Financial Risk Manager (FRM) Valuation and Risk Models Syllabus

How many VaR exceptions would you expect at 99% confidence over 250 trading days?

Expected exceptions $= (1-c)\times T = 0.01 \times 250 = 2.5$ exceptions per year.

What is Kupiec's unconditional coverage test used for in backtesting?

It tests whether the observed number of VaR exceptions is statistically consistent with the model's expected exception rate $p=1-c$, using a log-likelihood ratio statistic that follows a $\chi^2$ distribution with 1 degree of freedom.

In the Basel backtesting framework, what are the three zones for 99% VaR over 250 days?

Green zone: 0–4 exceptions (model acceptable). Yellow zone: 5–9 exceptions (increasing multiplier penalty). Red zone: 10 or more exceptions (model rejected, regulatory action).

Distinguish unconditional coverage from conditional coverage in VaR backtesting.

Unconditional coverage checks only the total number of exceptions. Conditional coverage (e.g. Christoffersen's test) also checks that exceptions are independent over time—clustered exceptions signal the model fails to react to changing volatility.

Define duration of a bond (Macaulay duration).

Macaulay duration is the weighted-average time to receipt of a bond's cash flows, weighted by the present value of each cash flow: $D_{Mac} = \frac{\sum_t t\, \frac{CF_t}{(1+y)^{t}}}{P}$, measured in years.

Define modified duration and its relation to price sensitivity.

$D_{mod} = \frac{D_{Mac}}{1+y/k}$ (for $k$ compounding periods). It gives the approximate percentage price change for a small yield change: $\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y$.

What is effective (DV01-based) duration and when is it used?

Effective duration estimates price sensitivity using a parallel shift of the yield curve and full repricing: $D_{eff} = \frac{P_{-} - P_{+}}{2\,P_0\,\Delta y}$. It is used for bonds with embedded options whose cash flows change with yields.

What is DV01 (dollar value of a basis point / PVBP)?

DV01 is the change in a bond's price for a one-basis-point change in yield: $\text{DV01} = -\frac{\Delta P}{\Delta y} \times \frac{1}{10000}$, equivalently $\text{DV01} = D_{mod}\times P \times 0.0001$.

Define convexity and write the duration-plus-convexity price approximation.

Convexity $C$ measures the curvature of the price-yield relationship (second derivative). The price change is $\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y + \tfrac{1}{2}C\,(\Delta y)^{2}$.

Why does duration alone underestimate the true price change for large yield moves, and how does convexity correct it?

The price-yield curve is convex (curved), but duration assumes a linear relationship. For large moves duration overstates price drops and understates gains; the positive convexity term $\tfrac{1}{2}C(\Delta y)^2$ adds back the curvature, improving the estimate.

Is positive convexity beneficial to a bondholder, and why?

Yes. With positive convexity, prices rise more when yields fall than they fall when yields rise by the same amount—an asymmetry favorable to the holder. Investors pay for it via lower yields.

What are key rate durations (partial durations) and why are they needed?

Key rate durations measure a bond's price sensitivity to a change in the yield at a single specific maturity (key rate) while holding other rates constant. They capture exposure to non-parallel yield-curve shifts (steepening, flattening, twists) that a single effective duration misses.

How do key rate durations relate to effective duration?

The sum of all key rate durations approximately equals the effective (overall) duration: $D_{eff} \approx \sum_i \text{KRD}_i$. Each KRD decomposes the total sensitivity across the curve.

What limitation of duration do partial/key rate durations address?

Standard duration assumes parallel shifts of the yield curve. Real curves move non-parallel (slope and curvature changes); key rate/partial durations measure sensitivity to localized changes at different maturities.

How do you hedge a bond position's interest rate risk using duration matching (dollar duration)?

Choose a hedge position so the combined dollar durations net to zero: $D_{mod}^{P}\times P + D_{mod}^{H}\times H = 0$. Using DV01, the hedge ratio is $N = -\frac{\text{DV01}_{portfolio}}{\text{DV01}_{hedge}}$.

What residual risk remains after a duration-matched hedge, and how can it be reduced?

Duration matching only neutralizes first-order (parallel) risk; residual exposure to large moves (convexity mismatch) and to non-parallel shifts remains. It is reduced by also matching convexity and by using key-rate/bucket hedging.

State the Black-Scholes-Merton formula for a European call option.

$C = S_0 N(d_1) - K e^{-rT} N(d_2)$, where $d_1 = \frac{\ln(S_0/K) + (r + \sigma^{2}/2)T}{\sigma\sqrt{T}}$ and $d_2 = d_1 - \sigma\sqrt{T}$.

List the key assumptions of the Black-Scholes-Merton model.

The underlying follows geometric Brownian motion with constant volatility and drift; no dividends (in the basic form); constant risk-free rate; no transaction costs or taxes; continuous trading; no arbitrage; and lognormally distributed prices (normally distributed returns).

What is the put-call parity relationship for European options?

$C + K e^{-rT} = P + S_0$ (no dividends). Rearranged: $C - P = S_0 - K e^{-rT}$. It links the prices of a European call and put with the same strike and maturity.

How is a European put priced under Black-Scholes-Merton?

$P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$, using the same $d_1$ and $d_2$ as the call. It can also be obtained from put-call parity.

Describe how a one-step binomial tree prices an option using risk-neutral valuation.

The option value is the discounted expected payoff under risk-neutral probabilities: $f = e^{-rT}[p f_u + (1-p) f_d]$, where $p = \frac{e^{rT} - d}{u - d}$ is the risk-neutral up-probability, and $u$, $d$ are the up/down factors.

In a Cox-Ross-Rubinstein binomial tree, how are the up and down factors set?

$u = e^{\sigma\sqrt{\Delta t}}$ and $d = e^{-\sigma\sqrt{\Delta t}} = 1/u$, where $\sigma$ is volatility and $\Delta t$ is the length of each time step. This matches the variance of returns.

Why are binomial trees preferred over Black-Scholes for American options?

Binomial trees allow checking the early-exercise condition at each node (taking the max of the intrinsic value and the continuation value), so they can price American-style early exercise, which the closed-form Black-Scholes cannot handle.

Define delta ($\Delta$) of an option and its value for a Black-Scholes call.

Delta is the sensitivity of the option price to the underlying price: $\Delta = \frac{\partial f}{\partial S}$. For a non-dividend BSM call, $\Delta = N(d_1)$, ranging from 0 to 1; for a put, $\Delta = N(d_1) - 1$.

Define gamma ($\Gamma$) and explain its significance.

Gamma is the rate of change of delta with respect to the underlying: $\Gamma = \frac{\partial^{2} f}{\partial S^{2}}$. It measures convexity of the option value; high gamma means delta changes rapidly, so a delta hedge must be rebalanced frequently. Gamma is largest for at-the-money, near-expiry options.

Define theta, vega, and rho.

Theta ($\Theta = \partial f/\partial t$) is time decay—sensitivity to passage of time. Vega ($\partial f/\partial\sigma$) is sensitivity to volatility. Rho ($\partial f/\partial r$) is sensitivity to the risk-free interest rate.

What does it mean for a portfolio to be delta-neutral, and why is it not sufficient alone?

Delta-neutral means the net delta is zero, so it is immune to small first-order moves in the underlying. It is insufficient because it ignores gamma: for larger moves the delta changes, so the hedge breaks down unless the portfolio is also gamma-hedged.

What is the volatility smile (and skew)?

A plot of Black-Scholes implied volatility against strike price for options of the same maturity. A 'smile' shows higher implied vol for deep ITM and OTM strikes; equity options typically show a 'skew/smirk' with higher implied vol for low strikes (out-of-the-money puts), reflecting crash fear and fat left tails.

What does the existence of a volatility smile imply about the Black-Scholes assumptions?

It implies the true distribution of returns is not lognormal with constant volatility—markets price in fat tails and skewness. If BSM held exactly, implied volatility would be constant (flat) across all strikes.

What is the volatility surface?

A three-dimensional representation of implied volatility as a function of both strike (or moneyness) and time to maturity. Traders interpolate across this surface to price and risk-manage options consistently with quoted market prices.

How is default probability commonly measured, and distinguish PD from cumulative/marginal default probability.

Default probability (PD) is the likelihood a borrower defaults over a horizon. It can be derived from credit spreads, ratings/historical transition data, or structural models (e.g. Merton). Marginal PD is the chance of default in a given period given survival to its start; cumulative PD is the probability of defaulting by some horizon.

Write the formula for Expected Loss (EL) on a credit exposure.

$\text{EL} = \text{PD} \times \text{LGD} \times \text{EAD}$, where PD is probability of default, LGD is loss given default ($=1-\text{recovery rate}$), and EAD is exposure at default.

Distinguish Expected Loss from Unexpected Loss in credit risk.

Expected Loss is the average anticipated loss, covered by loan-loss provisions/pricing. Unexpected Loss is the volatility (standard deviation) of credit losses around the mean—the potential for losses to exceed EL—and is covered by economic/regulatory capital.

How does the Merton structural model define default?

Equity is modeled as a call option on the firm's assets with strike equal to the face value of debt $D$. Default occurs at maturity if asset value $V_T < D$; the default probability is $N(-d_2)$, and distance to default measures how many standard deviations assets are above the default point.

Compare external and internal credit ratings.

External ratings are issued by agencies (Moody's, S&P, Fitch) using public scales and are 'through-the-cycle' (stable across the business cycle). Internal ratings are a bank's own assessments, often 'point-in-time' (reflecting current conditions), used for internal capital and lending decisions.

What is a rating transition (migration) matrix?

A table giving the probabilities that an issuer rated in a given category migrates to each other rating category (including default) over a fixed horizon, typically one year. Rows are starting ratings, columns are ending ratings, and each row sums to 100%.

What key empirical properties are observed in rating transition matrices?

The largest probabilities lie on the diagonal (ratings are sticky/stable); default probabilities rise sharply as credit quality falls; transition probabilities are higher for adjacent ratings; and migrations exhibit some momentum/autocorrelation (a downgrade makes further downgrade more likely).

Distinguish 'through-the-cycle' from 'point-in-time' rating philosophies.

Through-the-cycle ratings assess default risk over a full economic cycle and are stable, avoiding frequent changes due to short-term conditions. Point-in-time ratings reflect current economic conditions and a near-term horizon, so they are more volatile and migrate more with the cycle.

How can default probability be inferred from a bond's credit spread?

Approximately, credit spread $\approx \text{PD} \times \text{LGD}$ (spread $\approx$ PD $\times (1-\text{recovery})$). Solving gives an implied (risk-neutral) PD $\approx \frac{\text{spread}}{1-\text{recovery rate}}$. Risk-neutral PDs from spreads typically exceed real-world historical PDs.

What is recovery rate and how does it relate to LGD?

Recovery rate is the fraction of exposure recovered after default. Loss given default is its complement: $\text{LGD} = 1 - \text{recovery rate}$. Recovery depends on seniority and collateral, with senior secured debt recovering more than subordinated/unsecured debt.