VAR and Auto corelation

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Hi David,

Below is the question from Jorion:

Q. Consider a portfolio with a one-day VAR of $1 million. Assume that the
market is trending with an autocorrelation of 0.1. Under this scenario, what
would you expect the two-day VAR to be?
a. $2 million
b. $1.414 million
c. $1.483 million
d. $1.449 million

Ans:c) Knowing that the variance is V(2-day) = V(1-day) [2 + 2ρ], we find
VAR(2-day) = VAR(1-day)

2 + 2ρ = $1
√
2 + 0.2 = $1.483, assuming the same
distribution for the different horizons.


Can you please throw some light on this concept . the formula is V(2-day) = V(1-day) [2 + 2ρ], here 2+ will be the number of days to which VAR is extended or it is constant? In question if autocorrelation is not give, so do we assume it to be 0.??

 

[David Harper CFA FRM]

Hi snigdha,

It's just a different way to (IMO) the easier way to view this: as a two-asset portfolio VaR, but instead we are just combining the two assets over time.

2-asset VaR = SQRT[VaR(1)^2 + VaR(1)^2 + 2*VaR(1)*VaR(2)*correlation]. In this case,
2-asset VaR = SQRT[1 + 1 + 2*1*1*0.1] = 1.483

i.e., since it's just two periods, autocorrelation is the same as regular correlation

David