In the univariate Gaussian model, each risk factor is modeled separately independent of the others, and the dependence between the risk factors is captured by the covariance matrix (or its equivalent combination of the correlation matrix and the variance matrix). Risk factors could include interest rates of different tenors, different equity market levels etc.

While this is a simple enough model, it has a number of limitations.

First, it fails to fit to the empirical distributions of risk factors, notably their fat tails and skewness. Second, a single covariance matrix is insufficient to describe the fine codependence structure among risk factors as non-linear dependencies or tail correlations are not captured. Third, determining the covariance matrix becomes an extremely difficult task as the number of risk factors increases. The number of covariances increases by the square of the number of variables.

But an inability to capture linear relationships between the factors is not one of the limitations of the univariate Gaussian approach - in fact it is able to do that quite nicely with covariances.

A way to address these limitations is to consider joint distributions of the risk factors that capture the dynamic relationships between the risk factors, and that correlation is not a static number across an entire range of outcomes, but the risk factors can behave differently with each other at different intersection points.

Economic capital is often calculated with a view to maintaining the credit ratings for a firm. It is the capital available to absorb unexpected losses, and credit ratings are also based upon a certain probability of default. Economic capital is often calculated at a level equal to the confidence required for the desired credit rating. For example, if the probability of default for a AA rating is 0.02%, and the firm desires to hold an AA rating, then economic capital maintained at a confidence level of 99.98% would allow for such a rating. In this case, economic capital set at a 99.8% level can be thought of as the level of losses that would not be exceeded with a 99.8% probability, and would help get the firm its desired credit rating.

Choice 'c' is the correct answer. Economic capital does not target minimizing the cost of capital, nor is it a provision for losses arising from market risk. The concept of economic capital is unrelated to where an institution or firm is based, therefore Choice 'a' is incorrect as well.

The easiest way to answer this question is to ignore the joint probability of default as that is irrelevant to expected losses. The joint probability of default impacts the volatility of the losses, but not the expected amount. One way to think about it is to think of asset portfolios, where diversification reduces risk (ie standard deviation) but the expected returns are nothing but the average of the expected returns in the portfolio. Just as the expected returns of the portfolio are not affected by the volatility or correlations (these affect standard deviation), in the same way the joint probability of default does not affect the expected losses. Therefore the expected losses for this portfolio are simply $1m x 10% + $1m x 15% = $250,000.

This can also be seen from the lens of a joint probability distribution as follows:

There are four possibilities for this portfolio:

- Only loan A defaults: loss of $1m: 9% probability

- Only loan B defaults: loss of $1m: 14% probability

- Both loan A and B default: loss of $2m: 1% probability

- Neither A nor B default: loss of $0m: 76% probability

Therefore the expected losses on the portfolio are ($1m x 9%) + ($1m x 14%) + ($2m x 1%) + ($0m x 76%) = $250,000.

(Notes: How is the above table calculated? The totals (10%, 90%, 15% and 85%) are filled in first. The top left cell (both A & B default) is given as 1%. We can now calculate the rest of the cells as the totals are known.)

The Earnings Volatility approach to calculating economic capital is a top down approach that considers economic capital as being the capital required to make for the worst case fall in earnings, and calculates EC as equal to the worst case decrease in earnings capitalized at the rate of return expected of the firm. The worst case decrease in earnings, or the earnings-at-risk can only be stated at a given confidence level, and is equal to the Expected Earnings less Earnings under the worst case scenario.

This is a tricky question in the sense no risk management professional can be expected to know the answer to this one unless they have read Chapter 2 of the PRMIA handbook. So this question appears purely for the sake of something you would need to know purely for the sake of the exam.

PRMIA's handbook classifies governance sructures as reactive, preventative and active. Reactive structures involve monitoring signals after the event leading to corrective actions. Preventative structures are forward looking and anticipate issues before they arise. Active structures include considerations of operational efficiency and not just governance. All other answers are made up phrases and are incorrect.

In reality, corporations employ all structures together without worrying about the boundary between the three, and these distinctions do not exist except in textbooks.

Units of Measure for operational risk are homogenous groupings of risks to allow sensible modeling decisions to be made. For example, some risks may be fat-tailed, for example the risk of regulatory fines. Other risks may have finite tails - for example damage to physical assets risk (DPA) may be limited to the value of the asset in the question.

Additionally, risk reporting may need to be done at the line of business, legal entity or regional basis, and in order to be able to do, so the right level of granularity needs to be captured in the risk modeling exercise. The level of granularity applied is called the 'unit of measurement' (UoM), and it is okay to adopt all of the choices listed above as the dimensions that describe the unit of measure.

Note that it is entirely possible, even likely, to use legal entity, risk type, region, business and other dimensions simultaneously, though doing so is likely to result in an extremely large number of UoM combinations. That can be addressed by then subsequently grouping the more granular UoMs into larger UoMs, which may ultimately be used for frequency and severity estimation.

Choice 'a' is the correct answer. Refer to the detailed loss event type classification under Basel II (see Annex 9 of the accord). You should know the exact names of all loss event types, and examples of each.

Choice 'd' represents a risk that does not arise from its holdings of mortgages. Therefore Choice 'd' is the correct answer.

All the other risks identified are correct - the bank faces interest rate, default and prepayment risks on its mortgages.

Unexpected loss for a credit portfolio refers to the excess of the VaR estimate over the average expected loss. The term 'unexpected loss' has this specific meaning in the context of credit risk, and not any other intuitive meaning. So if for a portfolio worth $100m expected losses are 4%, and the credit VaR at 99% is $12m, then unexpected losses at that VaR quintile are $8m. This is unrelated to actual realized losses versus expected losses.

Therefore Choice 'd' is the correct answer and the others are not.

Unexpected loss is used to determine the capital reserves to be maintained against a credit portfolio at a certain level of confidence.

One may use separate UoMs for frequency and severity modeling, for example, a combined UoM may be used for estimating the frequency of cyber attacks in a scenario, while the severity may be modeled using a more granular line-of-business UoM. Therefore statement I is false, while statement IV is true. Statement II is correct, UoMs can be grouped together into larger units bas ed on the facts relating to the business, controls and the business environment. Similarly, UoMs can be grouped together based on statistical clustering techniques using the 'distance' between the units of measure and combining UoMs that are closer to each other. In addition, it is also possible to combine both business knowledge and statistical algorithms to combine UoMs.