This document discusses applying "square root of time" scaling to calculate value at risk (VaR) over longer time horizons when direct calculation is not possible due to insufficient historical data. It explains that regulators may request VaR measures over periods like 2 weeks, but with only 4 years of data there would be too few samples to directly calculate a 99% VaR. Therefore, 1-day VaR is computed and multiplied by a scaling factor based on the square root of time. While theoretically incorrect due to non-constant volatility, this approach is examined to determine if it provides an acceptable approximation, especially for tail distributions like 95% or 99% VaR. MATLAB code is presented to automate the calculations and visually examine the

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

2. 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%?

3. 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.

4. 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.

5. Mathematical definition
Given 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.:

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

7. Example: Excel snapshot 1-day VaR

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

9. Where the 𝐧 (square root of time) comes
from?
If Xi I.I.D → Ϭ( Xi )= 𝑛 Ϭ(Xi )
Sum P&L’s:
𝐷2
𝑋 + 𝑌 = 𝐸(𝑋 + 𝑌)2
−(𝐸(𝑋 + 𝑌))2
= 𝐸 𝑋 2
+ 2𝐸𝑋𝑌 + 𝐸 𝑌 2
− 𝐸𝑋 + 𝐸𝑌 2
= 𝐸𝑋2
+ 𝐸𝑌2
+ 2𝐸𝑋𝑌 − 𝐸𝑋 2
− 2 𝐸𝑋 𝐸𝑌 − 𝐸𝑌 2
= 𝐸𝑋2
− 𝐸𝑋 2
+ 𝐸𝑌2
− 𝐸𝑌 2
+ 2𝐸𝑋𝑌 − 2 𝐸𝑋 𝐸𝑌 .
Now X,Y are independent, so: EXY = (EX)(EY).
→ 2(𝐸𝑋2
− (𝐸𝑋)2
) → D(X + Y) = 2𝐷𝑋
Notice: Same procedure for D(PnL1+…+PnLn) = 𝑛𝐷(𝑃𝑛𝐿)
If P&L is I.I.D.:
 Weekly VaR = 𝑉𝑎𝑅 𝑑𝑎𝑖𝑙𝑦* 5
 Bi-weekly VaR = 𝐕𝐚𝐑 𝐝𝐚𝐢𝐥𝐲* 𝟏𝟎
 Monthly VaR = 𝑉𝑎𝑅 𝑑𝑎𝑖𝑙𝑦* 20

10. 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.

12. 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 10 is acceptably conservative
but it necessarily true in all market conditions. ( As we see , there are breaches in
some cases ).