Risk Analytics

Monte Carlo Student-t

Estimate portfolio risk by simulating correlated asset returns using a multivariate Student-t model.

Overview

The Monte Carlo Student-t method estimates portfolio risk by generating multiple possible return scenarios for the assets in a portfolio using a Student-t distribution.

The API calculates historical returns from the provided price data, estimates the mean return and covariance structure of the assets, and uses these values to generate correlated simulated returns.

Unlike the Monte Carlo Normal method, the Student-t distribution can produce heavier tails, allowing the simulation to generate more extreme return outcomes.

Each simulation produces a possible portfolio outcome over the selected time horizon. The resulting distribution of portfolio profit and loss is then used to calculate risk metrics such as Value at Risk (VaR) and Conditional Value at Risk (CVaR).

Input Data

Monte Carlo Student-t uses historical price data from the selected dataset.

For a multi-asset portfolio, the dataset contains price observations for each asset included in the simulation. The API calculates a return series for each asset and uses the resulting observations to estimate the statistical inputs required for the simulation.

Calculate Historical Returns

The API converts historical prices into simple returns before estimating the simulation parameters.

R_t = \frac{ P_t - P_{t-1} }{ P_{t-1} }

Where:

$R_t$ is the return for the current observation.
$P_t$ is the current price.
$P_{t-1}$ is the price from the previous observation.

The API calculates a return series for each asset in the dataset. These return series are used in the following parameter estimation and simulation steps.

Estimate Mean and Covariance

For each asset, the API calculates the mean of the historical return series.

\mu = \frac{1}{n} \sum_{i=1}^{n} R_i

The API also calculates a covariance matrix across the assets in the dataset.

For two assets A and B, covariance is calculated from the deviations of their historical returns from their respective mean returns.

\operatorname{Cov}(A, B) = \frac{ \sum_{i=1}^{n} (R_{A,i}-\mu_A) (R_{B,i}-\mu_B) }{ n-1 }

The resulting covariance matrix is used to model the relationship between simulated returns across multiple assets.

Cholesky Decomposition

For portfolios containing multiple assets, the API uses the covariance matrix to model the relationship between asset returns.

The covariance matrix is decomposed using Cholesky decomposition before generating correlated random shocks for the simulation.

\Sigma = LL^T

$\Sigma$ is the covariance matrix of asset returns.
$L$ is a lower triangular matrix.
$L^T$ is the transpose of $L$ .

The resulting matrix is used in the next step to transform independent random variables into correlated shocks that follow the covariance structure estimated from the input data.

Generate Standardized Student-t Variables

The API generates Student-t random variables to model a greater probability of extreme return outcomes than the normal distribution.

The Student-t variables are standardized to have an identity covariance matrix, allowing the Cholesky factor of the historical covariance matrix to introduce the desired covariance structure.

Standard Normal Variables

First, the API generates a vector of independent standard normal random variables.

Z \sim \mathcal{N}(0, I)

Chi-Square Scaling Variable

The API also generates an independent chi-square scaling variable with the specified number of degrees of freedom. The same scaling variable is applied to the Student-t random variable vector for a given simulated observation.

S \sim \chi^2_{\nu}

$\nu$ represents the degrees of freedom of the Student-t distribution.
$S$ represents the chi-square scaling variable.
$S$ is generated independently of $Z$ .

Standardized Student-t Variables

The API combines the normal random variable vector and the chi-square scaling variable to generate standardized Student-t variables.

Y = \sqrt{ \frac{\nu - 2}{S} } Z

$Y$ represents the standardized Student-t random variable vector.
$Z$ represents the vector of independent standard normal random variables.
$S$ represents the chi-square scaling variable applied to the random variable vector.
$\nu$ represents the degrees of freedom.

The random scaling factor produces heavier tails than the normal distribution. When $\nu > 2$ , the variables are standardized so that their covariance matrix is the identity matrix.

\operatorname{Cov}(Y) = I

The standardized Student-t variables are transformed in the next step using the Cholesky factor to introduce the covariance structure between asset returns.

Generate Correlated Shocks

The API transforms the standardized Student-t variables using the Cholesky factor of the covariance matrix to generate correlated shocks.

X = LY

$Y$ represents the standardized Student-t random variable vector.
$L$ is the lower triangular matrix obtained from the Cholesky decomposition of the covariance matrix.
$X$ represents the resulting correlated Student-t shocks.

Because the standardized Student-t variables have an identity covariance matrix, the transformation produces shocks with the covariance structure estimated from the historical return data.

\operatorname{Cov}(X) = L \operatorname{Cov}(Y) L^T = LIL^T = \Sigma

The resulting shocks preserve the heavier-tailed behavior of the Student-t distribution while incorporating the relationships between asset returns.

Calculate Simulated Returns

The API calculates simulated returns by combining the estimated mean return of each asset with the correlated Student-t shocks generated during the simulation.

R_{\text{simulated}} = \mu + X

$R_{\text{simulated}}$ represents the simulated return.
$\mu$ represents the estimated mean return of each asset.
$X$ represents the correlated Student-t shock.

A simulated return is generated for each asset at every simulated time step across all requested simulation scenarios.

Compared with the Monte Carlo Normal method, the simulated returns incorporate heavier-tailed shocks, increasing the probability of extreme outcomes.

Simulate Asset Value Paths

The API applies the simulated returns to the initial value of each asset position to generate simulated value paths across the requested time horizon.

V_{i,t} = V_{i,t-1}(1 + R_{i,t})

$V_{i,t}$ represents the simulated value of asset position $i$ at time step $t$ .
$V_{i,t-1}$ represents the value of the asset position at the previous time step.
$R_{i,t}$ represents the simulated return of asset $i$ for the current time step.

The calculation is applied sequentially across the requested time horizon, producing a simulated value path for each asset position in every simulation scenario.

Equivalently, the value at each time step can be expressed as the cumulative product of the simulated growth factors.

V_{i,t} = V_{i,0} \prod_{k=1}^{t} (1 + R_{i,k})

For the first simulated time step, the calculation starts from the initial value of each asset position provided in the portfolio input.

Calculate Portfolio Value

For each simulation scenario, the API calculates the total portfolio value by summing the simulated values of all asset positions.

V_t = \sum_{i=1}^{n} V_{i,t}

$V_t$ represents the total portfolio value at time step $t$ .
$V_{i,t}$ represents the simulated value of asset position $i$ at time step $t$ .
$n$ represents the number of asset positions in the portfolio.

The calculation is performed for every simulation scenario across the requested time horizon.

The terminal portfolio value is the total portfolio value at the final simulated time step.

V_{\text{terminal}} = V_T

Calculate Profit and Loss

After calculating the terminal portfolio value for each simulation scenario, the API calculates the corresponding profit or loss relative to the initial portfolio value.

\operatorname{PnL} = V_{\text{terminal}} - V_{\text{initial}}

$\operatorname{PnL}$ represents the profit or loss for a simulation scenario.
$V_{\text{terminal}}$ represents the total portfolio value at the final simulated time step.
$V_{\text{initial}}$ represents the initial total portfolio value.

A positive P&L represents a simulated gain, while a negative P&L represents a simulated loss.

The resulting P&L values from all simulation scenarios form the distribution used to calculate Value at Risk and Conditional Value at Risk.

Calculate Value at Risk

The API calculates Value at Risk from the simulated portfolio profit and loss distribution at the requested confidence level.

P&L Quantile

First, the API identifies the lower-tail quantile of the simulated P&L distribution.

q_{\mathrm{PnL}} = Q_{1-c}(\mathrm{PnL})

$q_{\mathrm{PnL}}$ represents the lower-tail P&L quantile.
$c$ represents the requested confidence level.
$Q_{1-c}$ represents the quantile at probability $1-c$ .

For example, at a 95% confidence level, the API evaluates the 5th percentile of the simulated P&L distribution.

Value at Risk

The API converts the lower-tail P&L quantile into a non-negative loss value.

\operatorname{VaR} = \max\left( 0, -q_{\mathrm{PnL}} \right)

The resulting VaR value represents the estimated loss threshold at the requested confidence level over the simulated time horizon.

Because the P&L distribution is generated using Student-t shocks, the simulation assigns greater probability to extreme outcomes than a comparable normal-distribution model.

The API also returns the underlying P&L quantile so clients can inspect the threshold directly.

Calculate Conditional Value at Risk

Conditional Value at Risk is calculated from simulated P&L outcomes in the lower tail of the distribution, defined by outcomes at or below the lower-tail P&L quantile used in the VaR calculation.

Tail Outcomes

First, the API identifies simulated P&L outcomes that fall at or below the lower-tail P&L quantile.

\mathrm{PnL} \leq q_{\mathrm{PnL}}

These outcomes represent the simulated scenarios in the selected lower tail of the P&L distribution.

Average Tail P&L

The API calculates the average P&L across all outcomes in the selected tail.

\overline{\mathrm{PnL}}_{\mathrm{tail}} = \frac{1}{N_{\mathrm{tail}}} \sum_{i \in \mathrm{tail}} \mathrm{PnL}_i

$\overline{\mathrm{PnL}}_{\mathrm{tail}}$ represents the average P&L of the selected tail outcomes.
$N_{\mathrm{tail}}$ represents the number of simulated outcomes in the tail.
$\mathrm{PnL}_i$ represents an individual simulated P&L outcome.

Conditional Value at Risk

The average tail P&L is converted into a non-negative loss value.

\operatorname{CVaR} = \max\left( 0, - \overline{\mathrm{PnL}}_{\mathrm{tail}} \right)

The resulting CVaR represents the average loss across simulated outcomes in the selected lower tail of the P&L distribution.

Because the P&L distribution is generated using Student-t shocks, the lower tail may contain more extreme outcomes than a comparable normal-distribution simulation.

Method Summary

Monte Carlo Student-t in the Quant Risk API follows these steps:

Convert historical prices into simple returns.
Estimate mean returns and the covariance matrix.
Apply Cholesky decomposition to the covariance matrix.
Generate independent standard normal variables.
Generate an independent chi-square scaling variable for each simulated observation using the specified degrees of freedom.
Combine the normal variables and chi-square scaling variable to generate standardized Student-t variables.
Transform the standardized Student-t variables into correlated shocks.
Generate simulated returns and asset value paths.
Calculate portfolio profit and loss for each simulation.
Calculate VaR and CVaR from the resulting P&L distribution.