Final answer:
The 95% Value at Risk (VaR) for investments A and B is $2 million each. For the combined portfolio, the 95% VaR remains $2 million due to the low probability of simultaneous large losses. This example illustrates that diversification can reduce risk, as the combined VaR is less than the sum of individual VaRs.
Step-by-step explanation:
The 95% Value at Risk (VaR) for investment A is the largest loss not exceeded with a 95% confidence level. Given a 5% chance of a $2 million loss and a 4% chance of a $10 million loss, the VaR at 95% confidence is $2 million, as it's the smallest loss not exceeded by the remaining 5% of the distribution. The same calculation applies to investment B since they have identical distributions. For the portfolio consisting of both A and B, independence means we must consider the joint probability distribution of the two investments. The worst 5% of outcomes are calculated by combining the probabilities of each investment's losses. However, due to the independence and the small probabilities of loss, the 95% VaR of the portfolio is still $2 million, simply because the probability of both investments incurring the $10 million loss simultaneously is very small (0.16%). The summation of the 95% VaRs of the individual investments ($4 million) is greater than the 95% VaR of the portfolio ($2 million), showing that diversification can indeed decrease risk.