Python for Finance

Value at Risk
In finance, implicitly or explicitly, rational investors always consider a tradeoff
between risk and returns. Usually, there is no ambiguity to measure returns.
However, in terms of risk, we have numerous different measures such as using
variance and standard deviation of returns to measure the total risk, individual
stocks' beta, or portfolio beta to measure market risk. In the previous chapters, we
know that the total risk has two components: market risk and firm-specific risks. To
balance between the benefit of return and the cost of risk, many measures can be
applied, such as the Sharpe ratio, Treynor ratio, Sortino ratio, and M2 performance
measure (Modigliani and Modigliani performance measure). All of those risk
measures or ratios have a common format: a trade-off between benefits expressed as
risk-premium and risk expressed as a standard deviation, or beta, or Lower Partial
Standard Deviation (LPSD). On the other hand, those measures do not consider a
probability distribution. In this chapter, a new risk measure called Value at Risk
(VaR) will be introduced and applied by using real-world data. In particular, the
following topics will be covered:
• Introduction to VaR
• Review of density and cumulative functions of a normal distribution
• Method I—Estimating VaR based on the normality assumption
• Conversion from 1-day risk to n-day risk, one-day VaR versus n-day VaR
• Normality tests
• Impact of skewness and kurtosis
• Modified VaR measure by using including skewness and kurtosis
• Method II—Estimating a VaR based on historical returns
• Linking two methods by using Monte Carlo simulation
• Backtesting and stress testing
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