# Moving average variance [closed]

I have generated a random series of returns drawn from a normal distribution and generated a random price series by compounding these returns (X) so $P_i = P_1(1+X)^i$. I want to show the analytic comparison between the variance of this price series to the variance of a k-period moving average of the price.

What I am finding is that the variance of the price series (not price returns) converges to the variance of k period average price for $all$ values of k (so the variance of moving average price = price variance regardless of the averaging period) for large data sets. This surprised me as I would have expected the variance of the moving average to be scaled down by k regardless of the period.

I have tried to derive this result mathematically but without success. Here is the starting point of my calculation: $$MA_k = 1/k . {\sum^k_{i=1}P_i} =$$ $$=1/k . (P_1 + P_1(1+x) + ... + P_1(1+x)^{k-1})$$ $$P_1/k.{\sum^{k-1}_{i=0}}(1+x)^i$$

From here I use a power series expansion, ignore higher order terms, and take the variance of the expression. I think I should find this expression $var(MA_k) = P_1^2.var(x)$ but that's a guess. I can see I am not incorporating the dependency on the size of the data because there appears to convergence in variances between the rolling average of the price and the price, but I am not sure how to do that.

Any assistance in filling in the missing steps would be very welcome here.

## closed as off-topic by Richard Hardy, LocalVolatility, zer0hedge, noob2, lehalleJun 29 '17 at 20:24

• This question does not appear to be about quantitative finance within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

• Shouldn't it be $P_i=P_{i-1}(1+x_i)$ ? – noob2 Jun 25 '17 at 20:52
• Hi Noob, yes I believe that would make the formula recursive. I did not know how to solve the recursive formula. Instead I made P a constant and tried to solve for the variance as function of x through an expansion (which I treat as a random variable) and by ignoring $O(x^2)$ terms. – markm Jun 25 '17 at 21:03
• Hi - I have edited the question to show my approach in more details and reduced to a single goal as opposed to the dual goals originally stated in the question – markm Jun 26 '17 at 8:42
• I'm voting to close this question as off-topic because it belongs on the Cross Validated site. – Richard Hardy Jun 26 '17 at 19:06
• Good suggestion Richard. The question does have a quantitative finance application in mind. However I can see that those with statistical/mathematical bent might find it more interesting (and easier to solve). – markm Jun 27 '17 at 8:40