Suppose that the daily change in the value of a portfolio...

Suppose that the daily change in the value of a portfolio is, to a good approximation, linearly dependent on two factors, calculated from a principal components analysis. The delta of a portfolio with respect to the first factor is 6 and the delta with respect to the second factor is -4 . The standard deviations of the factors are 20 and 8 , respectively. What is the 5-day $90 \% \mathrm{VaR}$ ?

Key Concepts

Value at Risk (VaR)

VaR is a statistical measure used in risk management to estimate the potential loss in value of a portfolio over a specified time period for a given confidence interval. It provides a threshold value such that the probability of a loss exceeding that value is at a predefined level, making it a critical tool for assessing market risk.

Linear Factor Model

A linear factor model expresses the returns or changes in the value of a portfolio as a linear combination of several risk factors. Each factor is associated with a sensitivity coefficient, often called delta, and the model captures how the portfolio's value responds to variations in these underlying factors, simplifying the risk analysis.

Portfolio Sensitivity (Delta)

Delta represents the sensitivity of a portfolio’s value to changes in an underlying risk factor. It quantifies the linear relationship between movements in the factor and the corresponding change in portfolio value, allowing risk managers to determine how individual components contribute to overall risk.

Principal Components Analysis (PCA)

PCA is a statistical technique used to reduce the dimensionality of a dataset by identifying new uncorrelated variables (principal components) that capture the most variance in the data. In finance, PCA helps in identifying the key underlying factors that drive asset returns, which simplifies the analysis of complex risk exposures.

Volatility Scaling and Time Aggregation

Volatility scaling refers to the adjustment of risk measures, such as VaR, from one time horizon to another, typically using the square root of time rule under the assumption of independent and identically distributed returns. This adjustment is crucial when translating daily risk estimates into multi-day risk measures for practical risk assessment.

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