Independent & Identically Distributed

[Kashif Khalid]

Hi guys

can you kindly explain the relevance of i.i.d and how this relates to the SQRT of time rule?

Many thanks

95kaskha

 

[ami44]

Hi 95kashka.

Can you provide some context?

 

[ShaktiRathore]

Hi
iid means variable which is normally distributed(same distribution) with constant variance and mean(identical) and with no autocorrelation(independent). If over a small time interval t1 the variance is sigma^2 then since variance is constant the variance over successive equal time interval t2 is sigma^2 ,in this way successsive variances add up since there is no autocorrelation in the successive returns. If all the succesive time intervals are equal to say 1unit then Var(t1)=Var(t2)=Var(t3)=....=Var(tn)=sigma^2 if t1+t2+..tn=T units,Var(T)=Var(t1+t2+...+tn)=Var(t1)+Var(t2)+....+Var(tn) variances add up because of no autocorrelation=> Var(T)=sigma^2+sigma^2+...sigma^2=sigma^2*T=>Var(T)=sigma^2*T=>stdDev(T)=sigma*sqrt(T). If we are given 1 unit of time std Dev and need to find stdDev over time T units then we simply scale stdDev over time 1 unit with square root of time(SRR).In this way the volatility scales over square root of time since variance scale over time. If time period were T1 we would add up constant variance sigma^2 T2 times therefore variance over T2 is T2*sigma^2 or stdDev(T2)=sigma*sqrt(T2).
Thanks