Core Risk Disciplines

A charity's endowment holds 70% in global equities (volatility 16% a year) and 30% in government bonds (volatility 6%). The correlation between their returns is −0.1. What is the portfolio's volatility, in % a year?

Asset X has a volatility of 25% a year and asset Y of 15%; their correlation is 0.30. With short sales allowed, what weight in X gives the two-asset portfolio with the lowest variance, in %?

A fund holds 50% in a bank share (β 1.3), 30% in a utility (β 0.8) and 20% in a food retailer (β 0.5). The risk-free rate is 3% and the market is expected to return 8%. The manager forecasts 9.2% for the fund. By how much does the forecast beat the return CAPM requires, in percentage points?

The tangency portfolio has an expected return of 8.5% and a volatility of 12.5%; the risk-free rate is 3%. A client wants a volatility of 10%. Using only the tangency portfolio and cash, what expected return can they get, in %?

An investment team runs a mean-variance optimiser on 8 asset classes. Raising one expected return by 0.5 percentage points moves that asset from 12% of the portfolio to 55%. What is the best reading, and response?

A desk's eight worst daily losses over the last 250 trading days were, in £ thousands, 4,120, 3,560, 3,010, 2,740, 2,480, 2,350, 2,200, 2,050. Every other day lost less. Using historical simulation, what is its one-day 97.5% expected shortfall, in £ thousands?

A book that is short index put options has a delta of £2,500 per index point, a gamma of −£12 per point² and a vega of −£30,000 per volatility point. The index falls 150 points and implied volatility rises 4 points. Using delta, gamma and vega, what is the book's P&L, in £? (Enter a loss as a negative number.)

A desk holds £20m of UK shares (daily volatility 1.2%) and a bond position with a DV01 of £30k per basis point, where yields have a daily standard deviation of 6 basis points. The correlation between the two P&Ls is −0.3. With a normal model and zero mean (z = 2.3263), what is the one-day 99% VaR, in £ thousands?

A bank's one-day 99% VaR model had 7 exceptions in the last 250 trading days. What follows under the Basel traffic light (with the 1996 multipliers)?

Why did the Basel market risk rules (FRTB) replace 99% VaR with 97.5% expected shortfall?

Brackenholt Foods Ltd has a £40m committed facility, £25m drawn, with a credit conversion factor of 50% on the undrawn part. It is secured on a warehouse worth £20m, taken at a 30% haircut. Under the bank's rules, the covered part of EAD loses 10% and the rest 45%. The borrower is grade 5 (PD 0.57%). What is its expected loss, in £?

A bank has £200m of loans to small businesses, spread over many borrowers, each with a PD of 2% and an LGD of 40%. The asset correlation is 0.12. Using the one-factor (Vasicek) model at 99.9%, as the IRB formula does, how much capital does the pool need for unexpected loss, in £ millions?

A recession starts. A bank's point-in-time PD for a borrower rises from 1.0% to 2.5%; its through-the-cycle grade does not change. Which use of the two is right?

The bank's rating model gives a borrower a PD of 5.0%. Its master scale has bands that double from grade to grade (grade 7 is 1.60% to 3.20%). Which grade, and what does it mean?

A borrower on the watchlist owes £10m and has stopped paying. Enforcing the security now would bring in £5.5m (present value, after costs). The workout team proposes a restructuring instead: if it works, the bank receives payments worth £9.0m in present value; if it fails, the security is sold later for £4.5m in present value. Above what probability of success is restructuring worth more than enforcing now, in %?

A bank has five derivatives with Skelbrook Shipping plc, worth to the bank (£ millions) 9, −5, 4, −6, 3. Skelbrook defaults and the bank recovers 35% of any claim. How much less does the bank lose if close-out netting is enforceable than if it is not, in £ millions?

Under a two-way CSA with a dealer (threshold £2,000k each way, minimum transfer amount £250k, transfers rounded to £100k), the bank's trades are worth −£6,380k to it today. It has already posted £3,500k of collateral. How much more must it deliver today, in £ thousands?

A new uncollateralised swap with Skelbrook has expected exposures at the middle of years 1, 2 and 3 of £4m, £5m, £3m. Skelbrook's CDS spread is 240bp and recovery is 40%. Using the credit triangle for a flat hazard rate, one term a year, and discounting at 4% a year to mid-year, what is the CVA, in £ thousands?

Which of these trades carries specific wrong-way risk?

A clearing member defaults and closing out its portfolio costs more than its initial margin. In what order does a typical CCP's default waterfall absorb the rest?

Thistleby Bank holds unencumbered Level 1 assets of £400m, Level 2A of £150m and Level 2B of £60m (haircuts 15% and 50%). Its 30-day stressed outflows are stable retail deposits £2,000m at 5%; less stable retail deposits £800m at 10%; operational corporate deposits £400m at 25%; other corporate deposits £300m at 40% and undrawn corporate credit facilities £500m at 10%. Contractual inflows from performing loans due within 30 days are £240m, counted at 50%.

What is its liquidity coverage ratio, in %?

Thistleby's funding (£ millions, with its available stable funding factor): capital 300 (100%), stable retail deposits 2,000 (95%), less stable retail deposits 800 (90%), corporate deposits (under a year) 700 (50%). Its assets needing stable funding (with the required factor): residential mortgages 2,400 (65%), corporate loans (under a year) 600 (50%), corporate loans (over a year) 500 (85%), Level 1 government bonds 400 (5%). What is its net stable funding ratio, in %?

In a combined stress, a bank's counterbalancing capacity is £950m. Net outflows are £60m a day for the first 5 days, then £20m a day. For how many full days does the capacity cover the outflows?

Which of these is the soundest contingency funding plan?

Two branches each raise £100m of 30-day deposits at the same interest rate: one from stable retail customers, the other from non-operational corporate deposits. How should treasury's funds transfer pricing treat them?

Wrenmoss Building Society's repricing gaps (assets less liabilities repricing in each band) are 150 in 0–3 months, −400 in 3–6 months and −250 in 6–12 months, in £ millions. Rates rise 100bp today and stay there. Weighting each band by the time left in the year after its mid-point, what is the change in next year's net interest income, in £ millions? (Enter a fall as a negative number.)

Wrenmoss has £5,000m of rate-sensitive assets with a modified duration of 2.0 and £4,600m of liabilities (with deposits slotted by behaviour) with a duration of 0.9. Using durations only, what is the change in its economic value of equity for a parallel rise of 200bp, in £ millions?

A bank has £1,000m of loans that reprice straight away with Bank Rate, funded partly by £800m of instant-access savings whose rate it moves with a beta of 0.4. Bank Rate rises 150bp. Ignoring everything else, how much does a year's net interest income change, in £ millions?

Across the six standard shock scenarios, Wrenmoss's worst ΔEVE is the parallel-up loss of £117.2m (question 27). Its Tier 1 capital is £400m. Which is right?

A bank hedged its five-year fixed-rate mortgages with pay-fixed swaps matched to their scheduled amortisation. Rates then fall sharply. What happens?

A bank's internal fraud events above its reporting threshold arrive as a Poisson process with a rate of 0.5 a year. What is the probability that it has none at all in the next 3 years, in %?

External fraud losses are fitted as 6 events a year (Poisson) with lognormal severity: median £200k and σ = 1.5 (of the log loss). What is the expected annual loss, in £ millions?

Under the Basel standardised approach, a bank's business indicator is €2,400m and its average annual operational loss over ten years is €26m. The marginal coefficients are 12% up to €1,000m and 15% from €1,000m to €30,000m. What is its operational risk capital, in € millions?

Which of these is the best leading key risk indicator for the risk of a major payments outage?

A bank sets an impact tolerance of 24 hours for its personal payments service. In a severe but plausible scenario test (its main data centre lost), the service is restored in 30 hours. What does operational resilience require?

A motor insurer reports net earned premium of €400m, net claims incurred of €268m and expenses of €116m: a combined ratio of 96%. The claims figure includes a release of €20m from reserves set up for earlier years. What is the combined ratio for the current accident year alone, in %?

A liability insurer's cumulative paid claims (€ millions) by accident year and development year are: year 1: 100, 150, 165; year 2: 120, 174; year 3: 140. Claims are fully paid by development year 3.

Using the volume-weighted chain ladder, what reserve is needed for years 2 and 3 together, in € millions?

A property insurer buys a catastrophe excess of loss treaty of €200m xs €100m for a premium of €18m, with one reinstatement at 100% (pro rata to the amount recovered). A windstorm causes €260m of claims. What does the event cost the insurer, net of the recovery and including the reinstatement premium, in € millions?

Kestwick Life writes both term assurance and annuities. Its mortality risk charge (deaths 15% higher) is €40m and its longevity charge (deaths 20% lower) is €100m. The standard formula correlation between them is -0.25. Which is right?

A composite insurer's standard formula charges are market risk €300m, non-life underwriting €200m and life underwriting €150m. The correlations are 0.25 between market and non-life, 0.25 between market and life, and 0 between life and non-life. What is its basic SCR from these three charges, in € millions?

A bank's liquidity buffer is £300m of central bank reserves and £1,200m of fixed-rate gilts with a modified duration of 6, held at fair value. Its net stressed outflows are £1,000m, for an LCR of 150%. In the IRRBB parallel-up scenario, gilt yields rise 200bp. With outflows unchanged, what is its LCR after the shock, in %?

Orrinsby Mutual, a non-life insurer, has own funds of €900m. Its SCR combines a market charge of €360m and a non-life charge of €240m with a correlation of 0.25. It holds €600m of shares with a beta of 1.1 to the equity index. In its ORSA it tests an index fall of 25%. Holding the SCR unchanged, what is its solvency ratio after the fall, in %?

Fallowmoor Capital LP, a hedge fund, defaults. Its trades with the bank are worth £22m and −£8m to the bank under one netting agreement, and the bank holds £6m of collateral. A margin call of £5m due the day before was never sent, because a reconciliation job failed and no one checked. The bank recovers 30% of its claim. Which is right?

A bank is downgraded two notches, from A to BBB+. Its CSAs with dealers set a threshold of £50m while it is rated A or better, and zero below. It owes the dealers £180m on its trades and has posted £130m, so the downgrade brings a call that it pays from Level 1 HQLA.

Before the downgrade its LCR was 150%: HQLA £1,500m, net outflows £1,000m. The outflows included this call, because the LCR counts collateral due on a downgrade of up to three notches. Three notches below BBB+ a funding agreement adds a further £40m trigger. What is the LCR after the downgrade, in %?

Kestwick Life holds €2,200m of bonds with a modified duration of 8 against annuity best estimate liabilities of €2,000m with a duration of 10. Rates fall 50bp across the curve. In the same quarter, new longevity data raise the best estimate by 3% (measured at today's rates). Using durations and ignoring cross effects, what are its own funds afterwards, in € millions?

Quenwick Payments, a payment firm whose customers' money arrives at a bank at about £90m an hour during the day, is hit by a cyber attack and sends nothing for 5 hours. The bank keeps paying out as normal. On a normal day its peak intraday liquidity usage is £250m. It has £300m of reserves and £400m of gilts it can pledge to the central bank at a 5% haircut. Which is right?

Stay with the CVA on the Skelbrook swap in question 18. Its CS01 (the change in CVA for a 1bp rise in Skelbrook's spread) is £1,016. A three-year CDS on Skelbrook has a risky annuity (RPV01) of 2.5658. What notional of CDS protection should the bank buy to hedge the CS01, in £ millions?

A group's stand-alone capital is £600m for credit risk, £200m for market risk and £200m for operational risk. It aggregates with correlations of 0.5 (credit and market), 0.2 (credit and operational) and 0.2 (market and operational). Using Euler (marginal) allocation, how much of the diversified total belongs to credit risk, in £ millions?