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Keep the covariance identity broad
Portfolio variance can be calculated from compatible weights and covariances without assuming a normal distribution. That second-moment identity does not establish the full shape of the portfolio loss law.
State the extra distribution assumption
If component losses are jointly normal, a fixed linear combination has a normal law. Its mean and variance then identify that normal distribution, permitting the parametric tail formulas. Independence of normal components is one sufficient setup, but normal-looking individual marginals alone are not.
Inspect an original dependence construction
Let X be standard normal and S be an independent equally likely +1/−1 sign. Set Y=SX. Both X and Y have standard normal marginals, and their covariance is zero. Yet X+Y is exactly zero whenever S=−1, giving a 50% atom that a positive-dispersion normal law cannot have.
Retain the exposure transformation
Nonlinear options or changing positions can also invalidate a simple normal linear-loss interpretation. Record the joint assumptions and pricing basis instead of converting a covariance result into an automatic normality claim.