Risk Metrics Derivation

Risk Metrics: Mathematical Foundations

This document provides rigorous mathematical derivations for Value at Risk (VaR), Conditional Value at Risk (CVaR), tail risk measures, and backtesting procedures.

1. Value at Risk (VaR)

1.1 Definition

Definition 1 (Value at Risk)

For a loss random variable $L$ and confidence level $\alpha \in (0, 1)$, Value at Risk is defined as:

\[\text{VaR}_\alpha(L) = \inf\{x \in \mathbb{R} : \mathbb{P}(L \leq x) \geq \alpha\} = F_L^{-1}(\alpha) \tag{1}\]

Equivalently, VaR is the $\alpha$-quantile of the loss distribution.

Interpretation: With probability $\alpha$, losses will not exceed $\text{VaR}_\alpha$.

Example: $\text{VaR}_{0.99} = $50\text{M}$ means there is a 1% chance of losing more than $50M.

1.2 Properties

Theorem 1 (VaR Properties)

VaR satisfies:

  1. Monotonicity: If $L_1 \leq L_2$ a.s., then $\text{VaR}\alpha(L_1) \leq \text{VaR}\alpha(L_2)$
  2. Translation invariance: $\text{VaR}\alpha(L + c) = \text{VaR}\alpha(L) + c$
  3. Positive homogeneity: $\text{VaR}\alpha(\lambda L) = \lambda \text{VaR}\alpha(L)$ for $\lambda > 0$

Theorem 2 (VaR is NOT Subadditive)

VaR does not generally satisfy:

\[\text{VaR}_\alpha(L_1 + L_2) \leq \text{VaR}_\alpha(L_1) + \text{VaR}_\alpha(L_2) \tag{2}\]
Counterexample Consider two binary losses with $\alpha = 0.95$: - $L_1 = \begin{cases} 0 & \text{w.p. } 0.96 \\ 100 & \text{w.p. } 0.04 \end{cases}$ - $L_2 = \begin{cases} 0 & \text{w.p. } 0.96 \\ 100 & \text{w.p. } 0.04 \end{cases}$ Assume $L_1, L_2$ are independent. Individual VaRs: - $\text{VaR}_{0.95}(L_1) = \text{VaR}_{0.95}(L_2) = 0$ Sum distribution: - $\mathbb{P}(L_1 + L_2 = 0) = 0.96^2 = 0.9216$ - $\mathbb{P}(L_1 + L_2 = 100) = 2 \times 0.96 \times 0.04 = 0.0768$ - $\mathbb{P}(L_1 + L_2 = 200) = 0.04^2 = 0.0016$ Therefore: - $\mathbb{P}(L_1 + L_2 \leq 0) = 0.9216 < 0.95$ - $\text{VaR}_{0.95}(L_1 + L_2) = 100 > 0 + 0 = \text{VaR}_{0.95}(L_1) + \text{VaR}_{0.95}(L_2)$ $\square$

1.3 Estimation Methods

Definition 2 (Historical VaR)

Given $n$ historical losses ${L_1, \ldots, L_n}$, let $L_{(k)}$ be the $k$-th order statistic. Then:

\[\widehat{\text{VaR}}_\alpha = L_{(\lceil n\alpha \rceil)} \tag{3}\]

Definition 3 (Parametric VaR)

Assuming $L \sim \mathcal{N}(\mu, \sigma^2)$:

\[\text{VaR}_\alpha = \mu + \sigma \Phi^{-1}(\alpha) \tag{4}\]

where $\Phi^{-1}$ is the standard normal quantile function.

For $\alpha = 0.99$: $\Phi^{-1}(0.99) \approx 2.326$

2. Conditional Value at Risk (CVaR)

2.1 Definition

Definition 4 (Conditional VaR / Expected Shortfall)

\[\text{CVaR}_\alpha(L) = \mathbb{E}[L \mid L \geq \text{VaR}_\alpha(L)] \tag{5}\]

Alternative representation:

\[\text{CVaR}_\alpha(L) = \frac{1}{1-\alpha}\int_\alpha^1 \text{VaR}_u(L) \, du \tag{6}\]

2.2 Coherent Risk Measure

Theorem 3 (CVaR Coherence)

CVaR is a coherent risk measure, satisfying:

  1. Monotonicity: $L_1 \leq L_2$ a.s. $\Rightarrow$ $\text{CVaR}\alpha(L_1) \leq \text{CVaR}\alpha(L_2)$
  2. Translation invariance: $\text{CVaR}\alpha(L + c) = \text{CVaR}\alpha(L) + c$
  3. Positive homogeneity: $\text{CVaR}\alpha(\lambda L) = \lambda \text{CVaR}\alpha(L)$ for $\lambda > 0$
  4. Subadditivity: $\text{CVaR}\alpha(L_1 + L_2) \leq \text{CVaR}\alpha(L_1) + \text{CVaR}_\alpha(L_2)$
Proof of Subadditivity Using the representation: $$ \text{CVaR}_\alpha(L) = \min_{\xi \in \mathbb{R}}\left\{\xi + \frac{1}{1-\alpha}\mathbb{E}[(L - \xi)^+]\right\} $$ Let $\xi_1^*, \xi_2^*$ be the optimizers for $L_1, L_2$ respectively. For the sum: $$ \text{CVaR}_\alpha(L_1 + L_2) \leq (\xi_1^* + \xi_2^*) + \frac{1}{1-\alpha}\mathbb{E}[(L_1 + L_2 - \xi_1^* - \xi_2^*)^+] $$ Since $(a + b)^+ \leq a^+ + b^+$: $$ \leq \xi_1^* + \frac{1}{1-\alpha}\mathbb{E}[(L_1 - \xi_1^*)^+] + \xi_2^* + \frac{1}{1-\alpha}\mathbb{E}[(L_2 - \xi_2^*)^+] $$ $$ = \text{CVaR}_\alpha(L_1) + \text{CVaR}_\alpha(L_2) $$ $\square$

2.3 Closed-Form for Normal Distribution

Lemma 2.1 (Normal CVaR)

For $L \sim \mathcal{N}(\mu, \sigma^2)$:

\[\text{CVaR}_\alpha(L) = \mu + \sigma \frac{\phi(\Phi^{-1}(\alpha))}{1 - \alpha} \tag{7}\]

where $\phi$ is the standard normal PDF.

Proof For $Z = (L - \mu)/\sigma \sim \mathcal{N}(0, 1)$: $$ \text{CVaR}_\alpha(L) = \mu + \sigma \mathbb{E}[Z \mid Z \geq \Phi^{-1}(\alpha)] $$ Let $z_\alpha = \Phi^{-1}(\alpha)$. Then: $$ \mathbb{E}[Z \mid Z \geq z_\alpha] = \frac{\int_{z_\alpha}^\infty z\phi(z)dz}{\mathbb{P}(Z \geq z_\alpha)} $$ The numerator: $$ \int_{z_\alpha}^\infty z\phi(z)dz = \int_{z_\alpha}^\infty z \frac{1}{\sqrt{2\pi}}e^{-z^2/2}dz = \frac{1}{\sqrt{2\pi}}e^{-z_\alpha^2/2} = \phi(z_\alpha) $$ The denominator is $1 - \alpha$. Therefore: $$ \text{CVaR}_\alpha(L) = \mu + \sigma \frac{\phi(\Phi^{-1}(\alpha))}{1 - \alpha} $$ $\square$

Example: For $\mu = 10\text{M}$, $\sigma = 5\text{M}$, $\alpha = 0.99$:

  • $\Phi^{-1}(0.99) = 2.326$
  • $\phi(2.326) = 0.0267$
  • $\text{CVaR}_{0.99} = 10 + 5 \times \frac{0.0267}{0.01} = 10 + 13.35 = 23.35\text{M}$

2.4 Monte Carlo Estimation

Algorithm 1: CVaR Estimation

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Input: N samples {L_1, ..., L_N}, confidence level α
Output: CVaR estimate

1. Sort samples: L_(1) ≤ L_(2) ≤ ... ≤ L_(N)
2. k = ceil(N * α)
3. CVaR = mean(L_(k), L_(k+1), ..., L_(N))

Theorem 4 (Estimation Convergence)

The MC estimator $\widehat{\text{CVaR}}_\alpha$ satisfies:

\[\sqrt{N}(\widehat{\text{CVaR}}_\alpha - \text{CVaR}_\alpha) \xrightarrow{d} \mathcal{N}(0, \sigma_{\text{CVaR}}^2) \tag{8}\]

where:

\[\sigma_{\text{CVaR}}^2 = \frac{1}{(1-\alpha)^2}\text{Var}(L \cdot \mathbf{1}_{L \geq \text{VaR}_\alpha}) \tag{9}\]

3. Tail Risk Measures

3.1 Tail Conditional Expectation

Definition 5 (TCE)

\[\text{TCE}_\alpha(L) = \mathbb{E}[L \mid L > \text{VaR}_\alpha(L)] \tag{10}\]

Note: TCE and CVaR differ when the distribution has atoms at VaR.

3.2 Expected Tail Loss

Definition 6 (ETL)

\[\text{ETL}_\alpha(L) = \frac{1}{1-\alpha}\mathbb{E}[L \cdot \mathbf{1}_{L \geq \text{VaR}_\alpha(L)}] \tag{11}\]

3.3 Relationship

Lemma 3.1 (Equivalence for Continuous Distributions)

If $F_L$ is continuous:

\[\text{CVaR}_\alpha(L) = \text{TCE}_\alpha(L) = (1-\alpha)^{-1} \text{ETL}_\alpha(L) \cdot (1-\alpha) = \text{ETL}_\alpha(L) \tag{12}\]

4. Sample Size Requirements

4.1 VaR Confidence Intervals

Theorem 5 (Binomial Confidence for VaR)

The empirical quantile $\hat{q}_\alpha$ has distribution:

\[\mathbb{P}(\hat{q}_\alpha \leq x) = \sum_{k=0}^{\lfloor n\alpha \rfloor} \binom{n}{k} F(x)^k (1-F(x))^{n-k} \tag{13}\]

Lemma 4.1 (Sample Size for VaR Precision)

To estimate $\text{VaR}_\alpha$ with relative precision $\epsilon$ at confidence $1-\delta$:

\[N \geq \frac{z_{1-\delta/2}^2 \alpha(1-\alpha)}{\epsilon^2 f(\text{VaR}_\alpha)^2} \tag{14}\]

where $f$ is the PDF of $L$.

Example: For $\alpha = 0.99$, $\epsilon = 0.05$, $\delta = 0.05$:

  • Need $N \approx 10,000$ samples for 5% precision at 95% confidence

4.2 CVaR Confidence Intervals

Theorem 6 (CVaR Standard Error)

For i.i.d. samples:

\[\text{SE}(\widehat{\text{CVaR}}_\alpha) = \frac{\sigma_{\text{tail}}}{\sqrt{N(1-\alpha)}} \tag{15}\]

where $\sigma_{\text{tail}} = \sqrt{\text{Var}(L \mid L \geq \text{VaR}_\alpha)}$.

4.3 Effective Sample Size

Definition 7 (Tail Sample Count)

The effective number of samples in the tail:

\[N_{\text{eff}} = N(1 - \alpha) \tag{16}\]

Minimum Requirements:

Confidence Tail Probability Min $N$ for 30 tail samples
95% 5% 600
99% 1% 3,000
99.9% 0.1% 30,000

5. Backtesting

5.1 Kupiec Test

Definition 8 (Kupiec Test)

For $n$ observations and $x$ VaR exceedances (violations), test:

  • $H_0$: True violation rate = $1 - \alpha$
  • $H_1$: True violation rate $\neq 1 - \alpha$

Test Statistic:

\[\text{LR}_{\text{uc}} = -2\log\left[\frac{(1-\alpha)^{n-x}\alpha^x}{\hat{p}^{n-x}(1-\hat{p})^x}\right] \tag{17}\]

where $\hat{p} = x/n$.

Distribution: Under $H_0$, $\text{LR}_{\text{uc}} \sim \chi^2_1$.

Decision Rule: Reject $H_0$ if $\text{LR}{\text{uc}} > \chi^2{1,1-\delta}$.

5.2 Christoffersen Test

Definition 9 (Independence Test)

Test whether violations cluster:

\[\text{LR}_{\text{ind}} = -2\log\left[\frac{(1-\pi)^{n_{00}+n_{10}}\pi^{n_{01}+n_{11}}}{(1-\pi_0)^{n_{00}}\pi_0^{n_{01}}(1-\pi_1)^{n_{10}}\pi_1^{n_{11}}}\right] \tag{18}\]

where:

  • $n_{ij}$ = transitions from state $i$ to state $j$
  • $\pi_i = n_{i1}/(n_{i0} + n_{i1})$
  • $\pi = (n_{01} + n_{11})/n$

Distribution: Under $H_0$ (independence), $\text{LR}_{\text{ind}} \sim \chi^2_1$.

5.3 Combined Test

Definition 10 (Conditional Coverage Test)

\[\text{LR}_{\text{cc}} = \text{LR}_{\text{uc}} + \text{LR}_{\text{ind}} \tag{19}\]

Distribution: Under $H_0$, $\text{LR}_{\text{cc}} \sim \chi^2_2$.

5.4 Traffic Light System

Definition 11 (Basel Traffic Light)

Zone Exceedances (250 days, 99% VaR) Action
Green 0-4 No action
Yellow 5-9 Increased monitoring
Red 10+ Capital add-on

Probabilities (under correct model):

  • Green: $\sum_{k=0}^{4} \binom{250}{k}(0.01)^k(0.99)^{250-k} \approx 89\%$
  • Yellow: $\approx 10.5\%$
  • Red: $\approx 0.5\%$

6. Variance Decomposition

6.1 Expected Loss Components

Definition 12 (Loss Decomposition)

\[L = \underbrace{EL}_{\text{Expected}} + \underbrace{UL}_{\text{Unexpected}} \tag{20}\]

where:

  • $EL = \mathbb{E}[L]$ (expected loss)
  • $UL = L - EL$ (unexpected loss)

6.2 Capital Requirements

Definition 13 (Economic Capital)

\[EC_\alpha = \text{VaR}_\alpha(L) - \mathbb{E}[L] \tag{21}\]

Definition 14 (Risk Capital via CVaR)

\[RC_\alpha = \text{CVaR}_\alpha(L) - \mathbb{E}[L] \tag{22}\]

6.3 Portfolio Variance Decomposition

Theorem 7 (Marginal Contribution to VaR)

For portfolio loss $L = \sum_i w_i L_i$:

\[\text{MVaR}_i = \frac{\partial \text{VaR}_\alpha(L)}{\partial w_i} \approx \frac{\text{Cov}(L_i, L)}{\sigma_L} \cdot \Phi^{-1}(\alpha) \tag{23}\]

under normality assumption.

Euler Decomposition:

\[\text{VaR}_\alpha(L) = \sum_i w_i \cdot \text{MVaR}_i \tag{24}\]

7. Numerical Example

7.1 Portfolio Setup

  • 500 loans, total exposure $500M
  • Expected default rate: 3%
  • LGD: 40%
  • Asset correlation: 15%

7.2 Risk Metric Computation

Monte Carlo (10,000 simulations):

Metric Value As % of Exposure
Expected Loss $6.0M 1.2%
Std Dev $4.5M 0.9%
VaR 95% $12.5M 2.5%
VaR 99% $18.2M 3.6%
CVaR 99% $22.8M 4.6%

Capital Calculations:

  • Economic Capital (99%): $18.2M - $6.0M = $12.2M$
  • Risk Capital (99%): $22.8M - $6.0M = $16.8M$

7.3 Backtest Results

After 250 days with 99% VaR:

  • Expected violations: 2.5
  • Observed violations: 4

Kupiec Test: \(\text{LR}_{\text{uc}} = -2\log\left[\frac{0.01^4 \cdot 0.99^{246}}{(4/250)^4 \cdot (246/250)^{246}}\right] = 1.47\)

$\chi^2_1(0.95) = 3.84$

Conclusion: Do not reject $H_0$ (model acceptable).

8. Implementation

8.1 Python Functions

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import numpy as np
from scipy import stats

def var_historical(losses, alpha):
    """Historical VaR."""
    return np.percentile(losses, alpha * 100)

def cvar_historical(losses, alpha):
    """Historical CVaR."""
    var = var_historical(losses, alpha)
    return losses[losses >= var].mean()

def var_parametric(mu, sigma, alpha):
    """Parametric VaR (Normal)."""
    return mu + sigma * stats.norm.ppf(alpha)

def cvar_parametric(mu, sigma, alpha):
    """Parametric CVaR (Normal)."""
    z = stats.norm.ppf(alpha)
    return mu + sigma * stats.norm.pdf(z) / (1 - alpha)

8.2 Backtest Implementation

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def kupiec_test(violations, n_obs, alpha):
    """Kupiec unconditional coverage test."""
    p_model = 1 - alpha
    x = violations
    p_hat = x / n_obs

    lr_num = (p_model ** (n_obs - x)) * ((1 - p_model) ** x)
    lr_den = (p_hat ** (n_obs - x)) * ((1 - p_hat) ** x)

    lr_stat = -2 * np.log(lr_num / lr_den)
    p_value = 1 - stats.chi2.cdf(lr_stat, 1)

    return lr_stat, p_value

References

  1. Artzner, P., et al. (1999). Coherent measures of risk. Mathematical Finance.
  2. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk.
  3. Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives.
  4. Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review.
  5. Basel Committee on Banking Supervision. (2019). Minimum capital requirements for market risk.