FRM (Financial Risk Manager) Valuation and Risk Models (Part I) Syllabus

What are the main drawbacks of Monte Carlo VaR?

It is computationally intensive (many full revaluations), depends heavily on the chosen model and distributional assumptions (model risk), and may suffer sampling/simulation error if too few paths are used.

Define Expected Shortfall (ES) / Conditional VaR.

ES is the expected (average) loss conditional on the loss exceeding the VaR threshold: $$\text{ES} = E[L \mid L > \text{VaR}]$$ It measures the average severity of losses in the tail beyond VaR.

Why is Expected Shortfall often preferred to VaR for tail risk?

VaR tells you only the threshold but says nothing about how bad losses can get beyond it; ES captures the magnitude of tail losses and is a coherent risk measure (it satisfies subadditivity), making it more conservative and informative about extreme outcomes.

List the four properties a coherent risk measure must satisfy.

1) Monotonicity; 2) Subadditivity; 3) Positive homogeneity; 4) Translation invariance.

State the subadditivity property of a coherent risk measure.

$$\rho(A+B) \leq \rho(A) + \rho(B)$$ The risk of a combined portfolio is no greater than the sum of the individual risks; this reflects the benefit of diversification.

State the monotonicity property of a coherent risk measure.

If portfolio $A$ has returns that are always less than or equal to portfolio $B$ ($A \leq B$ in all states), then $\rho(A) \geq \rho(B)$. A portfolio that always does worse must carry more risk.

State the positive homogeneity property of a coherent risk measure.

For $\lambda \geq 0$: $$\rho(\lambda A) = \lambda\,\rho(A)$$ Scaling a position by a positive factor scales its risk by the same factor.

State the translation invariance property of a coherent risk measure.

Adding a risk-free amount of cash $n$ to a portfolio reduces its risk by that amount: $$\rho(A + n) = \rho(A) - n$$ Holding cash directly offsets risk.

Which coherence property does VaR generally violate, and why does this matter?

VaR violates subadditivity (it is not subadditive in general, especially with non-normal/fat-tailed distributions). This means VaR can penalize diversification—portfolio VaR can exceed the sum of component VaRs—which is its key theoretical flaw.

Is Expected Shortfall a coherent risk measure?

Yes. Expected Shortfall satisfies all four coherence properties, including subadditivity, which is the property VaR generally fails.

List three major limitations of VaR as a risk measure.

1) It says nothing about the size of losses beyond the threshold (tail blindness); 2) it is generally not subadditive (non-coherent); 3) it can give a false sense of security and is sensitive to estimation method and assumptions. It also may be gamed and ignores liquidity.

Define backtesting of a VaR model.

Backtesting is the process of comparing actual realized losses (or P&L) against the VaR forecasts over time to verify that the frequency of exceptions (losses exceeding VaR) is consistent with the model's stated confidence level.

What is a VaR 'exception' (exceedance)?

An exception occurs when the actual loss on a day exceeds the VaR estimate for that day. For a 99% one-day VaR model, exceptions are expected on about 1% of days.

For a 99% VaR over 250 trading days, what is the expected number of exceptions?

$$250 \times (1 - 0.99) = 2.5 \text{ exceptions}$$ The Basel framework uses a 250-day window where about 2 to 3 exceptions are expected for a sound model.

What statistical test is commonly used to test whether the observed number of VaR exceptions is consistent with the model?

Kupiec's proportion-of-failures (POF) test, an unconditional coverage test based on the likelihood ratio of observed exceptions $x$ in $T$ days versus the expected failure rate $p = 1-c$.

Distinguish unconditional coverage from conditional coverage in backtesting.

Unconditional coverage tests whether the total number of exceptions matches the expected number. Conditional coverage additionally tests whether exceptions are independent over time (not clustered)—e.g., the Christoffersen test combines both.

What are Type I and Type II errors in the context of VaR backtesting?

Type I error: rejecting a correct (accurate) VaR model. Type II error: failing to reject an incorrect (inaccurate) VaR model. Basel's traffic light zones balance these two errors.

Describe the three zones of the Basel traffic light backtesting approach (99% VaR, 250 days).

Green zone: 0–4 exceptions (model acceptable). Yellow zone: 5–9 exceptions (possible problem, increased scrutiny and capital multiplier). Red zone: 10 or more exceptions (model deemed inaccurate, must be revised).

How does the Basel backtesting result affect the market-risk capital multiplier?

The supervisory multiplier (base $k=3$) increases with the number of exceptions: green zone adds 0, yellow zone adds a plus-factor from 0.40 up to 0.85, and red zone adds 1.00 (multiplier $= 4$) and may trigger model rejection.

What is the Basel formula linking the capital charge to VaR and the multiplier?

$$\text{Market Risk Capital} = \max\!\left(\text{VaR}_{t-1},\; (k+\text{plus factor}) \cdot \frac{1}{60}\sum_{i=1}^{60}\text{VaR}_{t-i}\right)$$ where $k$ is the supervisory multiplier (minimum 3).

In a binomial tree option-pricing model, what do the parameters $u$ and $d$ represent?

$u$ is the up-move multiplier and $d$ is the down-move multiplier of the underlying asset price over one step ($d<1<u$, often $d = 1/u$). The price goes to $S_0 u$ or $S_0 d$ each step.

State the risk-neutral probability of an up move in a one-step binomial tree.

$$p = \frac{e^{r\,\Delta t} - d}{u - d}$$ where $r$ is the risk-free rate, $\Delta t$ the step length, and $u,d$ the up/down multipliers. The down-move probability is $1-p$.

How is an option valued in a one-step binomial tree using risk-neutral valuation?

$$f = e^{-r\,\Delta t}\left[p\,f_{u} + (1-p)\,f_{d}\right]$$ where $f_u$ and $f_d$ are the option payoffs after an up and down move and $p$ is the risk-neutral up probability.

What is the core principle of risk-neutral valuation?

In a risk-neutral world, all assets earn the risk-free rate, so derivatives can be priced by computing the expected payoff using risk-neutral probabilities and discounting at the risk-free rate. The actual (real-world) probabilities and risk preferences are irrelevant to the price.

How is a multi-step binomial tree used to value an option?

Build the tree forward to expiration, compute payoffs at the terminal nodes, then work backward node-by-node discounting expected risk-neutral values at each step; for American options, at each node take the maximum of the discounted continuation value and the immediate exercise value.

How are American-style options handled differently from European options in a binomial tree?

At each node the American option value is $\max(\text{intrinsic value},\ \text{discounted continuation value})$, allowing for early exercise; European options use only the discounted continuation value (exercise only at expiry).

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

$$C = S_{0}N(d_{1}) - K e^{-rT}N(d_{2})$$ where $$d_{1} = \frac{\ln(S_{0}/K) + (r + \sigma^{2}/2)T}{\sigma\sqrt{T}},\quad d_{2} = d_{1} - \sigma\sqrt{T}$$

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

$$P = K e^{-rT}N(-d_{2}) - S_{0}N(-d_{1})$$ with $d_1$ and $d_2$ defined as in the call formula. It can also be obtained from put-call parity.

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

The underlying follows geometric Brownian motion with constant volatility and drift (lognormal prices); constant risk-free rate; no dividends (in the basic form); frictionless markets (no taxes/transaction costs); continuous trading; no arbitrage; and short selling permitted. The option is European-style.

What are the five inputs required to price an option with the Black-Scholes-Merton model, and which is not directly observable?

Inputs: current stock price $S_0$, strike price $K$, time to maturity $T$, risk-free rate $r$, and volatility $\sigma$. Volatility $\sigma$ is the only input that is not directly observable and must be estimated (e.g., implied or historical volatility).