How it works
Illustrative dataRiskAtoms Bank plc's Markets desk
The module's trading book: eight positions in four books (Equities, Rates, FX and commodities, Credit) driven by eight risk factors, with 750 days of seeded factor moves that include a stressed spell (days 480 to 539) and four jump days. Every position is revalued in full. The numbers are made up for teaching, in pounds, on one valuation day; they are not market data.
The problems use new numbers: smaller books, sub-books and later snapshots of the desk, on the module's market levels and, where a covariance matrix is needed, its calm and stressed covariance matrices. The tail losses, dealer quotes, liquidity data and backtest counts are made up too.
Problem 1Mapping a new desk onto the risk system
L3.2.02 · L3.2.04 · L3.2.05 · L3.2.07Market risk analyst, RiskAtoms Bank plc
The bank is setting up a smaller Sterling desk beside the Markets desk. Before it trades, you check how the risk system will see its book. On the module's levels GBP/USD is 1.30 dollars per pound, and the swap curve is 4.00% at 2 years and 4.40% at 10 years. As on the Markets desk, the system has only those two rate points: anything in between is mapped onto them by linear interpolation in maturity.
| Position | Terms | Risk system says |
|---|---|---|
| US equity basket | $30m, beta 1 to the US index | US index, GBP/USD |
| FX forward | Sell $24m for pounds in 3 months at 1.30 | GBP/USD |
| 10-year gilt | £60m face, 4.25% semi-annual coupon, yield 4.40% | −£47,780 per bp (10-year) |
| 2-year swap, pay fixed | £200m at 4.00% (par) | +£37,720 per bp (2-year) |
| Corporate bonds (one issuer) | £40m face, 7 years, 5.00% annual coupon, spread 150bp over the 7-year swap rate; value £38.31m | −£21,930 per bp rise in its yield |
Lessons: L3.2.02: Risk factors and position mapping · L3.2.04: Interest rate risk in the trading book · L3.2.05: FX and commodity risk · L3.2.07: Credit spread risk
(a)What is the first-order P&L of the basket and the forward together for a 1% rise in GBP/USD (a stronger pound)? Answer in £.
£(b)Map the corporate bonds' rate risk onto the two curve points. What P&L per bp lands on the 2-year point?
£ per bp(c)Using the mapped sensitivities, what does the book make or lose if the curve bear-steepens: 2-year rate +10bp, 10-year rate +25bp, spreads unchanged? Answer in £ (negative for a loss).
£(d)How much more 2-year pay-fixed swap (same terms) would bring the book's parallel DV01 (2-year plus 10-year) to zero? Answer in £m of notional.
£m(e)After that trade, which curve move would still lose the book money?
(f)The bond issuer defaults overnight and bondholders expect to recover 40% of face. What is the jump-to-default loss? Answer in £m.
£m
Problem 2A short call position, hedged, on a bad day
L3.2.03 · L3.2.06 · L3.2.08Equity derivatives risk manager
The Equities book has sold 500 UK index call contracts to a client: £10 per index point, strike 8,200, two months to expiry. The index is at 8,000, implied volatility 16%, the rate 4% and the dividend yield 3.5%. Black-Scholes gives, per option bought, in index points:
| Price | Delta | Gamma (per point) | Vega (per vol point) | Theta (per calendar day) |
|---|---|---|---|---|
| 127.53 | 0.3676 | 0.000718 | 12.26 | −1.638 |
The desk buys UK index futures to make the position delta-neutral (a future gains its notional times the index return). Next day the index falls 3% to 7,760, implied volatility rises 3 points to 19%, and one calendar day passes. Full revaluation of the calls and the futures gives a hypothetical P&L of −£234,400.
Lessons: L3.2.03: Equity risk · L3.2.06: Option Greeks across a book · L3.2.08: P&L attribution and explain
(a)What notional of index futures makes the short calls delta-neutral? Answer in £m.
£m(b)With the delta hedged, what does the gamma term alone give for the day? Answer in £.
£(c)Add the vega and theta terms. What is the delta-gamma-vega-theta estimate of the day's P&L? Answer in £.
£(d)What is the unexplained P&L (actual minus explained)? Answer in £.
£(e)What is most of that unexplained P&L?
Problem 3Where the VaR of a three-factor book sits
L3.2.09Market risk analyst, Markets desk
A sub-book of the Markets desk has three exposures. Use the module's calm daily volatilities and correlations, delta-normal VaR, one day, 99% (z = 2.3263), mean zero.
| Factor | Sensitivity | Daily volatility |
|---|---|---|
| UK index | £25m per 1.00 log return (£250,000 per 1%) | 1.0% |
| 10-year swap rate | −£50,000 per bp | 7bp |
| IG credit spread | −£30,000 per bp | 2.5bp |
Correlations: UK index with the 10-year rate 0.15, UK index with the IG spread −0.50, 10-year rate with the IG spread −0.15.
(a)What is the standard deviation of the book's daily P&L? Answer in £.
£(b)What is the 99% one-day VaR? Answer in £.
£(c)What is the component (Euler) VaR of the 10-year rate exposure? Answer in £.
£(d)The desk wants to buy gilts that add −£10,000 per bp at 10 years. Using the marginal VaR, by how much will VaR change? Answer in £ (negative if it falls).
£(e)If you recompute VaR in full after the trade, the change is…
Problem 4A sell-off six weeks ago
L3.2.10 · L3.2.11 · L3.2.12VaR model owner
The desk's historical simulation revalues today's positions in full on each of the last 500 days. A sell-off between 37 and 44 days ago produced several of the worst days. The 15 worst losses, in £ thousands, with how many days ago each happened (1 = yesterday):
| Rank | Loss (£k) | Days ago |
|---|---|---|
| 1 | 3,420 | 41 |
| 2 | 2,760 | 39 |
| 3 | 2,310 | 212 |
| 4 | 2,050 | 44 |
| 5 | 1,940 | 380 |
| 6 | 1,880 | 37 |
| 7 | 1,810 | 455 |
| 8 | 1,760 | 120 |
| 9 | 1,705 | 43 |
| 10 | 1,660 | 300 |
| 11 | 1,630 | 8 |
| 12 | 1,595 | 260 |
| 13 | 1,570 | 470 |
| 14 | 1,540 | 90 |
| 15 | 1,515 | 160 |
The desk also runs a Monte Carlo model with 10,000 paths; the simulated P&L has a standard deviation of £860,000.
Lessons: L3.2.10: Historical simulation in practice · L3.2.11: Monte Carlo VaR · L3.2.12: Expected shortfall in practice
(a)What is the 99% one-day historical VaR? Answer in £k.
£k(b)What is the 97.5% one-day expected shortfall? Answer in £k.
£k(c)The desk head asks to drop the worst day as a one-off, leaving 499 days. What would the 97.5% ES be? Answer in £k.
£k(d)With age weights (Boudoukh, Richardson and Whitelaw) and λ = 0.99 over the 500 days, what is the 99% VaR? Answer in £k.
£k(e)For the Monte Carlo model, what is the standard error of its 99% VaR estimate (normal approximation)? Answer in £k.
£k
Problem 5From books to the desk, from one day to ten
L3.2.13Head of market risk
A smaller version of the Markets desk has three books. Their stand-alone one-day 99% VaRs are Equities £1.30m, Rates £0.90m and Credit £0.50m. The correlations of their daily P&L are −0.20 (Equities, Rates), 0.60 (Equities, Credit) and 0.10 (Rates, Credit). Treat the P&L as normal with mean zero, so VaRs aggregate like standard deviations.
(a)What is the desk's one-day 99% VaR? Answer in £m.
£m(b)What is the diversification benefit against the sum of the books' VaRs? Answer in £m.
£m(c)The desk's daily P&L has autocorrelation 0.15 at lag 1, 0.15² at lag 2 and so on. What is the 10-day 99% VaR, allowing for it? Answer in £m.
£m(d)In a stress the correlations rise to 0.30, 0.85 and 0.40 (same order), with the books' VaRs unchanged. What is the desk's VaR then? Answer in £m.
£m(e)Which of these would also make √10 × the one-day VaR understate the desk's ten-day risk?
Problem 6Six exceptions, or eight?
L3.2.14Model validation, RiskAtoms Bank plc
You backtest the Markets desk's 99% one-day VaR over the last 250 days. The hypothetical P&L (yesterday's positions revalued on today's moves) lost more than the VaR on 6 days. The actual P&L, which also includes intraday trading and new deals, did so on 8 days. Use the binomial model with p = 1%, Kupiec's test at 5% (critical value 3.841) and the Basel traffic light (green 0 to 4, amber 5 to 9, red 10 or more; the 1996 framework called amber “yellow”).
Lessons: L3.2.14: Backtesting
(a)If the model is right, what is the probability of 6 or more exceptions in 250 days? Answer in %.
%(b)What is Kupiec's likelihood ratio for the 6 hypothetical exceptions?
(c)Which statement is right?
(d)The supervisor counts the larger of the two figures. Under the 1996 rules, what multiplier applies to the bank's market risk capital?
(e)Suppose the model is in fact too optimistic and the true chance of an exception is 2% a day. What is the probability that a 250-day backtest still lands in the green zone? Answer in %.
%
Problem 7A stagflation scenario against the limits
L3.2.15 · L3.2.16Market risk manager, Markets desk
After a month of trading, the Markets desk's risk report shows these sensitivities:
| Sensitivity | Value |
|---|---|
| UK equity delta | £350,000 per 1% |
| UK equity gamma (long puts) | delta rises by £18,000 per 1% for each 1% the index moves |
| 2-year rate | +£36,000 per bp |
| 10-year rate | −£95,000 per bp |
| CS01 (IG spread) | −£34,000 per bp |
| Vega (UK index vol) | +£120,000 per vol point |
The stress committee's “UK stagflation” scenario: UK index -12%, 2-year rate +80bp, 10-year rate +60bp, IG spread +50bp, implied volatility +8 points; other factors unchanged. The desk's stress limit is £7.5m and its 10-year bucket DV01 limit £110,000 per bp. From the same sensitivities, the desk's delta-normal daily P&L standard deviation is £574,000 with the module's calm covariance and £1,282,000 with its stressed covariance.
Lessons: L3.2.15: Stress testing the trading book · L3.2.16: Market risk limits
(a)What is the scenario's P&L from the sensitivities, including the gamma term? Answer in £m (negative for a loss).
£m(b)What is the utilisation of the stress limit? Answer in %.
%(c)A reverse stress test asks how far away a £10m one-day loss is. Under the stressed covariance, how many standard deviations of the desk's P&L is it?
σ(d)A trader wants to buy more 10-year gilts, −£790 per bp per £1m face. What is the most the desk can buy without breaching the 10-year DV01 limit or the stress limit? Answer in £m face.
£m(e)Under the calm covariance the same £10m loss is 17.4 standard deviations away. What should the desk take from the two distances?
Problem 8What the bonds are really worth, and what it costs to get out
L3.2.17 · L3.2.18Product control and market risk, joint review
A review of the Credit and Rates positions on a smaller desk. Its one-day 99% VaR is £1.45m. The liquidity data (full bid-ask spreads as a share of value, their volatility, and average daily volume):
| Position | Value | Spread | Spread sd | ADV |
|---|---|---|---|---|
| Corporate bonds | £45m | 60bp | 20bp | £6m |
| 10-year gilts | £90m | 4bp | 1bp | £1.2bn |
| UK equity basket | £20m | 8bp | 4bp | £150m |
The corporate bonds' stand-alone one-day 99% VaR is £150,000 and their CS01 −£27,000 per bp. The front office marks them at a 150bp spread. Three dealers quote 154, 157, 163bp; the independent level is the median quote, and the tolerance is 3bp.
Lessons: L3.2.17: Market liquidity risk · L3.2.18: Valuation control and the daily risk report
(a)What is the liquidity-adjusted VaR, adding Bangia's cost of liquidity with a multiple of 3 spread volatilities? Answer in £m.
£m(b)Selling the corporate bonds evenly at 25% of daily volume, what is their VaR over the whole exit? Answer in £k.
£k(c)What valuation adjustment does independent price verification make to the bonds? Answer in £ (negative if it lowers their value).
£(d)For prudent valuation, what market price uncertainty AVA do the quotes imply at 90% confidence (normal quote errors)? Answer in £.
£(e)Which other additional valuation adjustment does the bonds' long exit feed most directly?