VaR, Credit Risk, and Interest Rate Models: LMM vs. Hull-White

Finance · Medium · Free problem

Three core concepts from quantitative finance:

(a) Value at Risk (VaR): Define VaR and explain the three standard methods for computing it (parametric, historical simulation, Monte Carlo). What does $\text{VaR}_{95}$ of $\

{,}000{,}000$ mean in practice?

(b) Credit Risk: Define the components of expected credit loss: PD, LGD, and EAD. How do they combine into an expected loss figure?

(c) Interest Rate Modeling: Compare the LIBOR Market Model (LMM) and the Hull-White (HW) model. What is each model designed for, and what are the key trade-offs?

Hints

  1. For VaR: distinguish between what VaR *is* (a quantile of the loss distribution) and what it is *not* (an expectation or a bound on worst-case loss).
  2. For credit risk: the three factors PD, LGD, and EAD each measure a different dimension of credit loss -- probability, severity, and size; they multiply because the events are treated as independent in the simple model.
  3. For LMM vs. HW: the key distinction is what random variable each model directly specifies -- LMM models the same forward rates that appear in swap and cap payoffs, while HW models the unobservable instantaneous short rate.

Worked Solution

How to Think About It: These are three independent conceptual blocks that a quant interviewer might bundle together to test breadth. Answer each crisply. VaR is a risk threshold, not an expectation -- a common mistake is treating it as the expected loss. Credit risk decomposes loss into three factors that you can estimate separately. LMM vs. HW is a model complexity vs. tractability trade-off that any rates quant needs to be able to articulate.

Key Insight: For each topic, know the definition, the formula, and one practical limitation or trade-off.

The Method:

Part (a): Value at Risk

VaR at confidence level $\alpha$ is the threshold loss $L^{*}$ such that the probability of losing more than $L^{*}$ over a given horizon is at most

- \alpha$:

$P(\text{Loss} > \text{VaR}_\alpha) = 1 - \alpha$

$\text{VaR}_{95} = \

{,}000{,}000$ means: there is a $5\%$ chance of losing more than $\
{,}000{,}000$ in the next trading day (or whatever horizon is specified). It says nothing about *how much* more you might lose beyond that threshold.

Three computation methods:

  1. Parametric (variance-covariance): Assume returns are normally distributed. Then $\text{VaR}_\alpha = \mu - z_\alpha \sigma$ where $z_\alpha$ is the $\alpha$-quantile of the standard normal. Fast and analytically tractable. Fails when returns are fat-tailed or the portfolio has non-linear payoffs (options).
  1. Historical simulation: Use the actual historical return distribution -- sort past $n$-day returns, take the $(1-\alpha)$-percentile. Non-parametric and captures fat tails. But relies entirely on the historical window; rare events not in the window are invisible.
  1. Monte Carlo: Simulate thousands of return scenarios from a model, compute the portfolio value for each, take the percentile. Most flexible (handles non-linearities, correlations, fat tails). Most expensive computationally.

Practical limitations of VaR: It is not sub-additive in general (a portfolio of two positions can have higher VaR than the sum, which violates the intuition that diversification reduces risk). Expected Shortfall (CVaR, $ES$) -- the expected loss *given* that you exceed VaR -- is a better coherent risk measure and is now preferred by regulators under Basel III.

Part (b): Credit Risk -- Expected Loss

Expected credit loss (EL) decomposes as:

$EL = PD \times LGD \times EAD$

  • PD (Probability of Default): The probability that the counterparty fails to meet its obligations. Estimated from credit ratings, credit spreads, or structural models (e.g., Merton model).
  • LGD (Loss Given Default): The fraction of exposure you lose conditional on default (i.e.,
    - \text{recovery rate}$). Typical values: $40-60\%$ for senior unsecured corporate debt.
  • EAD (Exposure at Default): The notional amount outstanding at the time of default. For a loan, this is straightforward. For derivatives, it requires modeling future mark-to-market exposure (expected exposure profile).

Example: A $\

0{,}000{,}000$ loan with $PD = 2\%$, $LGD = 50\%$: $EL = 0.02 \times 0.50 \times \
0{,}000{,}000 = \
00{,}000$

Note: EL is the *average* -- in any given year you either lose $\$5{,}000{,}000$ (if default) or nothing. The capital question is about unexpected loss (UL), the tail above EL.

Part (c): LMM vs. Hull-White

| Feature | LIBOR Market Model (LMM) | Hull-White (HW) | |---|---|---| | What it models | Forward LIBOR rates $F(t, T_i, T_{i+1})$ | Short rate $r(t)$ | | Distribution | Log-normal (each forward rate) | Normal (short rate) | | Calibration | Caps/floors and swaptions simultaneously | Fits initial yield curve exactly | | Negative rates | Cannot model (log-normal) | Can model (normal) | | Closed-form formulas | Black's formula for caps/floors (by construction) | Bonds, European options | | Complexity | High -- high-dimensional SDE system | Low -- one-factor mean-reverting SDE | | Best for | Pricing and hedging exotic interest rate derivatives (Bermudan swaptions, CMS products) | Simpler rate products: bonds, vanilla swaptions, CVA calculations |

The core trade-off: LMM is market-consistent and rich (it directly models the rates that appear in payoff formulas), but it has no closed-form bond prices and is expensive to simulate. HW is analytically tractable (closed-form bond prices, affine term structure) and fast, but assumes normally distributed rates and has limited flexibility for fitting volatility surfaces.

Post-LIBOR note: LMM has been adapted to SOFR and other risk-free rates ("RFR Market Models"), but the same conceptual framework applies.

Answer: VaR is a loss quantile, not an expectation. Expected credit loss is $PD \times LGD \times EAD$. LMM models forward rates directly (complex, market-consistent, for exotics); HW models the short rate (tractable, analytical, for vanilla products).

Intuition

VaR is one of the most widely used and widely misunderstood risk metrics in finance. The phrase "we have a 95% VaR of

million" sounds reassuring, but it tells you nothing about what happens in the worst 5% of scenarios. A portfolio with VaR =
M could have losses of
.01M in bad scenarios, or losses of
00M -- VaR cannot distinguish between these. This is why Expected Shortfall (the mean loss in the tail) has replaced VaR as the headline risk measure in modern regulatory frameworks. Understanding the difference is not just academic -- it changes how you think about tail hedging.

The LMM vs. HW comparison is a microcosm of a broader trade-off in quantitative modeling: market-consistent complexity vs. analytical tractability. LMM was developed precisely because practitioners needed models that directly corresponded to observable market prices (forward LIBOR rates, cap volatilities). HW remains popular for anything where you need to run many simulations quickly (CVA/XVA calculations, Monte Carlo scenario generation) because its closed-form bond prices dramatically cut computational cost. Both models remain in use because different problems demand different tools -- no single model dominates.

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