Five-period normal loss under two dependence assumptions

State the original period law

Each period’s loss has normal marginal mean 2 and dispersion 10. The total is the sum of five period losses. In Model A the periods are independent. In Model B every period is 2+10Z with the same standard normal shock Z.

Construct each aggregate

Model A has aggregate mean 10 and dispersion √5×10. Model B is 10+50Z, with mean 10 and dispersion 50. Both sums are normal under their stated joint constructions, but their variances differ.

Compare the 95% summaries

Retain the dependence assumption

The same marginal dispersions and period count do not identify a unique aggregate risk record. Cross-period covariance matters. This teaching comparison does not infer independence or perfect correlation from observed returns.