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Convert normal return moments to loss moments

For a fixed starting value V and loss L=−Vr, mean loss is −Vμr and loss standard deviation is Vσr.

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Define the transformation before calculating

Assume r is a simple same-horizon return and V is a fixed positive starting value, with no intervening cash flows in this original setup. Define value loss L=−Vr. This is a linear transformation; it is not a compounded multi-period return model.

Transform location and scale

μL=−Vμr,σL=Vσr(V>0)\mu_L=-V\mu_r,\qquad\sigma_L=V\sigma_r\quad(V>0)

The negative sign reverses the mean; standard deviation remains nonnegative.

Use the original return assumptions

For V=1,000,000, expected return 1% and return standard deviation 2%, mean loss is −10,000 and loss standard deviation is 20,000. A positive mean return becomes a negative mean loss, not a negative standard deviation.

Keep the distribution claim separate

If the return law is normal, this linear transformation produces a normal loss law. If only the return mean and dispersion are supplied, those moments alone do not establish normality. Nonlinear repricing or cash-flow effects need a different setup.

Further references