Foundations of Risk

Harbour Bank has a €1,500m book of small business loans. It expects credit losses of 0.6% of the book a year and has priced them into its loan rates. It has €90m of capital and must keep at least €66m. What is the highest loss rate, as a % of the book, it could suffer in one year without falling below that floor?

A multi-asset fund's return next year will be 25% with probability 30%, 8% with probability 50% or −15% with probability 20%. The risk-free rate is 3%. What is the fund's Sharpe ratio, to two decimal places?

Which of these is closest to Knightian uncertainty rather than measurable risk?

An airline's shareholders hold diversified portfolios and could hedge oil prices themselves. Why might the airline still be right to hedge part of its fuel bill?

A hedge fund has $3bn of capital and positions worth 30 times that, spread across many markets. In a flight to quality all its trades lose 2.5% together, and forced sales cost a further 0.5% of its positions. What share of its capital does it lose, in %?

A UK charity's endowment holds £50m of US shares and £30m of gilts with a modified duration of 8. In a risk-off week US shares fall 6% in dollars, the dollar rises 4% against the pound, and gilt yields fall 0.25 percentage points. Using first-order sensitivities and ignoring cross effects, what is the change in the endowment's value, in £m? (Enter a loss as a negative number.)

A fund holds £40m of one company's BBB bonds, with a duration of 4.5 and a credit spread of 1.2%. In March, spreads on all BBB bonds widen by 0.3 percentage points on recession fears, with no news about the company. In April the company is downgraded to BB and its spread rises to 2.8%. Every coupon is paid and government bond yields do not move. Of the total loss, how much is credit risk (migration), in £m?

A broker-dealer holds £3bn of cash and £10bn of bonds that, in a stress, could only be sold at a 15% discount. It expects stressed net outflows of £0.8bn a day. How many days could it survive on these assets alone, to one decimal place?

A trader hides losing positions by booking fictitious offsetting trades. When the bank finds out, markets have moved against the real positions and closing them costs $700m. How is the loss classified?

Which pairing of event and risk type is right?

Kestrel Bank has €9.1bn of CET1 capital and €70bn of risk-weighted assets. Its risk appetite sets an early warning at a CET1 ratio of 12.5% and a limit at 11.5%; its capacity (the regulatory requirement with buffers) is 10%. The board asks a reverse question: in a stress where risk-weighted assets grow 5%, what is the largest loss of CET1 the bank could take and still be at or above its appetite limit, in € millions?

At a mid-sized insurer, the underwriting teams price and accept risks. The risk function sets underwriting limits and challenges exceptions. The head of internal audit reports to the chief financial officer, who also sets internal audit's budget and the head of audit's bonus. What is the main weakness?

A bank's Chief Risk Officer reports to the CEO and meets the board risk committee privately every quarter. But the CEO alone can dismiss the CRO, and half the CRO's bonus depends on the investment bank's revenue. Which pair of changes would best protect the CRO's independence?

A trader is awarded a bonus of €900,000: 40% paid at once and the rest deferred, vesting in equal slices at the end of each of the next 3 years. The profits behind it later turn out to rest on mis-marked positions. The bank will apply malus to anything not yet vested, and can claw back a third of whatever has already been paid. How much more does the bank recover if the problem comes to light just after the first deferred slice vests, rather than just after the last one, in euros?

A bank pays its loan advisers a bonus for every payment protection policy they sell with a loan. Complaints are rising from customers who could never have claimed on the cover. The board's first idea is to retrain the advisers. Why is that unlikely to be enough?

Larkfield Bank scores risks on a 5 × 5 heat map. Likelihood bands (chance in a year): under 5%, 5% to 20%, 20% to 50%, 50% to 80%, over 80%. Impact bands: under €1m, €1m to €5m, €5m to €20m, €20m to €100m, over €100m. Score = likelihood band × impact band.

• R-A, card payments declined for a few hours: 70% a year, €0.8m if it happens.
• R-B, a mis-sold savings product needs redress: 30% a year, €4m if it happens.
• R-C, a cyber attack takes online banking down for a week: 3% a year, €60m if it happens.
• R-D, a fire closes the main operations centre: 10% a year, €12m if it happens.

Which statement is right?

On a bank's board dashboard, operational losses so far this year are €9.6m, against an annual appetite limit of €15m and an amber trigger of €12m. Last quarter's report showed €6.4m. If the fourth quarter brings the same losses as the third, what share of the year's appetite will have been used by year end, in %?

Over the last year the key risk indicator "payments repaired by hand" has run between 1.8% and 2.6% a month, with no losses of note. The risk owner is setting thresholds for the first time. Which choice is best?

Which risk register row is written properly?

In its risk and control self-assessment, the payments team rates its "second person checks every payment over €100,000" control as effective. The risk function samples 40 such payments and finds the check was skipped on 9 of them. What should happen?

A new fund has five daily returns: 0.8%, −1.2%, 0.5%, −0.3%, 1.4%. Estimate its daily volatility with the sample standard deviation (dividing by n − 1) and annualise it using 252 trading days. What is the annual volatility, in %, to one decimal place?

A risk report says a hedge fund's daily returns have a skewness of −0.9 and a kurtosis of 7.5 (a normal has 0 and 3). The fund's VaR is calculated with the normal formula. What should the risk manager conclude?

A portfolio is 70% in an equity fund with annual volatility 18% and 30% in a bond fund with annual volatility 6%. The correlation between them is -0.2. What is the portfolio's annual volatility, in %, to two decimal places?

A fund holds:

• $12m of shares with a beta of 0.9 to the index;
• bonds with a DV01 of $4,200;
• 30 index put contracts, each worth $100 per index point, with a delta of -0.35.

The index, now at 4,000, falls 4% and yields fall 20 basis points. Using sensitivities only, what is the fund's P&L, in dollars? (Enter a loss as a negative number.)

A €25m holding of a European equity index fund has an annualised volatility of 18%. Assuming normal returns with zero mean, independent days and 252 trading days a year, what is its 5-day 95% VaR, in euros? (z at 95% is 1.645.)

A desk has 300 days of one-day P&L. Its eight worst losses, in $ thousands, are 640, 515, 470, 455, 390, 360, 350, 320. Using this course's convention (k = n(1 − α) rounded up), what is its one-day 99% historical expected shortfall, in $ thousands?

A pension fund runs three tests on today's holdings. (1) Equities fall 25% and nothing else moves. (2) The market moves of autumn 2008 are replayed. (3) It searches for the combinations of market moves that would push its funding below the regulator's minimum. Which labels fit, in that order?

A chocolate maker must buy 2,500 tonnes of cocoa in three months. Cocoa futures are for 10 tonnes each. Monthly changes in the price it pays have a standard deviation of $75 a tonne; those of the futures price, $90; their correlation is 0.85. Using the minimum-variance hedge ratio, how many futures contracts should it buy, to the nearest whole contract?

A pension fund holds 500,000 units of an equity index fund, bought at €100. It buys six-month puts with a strike of €95 for €3.20 and sells six-month calls with a strike of €110 for €2.10. At expiry the fund is at €82. What is the profit or loss on the hedged holding, in € millions? (Enter a loss as a negative number.)

A bank holds $120m (market value) of fixed-rate bonds with a modified duration of 6.2. A new pay-fixed interest rate swap of similar maturity has a DV01 of $780 per $1m of notional. What notional, in $ millions, makes the position DV01-neutral? Two decimals.

A bank has five trades with one hedge fund under an ISDA Master Agreement with close-out netting. From the bank's side they are worth +6.0, -4.0, +9.0, -3.0, +1.5 (US$ millions). The CSA has a threshold of 2 and a minimum transfer amount of 0.5, and the bank already holds 5 of collateral. How much collateral does the bank call today, in US$ millions?

A property insurer buys two catastrophe excess of loss layers: €30m xs €20m and €50m xs €50m. A storm causes €90m of claims. How much of the storm does the insurer keep, in € millions?

A global systemically important bank in the 1.5% surcharge bucket operates where the countercyclical buffer is 0.5%. It has €50bn of CET1 capital and €600bn of risk-weighted assets. Under Basel III's capital conservation rules, at most what share of its profits may it pay out as dividends, buybacks and bonuses, in %?

A bank holds €200m of loans to companies rated BBB, €100m of loans to companies rated AA and €300m of residential mortgages. By how much does its minimum total capital for credit risk (8% of risk-weighted assets) fall when it moves from Basel I to the Basel II standardised approach, in € millions?

A life insurer's Solvency II balance sheet, in € millions: assets 6,000, of which 800 in listed shares; best estimate of liabilities 4,900; risk margin 250; other liabilities 150. Its SCR is 450. The shares then fall 30%. Holding the liabilities and the SCR unchanged, what is its solvency ratio after the fall, in %?

A bank lends €2m to a company. The loan starts in IFRS 9 stage 1, with a 12-month PD of 1.5% and an LGD of 45%. Later its credit risk rises significantly and its lifetime PD is now 9%. Ignoring discounting and with the exposure unchanged, by how much does the provision rise, in € thousands?

Which description of who supervises whom is right?

A bank's government bond desk holds US Treasuries with a DV01 of $85,000. Over the last year the daily change in the yields it holds has a sample standard deviation of 7 basis points. Its limit is a one-day 99% VaR of $1.2m. Treating the daily P&L as normal with zero mean (z = 2.3263), and using the bank's escalation ladder (early warning at 80% of the limit (head of desk and market risk), breach above 100% (head of business and head of market risk, same day), serious breach above 110% (the Chief Risk Officer decides), and above 125% the CEO and the chair of the board risk committee), which is right?

Stay with that desk. The CRO decides to keep the bonds but hedge part of the rate risk with a pay-fixed interest rate swap that has a DV01 of $850 per $1m of notional, so that the desk's VaR falls back to the early-warning level of 80% of its limit. Assuming the swap moves with the same yields, what notional does the desk need, in $ millions, to one decimal place?

A bank has €60bn of fixed-rate mortgages with a modified duration of 4, fully hedged with cleared pay-fixed swaps of the same DV01. Its liquidity coverage ratio is 133.3%: €48bn of high-quality liquid assets against €36bn of net stressed outflows. Its risk appetite has an early warning at an LCR of 130% and a limit at 115%. Rates fall 75 basis points in a week; the swaps lose value and the bank pays the loss as variation margin from its HQLA. With net outflows unchanged, what is its LCR now, in %?

A property insurer has €1,200m of own funds and a Solvency Capital Requirement of €800m. It holds a catastrophe excess of loss treaty of €400m xs €150m. A hurricane causes €650m of claims across thousands of its policies. Holding the SCR unchanged, what is its Solvency II ratio after the hurricane, in %?

A trading business has two desks. The equity desk's one-day 99% VaR is $4m and the credit desk's is $3m; both are normal VaRs with zero mean. In calm markets the correlation between their daily P&L is 0.2. The business VaR limit is $6m. In a crisis the correlation jumps to 0.8 while each desk's VaR stays the same. What is the business's one-day 99% VaR in the crisis, in $ millions, to two decimal places?

A software change fails and takes a bank's card and app payments down for a whole day. It pays its customers €2m in compensation. For three months before, its key risk indicator for failed or rolled-back changes had been red, but no one escalated it. Which response best puts the event in its place?