h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I have stumbled upon another paradox today.
Suppose that you throw two dice. Every step of the way you notice the numbers that are on them and ,thereafter, multiply those numbers , viz. my current iteration of throwing produces 2 on one dice and 6 on the another, I record the result of 2*6 =12 in my logbook. With a number of repeated attempts (iterations) increasing, it is becoming clear, beyond peradventure,that what I have in my log book is the descrete stochastic variable(Z) that can be described by its mean (M) and dispersion(D). On the other hand,this varible represents nothing more than multiplication of two (independent!)discrete varibles X and Y represented by each of the dice, thus Z=X*Y. It is clear that Z and X*Y are sinonymous on account of the mutual independence beetween X and Y.
Now the dispersion of two multipled variables X*Y is known as their covariance(COV XY). Textbooks on statistics usually contain the demonstration of the fact,that covariance of two independent variables equals 0. Yet,it is equally clear that dispesion of Z=X*Y is anything but zero! What shall I do about it?

This paradox elicits itself in a variety of situations. For example, assume that a company that has its revenues denominated in Euros invests in American stocks,that are denominated in USDs. Assume that returns on both markets are normally destributed. On a certain day the company has registered that return on stocks in its portfolio of investments were =5%, on the same day Euro has weakened against the dollar, thus producing negative return on EUR/USD exchange rate of -0,2%.The company should record its total return of 1,05*0,998=1,479 (+4,79%). Now if the company wants to know the volatility(dispersion) of its investments,there are two ways about it, either believing that volatility of its investments is zero (perfect hedging is achieved!), Forex and stock markets being for the purposes of the arguement independent, or measuring the volatility of investments aposteriori (in this case it will never be zero!).
Guidance books on measuring Values-at-risk (VaR),the calculation of which requires the derivation of volatility, usually get around the problem by employing the portfolio theory and saying that in such instances what we have is the superimposed portfolio of stock and forex markets.i.e. same investments are kept 100% in stocks and 100% in Forex, thus D(investments)= (1*D(stocks))+(1*D(Forex)).(Again, we neglect the correlation components of the equation by stating that both markets are independent). Such treatment of the issue,in my view , violates basic premises of the portfolio theory by claiming that an investment portfolio could have 200% capicity.(it is known that the sum of portfolio weights does never exceed 100%). Moreover, we inadvertantly supplant the derived expression of R= X*Y for market returns by the erroneous view that R=X+Y

Covariance is E(XY)-E(X)E(Y) though, whereas the variance, what you're calling dispersion, of XY is E(XYXY)-E(XY)E(XY), so the first can be zero and the second nonzero without any problem.