The exact formula for VaR is = -(Z α σ + μ ), where Z α is the z-multiple for the desired confidence level, and μ is the mean return. Now Z α 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 -Z ασ , 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 -Z ασ = 250,000 (as μ =0), and therefore -Z ασ - 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.

This question is based upon a weather derivative contract traded on the CME in the US. For each day, 'Heating-Degree-Days' (HDD) is calculated as equal to 65 degrees Fahrenheit minus the daily average temperature. The daily average temperature is based upon the temperature reported by the Earth Satellite Corporation using a specified automated weather station. Based upon daily values of HDD, an aggregated number called the 'CME degree days index' is calculated for each contract month. In other words, the index for a particular month is just the aggregation of the 'HDD' value for each of the days of that month. Each contract settles at the end of the month at a value equal to $20 x Degree Days Index. (In a similar way, 'Cooling Degree Days' are also calculated and a futures contract offered, except that CDD is equal to the average daily temperature minus 65 degrees). (Source: CME's website at CMEGroup.com)

In the given question, we are interested in hedging against the possibility of the temperature being too low. This means we should buy the HDD futures contract (the lower the temperature, the higher the difference of the average temperature from 65 degrees, and the higher the settlement). Therefore Choice 'b' is the correct answer. The lower the actual temperature turns out to be, the higher the payout to the grower. It would not be wise to wait till January to buy the contract as by then the prices of the contract would have already risen if the grower's fears of a colder January appear to be coming true. He can hedge his exposure by immediately locking in the January prices.

A recap would be in order here: calculating operational risk capital is a multi-step process.

First, we fit curves to estimate the parameters to our chosen distribution types for frequency (eg, Poisson), and severity (eg, lognormal). Note that these curves are fitted at the UoM level - which is the lowest level of granularity at which modeling is carried out. Since there are many UoMs, there are are many frequency and severity distributions. However what we are interested in is the loss distribution for the entire bank from which the 99.9th percentile loss can be calculated. From the multiple frequency and severity distributions we have calculated, this becomes a two step process:

- Step 1: Calculate the aggregate loss distribution for each UoM. Each loss distribution is based upon and underlying frequency and severity distribution.

- Step 2: Combine the multiple loss distributions after considering the dependence between the different UoMs. The 'dependence' recognizes that the various UoMs are not completely independent, ie the loss distributions are not additive, and that there is a sort of diversification benefit in the sense that not all types of losses can occur at once and the joint probabilities of the different losses make the sum less than the sum of the parts.

Step 1 requires simulating a number, say n, of the number of losses that occur in a given year from a frequency distribution. Then n losses are picked from the severity distribution, and the total loss for the year is a summation of these losses. This becomes one data point. This process of simulating the number of losses and then identifying that number of losses is carried out a large number of times to get the aggregate loss distribution for a UoM.

Step 2 requires taking the different loss distributions from Step 1 and combining them considering the dependence between the events. The correlations between the losses are described by a 'copula', and combined together mathematically to get a single loss distribution for the entire bank. This allows the 99.9th percentile loss to be calculated.

According to the Basel II framework, Tier 1 capital, also called core capital or basic equity, includes equity capital and disclosed reserves.

Tier 2 capital, also called supplementary capital, includes undisclosed reserves, revaluation reserves, general provisions/general loan-loss reserves, hybrid debt capital instruments and subordinated term debt issued originally for 5 years or longer.

Tier 3 capital, or short term subordinated debt, is intended only to cover market risk but only at the discretion of their national authority. This only includes short term subordinated debt originally issued for 2 or more years.

An interesting thing to note is the difference between 'subordinated term debt' under Tier 2 and the 'short term subordinated debt' under Tier 3. The distinction is based upon the years to maturity at the time the debt was issued. The remaining time to maturity is not relevant. For the subordinated term debt included under Tier 2, the amount that can be counted towards capital is reduced by 20% for every year when the debt is due within 5 years. This takes care of the time to maturity problem for Tier 2 subordinated debt. For Tier 3 short term subordinated debt, this is not an issue because debt will only qualify for Tier 3 if it has a lock-in clause stipulating that the debt is not required to be repaid if the effect of such repayment is to take the bank below minimum capital requirements.

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.

'In store' delivery is for delivery in a standardized location, and the buyer is handed a 'warrant' that allows him to pick the goods up. This is the cheapest means of physical delivery. The other prices will be higher as they involve more costs for the seller who has to get the goods on board a ship, or to the docks, or insurance and freight as well. Choice 'c' is the correct answer.