I have a probability distribution that can be described by four momente: return, standard deviation, skewness, and kurtosis. I have annual 1-year measures for these four moments. Now, I want to compute these as 2-year measures. I know how to do this with return and standard deviation. For example, if I know the expected return is 10% per year, in 2 years the expected return is simply 2*10% = 20%. And, if the standard deviation for one year is 20%, the 2-year standard deviation is simply 20% *SQRT (2). Now, how does this work for the higher order moments?

If the two year measure is the sum of two independent and identically distributed one-year measures then the two year central moments are twice the one year values. That means variance is multiplied by 2, so the standard deviation is multiplied by the square root of two, as you say. Skewness and kurtosis are not similarly transformed, so if you have unstandardized central moments, you just multiply by 2.However, skewness is sometimes standardized by dividing by the standard deviation cubed. In that case, the two year standardized central skewness is the one year value divided by the square root of two.Kurtosis is sometimes standardized by dividing by the standard deviation squared, and also sometimes defined as excess kurtosis by subtracting three afterwards. In the first case, two year kurtosis will be half the one-year value, in the second case, it will half the one-year value minus 1.5.So make sure you get the precise definitions of your moments before you convert.

Hey, thanks for all your help.The skewness and kurtosis that I am using are as defined in Excel.Namely, Skewness is normalized, by dividng by standard deviation cubedKurtosis is normalized by dividing by standard deviation to the 4th power and then subtracting 3 (in other words, excess kurtosis). (I think you may have a typo in your reply, as you indivate standard deviation squared as the normalization. I am sure 4th power is what you meant.)Anyway, I worked through the math, and verified the results:skewness in period 2 = [1 / SQRT(2) ] * (Skewness in period 1)excess kurtosis in period 2 = (1/2) * (Kurtosis in period 1) - 1.5I just want to be sure that I have this correct for a general time period t:skewness(t) = [1 / SQRT(t)]* (Skewness in period 1)Kurtosis(t) = (1/t) * (Kurtosis in period 1) - 1.5Could you please verify that I have this correct?Much thanks for helping me along

You're right about the typo, I meant to say variance squared, or standard deviation to the fourth, not standard deviation squared.Your formulae are almost correct but you subtract 3*[ 1 - (1/t) ] in general, not 1.5, for kurtosis.

Thanks again for your help.I am buidling a mote carlo portfolio simulator in Excel/VBA and need that for a static portfolio I have the right return distribution in the future years.

I'm struggling with the Excess Kurtosis measure:If Excess Kurtosis in period 1 is 0:Then in period t:Excess Kurtosis in period t = (1/2) Kurtosis in period 1 - [3 *(1- 1/t)]That means in period 2:Excess Kurtosis in period 2 = 0.5 *0 - [3* (1-0.5)] = 0 - 3*0.5 = 0-1.5 = -1.5This does not seem right???? If Excess Kurtosis is zero in period 1, it should be zero in all future periods???Please let me know where my thinking is astray.

If excess kurtosis in period 1 is 0, the kurtosis is 3. Excess kurtosis in the first t periods is 3/t - 3 + 3*[ 1 - (1/t) ] = 0.

Thanks, but I am still a little confused.What is the general forumula for excess kurtosis in period t given excess kurtosis in period 1?I just want to be sure tha tI have this all straight in my head. Much thanks in advance

When you aggregate variables, the moment of order k of the sum depends on the moments of all orders up to k of the individual variates. Since excess kurtosis is standardized by mean and variance, those cancel out. But the excess kurtosis of the sum still depends on the skewness of the individual variates, in addition to the kurtosis.

I must really be missing something. I thought the general formula for excess Kurtosis in period t was:1/t (Excess Kurtosis in Period 1) - 3 (1 - 1/t)I must have the general formula wrong.Sorry for the frustration. This is a bit new to me

Try this link.