (2) Seasoning/age of the loan (burnout and ramp-up); (3) Seasonality (housing turnover peaks in summer); (4) Burnout (pools that already prepaid heavily slow down); plus housing turnover, defaults, and the path of rates.

Define Option-Adjusted Spread (OAS) and what it represents.

OAS is the constant spread added to all the spot rates on the model's interest-rate tree such that the model price equals the security's market price, AVERAGED over many simulated rate paths. It strips out the value of the embedded prepayment option, isolating compensation for liquidity/credit/model risk: $$\text{Market Price} = \frac{1}{N}\sum_{i=1}^{N} PV(\text{path } i \mid \text{spot} + \text{OAS})$$

How does OAS relate to the Z-spread (zero-volatility spread)?

$$\text{Z-spread} = \text{OAS} + \text{Option Cost}$$ The Z-spread is a single spread over the spot curve with no rate volatility/optionality. OAS removes the embedded option's value, so for a security where the investor is SHORT the option (like an MBS), Option Cost $>0$ and OAS $<$ Z-spread.

What is a Collateralized Mortgage Obligation (CMO) and what is tranching meant to accomplish?

A CMO redirects the cash flows of mortgage pass-throughs into multiple bond classes (tranches) with different priorities and risk profiles. Tranching redistributes prepayment (extension/contraction) risk across tranches to meet different investor maturity/risk preferences, even though total prepayment risk is unchanged.

Describe a sequential-pay CMO structure.

In a sequential-pay CMO, all principal (scheduled + prepayments) is paid first to Tranche A until fully retired, then to B, then C, etc., while every outstanding tranche receives interest. Earlier tranches (A) have short, more certain lives (low extension risk); later tranches absorb more extension risk and have longer average lives.

Explain PAC and Support (companion) tranches in a CMO.

A Planned Amortization Class (PAC) receives principal on a fixed schedule as long as prepayment speeds stay within a PAC collar (e.g. 100-300% PSA), giving stable, predictable cash flows and low prepayment risk. The Support/companion tranche absorbs prepayment variability — getting excess principal in fast-pay periods and being deferred in slow periods — so it bears amplified extension and contraction risk.

Describe an Interest-Only (IO) and Principal-Only (PO) strip and how each reacts to falling rates.

A PO receives only principal payments; its value RISES when rates fall (faster prepayments return principal sooner) — positive duration-like behavior. An IO receives only interest on the outstanding balance; its value FALLS when rates drop because faster prepayments shrink the balance generating interest. IOs often have negative effective duration.

How is the effective (option-adjusted) duration of an MBS computed?

$$D_{\text{eff}} = \frac{P_{-} - P_{+}}{2 \times P_0 \times \Delta y}$$ where $P_{-}$ and $P_{+}$ are prices from re-running the prepayment/OAS model after shifting rates down and up by $\Delta y$, holding OAS constant. Because prepayments are re-estimated at each shift, it captures the option's effect, unlike modified duration.

What is negative convexity in MBS, and write effective convexity.

Negative convexity means price appreciation when rates fall is LESS than price depreciation when rates rise — caused by the homeowner's prepayment option capping upside. $$C_{\text{eff}} = \frac{P_{+} + P_{-} - 2P_0}{P_0 (\Delta y)^{2}}$$ For MBS this is often negative over the refinancing region; the price/yield curve bends the 'wrong' way.

Define implied volatility and how it is obtained.

Implied volatility is the volatility input that, when plugged into an option pricing model (e.g. Black-Scholes), makes the model price equal the observed market price. It is the market's forward-looking expectation of volatility, backed out by numerical inversion since the model is not analytically invertible for $\sigma$.

Define the implied volatility term structure and what an upward- vs downward-sloping structure implies.

It is the plot of implied volatility against option maturity (for a fixed moneyness). An upward slope implies the market expects volatility to RISE over time (often when current vol is low/mean-reverting up); a downward slope implies expected falling volatility (often after a vol spike). It reflects mean reversion of volatility.

Define the volatility skew/smile and the typical equity-index pattern.

The volatility skew/smile is implied volatility plotted against strike (or moneyness) for fixed maturity. For equity indices, there is a pronounced skew: low-strike (OTM puts) have HIGHER implied vol than high-strike calls, reflecting crash fear and the leverage effect — fat left tail. This contradicts Black-Scholes' constant-vol assumption.

How does the volatility smile reveal the market's implied return distribution versus lognormal?

A symmetric smile implies fatter tails than lognormal (excess kurtosis) — both deep OTM puts and calls priced richer. An equity skew (higher left-side vol) implies a left-skewed distribution: a heavier left tail and thinner right tail than Black-Scholes lognormal assumes. Currency options typically show a more symmetric smile.

What is correlation trading and which products provide direct exposure to correlation?

Correlation trading takes positions on the realized vs implied correlation among assets. Key vehicles: dispersion trades (sell index volatility, buy single-name volatility — long correlation if index is rich), correlation swaps, and basket options. Index implied volatility embeds an implied correlation linking constituent vols to the index vol.

Give the relationship linking index variance to constituent variances and average correlation in a basket.

For weights $w_i$, single-name vols $\sigma_i$, and average correlation $\bar{\rho}$: $$\sigma_{\text{index}}^{2} = \sum_i w_i^{2}\sigma_i^{2} + \sum_{i\neq j} w_i w_j \rho_{ij}\sigma_i\sigma_j$$ Higher correlation raises index variance toward the weighted-average of single-name vols; lower correlation creates a larger diversification gap (the basis dispersion traders exploit).

Define a variance swap payoff and explain why variance (not volatility) swaps are commonly traded.

$$\text{Payoff} = N_{\text{var}} \times \left(\sigma_{\text{realized}}^{2} - K_{\text{var}}^{2}\right)$$ where $N_{\text{var}}$ is the variance notional and $K_{\text{var}}$ the strike (variance). Variance swaps are favored because their payoff in variance is exactly replicable with a static portfolio of options across all strikes plus dynamic delta-hedging, whereas volatility (the square root) is not.

Relate variance notional to vega notional and explain the convexity of a variance swap.

$$N_{\text{vega}} = 2 K_{\text{var}} \times N_{\text{var}}$$ A variance swap is long convexity in volatility: because payoff is in $\sigma^{2}$, gains from rising vol exceed losses from equal falling vol, so realized variance swaps tend to price above (at a premium to) at-the-money implied volatility (the variance risk premium).

Define vega, gamma, and vanna as higher-order option risks in an options book.

Vega: sensitivity of value to implied volatility, $\frac{\partial V}{\partial \sigma}$. Gamma: $\frac{\partial^{2} V}{\partial S^{2}}$, the rate of change of delta. Vanna: $\frac{\partial^{2} V}{\partial S \partial \sigma}$, cross-sensitivity of delta to vol (or vega to spot). Volga (vomma): $\frac{\partial^{2} V}{\partial \sigma^{2}}$, vega's sensitivity to vol — drives skew/smile hedging.

Why can't a single vanilla option hedge all higher-order risks, and how are skew/smile risks hedged?

Delta-hedging neutralizes only first-order spot risk; gamma, vega, vanna, and volga remain. Because implied vol varies by strike (skew), hedging vega at one strike leaves exposure at others. Risk managers hedge with a portfolio of options across multiple strikes/maturities (e.g. risk reversals to hedge vanna/skew, butterflies/strangles to hedge volga/smile curvature).

Outline the evolution of the Basel market risk framework from 1996 to Basel 2.5.

1996 Market Risk Amendment introduced capital for trading-book market risk via Standardized and Internal Models (10-day, 99% VaR) approaches. After the 2007-09 crisis, Basel 2.5 (2009) added Stressed VaR, the Incremental Risk Charge (IRC) for default/migration, and a Comprehensive Risk Measure (CRM) for correlation trading — roughly tripling trading-book capital.

What were the key deficiencies of the pre-FRTB (VaR-based) regime that motivated reform?

(1) VaR is not coherent (not sub-additive) and ignores tail severity; (2) a fuzzy, arbitrageable boundary between banking and trading books allowing regulatory capital arbitrage; (3) VaR uses a single liquidity horizon, ignoring varying market liquidity; (4) procyclicality and inadequate capture of tail/credit-spread risk during stress.

What is the Fundamental Review of the Trading Book (FRTB) and its two capital approaches?

FRTB (Basel III final market-risk standard) overhauls trading-book capital. Two approaches: the Standardized Approach (SA) — sensitivities-based method plus default risk charge and residual risk add-on; and the Internal Models Approach (IMA) — requires regulatory approval. FRTB makes the SA mandatory to compute (as a floor/fallback) for all banks.

Which two major risk-measurement changes did FRTB introduce in the Internal Models Approach?

(1) It replaces 99% VaR with 97.5% Expected Shortfall (a coherent, tail-sensitive measure) calibrated to a stress period. (2) It replaces a single 10-day horizon with five liquidity horizons (10, 20, 40, 60, 120 days) assigned by risk-factor liquidity, scaling ES for varying market liquidity.

How does FRTB redefine the boundary between the banking book and trading book?

FRTB sets a stricter, more objective boundary based on intent and instrument type to curb arbitrage. Instruments are presumptively assigned (e.g. trading intent, market-making, equities go to trading book); reclassification between books is heavily restricted and any capital benefit from switching is disallowed, reducing regulatory capital arbitrage.

Under the FRTB Internal Models Approach, what tests must each trading desk pass to retain model approval?

Each desk must pass: (1) Backtesting of 1-day VaR against actual and hypothetical P&L (limits on exceptions), and (2) P&L Attribution (PLA) test comparing the risk model's P&L (RTPL) to the front-office hypothetical P&L (HPL) using Spearman correlation and Kolmogorov-Smirnov metrics. Failing desks fall back to the Standardized Approach. Non-modellable risk factors (NMRFs) get a separate stressed capital add-on.

In FRTB's sensitivities-based Standardized Approach, what three risk components are aggregated?

(1) Delta (linear sensitivity to risk factors), (2) Vega (sensitivity to implied volatility), and (3) Curvature (capturing non-linear/gamma risk from large up/down shocks). These are computed per risk class with regulatory risk weights and prescribed correlations, then added to a Default Risk Charge (DRC) and a Residual Risk Add-On (RRAO).

Why is Expected Shortfall considered superior to VaR as a risk measure, referencing coherence?

ES is the expected loss conditional on exceeding VaR: $$\mathrm{ES}_{\alpha} = E[L \mid L \geq \mathrm{VaR}_{\alpha}] = \frac{1}{1-\alpha}\int_{\alpha}^{1}\mathrm{VaR}_u\,du$$ ES is a coherent risk measure — it satisfies sub-additivity (diversification never increases risk), which VaR can violate. ES also accounts for the magnitude of tail losses beyond the VaR threshold, which VaR ignores.