PRMIA 8008 PRM Certification - Exam III: Risk Management Frameworks, Operational Risk, Credit Risk, Counterparty Risk, Market Risk, ALM, FTP - 2015 Edition Exam Practice Test

Statement I is correct as credit VaR calculations often use a one year time horizon. This is primarily because the cycle in respect of credit related activities, such as loan loss reviews, accounting cycles for borrowers etc last a year.

Statement II is false. There are two ways in which loss assessments in respect of credit risk can be made: default mode, where losses are considered only in respect of default, and no losses are recognized in respect of the deterioration of the creditworthiness of the borrower (which is often expressed through a credit rating transition matrix); and the mark-to-market mode, where losses due to both defaults and credit quality are considered. The default mode is used for the loan book where the institution has lent moneys and generally intends to hold the loan on its books till maturity. The mark to market mode is used for traded securities which are not held to maturity, or are held only for trading.

Statement III is correct. The confidence interval, or the quintile of losses used for maintaining credit ratings tends to be very high as the possibility of the institution's default needs to be remote.

Statement IV is correct too, for the reasons explained earlier.

Statement I is correct. Retail Risk Based Pricing (RRBP) involves the use of borrower specific data (such as FICO scores, average balances etc) to arrive at credit decisions. These 'retail' credit decisions may include decisions on whether to grant a line of credit, a mortgage, issue a credit card, or any of the various other retail activities a bank may be dealing with. At the same time, this data can also be used to price the product, in addition to providing a yes or no credit decision so that risky borrowers are charged more than less risky borrowers.

Statement II is not correct, because an integrated Risk Information Management Environment includes three elements - people, processes and technology (and not just people and processes).

Statement III is correct. An LDM is a blue print of an organization's data, and describes the relationships between the various data elements.

Statement IV is not correct because reference data and metadata are not the same thing. Reference data refers to relatively static data, such as customer name (while actual transactions may not be so static). Metadata refers to data about data, and is stored in a data dictionary.

Therefore Choice 'b' is the correct answer and the rest are incorrect.

A standard model for representing asset returns in finance is the Geometric Brownian Motion process, and returns according to this model can be estimated by the expression given in Choice 'b'. Note that prices according to this model are log-normally distributed, and returns are normally distributed.

Concentration risk in a credit portfolio arises due to a high degree of correlation between the default probabilities of the issuers of securities in the portfolio. For example, the fortunes of the issuers in the same industry may be highly correlated, and an investor exposed to multiple such borrowers may face 'concentration risk'.

A low degree of correlation, or independence of individual defaults in the portfolio actually reduces or even eliminates concentration risk.

The fact that issuers are from the same country may not necessarily give rise to concentration risk - for example, a bank with all US based borrowers in different industries or with different retail exposure types may not face practically any concentration risk. What really matters is the default correlations between the borrowers, for example a lender exposed to cement producers across the globe may face a high degree of concentration risk.

Expected value = $90 ($100 - 10%)

Value at 99% confidence level = $65

Therefore economic capital required at this level of confidence = $90 - $65 = $25.

Choice 'a' is the correct answer, the other choices are not.

(We can also look at it this way as explained in section III.B.6.2.2 of the handbook: Economic capital is designed to absorb unexpected losses, which are equal to total losses at a given confidence level minus expected losses. (Expected losses are to be covered by credit reserves). Total losses are $100-$65=$35, and expected losses are 10%*$100=$10, therefore economic capital should be $35-$10=$25.)

Choice 'a' is the correct answer. Exposures such as those to a line of credit of which only a part (or none) may be drawn at the time of assessment present a difficulty when attempting to quantify credit risk. It is not correct to take the entire amount of the line as the exposure at default, and likewise the current exposure is likely to be too aggressively low a number to consider.

While the borrower has an information advantage in that he would be aware of the deterioration in credit standing before the bank and would probably draw cash prior to default, it is unlikely that the entire amount of the line of credit would be drawn in all cases. In some cases, none may be drawn. In other cases, the bank would become aware of the situation and curtail or cancel access to the credit line in a timely fashion.

Therefore a fixed proportion of existing credit lines is considered a reasonable approximation of the exposure at default against credit lines.

All the statements in this question are directly based on the principles for effective stress testing as laid down in the BCBS document on stress testing issued in May 2009. Statement 1 is correct and is an almost verbatim reproduction of principle 10 as laid down in that document. Statement II is incorrect as it is contrary to principle 11 laid down in the same document. Statement III is correct as discovering hidden vulnerabilities and inconsistencies in hedging strategies is one of the objectives of reverse stress tests. Similarly, even though reputational risk is not really covered under any risk category under Basel II (as it is not a part of either market, credit or operational risk), principle 14 of this paper requires the mitigation of spill-over effects on market confidence of reputational risk when thinking about stress tests.

Thus statements I, III and IV are correct and statement II is incorrect.

The recovery rate, the default hazard rate (also called the average default intensity) and the spread on debt are linked by the equation Hazard Rate = Spread/(1 - Recovery Rate). Therefore, the recovery rate implicit in the given data is = 1 - 8%/10% = 20%.

Once we have a number of distributions in the model space, the task is to select the "best" distribution that is likely to be a good estimate of true severity. We have a number of distributions to pick from, an empirical dataset (from internal or external losses), and we can estimate the parameters for the different distributions. We then have to decide which distribution to pick, and that generally requires considering both approximation and fitting errors.

There are three methods that are generally used for selecting a model:

1. The cross-validation method: This method divides the available data into two parts - the training set, and the validation set (the validation set is also called the 'testing set'). Parameter estimation for each distribution is done using the training set, and differences are then calculated based on the validation set. Though the temptation may be to use the entire data set to estimate the parameters, that is likely to result in what may appear to be an excellent fit to the data on which it is based, but without any validation. So we estimate the parameters based on one part of the data (the training set), and check the differences we get from the remaining data (the validation set).

2. Complexity penalty method: This is similar to the cross-validation method, but with an additional consideration of the complexity of the model. This is because more complex models are likely to produce a more exact fit than simpler models, this may be a spurious thing - and therefore a 'penalty' is added to the more complex models as to favor simplicity over complexity. The 'complexity' of a model may be measured by the number of parameters it has, for example, a log-normal distribution has only two parameters while a body-tail distribution combining two different distributions may have many more.

3. The bootstrap method: The bootstrap method estimates fitting error by drawing samples from the empirical loss dataset, or the fit already obtained, and then estimating parameters for each draw which are compared using some statistical technique. If the samples are drawn from the loss dataset, the technique is called a non-parametric bootstrap, and if the sample is drawn from an estimated model distribution, it is called a parametric bootstrap.

4. Using goodness of fit statistics: The candidate fits can be compared using MLE based on the KS distance, for example, and the best one selected. Maximum likelihood estimation is a technique that attempts to maximize the likelihood of the estimate to be as close to the true value of the parameter. It is a general purpose statistical technique that can be used for parameter estimation technique, as well as for deciding which distribution to use from the model space.

All the choices listed are the correct answer.

As part of a reverse stress test, a firm has to identify and assess the scenarios most likely to cause it to fail, or in other words using the language used by the FSA in the UK, for its current business plan to become unviable. A firm's business plan should be considered to become unviable at the point that crystallizing risks cause the market to lose confidence it it, with the consequence that counterparties and other stakeholders are unwilling to transact with it or provide capital to the firm and, where releant, that existing counterparties may seek to terminate their contracts. Recent experience suggests that this point is reached well before a firm's regulatory capital is exhausted.

Large known losses, or negative EBITDA (earnings before interest , tax, depreciation and amortization) may be indicators or contribute to the loss of confidence, but do not of themselves make the current business plan unviable. Therefore Choice 'd' is the correct answer.

Generally, the probability of default is an input into determining the price of a security. However, if we know the market price of a security, we can back out the probability of default that the market is factoring into pricing that security. Market implied default probabilities are the probabilities of default priced into security prices, and can be determined from both bond prices and CDS spreads. Credit ratings issued by a credit agency do not give us 'market implied default probabilities', and neither does an internal scoring model like Altman's as these do not consider actual market prices in any way.

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

Current replacement values are a very poor measure of the credit exposure from a derivative contract, because the future value of these instruments is unpredictable, ie is stochastic, and the range of values it can take increases the further ahead in the future we look. Therefore it is common for credit exposures for derivatives to be measured using forward looking exposure profiles, which are distributions of the expected value of the derivative at the time horizon for which credit risk is being measured. To be conservative, a high enough quintile of this distribution is taken as the 'loan equivalent value' of the derivative as the exposure. Choice 'c' is the correct answer.

The notional value of derivative contracts generally tends to be quite high and unrelated to their economic value or the counterparty exposure. Therefore notional value is irrelevant.

Statement I is true - capital adequacy indeed is a reference to the ability of the firm to stay a 'going concern'. (Going concern is an accounting term that means the ability of the firm to continue in business without the stress of liquidation.)

Statement II is not true because even though the stated objective of regulatory capital requirements is similar to the purposes for which economic capital is calculated, regulatory capital calculations are based upon a large number of ad-hoc estimates and parameters that are 'hard-coded' into regulation, while economic capital is generally calculated for internal purposes and uses an institution's own estimates and models. They are rarely identical.

Statement II is not true as the purpose of economic capital is to provide a buffer against unexpected losses. Expected losses are covered by the P&L (or credit reserves), and not capital.

Statement IV is incorrect as even though economic capital may be calculated at very high confidence levels, that is never 100% which would require running a 'risk-free' business, which would mean there are no profits either. The level of confidence is set at a level which is an acceptable balance between the interests of the equity providers and the debt holders.

The exact formula for VaR is = -(Zα σ + μ), where Z α is the z-multiple for the desired confidence level, and μ is the mean return. Now Zα is always a negative number, or at least will certainly be provided the desired confidence level is greater than 50%, and μ is often assumed to be zero because generally for the short time periods for which market risk VaR is calculated, its value is very close to zero.

Therefore in practice the formula for VaR just becomes -Zασ, and since Z is always negative, we normally just multiply the Z factor without the negative sign with the standard deviation to get the VaR.

For this question, there are two ways to get the answer. If we use the formula, we know that -Zασ= 250,000 (as μ=0), and therefore -Zασ - μ = 250,000 - 10,000 = $240,000.

The other, easier way to think about this is that if the mean changes, then the distribution's shape stays exactly the same, and the entire distribution shifts to the right by $10,000 as the mean moves up by $10,000. Therefore the VaR cutoff, which was previously at -250,000 on the graph also moves up by 10k to -240,000, and therefore $240,000 is the correct answer.

The other choices are intended to confuse by multiplying the z-factor for the 99% confidence level with 10,000 etc.

For EVT, we use the block maxima or the peaks-over-threshold methods. These provide us the data points that can be fitted to a GEV distribution.

Least squares and maximum likelihood are methods that are used for curve fitting, and they have a variety of applications across risk management.

Loss provisioning is intended to cover expected losses. Economic capital is expected to cover unexpected losses. No capital or provisions are set aside for losses in excess of unexpected losses, which will ultimately be borne by equity.

Choice 'd' is the correct answer.

Model risk is the risk that a model built for estimating a variable will produce erroneous estimates. Model risk is caused by a number of factors, including:

a) Misspecifying the model: For example, using a normal distribution when it is not justified.

b) Model misuse: For example, using a model built to estimate bond prices to estimate equity prices

c) Parameter estimation errors: In particular, parameters that are subjectively determined can be subject to significant parameter estimation errors

d) Programming errors: Errors in coding the model as part of computer implementation may not be detected by end users

e) Data errors: Errors in data used for building the model may also introduce model risk

Therefore the correct answer is d, as all the choices are a source of model risk.

In the February of 2009, the Federal Reserve (which is the US central bank system) and other US banking regulators embarked on a simultaneous assessment of the capital held by the 19 largest US bank holding companies. This was an unprecedented exercise of a kind never undertaken before, and was known as the Supervisory Capital Assessment Program (SCAP). The purpose of the exercise was to determine the amount of additional capital (called the 'capital buffer') each of the institutions covered would need to ensure that it would have sufficient capital if the economy weakened more than was then expected. The idea was that these financial institutions would then raise additional capital equal to their respective capital buffers by the fourth quarter of 2009.

Statement I is false on two counts: firstl the SCAP was conducted by the US central bank and other regulators, and not the 'Treasury Department' (the Treasury Department in the US is the equivalent of the Ministry of Finance in may other countries). Second, the SCAP was a one time exercise, and not annual.

Statement II is correct. The regulators prescribed rates of losses on credit assets of different kinds and other macro-economic assumptions, and asked the banks to determine the extent of losses they would need to bear (in addition to calculating them independently too). Therefore the SCAP was a stress test where the scenario was prescribed by the regulators.

Statement III is false. Capital buffer under the SCAP referred to the additional capital the banks would need to have certain ratios of capital, and not 'excess' capital.

Statement IV is correct. The SCAP envisaged two capital targets: a Tier 1 capital ratio in excess of 6% at the end of 2010; and a Tier 1 common capital ratio in excess of 4%. Therefore both the total Tier 1 capital and Tier 1 common capital were targeted.

Therefore Choice 'c' is the correct answer.

Choice 'c' is the correct answer. The Markov property is the assumption that there is no ratings momentum, and that transition probabilities are dependent only upon where the rating currently is and where it is going to. Where it has come from, or what the past changes in ratings have been, have no effect on the transition probabilities. ('Herstatt risk' refers to settlement risk, and is irrelevant.)

In statistics, which is relevant to risk management, a distinction is often drawn between 'frequentists' and 'Bayesians'. Frequentists rely upon data to draw conclusions as to probabilities. Bayesians consider conditional probabilities, ie, take into account what things are already known, and inject sometimes subjective a-priori probabilities into the calculations. Statement I describes Bayesians, and not frequentists. In reality however, the difference is merely academic. Risk managers use whichever technique best applies to the given situation without making it about ideology.

The difference between 'Knightian uncertainty' and 'Risk' is similarly academic. Knightian uncertainty refers to risk that cannot be measured or calculated. 'Risk' on the other hand refers to things for which past data exists and calculations of exposure can be made. To give an example in the context of the financial world, the risk from a pandemic creating systemic failures from a failure of payment and settlement systems and the like is 'Knightian uncertainty', but the market risk from equity price movements can be modeled (albeit with limitations) and is calculable. Statement II is therefore correct.

Once a risk is identified, it can be mitigated, accepted, avoided or eliminated, or transferred by way of insurance. Therefore statement III is correct.

Confidence accounting is a conceptual idea that suggests that accounting statements make reference to ranges as opposed to point estimates in financial statements. For example, instead of saying that the pension obligation is $xx million, the company should say the pension obligation is in a range of $xx m - $yy m with a certain confidence level. Statement IV is therefore inaccurate.

Marginal liquidity gap measures the expected net change in liquidity over, say, a day. It is just equal to the liquidity inflow minus liquidity outflow. The cumulative liquidity gap measures the aggregate change in liquidity from a point in time, in other words it is just the summation of the marginal liquidity gap for each of the days included in the period under consideration. The residual liquidity gap goes one step further and adds available 'opening balance' of liquidity to the cumulative liquidity gap to reveal the days or times when the net liquidity is most at risk.

Liquidity at Risk measures the expected time to survival at a certain confidence level applied to the firm's cash flows - and is not a measure of the liquidity gap.

Therefore Choice 'a' is the correct answer.

If the hedge performs as expected, then any P&L on the long bond position will be offset by identical losses (or gains) on the hedge.

Since hedges are never perfect, and some residual risk such as basis risk, the inability to enter into an unrounded number of futures contracts will remain. However, the bulk of the risk would be mitigated, and the investor will be able to avoid any losses but will also forgo any gains. Therefore choice b is the correct answer and the rest are incorrect.

Generally, stress tests consider a shock or a severe scenario in order to determine the 'what-if' that circumstance were to materialize. They focus on the outcome based upon a set of shocks. In a reverse stress test, the outcome is assumed to be known (generally something as severe as bankruptcy, non-compliance with capital requirements etc), and the test is intended to work out what shocks or events would lead to such an outcome.

Reverse stress tests therefore start from a known stress test outcome (such as breaching regulatory captial ratios, illiquidity or insolvency) and then asking what events could lead to such an outcome for the bank. This can be quite a challenging task. Principle 9 laid out in the BCBS document on stress testing (May 2009) (which is part of the PRM syllabus effective March 1, 2010) lays down the expectations relating to reverse stress tests.

Therefore Choice 'a' is the correct answer. All the other choices are nonsensical.

High frequency and low severity risks, for example the risks of fraud losses for a credit card issuer, may have high expected losses, but low unexpected losses. In other words, we can generally expect these losses to stay within a small expected and known range. The capital requirement will be the worst case losses at a given confidence level less expected losses, and in such cases this can be expected to be low.

On the other hand, medium severity medium frequency risks, such as the risks of unexpected legal claims, 'fat-finger' trading errors, will have low expected losses but a high level of unexpected losses. Thus the capital requirement for such risks will be high.

It is also worthwhile mentioning high severity and low frequency risks - for example a rogue trader circumventing all controls and bringing the bank down, or a terrorist strike or natural disaster creating other losses - will probably have zero expected losses & high unexpected losses but only at very high levels of confidence. In other words, operational risk capital is unlikely to provide for such events and these would lie in the part of the tail that is not covered by most levels of confidence when calculating operational risk capital.

Note that risk capital is required for only unexpected losses as expected losses are to be borne by P&L reserves. Therefore the operational risk capital requirements for a low severity high frequency risk is likely to be low when compared to other risks that are lower frequency but higher severity.

Thus Choice 'c' is the correct answer.

Gross income is defined by Basel II (see para 650 of the Basel standard) as net interest income plus net non-interest income. It is intended that this measure should: (i) be gross of any provisions (e.g. for unpaid interest); (ii) be gross of operating expenses, including fees paid to outsourcing service providers; (iii) exclude realised profits/losses from the sale of securities in the banking book; and (iv) exclude extraordinary or irregular items as well as income derived from insurance.

What this means is that gross income is calculated without deducting any provisions or operating expenses from net interest plus non-interest income; and does not include any realised profits or losses from the sale of securities in the banking book, and also does not include any extraordinary or irregular item or insurance income.

Therefore operating expenses are to be not to be deducted for the purposes of calculating gross income, and neither are any provisions. Profits and losses from the sale of banking book securities are not considered part of gross income, and so isn't any income from insurance or extraordinary items.

Of the listed choices, only net non-interest income needs to be included for gross income calculations, and the others are to be excluded. Therefore Choice 'd' is the correct answer. Try to remember the components of gross income from the definition above because in the exam the question may be phrased differently.

Monte Carlo simulations can be effectively used in all cases where an analytical estimate of the VaR cannot be made for any reason - which may include complexity of portfolios, discontinuities or non-linearity in returns or just the plain unavailability of closed form analytical models. Therefore Choice 'd' is the correct answer.

The second order approximation of the VaR of an options position is given by [Option delta x Underlying's VaR - Option gamma/2 x (Underlying's VaR)^2]. Therefore, a higher gamma reduces VaR and a lower gamma increases VaR. Hence Choice 'b' is the correct answer.

When the economy is expanding, firms are less likely to default. Therefore the conditional probability of default, given an economic expansion, is likely to be lower than the unconditional probability of default. Therefore Choice 'a' is the correct answer and the other statements are incorrect.

The number of futures contracts to sell are equal to $10m x 1.1/$50,000 = 220. Liquidating his portfolio would reduce the beta to zero, but would also get rid of the bets he wants to play on. Therefore Choice 'c' is the correct answer.

(Note that futures and spot prices generally move together allowing futures positions to be used for hedging the risk against movement in spot prices. However there is a basis risk between spot and futures, therefore the a perfect hedge is never possible with futures. If interest rates move a great deal, spot and futures prices may diverge. Of course, this risk is generally quite low but may become amplified with large leveraged portfolios. Just something to be aware of.)

The correct answer is Choice 'c'. The other choices do not refer to 'regulatory arbitrage'.

When calculating volatility weighted VaR, returns are adjusted by a factor equal to the current volatility divided by the historical volatility, ie the volatility that existed during the time period the returns were earned. If the current volatility is greater than the historical volatility (also called contemporary volatility), then it has the effect of increasing the magnitude of any past returns (whether positive or negative). This in turn increases the VaR.

Consider an example: if the current volatility is 2%, and a return of -5% was earned at a time when the volatility was 0.8%, then the volatility weighted return would be 12.5% (=-5% x 2%/0.8%). Clearly, this has the effect of increasing the VaR.

Choice 'd' is therefore the correct answer.

If capital calculations are done in a standard way regardless of risk (as reflected by credit ratings), then it creates a perverse incentive for the lenders' employees to lend to the riskiest borrowers that offer the highest expected returns as there is no incentive to 'save' on economic capital requirements that are equal for both safe and unsafe borrowers. Therefore statement I is correct.

Given that the portfolio of such an institution is likely to then comprise poor quality borrowers, and economic capital would be based upon 'average' expected ratings, it is likely to carry lower economic capital given its exposures. Therefore any such economic risk capital model is likely to understate economic capital requirements. Therefore statement IV is correct.

Statements II and III are incorrect and Choice 'b' is the correct answer.

The VaR in this case will be the top quintile of observations. In this case, since there are exactly 100 observations, this would mean the worst return would become the VaR. Therefore Choice 'b' is the correct answer. Choice 'a' and Choice 'c' make no sense. This highlights that at higher confidence levels, fewer and fewer observations impact the VaR if we are using historical simulation based VaR.

High frequency and low severity risks, for example the risks of fraud losses for a credit card issuer, may have high expected losses, but low unexpected losses. In other words, we can generally expect these losses to stay within a small expected and known range. The capital requirement will be the worst case losses at a given confidence level less expected losses, and in such cases this can be expected to be low.

On the other hand, medium severity medium frequency risks, such as the risks of unexpected legal claims, 'fat-finger' trading errors, will have low expected losses but a high level of unexpected losses. Thus the capital requirement for such risks will be high.

It is also worthwhile mentioning high severity and low frequency risks - for example a rogue trader circumventing all controls and bringing the bank down, or a terrorist strike or natural disaster creating other losses - will probably have zero expected losses & high unexpected losses but only at very high levels of confidence. In other words, operational risk capital is unlikely to provide for such events and these would lie in the part of the tail that is not covered by most levels of confidence when calculating operational risk capital.

Note that risk capital is required for only unexpected losses as expected losses are to be borne by P&L reserves. Therefore the operational risk capital requirements for a low severity high frequency risk is likely to be low when compared to other risks that are lower frequency but higher severity.

Thus Choice 'c' 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.

According to a paper by the BCBS, here is an excellent summary of what happened. Based on this, Choice 'c' is the correct answer.

"At the outset of the crisis, mortgage default shocks played a part in the deterioration of market prices of collateralised debt obligations (CDOs). Simultaneously, these shocks revealed deficiencies in the models used to manage and price these products. The complexity and resulting lack of transparency led to uncertainty about the value of the underlying investment. Market participants then drastically scaled down their activity in the origination and distribution markets and liquidity disappeared. The standstill in the securitisation markets forced banks to warehouse loans that were intended to be sold in the secondary markets. Given a lack of transparency of the ultimate ownership of troubled investments, funding liquidity concerns were triggered within the banking sector as banks refused to provide sufficient funds to each other. This in turn led to the hoarding of liquidity, exacerbating further the funding pressures within the banking sector. The initial difficulties in subprime mortgages also fed through to a broader range of market instruments since the drying up of market and funding liquidity forced market participants to liquidate those positions which they could trade in order to scale back risk. An increase in risk aversion also led to a general flight to quality, an example of which was the high withdrawals by households from money market funds."

This situation represents one of the paradoxes in estimating severity that one needs to be aware of - the improvement in controls reduces the weight of the body/middle of the distribution and moves it towards the tails (as the total probability under the curve must stay at 100%) and the distribution becomes more heavy tailed. As a result, the 99.9th percentile loss actually increases. instead of decreasing, creating a counterintuitive result. Therefore the correct answer is that the capital required will increase.

If scenario analysis produces such a result, the analyst must question if the 1 in 100 year loss severity is still accurate. If the new control has reduced the severity in the body of the distribution, the question as to why the more extreme losses have not changed should be raised.

For a number of processes, including many in finance, while a distribution such as the normal distribution is a good approximation of the distribution near the modal value of the variable, the same normal distribution may not be a good estimate of the tails. For this reason, the Pareto distribution is one of the distributions that is often used to model the tails of another distribution. Generally, if you have a set of observations, and you discard all observations below a threshold, you are left with what are called 'exceedances'. The threshold needs to be reasonably far out in the tail. If from each value of the exceedances you subtract the threshold value, the resulting dataset is estimated by the generalized Pareto distribution.

The Pareto distribution has a 'shape parameter'. The average of two Pareto distributions with tail parameters ξ1 and ξ2 (ξ is a Greek character, pronounced as 'sai' (saa-eee)), is the weighted average of ξ1 and ξ2 with weights proportional to the number of observations in the datasets underlying the distributions.

Probability of the joint default of both A and B =

We know all the numbers except default correlation, and we can solve for it.

Default Correlation*SQRT(0.03*(1 - 0.03)*0.08*(1 - 0.08)) + 0.03*0.08 = 0.014.

Solving, we get default correlation = 25%

This is easy to calculate as follows: At the 99% confidence level, the VaR=2.326 * Std Deviation (remember the z values at the 95% and 99% levels, the PRMIA exam may not give you these values). Thus the 1-day standard deviation is $250m/2.326, and the 250-day standard deviation is √250 * ($250m/2.326) = $1,699.4m.

Remember: if you know the VaR, you know the standard deviation. Once you know the standard deviation for any period of time, you can convert it into standard deviation for another period using the square root of time rule. You can also calculate the VaR at a different confidence level too.

A cumulative accuracy plot measures the accuracy of credit ratings assigned by rating agencies by considering the relative rankings of obligors according to the ratings given. Choice 'd' is the correct answer.

The z value at the 95% confidence level is 1.64. Since the variance is 0.16, the annual volatility is 40%. Therefore the daily volatility is 40% x √10/250 = 8%. The VaR therefore is 8% x 1.64 x $240m = $31,488,000

The major problem with using return on equity as a measure of performance is that return on equity is not adjusted for risk. Therefore, a riskier investment will always come out ahead when compared to a less risky investment when using return on equity as a performance metric.

Return on equity does not ignore the effect of leverage (though return on assets does) because it considers the income attributable to equity, including income from leveraged investments.

Return on equity is generally measured after interest and taxes at the company wide level, though at business unit level it may use earnings before interest and taxes. However this does not create a problem so long as all performance being covered is calculated in the same way.

Cash flows being different from accounting earnings can create liquidity issues, but this does not affect the effectiveness of ROE as a measure of performance.

The three pillars under Basel II are minimum capital requirements, supervisory review process and market discipline. Therefore statement I is false. The other two statements are accurate. Therefore Choice 'd' is the correct answer.

Statement I is true. If a Monte Carlo simulation is based upon the same parameters as used for analytical VaR, and enough number of simulations are carried out, we would get the same results as with analytical VaR.

Statement II is false. We cannot use Monte Carlo simulations using parameters based upon a normal assumption when the underlying distribution is not normal. For example, if a return stream is based upon say a uniform distribution, we cannot use a simulation based upon drawings from a normal distribution even though we use the same mean and standard deviation.

Statement III is true. This is the advantage of historical simulations - no assumptions are necessary. (Historical simulations however often suffer from the great disadvantage of the paucity of data that would cover all possibilities.)

Statement IV is true. The results of historical simulations are limited to the data they are based upon.

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.

The cumulative probability of default for the security for the 5 years is [1 - (1 - probability of default upto year 4)*(1 - probability of default in year 5)]. An easier way to think about this is that the Probability of survival till year 5 = (Probability of survival till year 4 * Probability of survival during year 5). Using the relationship that probability of default = 1 - probability of survival, we can calculate the required probability in all cases.

In this case, the cumulative probability of default for the security for 5 years = 1 - (1 - 11.47%)*(1 - 5%) = 15.8695%, therefore Choice 'c' is the correct answer.

A fat tailed distribution has more weight in the tails, and therefore at a high level of confidence the VaR estimate will be higher for a distribution with heavier tails. At relatively lower levels of confidence however, the situation is reversed as the heavier tailed distribution will have a VaR estimate lower than a thinner tailed distribution.

A higher level of kurtosis implies a 'peaked' distribution with fatter tails. Among the given choices, a distribution with kurtosis equal to 8 will have the heaviest tails, and therefore a higher VaR estimate. Choice 'a' is therefore the correct answer. Also refer to the tutorial about VaR and fat tails.

Pre-settlement risk is the risk that one of the counterparties defaults prior to the date for the maturity of the transaction in question. This may be an unrelated default, in fact there may have been no default on that particular contract, but the party may have defaulted on its other obligations, or filed for bankruptcy. To deal with such cases and to protect the interests of both the parties, it is common to provide for immediate termination of positions and settlement based on the current replacement value of the contracts. Therefore statements I and II are correct.

Statement III is correct as well - the exposure from an OTC derivative contract derives from its current replacement value, and not the notional. If the current replacement value is negative, then the credit exposure is considered equal to zero.

Statement IV is correct as it is quite common to restate all exposures - those from credit lines, OTC derivatives etc - in loan equivalent terms prior to estimating credit risk.

Measures of risk sensitivity include delta, gamma, vega, PV01, convexity and CR01, among others. They allow approximating the change in the value of a portfolio from a change (generally small) in one of the underlying risk factors.

Risk sensitivity measures are derivatives of the value of the portfolio, calculated with respect to the risk factor. Some risk sensitivity measures are second derivatives, and allow a more precise calculation of the change in the value of the portfolio. Many risk sensitivities are represented by Greek letter, but not all.

Delta (δ) is a measure of the change in portfolio value based on a 1% change in the value of the underlying. Gamma (γ) is a second order derivative that improves the calculation as part of a Taylor expansion. CR01 is a measure of the change due to a 1 basis point change in the credit spread. PL01 is not a measure of any kind of risk sensitivity, it does not mean anything.

An error rate of 10 basis points means the number of errors expected in a day will be 15 (recall that 100 basis points = 1%). Therefore the total number of errors expected in a year will be 15 x 250 = 3750. Choice 'a' is the correct answer.

The correct answer is choice 'a'

Recall the formula for calculating variance under EWMA. See below. Therefore the correct answer is =SQRT((1 - 0.93)*(LN(11/10))^2 + 0.93*((3.5%^2))) = 4.21%. Other answers are incorrect. Note that continuous returns are to be used, ie ln(11/10) and not discrete returns (=1/10) - though generally the difference between the two is small over short time periods. (If in the exam the answer doesn't exactly match, try using discrete returns.)

When two bonds that appear identical in every respect trade at different prices, the difference is often due to differences in liquidity between the two bonds (the less liquid bond will be cheaper and yield higher), and also due to the fact that ratings from the major rating agencies do not generally react to day to day changes in the market. The market's perception of the differences in the two credits will cause a divergence in the prices. This has been an extremely visible phenomenon during the credit crisis of 2007-2009, where fixed income security prices have changed sharply for many securities without any changes in external credit ratings.

Bonds carrying 'different market risk' is meaningless, and so is the difference in convexity (because the calculated convexity would be identical for similar bonds).

Therefore Choice 'c' is the correct answer.

Each of these represent valid techniques for performing stress testing and building stress scenarios. Therefore d is the correct answer. In practice, elements of each of these techniques is used depending upon the portfolio and the exact situation.