Value at Risk measures the likelihood of losses to an asset or portfolio, over a defined period for a given confidence interval, due to market risk. Such a narrow definition of risk is further limited to the VaR focus on downside risk and potential losses in the short-term; indeed, VaR can be computed over a quarter or a year, but it is usually computed over a day, a week or a few weeks.
Damodaran
In general, VaR has been developed for commercial and investment banks to capture the potential loss in value of their traded portfolios from adverse market movements over a specified period in order to match such losses with their capital and cash reserves. Its use in banks reflects their fear of a liquidity crisis.
Value at Risk Assumptions
It is important to understand the assumptions underlying Value at Risk in order to establish the practical and real value of such risk measure.
- Because returns may not be normally distributed (fat tails), it is assumed that the standardized return (computed as the return divided by the forecasted standard deviation) is normally distributed. Thus, a large return (positive or negative) in a period of high volatility may result in a low standardized return, whereas the same return following a period of low volatility will yield an abnormally high standardized return. However, normalized standardized returns still may under-represent the risk of more frequent large outliers.
- The standard deviation in returns is assumed to not change over time (homoskedasticity). Thus, the non-stationary of variances and covariances across assets over time, due to their fundamentals changing over time, can lead to a breakdown in the computed VaR.
- Most methods are designed for portfolios where there is a linear relationship between risk and portfolio positions.
Value at Risk methods
There are multiple variations and definitions of VaR; however, to estimate the probability of the loss, with a confidence interval, we need to define the probability distributions of individual risks, the correlation across these risks, and the effect of such risks on value.
- Variance-Covariance approach. Every financial asset of the portfolio is mapped into a set of instruments that can easily represent the underlying market risks; then variances and covariances are estimated by looking at the historical data of such instruments. Finally, the Value at Risk for the portfolio is computed using the weights and the variances of the standardized instruments.
- Historical Simulation method. A hypothetical time series of returns is computed by running the portfolio through actual historical data and computing the changes that would have occurred in each period. Then, the daily price changes are grouped into positive and negative numbers: the negative VaR is the price change at the 99th percentile of the negative price changes. Note that there are no underlying assumptions of normality driving the conclusion, that each day in the time series has equal weight, and that history is considered a good proxy for the future.
- Montecarlo Simulation approach. As in the Variance-covariance method, individual portfolio assets are converted into positions in standardized instruments. Then, the probability distribution and the correlation with other instruments are specified for each of the market risk factors. Thus, alternate distributions and parameters can be used to adapt to subjective views. Note that as the number of market risk factors increases, the quality of the simulation may decrease and the computational efforts exponentially increase. The price series can be modeled depending on what it is expected in the future:
- The price evolution is simulated using Geometric Brownian Motion with drift assuming constant expected return and volatility
- Jump diffusion can be employed to model fatter tails than GBM and the occasional “jump” in prices appearing during crashes as Black Tuesday (1929), Black Monday (1987), Asian crisis (1997), Dot-com bubble (2001), Mortgage Bubble (2008), etc.
- Scenario Simulation method. Principal component analysis is applied to the portfolio to select the risk factors. Then, a number of discrete scenarios are produced by applying a limited number of pre-specified shocks to the portfolio’s risk factors.
Obviously, the results of the different methods are a function of the input parameters. The historical simulation and variance-covariance methods are equivalent if the historical returns data is normally distributed and is used to estimate the variance-covariance matrix. The historical, variance-covariance, and Monte Carlo approaches will converge if all of the inputs are normally distributed. However,
- the variance-covariance approach is appropriate for linear portfolios (i.e., ones that do not include options), over very short time periods (a day or a week)
- historical simulations are fine for stable risk sources with substantial historical data (commodity prices, for instance)
- Monte Carlo simulations, among the ones discussed above, are the only option for non-linear portfolios.
References
M. Choudhry (2003). The Bond and Money Markets.
A. Damodaran. Value at Risk.
Y. Hilpisch (2014). Python for Finance.
E. Marsden (2018). Estimating Value at Risk using Python.