Statistical Analysis Part III: Value at Risk

Sharpe Ratio
The Sharpe ratio calculates a risk Vs reward ratio.
It measures the excess return on an investment from its volatility. The higher the ratio the greater the return per unit of volatility the investment will yield. This ratio was developed by W.F Sharpe. Other similar ratio’s include Treynor ratio’s and Jensen’s alphas. Mathematically it is expressed as
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Value at Risk
Value at risk is most often used to asses the risk of a portfolio of assets. It is the maximum potential loss in a portfolio over a specific period of time given a certain probability. For example a VaR of $1 million given a horizon period of 1 week and a probability of 5% would indicate that there is a 5% chance that the portfolio’s loss will be greater than $1million. Alternatively it can also be phrased as there is a 95% probability that the portfolio will not loose more than $1 million in a week.

There are 3 methods to calculating value at risk. The first method is historical simulation, the second is variance/covariance and the third is known as Monte Carlo Simulation. Each of them has their benefits and drawbacks. .

The historical simulation is the simplest method as all one has to do is plot a chart of the daily P&L of the portfolio. The second method known as variance/covariance is more sophisticated and uses historical data of the individual assets in the portfolio to calculate individual asset variances and their correlations with each other.

In theory, the lower the correlation of two securities, the lower the risk of the portfolio. This is because correlation measures the strength if two or more securities moving in the same direction. The lower the correlation between two securities the lower the likelihood that if one security where to drop the other one will as well. This creates an effective hedge. For example airlines should be negatively correlated with the price of crude oil. If the price of oil increases airline companies will need to spend more money on fuel which increases costs which lowers net income. If a portfolio manager has oil and airline companies in his portfolio he will be hedged to an extent because if oil suddenly drops the airline company should more than likely rise in price.

The VaR of a portfolio should increase if the historical volatility of the assets rises. This is because historical volatility measures the amount of variation in a securities price and the higher the number the higher the likelihood that the security price will suddenly drop. (See section on historical volatility).  There are two methodologies to using historical volatility in VaR calculations. There is the equal weighted volatility which simply treats each historical point in the data set equally and there is the J.P Morgan model which puts a greater weight on more recent data than older ones.

The third model, Monte Carlo is a simulation model. Like the name suggests, a quantitative model is built by making assumptions on the price behaviors and distribution of the individual assets. The model then generates random numbers and creates a scenario analysis on how the P&L of the portfolio varies. The second method is the most practical one. As long one has the relevant historical data one can download them into Excel and perform the necessary calculations.

The great feature on VaR is that it works across asset classes. Before many portfolio analysts used varied methods to calculate risk such as beta and duration. This method allows us to use the same methodology to output a single consistent figure. The problem with VaR however is that there is no consistent method of calculating VaR, rather it is tweaked by the many assumptions. On the positive side however the proponents of VaR argue that whichever of the above three model one uses there is little difference in the VaR output compared to the difference created if the probability level and time horizon changes. 

Application
Due to his capital adequacy rules and laws created by the Financial Standards Board that banks mark to market their trading, banks have used VaR models more frequently than hedge funds. In fact the famous hedge fund Long Term Capital management used a VaR like method to calculate its risk and obviously the model was not successful, as correlations vary with time especially if financial crises hit the markets. During the crisis period spreads were moving many standard deviations away from the mean and correlations of different assets were approaching 1. Statistically speaking VaR’s failings appear because of two reasons. Firstly market movements outside the VaR probability distribution occur more often than many models take into account, implying that the markets actually contain non normal distributions of returns or “fat tails”. Secondly the predictions of financial crises are much more random than the probability of daily short term portfolio loss.

Value at Risk for FX
Calculating the VaR for foreign exchange can be thought of slightly different than other assets because an FX pair is simply a swap between oen currency to another. The fact remains that there is historical data for the FX rates and a volatility function can be calculated to make VaR calculations feasible.