Application of “Square Root of Time” Scaling Created: Annamária LakiSupervisor: Áron Varga (Morgan Stanley, Model Risk Management)

Sometimes regulators request market risk measures (eg. VaR) on a relatively long time horizon. The computations involving this length of time are meaningless, as there is no available good quality data. Imagine for example that a 2 week 99% VaR is requested. Even if we had 4 years of coherent data, we would have only 1 sample point for the 99% quantile. This is not sufficient. Therefore time-scaling techniques are used. This means that 1 day VaR is computed and it is multiplied by a scaling factor. Of course the sample will demonstrate a level of auto-correlation and will not have constant volatility, so this is theoretically incorrect. The question arises, how big of a mistake do we make if we use the square-root of time scaling. Can we convince ourselves and our audience (i.e. the regulators) that this approximation is acceptable? Can we present visually tangible / tractable diagrams to this effect? Can we at least say something about the tails of the distribution; say at 95% or 99%?

Value at Risk Definition

In financial mathematics and financial risk management, value at risk (VaR) is a widely used risk measure of the risk of loss on a specific portfolio of financial exposures. For a given portfolio, time horizon, and probability p, the p VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon exceeds this value is p. This assumes mark-to-market pricing, and no trading in the portfolio. For example, if a portfolio of stocks has a one-day 5% VaR of $1 million, there is a 0.05 probability that the portfolio will fall in value by more than $1 million over a one day period if there is no trading. Informally, a loss of $1 million or more on this portfolio is expected on 1 day out of 20 days (because of 5% probability). VaR has four main uses in finance: risk management, financial control, financial reporting and computing regulatory capital. VaR is sometimes used in non-financial applications as well.

We need

Portfolio eg.(S&P500) Initial capital : eg. $1mill Time-horizon : eg. 2013-2015 number of days ( 2 years 500 days ) Confidence level: eg. 1% 99%. The confidence level shows us where is

the minimum or maximum VaR takes place ( eg. 1% of 2 years 5th

place ) Important assumption on strategy: This is a constant portfolio.

Mathematical definitionGiven a confidence level between 0,1 the VaR of the portfolio at the confidence level α is given by the smallest number ℓ such that the probability that the loss L exceeds ℓ is at most (1- a). Mathematically, if L is the loss of a portfolio, then VaRα (L) is the level α-quantile i.e.:

How we can compute 1 day VaR? 1 day P&L: : VaR is a quantile of the P&L distribution  Ordered P&L ( from min. to max.)

Example: Excel snapshot 1-day VaR

We have 1-day VaR, but our task is to computing 10-day VaR as a request of regulatory capital.BUT! We Do not have enough data, because there are 250 trading days means only 25 observation in a year. 4 years we have 100 observations, so we can not get a good 1% evaluation. So we must use 1-day var to compute 10-day VaR.

10 day VaR10 day VaR is requested by regulators, which comes from Basel Committee on Banking Supervision . Link: http://www.bis.org/publ/bcbs158.htm

Where the (square root of time) comes from?

If Xi I.I.D Ϭ( )= Ϭ(Xi )Sum P&L’s:

Now X,Y are independent, so: . =

Notice: Same procedure for D(PnL1+…+PnLn) =

If P&L is I.I.D.: Weekly VaR = * Bi-weekly VaR = * Monthly VaR = *

VaR-model in MatLab

We made a VaR-model in Matlab and run it for S&P500. In this code, we can change: Time-horizon Confidence level The days of VaR (n)Each point in the figure is a ratio Var n / Var 1 day for a 2 year period.

Conclusion

Computing VaR with the historical method could be the simplest way. In this work I tried to introduce the meaning of VaR and after some definition computed with the chosen historical method, which uses observed P&L. After we got 1-day VaR we wanted to see 10-day VaR, as a regulatory capital. For this case we wrote a program in MatLab. Where we can see nowadays is acceptably conservative but it necessarily true in all market conditions. ( As we see , there are breaches in some cases ).