= Solution
Let annual total loss be $L=\sum_{i=1}^{N_1}X_i$, with an empty sum equal to zero. The assumptions give a <compound Poisson distribution>. Conditioning on the count,
$$
\mathbb E[L\mid N_1]=N_1\mu,\qquad
\operatorname{Var}(L\mid N_1)=N_1\sigma^2.
$$
The <tower property> and total-<variance> formula therefore give
$$
\boxed{\mathbb EL=\lambda\mu,\qquad\operatorname{Var}L=\lambda(\sigma^2+\mu^2).}
$$
The $\lambda\mu^2$ term is the extra uncertainty in the number of defaults. Replacing it by zero would incorrectly treat that count as deterministic.
For sufficiently many annual events and an adequate <normal approximation>, the <value at risk> at confidence level $p$ is the <compound-Poisson annual loss quantile>
$$
\boxed{\operatorname{VaR}_p(L)\approx\lambda\mu+z_p\sqrt{\lambda(\sigma^2+\mu^2)},\qquad z_p=\Phi^{-1}(p).}
$$
The supplied quantiles give $z_{0.90}=1.282$, $z_{0.95}=1.645$ and $z_{0.99}=2.326$. In particular a one-sided 95% annual loss quantile uses $1.645$, not the $1.96$ from the two-sided 95% rate interval. If the expected loss is covered separately, the approximate unexpected-loss reserve is the second term, rather than the full quantile.
Estimate the rate from a suitable historical count/exposure or gap sample, then insert $\widehat\lambda$ into the loss formula. The interval in part (d) measures rate-estimation uncertainty; for nonnegative loss <mean> it can be propagated to lower and upper approximate quantiles because the displayed expression is increasing in $\lambda$. Severity parameters also require estimation, so the rate interval alone is not a <confidence interval> accounting for every source of uncertainty.
<Mean> and <variance> do not determine the exact loss quantile. The actual distribution is
$$
\mathbb P(L\leq\ell)=e^{-\lambda}\sum_{n=0}^\infty\frac{\lambda^n}{n!}F_X^{*n}(\ell).
$$
For nonnegative losses it has a zero atom at least $e^{-\lambda}$, so if that mass exceeds $p$ its exact <value at risk> is zero. With a small rate or a strongly skewed severity distribution, a <normal approximation> can therefore be poor. A loan book with dependent defaults, rating migration or varying exposures also needs more than the constant-intensity independent-severity model. These are model qualifications, not deductions from the <credit rating> alone.
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