Chapter 11. Value at Risk

In finance, implicitly or explicitly, rational investors always consider a trade-off 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...