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This is a very basic question that I am trying to verify.I am trying to determine how to calculate confidence intervals for a log-normal portfolio (described by return (ret) and standard deviation(STD)). I construct a portfolio of assets and determine the resulting expected return and standard deviation and a Z-score corresponding to a given percentile. I want to verify the below equation is the one to use.For example, how would I calculate the 5% and 95% ranges of returns that the portfolio is EXPECTED to return 3 years from now?The second question: I encorporate higher order moemnts (i.e., skewness and kurtosis) and can get adjusted Z-scores by using a aproximation (e.g. Cornish Fischer expansion), then I should be able to get those same persentiles simply by plugging the new Z values intot he above?Thanks.

Normally by a return R we mean R = Change in Value/Initial Value.With that definition, returns can range from -1 to + infinity.Then, under a log-normal assumption, X = log (1+R) is normally distributed.The Z score eqn is Z = (X - m T)/(s Sqrt(T)) wherem and s are the annualized values of E[log(1+R)] and Sqrt[Var[log(1+R)]]However, often people will approximate m by mu - 1/2 sigma^2, wheremu = E[R] and sigma = Sqrt[Var[R]]. Similarly, people will approximate s = sigma.If you make those approximations and solve the Z score eqn for the critical R, you getR = e^{(mu - 0.5 sigma^2) T + Z sigma Sqrt(T)} - 1Note that this gives correctly that R -> -1 = -100% as Z -> -infinity You asked for a 3 year example. For mu = 0.06, sigma = 0.20, T = 3 and Z = +/- 1.64 (5%, 95% level), I getR(5%) = -32%R(95%) = +111%Please check my work.regards, p.s you might want to look at the difference between Sqrt[Var[log(1+R)]]and Sqrt[Var[R]] in a historical series that's similar to the portfolio you're projecting.Similarly for the mean approximation.

Much thanks for your explanation. I think I had it right, but needed some verification. Any thoughts to adjusting for higher order moments? (Excess) Skewness and Kurtosis?