The key to pricing loans is to make sure that the prices cover expected losses. The correct measure of expected losses is the mean, and not the median. To the extent the median is different from the mean, the loans would be over or underpriced.
The loss curve for credit defaults is a distribution skewed to the right. Therefore its mode is less than its median which is less than its mean. Since the median is less than the mean, the bank is pricing in fewer losses than the mean, which means over the long run it is underestimating risk and underpricing its loans. Therefore Choice 'd' is the correct answer.
If on the other hand for some reason the bank were overpricing risk, its loans would be more expensive than its competitors and it would lose market share. In this case however, this does not apply. Loan pricing decisions are driven by the rate of defaults, and not the other way round, therefore any pricing decisions will not reduce the rate of default.

The VaR for a spot FX position is merely a function of the standard deviation of the exchange rate. If V be the value of the position (in this case, EUR 15m x 1.5 = USD 22.5m), z the appropriate z value associated with the level of confidence desired, and # be the standard deviation of the portfolio, the VaR is given by Z#V.
In this case, the 10-day standard deviation is given by SQRT(10/250)*16%. Therefore the VaR is =1.
645*15*1.5*(16%*SQRT(10/250)) = USD 1.1844m. Choice 'c' is the correct answer.

An n x n positive definite square matrix will have n eigenvectors, and not n - 1. Therefore statement I is incorrect.
A correlation and covariance matrix are related to each other through the matrix of standard deviations. If the covariance matrix is represented by V, the correlation matrix by C and D is the diagonal matrix of standard deviations, then V = DCD. However, there is no simple relationship between the eigenvalues of the two matrices, and it is not possible to derive the eigenvalues for one given the eigenvalues for the other. Therefore statement II is false.
Principal components are uncorrelated to each other. That is correct, and in fact PCA is useful because of this being so. Statement III is therefore true.
PCA does not explain 100% of the variation in a system with just three components - statement IV is false.
(Remember though that most (though not 100%) of the variation in a system of term structures is explained by the first three components - trend, tilt and curvature).
Thus Choice 'd' is the correct answer.

Normal mixtures, EVT and the t-distribution are all possible solutions addressing the issue of heavy tails in financial returns.
EWMA and GARCH address volatility clustering, which is the other problem when doing risk calculations.
Therefore Choice 'b' is the correct answer as EWMA is not used to address heavy tails but volatility clustering.

Odds are the ratio between the probability of the occurence of an event to the probability that the event does not occur.
If odds are H, then p = H/(1 + H) and H = p/(1-p). In this case the odds are 1:5, or 1/5, therefore the correct answer is Choice 'a', equal to (1/5)/(1 + 1/5) = 1/6 = 16.67%. All other choices are incorrect.