Chapter 9 Market risk Answers to end-of-chapter questions Questions and problems 1 What is meant by ‘market risk’? Market risk is the uncertainty of the effects of changes in economy-wide systematic factors that affect earnings and stock prices of different firms in a similar manner. Some of these market-wide risk factors include volatility, liquidity, interest rate and inflationary expectation changes. 2 Why is the measurement of market risk important to the manager of a financial institution? Measurement of market risk can help an FI manager in the following ways: (i) Provides information on the risk positions taken by individual traders. (ii) Establishes limit positions on each trader based on the market risk of their portfolios. (iii) Helps allocate resources to departments with lower market risks and appropriate returns. (iv) Evaluates performance based on risks undertaken by traders in determining optimal bonuses. (v) Helps develop more efficient internal models so as to avoid using standardised regulatory models. 3 What is meant by ‘daily earnings at risk’ (DEAR)? What are the three measurable components? What is the price volatility component? Daily earnings at risk or DEAR is defined as the estimated potential loss of a portfolio’s value over a one-day unwind period as a result of adverse moves in market conditions, such as changes in interest rates, foreign exchange rates and market volatility. DEAR is comprised of (a) the dollar value of the position, (b) the price sensitivity of the assets to changes in the risk factor, and (c) the adverse move in the yield. The product of the price sensitivity of the asset and the adverse move in the yield provides the price volatility component. 4 Follow Bank has a $1 million position in a five-year, zero-coupon bond with a face value of $1 402 552. The bond is trading at a yield to maturity of 7 per cent. The historical mean change in daily yields is 0.0 per cent, and the standard deviation is 12 basis points.

a. What is the modified duration of the bond? MD = D/(1 + R) = 5/(1.07) = 4.6729 years b. What is the maximum adverse daily yield move given that we desire no more than a 1 per cent chance that yield changes will be greater than this maximum? Potential adverse move in yield at 1 per cent = 2.33 = 2.33 × 0.0012 = 0.002796 c. What is the price volatility of this bond? Price volatility = MD × potential adverse move in yield = 4.6729 × 0.002796 = 0.013065 or 1.3065 per cent d. What is the daily earnings at risk for this bond? DEAR = ($ value of position) × (price volatility) = $1 000 000 × 0.013065 = $13 065 5 How can DEAR be adjusted to account for potential losses over multiple days? What would be the VaR for the bond in problem 4 for a 10-day period? What statistical assumption is needed for this calculation? Could this treatment be critical? The DEAR can be adjusted to account for losses over multiple days using the formula N-day VaR = DEAR × [N]½, where N is the number of days over which potential loss is estimated. N-day VaR is a more realistic measure when it requires a longer period for an FI to unwind a position, that is, if markets are less liquid. The value for the 10-day VaR in problem 4 above is $13 065 × [10]½ = $41 315. According to the above formula, the relationship assumes that yield changes are independent and daily volatility is approximately constant. This means that losses incurred one day are not related to losses incurred the next day. Recent studies have indicated that this is not the case, but that shocks are autocorrelated in many markets over long periods of time. 6 The DEAR for a bank is $8500. What is the VaR for a 10-day period? A 20-day period? Why is the VaR for a 20-day period not twice as much as that for a 10-day period? For the 10-day period: VaR = 8500 × [10]½ = 8500 × 3.1623 = $26 879.36 For the 20-day period: VaR = 8500 × [20]½ = 8500 × 4.4721 = $38 013.16 The reason that VaR20 (2 × VaR10) is because [20]½ (2 × [10]½). The interpretation is that the daily effects of an adverse event become less as time moves farther away from the event. 7 The mean change in the daily yields of a 15-year, zero-coupon bond has been 5 basis points (bp) over the past year with a standard deviation of 15 bp. Use these data and assume the yield changes are normally distributed. (a) What is the highest yield change expected if a 99 per cent confidence limit is required; that is, adverse moves will not occur more than one day in 100? If yield changes are normally distributed, 98 per cent of the area of a normal distribution will be 2.33 standard deviations (2.33) from the mean—that is, 2.33—and 2 per cent of the area under the normal distribution is found beyond ± 2.33 (1 per cent under each tail, –2.33 and + 2.33, respectively). Thus, for a one-tailed distribution, the 99 per cent confidence level will represent adverse moves that not occur more than 1 day in 100. In this example, it means 2.33 × 15 = 34.95 bp. Thus, the maximum adverse yield change expected for this zero-coupon bond is an increase of 34.95 basis points, or 0.3495 per cent, in interest rates. (b) What is the highest yield change expected if a 95 per cent confidence limit is required? If a 95 per cent confidence limit is required, then 95 per cent of the area will be 1.96 standard deviations (1.96) from the mean. Thus, the maximum adverse yield change expected for this zero-coupon bond is an increase of 29.40 basis points (1.96 × 15) in interest rates. 8 In what sense is duration a measure of market risk? The market risk calculations are typically based on the trading portion of an FI’s fixed-rate asset portfolio because these assets must reflect changes in value as market interest rates change. As such, duration or modified duration provides an easily measured and usable link between changes in the market interest rates and the market value of fixed-income assets. 9 Bank Alpha has an inventory of AAA-rated, 15-year zero-coupon bonds with a face value of $400 million. The bonds are currently yielding 9.5 per cent in the over-the-counter market. (a) What is the modified duration of these bonds? Modified duration = (MD) = D/(1 + r) = 15/(1.095) = –13.6986. (b) What is the price volatility if the potential adverse move in yields is 25 basis points? Price volatility = (–MD) × (potential adverse move in yield) = (–13.6986) × (0.0025) = –0.03425 or –3.425 per cent. (c) What is the DEAR? Daily earnings at risk (DEAR) = ($ value of position) × (Price volatility) Dollar value of position = 400/(1 + 0.095)15 = $102.5293 million. Therefore, DEAR = $102.5293499 million × –0.03425 = –$3.5116 million or –$3 511 630. (d) If the price volatility is based on a 99 per cent confidence limit and a mean historical change in daily yields of 0.0 per cent, what is the implied standard deviation of daily yield changes? The potential adverse move in yields = confidence limit value × standard deviation value. Therefore, 25 basis points = 2.33 × , and = 0.0025/2.33 = 0.001073 or 10.73 basis points. 10 Bank Two has a portfolio of bonds with a market value of $200 million. The bonds have an estimated price volatility of 0.95 per cent. What are the DEAR and the 10-day VaR for these bonds? Daily earnings at risk (DEAR) = ($ value of position) × (Price volatility) = $200 million × 0.0095 = $1.9 million or $1 900 000 Value at risk (VaR) = DEAR × N = $1 900 000 × 10 = $1 900 000 × 3.1623 = $6 008 327.55 11 Bank of Southern Tasmania has determined that its inventory of 20 million euros (€) and 25 million UK pounds (£) is subject to market risk. The spot exchange rates are $1.25/€ and $1.60/£, respectively. The ’s of the spot exchange rates of the euro and the pound, based on the daily changes of spot rates over the past six months, are 65 bp and 45 bp, respectively. Determine the bank’s 10-day VaR for both currencies. Use adverse rate changes in the 99th percentile. FX position of € = €20m × 1.25 = $24 million FX position of £ = £25m × 1.60 = $40 million FX volatility € = 2.33 × 65bp = 151.45bp, or 1.5145% FX volatility £ = 2.33 × 45bp = 104.85bp, or 1.0485% DEAR = ($ value of position) × (Price volatility) DEAR of € = $24m × .015145 = $348 941 DEAR of £ = $40m × .010485 = $419 400 10-day VaR of € = $348 941 × 10 = $348 841 × 3.1623 = $1 103 448 10-day VaR of £ = $419 400 × 10 = $419 400 × 3.1623 = $1 326 259 12 Bank of Ayers Rock’s stock portfolio has a market value of $10 000 000. The beta of the portfolio approximates the market portfolio, whose standard deviation (m) has been estimated at 1.5 per cent. What is the 5-day VaR of this portfolio, using adverse rate changes in the 99th percentile? DEAR = ($ value of portfolio) × (2.33 × m) = $10m × (2.33 × 0.015) = $10m × 0.03495 = $0.3495m or $349 500 VaR = $349 500 × 5 = $349 500 × 2.2361 = $781 505.76 David Small, risk manager for Choice Bank, is estimating the aggregate DEAR of the bank’s portfolio of assets consisting of loans (L), foreign currencies (FX), and ordinary shares (EQ). The individual DEARs are $300 700, $274 000 and $126 700, respectively. If the correlation coefficients (ij) between L and FX, L and EQ, and FX and EQ are 0.3, 0.7 and 0.0, respectively, what is the DEAR of the aggregate portfolio? 14 Calculate the DEAR for the following portfolio with and without the correlation coefficients.

Assets Estimated DEAR S,FX S,B FX,B

Shares (S) $300 000 –0.10 0.75 0.20

Foreign exchange (FX) $200 000

Bonds (B) $250 000

What is the amount of risk reduction resulting from the lack of perfect positive correlation between the various asset groups? The DEAR for a portfolio with perfect correlation would be $750 000. Therefore, the risk reduction is $750 000 – $559 464 = $190 536. 15 What are the advantages of using the back simulation approach to estimate market risk? Explain how this approach would be implemented. The advantages of the back simulation approach to estimating market risk are that (a) it is a simple process, (b) it does not require that asset returns be normally distributed, and (c) it does not require the calculation of correlations or standard deviations of returns. Implementation requires the calculation of the value of the current portfolio of assets based on the prices or yields that were in place on each of the preceding 500 days (or some large sample of days). These data are rank-ordered from worst case to best and percentile limits are determined. For example, the 5 per cent worst case provides an estimate with 95 per cent confidence that the value of the portfolio will not fall more than this amount. 16 Export Bank has a trading position in Japanese yen and Swiss francs. At the close of business on 4 February, the bank had ¥300 000 000 and SF10 000 000. The exchange rates for the most recent six days are given below: Exchange rates per dollar at the close of business

4 Feb 3 Feb 2 Feb 1 Feb 29 Jan 28 Jan

Japanese yen 112.13 112.84 112.14 115.05 116.35 116.32

Swiss francs 1.4140 1.4175 1.4133 1.4217 1.4157 1.4123

(a) What is the foreign exchange (FX) position in dollar equivalents using the FX rates on 4 February? Japanese yen: ¥300 000 000/¥112.13 = $2 675 465.98 Swiss francs: SF10 000 000/SF1.414 = $7 072 135.78 (b) What is the definition of delta as it relates to the FX position? Delta measures the change in the dollar value of each FX position if the foreign currency depreciates by 1 per cent against the dollar. (c) What is the sensitivity of each FX position; that is, what is the value of delta for each currency on 4 February? Japanese yen: 1.01 × current exchange rate = 1.01 × ¥112.13 = ¥113.2513/$ Revalued position in $s = ¥300 000 000/113.2513 = $2 648 976.21 Delta of $ position to Yen = $2 648 976.21 – $2 675 465.98 = –$26 489.77 Swiss francs: 1.01 × current exchange rate = 1.01 × SF1.414 = SF1.42814 Revalued position in $s = SF10 000 000/1.42814 = $7 002 114.64 Delta of $ position to SF = $7 002 114.64 – $7 072 135.78 = –$70 021.14 (d) What is the daily percentage change in exchange rates for each currency over the five-day period?

Day Japanese yen Swiss franc

4 Feb –0.62921% –0.24691% % Change = [(Ratet/Ratet-1) – 1] × 100

3 Feb 0.62422% 0.29718%

2 Feb –2.52934% –0.59084%

1 Feb –1.11732% 0.42382%

29 Jan 0.02579% 0.24074%

(e) What is the total risk faced by the bank on each day? What is the worst-case day? What is the best-case day?

Japanese yen Swiss francs Total risk

Day Delta % Rate Risk Delta % Rate Risk

4 Feb –$26 489.77 –0.6292% $166.68 –$70 021.14 –0.2469% $172.88 $339.56

3 Feb –$26 489.77 0.6242% –$165.35 –$70 021.14 0.2972% –$208.10 –$373.45

2 Feb –$26 489.77 –2.5293% $670.01 –$70 021.14 –0.5908% $413.68 $1083.69

1 Feb –$26 489.77 –1.1173% $295.97 –$70 021.14 0.4238% –$296.75 –$0.78

29 Jan –$26 489.77 0.0258% –$6.83 –$70 021.14 0.2407% –$168.54 –$175.37

The worst-case day is 3 February, and the best-case day is 2 February. (f) Assume that you have data for the 500 trading days preceding 4 February. Explain how you would identify the worst-case scenario with a 95 per cent degree of confidence? The appropriate procedure would be to repeat the process illustrated in part (e) for all 500 days. The 500 days would be ranked on the basis of total risk from the worst-case to the best-case. The fifth percentile from the absolute worst-case situation would be day 25 in the ranking. (g) Explain how the 5 per cent value at risk (VaR) position would be interpreted for business on 5 February. Management would expect with a confidence level of 95 per cent that the total risk on 5 February would be no worse than the total risk value for the 25th worst day in the previous 500 days. This value represents the VaR for the portfolio. (h) How would the simulation change at the end of the day on 5 February? What variables and/or processes in the analysis may change? What variables and/or processes will not change? The analysis can be upgraded at the end of each day. The values for delta may change for each of the assets in the analysis. As such, the value for VaR may also change. 17 What is the primary disadvantage to the back simulation approach in measuring market risk? What effect does the inclusion of more observation days have as a remedy for this disadvantage? What other remedies can be used to deal with the disadvantage? The primary disadvantage of the back simulation approach is the confidence level contained in the number of days over which the analysis is performed. Further, all observation days typically are given equal weight, a treatment that may not reflect accurately changes in markets. As a result, the VaR number may be biased upward or downward depending on how markets are trending. Possible adjustments to the analysis would be to give more weight to more recent observations, or to use Monte Carlo simulation techniques. 18 How is Monte Carlo simulation useful in addressing the disadvantages of back simulation? What is the primary statistical assumption underlying its use? Monte Carlo simulation can be used to generate additional observations that more closely capture the statistical characteristics of recent experience. The generating process is based on the historical variance–covariance matrix of FX changes. The values in this matrix are multiplied by random numbers that produce results that pattern closely the actual observations of recent historic experience. 19. What is the difference between VaR and expected shortfall (ES) as measure of market risk? VaR corresponds to a specific point of loss on the probability distribution. It does not provide information about the potential size of the loss that exceeds it; that is, VaR completely ignores the patterns and the severity of the losses in the extreme tail. Thus, VaR gives only partial information about the extent of possible losses, particularly when probability distributions are non-normal. The drawbacks of VaR became painfully evident during the financial crisis as asset returns plummeted into the ‘fat tail’ region of non-normally shaped distributions. FI managers and regulators were forced to recognise that VaR projections of possible losses far underestimated actual losses on extreme bad days. Expected shortfall (ES), also referred to as conditional VaR and expected tail loss, is a measure of market risk that estimates the expected value of losses beyond a given confidence level; that is, it is the average of VaRs beyond a given confidence level. ES, which incorporates points to the left of VaR, is larger when the probability distribution exhibits fat tail losses. Accordingly, ES provides more information about possible market risk losses than VaR. For situations in which probability distributions exhibit fat tail losses, VaR may look relatively small, but ES may be very large. 20. Consider the following discrete probability distribution of payoffs for two securities, A and B, held in the trading portfolio of an FI: Probability A Probability B 50.00% $80m 50.00% $80m 49.00 60m 49.00 68m 1.00 –740m 0.40 –740m 0.60 –1 393m Which of the two securities will add more market risk to the FI’s trading portfolio according to the VaR and ES measures? The expected return on security A = 0.50($80m) + 0.49($60m) + 0.01(–$740m) = $62m The expected return on security B = 0.50($80m) + 0.49($68m) + 0.0040(–$740m) + 0.0060(–$1393m) = $62m For a 99% confidence level, VaRA = VaRB = –$740m For a 99% confidence level, ESA = –$740m, while ESB = 0.40(–$740m) + 0.60(–$1393m) = –$1131.8m While the VaR is identical for both securities, the ES finds that security B has the potential to subject the FI to much greater losses than security A. Specifically, if tomorrow is a bad day, VaR finds that there is a 1 per cent probability that the FI’s losses will exceed $740 million on either security. However, if tomorrow is a bad day, ES finds that there is a 1 per cent probability that the FI’s losses will exceed $740 million if security A is in its trading portfolio, but losses will exceed $1131.8 million if security B is in its trading portfolio. 21. Consider the following discrete probability distribution of payoffs for two securities, A and B, held in the trading portfolio of an FI: Probability (%) A Probability (%) B 55.00 $120m 55.00 $120m 44.00 95m 44.00 100m 1.00 –1 100m 0.30 –1 100m 0.70 –1 414m Which of the two securities will add more market risk to the FI’s trading portfolio according to the VaR and ES measures? The expected return on security A = 0.55($120m) + 0.44($95m) + 0.01(–$1100m) = $96.8m The expected return on security B = 0.55($120m) + 0.44($100m) + 0.0030(–$1100m) + 0.0070(–$1414m) = $96.8m For a 99% confidence level, VaRA = VaRB = –$1100m For a 99% confidence level, ESA = –$1100m, while ESB = 0.30(–$1100m) + 0.70(–$1414m) = –$1319.8m Thus, while the VaR is identical for both securities, the ES finds that security B has the potential to subject the FI to much greater losses than security A. Specifically, if tomorrow is a bad day, VaR finds that there is a 1 per cent probability that the FI’s losses will exceed $1100 million on either security. However, if tomorrow is a bad day, ES finds that there is a 1 per cent probability that the FI’s losses will exceed $1100 million if security A is in its trading portfolio, but losses will exceed $1319.8m if security B is in its trading portfolio. 22. An FI has ₤5 million in its trading portfolio on the close of business on a particular day. The current exchange rate of pounds for dollars is ₤0.6400/$, or dollars for pounds is $1.5625, at the daily close. The volatility, or standard deviation (σ), of daily percentage changes in the spot ₤/$ exchange rate over the past year was 58.5 bp. The FI is interested in adverse moves—bad moves that will not occur more than 1 per cent of the time, or 1 day in every 100. Calculate the one-day VaR and ES from this position. The first step is to calculate the dollar value position: Dollar value of position = pound value of position x dollar for pound exchange rate = ₤5 million × 1.5625 = $7 812 500 Using VaR, which assumes that changes in exchange rates are normally distributed, the exchange rate must change in the adverse direction by 2.33σ (2.33 × 58.5 bp) for this change to be viewed as likely to occur only 1 day in every 100 days: FX volatility = 2.33 × 58.5 bp = 136.305 bp In other words, using VaR during the last year the pound declined in value against the dollar by 136.305 bp 1 per cent of the time. As a result, the one-day VaR is: VaR = $7 812 500 × 0.0136305 = $106 488 Using ES, which assumes that changes in exchange rates are normally distributed but with fat tails, the exchange rate must change in the adverse direction by 2.665σ (2.665 × 58.5 bp) for this change to be viewed as likely to occur only 1 day in every 100 days: FX volatility = 2.665 × 58.5 bp = 155.9025 bp In other words, using ES during the last year the pound declined in value against the dollar by 155.9025 bp 1 per cent of the time. As a result, the one-day ES is: ES = $7 812 500 × 0.01559025 = $121 799 The potential loss exposure to adverse pound to dollar exchange rate changes for the FI from the ₤5 million spot currency holdings are higher using the ES measure of market risk. ES estimates potential losses that are $15 311 higher than VaR. This is because VaR focuses on the location of the extreme tail of the probability distribution. ES also considers the shape of the probability distribution once VaR is exceeded. 23. An FI has ¥500 million in its trading portfolio on the close of business on a particular day. The current exchange rate of yen for dollars is ¥80.00/$, or dollars for yen is $0.0125, at the daily close. The volatility, or standard deviation (σ), of daily percentage changes in the spot ¥/$ exchange rate over the past year was 121.6 bp. The FI is interested in adverse moves—bad moves that will not occur more than 1 per cent of the time, or 1 day in every 100. Calculate the one-day VaR and ES from this position. The first step is to calculate the dollar value position: Dollar value of position = yen value of position × dollar for pound exchange rate = ¥500 million × 0.0125 = $6 250 000 Using VaR, which assumes that changes in exchange rates are normally distributed, the exchange rate must change in the adverse direction by 2.33σ (2.33 × 121.6 bp) for this change to be viewed as likely to occur only 1 day in every 100 days: FX volatility = 2.33 × 121.6 bp = 283.328 bp In other words, using VaR during the last year the yen declined in value against the dollar by 283.328 bp 1 per cent of the time. As a result, the one-day VaR is: VaR = $6 250 000 × 0.0283328 = $177 080 Using ES, which assumes that changes in exchange rates are normally distributed but with fat tails, the exchange rate must change in the adverse direction by 2.665σ (2.665 × 121.6 bp) for this change to be viewed as likely to occur only 1 day in every 100 days: FX volatility = 2.665 × 121.6 bp = 324.064 bp In other words, using ES during the last year the yen declined in value against the dollar by 324.064 bp 1 per cent of the time. As a result, the one-day ES is: ES = $6 250 000 × 0.0324064 = $202 540 The potential loss exposure to adverse yen to dollar exchange rate changes for the FI from the ¥500 million spot currency holdings are higher using the ES measure of market risk. ES estimates potential losses that are $25 460 higher than VaR. This is because VaR focuses on the location of the extreme tail of the probability distribution. ES also considers the shape of the probability distribution once VaR is exceeded. 24. The Bank of Canberra’s stock portfolio has a market value of $250 million. The beta of the portfolio approximates the market portfolio, whose standard deviation (m) has been estimated at 2.25 per cent. What are the five-day VaR and ES of this portfolio using adverse rate changes in the 99th percentile? Daily VaR = ($ value of portfolio) × (2.33 × m ) = $250m × (2.33 × 0.0225) = $250m × 0.052425 = $13 106 250 5-day VaR = $13 106 250 × 5 = $13 106 250 × 2.2361 = $29 306 466 Daily ES = ($ value of portfolio) × (2.665 × m ) = $250m × (2.665 × 0.0225) = $250m × 0.0599625 = $14 990 625 5-day ES = $14 990 625 × 5 = $14 990 625 × 2.2361 = $33 520 057 25. Despite the fact that market risk capital requirements have been imposed on FIs since the 1990s, huge losses in value were recorded from losses incurred in FIs’ trading portfolios internationally. Why did this happen? What changes to capital requirements did regulators propose to prevent such losses from reoccurring? During the GFC, losses due to market risk were significantly higher than the minimum market risk capital requirements under BIS Basel I and Basel II rules for many banks in many countries. The GFC exposed a number of shortcomings in the way market risk was being measured in accordance with Basel II rules. Although the crisis largely exposed problems with the large-bank internal models approach to measuring market risk, the BIS also identified shortcomings with the standardised approach. These included a lack of risk sensitivity, a very limited recognition of hedging and diversification benefits, and an inability to sufficiently capture risks associated with more complex instruments. As a result, in July 2009 the BIS announced Basel 2.5, a final version of revised rules for market risk capital requirements. To address shortcomings of the standardised approach to measuring market risk, Basel III proposes a ‘partial risk factor’ approach as a revised standardised approach. Basel III also introduces a ‘fuller risk factor’ approach as an alternative to the revised partial risk factor standardised approach. To address shortcomings in the internal models approach, in addition to the risk capital charge already in place, an incremental capital charge is assessed which includes a ‘stressed value at risk’ capital requirement taking into account a one-year observation period of significant financial stress relevant to the FI’s portfolio. The introduction of stressed VaR in Basel 2.5 is intended to reduce the cyclicality of the VaR measure and alleviate the problem of market stress periods dropping out of the data period used to calculate VaR after some time. Basel 2.5 requires the following process be followed by large FIs using internal models to calculate the market risk capital charge. Basel III proposes to replace VaR models with those based on extreme value theory and expected shortfall (ES) (discussed above). The ES measure analyses the size and likelihood of losses above the 99th percentile in a crisis period for a traded asset and thus measures ‘tail risk’ more precisely. Thus, ES is a risk measure that considers a more comprehensive set of potential outcomes than VaR. The BIS change to ES highlights the importance of maintaining sufficient regulatory capital not only in stable market conditions, but also in periods of significant financial stress. Indeed, it is precisely during periods of stress that capital is vital for absorbing losses and safeguarding the stability of the banking system. Accordingly, the Committee intends to move to a framework that is calibrated to a period of significant financial stress. Two methods of identifying the stress period and calculating capital requirements under the internal models are the direct method and the indirect method. The direct method is based on the approach used in the Basel 2.5 stressed VaR. The FI would search the entire historical period and identify the period which produces the highest ES result when all risk factors are included. However, Basel III would require the FI to determine the stressed period on the basis of a reduced set of risk factors. Once the FI has identified the stressed period, it must then determine the ES for the full set of risk factors for the stress period. The indirect method identifies the relevant historical period of stress by using a reduced set of risk factors. However, instead of calculating the full ES model to that period the FI calculates a loss based on the reduced set of risk factors. This loss is then scaled using the ratio of the full ES model using current market data to the full ES model using the reduced set of risk factors using current market data. 26. In its trading portfolio, an FI holds 10 000 BHP Billiton (BHP) shares at a share price of $86.50 and has sold 5000 Woolworths (WOW) shares under a forward contract that matures in one year. The current share price for WOW is $20.50. The shift risk factor (i.e. standard deviation) for level I risk factor is 4 per cent, for level II risk factor is 6 per cent, for level III long positions is 9 per cent, for level III short positions is –9 per cent, and for non-hedgea ble risk is 1 per cent. Using the risk factors listed in Table 9.8, calculate the market risk capital charge on these securities. Step 1. Assign each instrument to applicable risk factors From Table 9.8, hedgea ble risk factors for these equities include level I worldwide equity index, level II equity index by broad industry category, and level III movements in the prices of individual equity. BHP and WOW have the same hedgea ble risk factors at levels I and II, that is, global and industry specific equity indices. However, movements in the prices of the two firms are unique. Thus, they do not have the same risk factor at level III and as a result they are mapped to different individual equity risk factors. There is also a non-hedgea ble risk factor for the GE equity price to capture basis risk from the forward contract. Step 2. Determine the size of the net risk position in each risk factor For each risk factor the FI determines a net risk position, calculated as the sum of gross risk positions for all instruments that are subject to that risk factor. The table below shows the gross and net positions for BHP and WOW equities for the equity risk factor. The size of the gross position in BHP for the three applicable risk factors is $865 000 (10 000 shares × $86.50) and for the short position in WOW is –$102 500 (5000 shares × $20.50). Note again that the two securities do not have the same risk factor at level III. Thus, they are mapped to different individual equity risk factors. Further, to capture basis risk from the forward contract, there is a non-hedgea ble risk factor for the WOW equity price, –$102 500. The net risk position of the two securities for each risk factor, listed in the last column of the table, is the sum of the gross risk factors for the securities at each level, that is, $762 500 for levels I and II, $865 000 and –$102 500, respectively, for level III, and –$102 500 for non-hedgea ble risk. BHP Gross WOW Gross Total size of Level Equity risk risk position risk position net risk position I Worldwide equity index $865 000 –$102 500 $762 500 II Industry equity index $865 000 –$102 500 $762 500 III BHP Billiton share price $865 000 - $865 000 Woolworths share price - –$102 500 –$102 500 N-h* Woolworths share price - –$102 500 –$102 500 Step 3. Aggregate overall risk position across risk factors The net risk positions are then converted into a capital charge by multiplying by regulator specified standard deviations (i.e. shift risk factors). The table below shows the calculations of the capital charge for market risk. The net risk positions (listed in column 3 for each risk level) are multiplied by the standard deviations assigned for each level (column 4) to produce the standard deviations of the net risk position. For example, the standard deviation of the net risk position for the level I worldwide equity index is equal to the net risk ($762 500) times the regulator set shift risk factor (4 per cent) to give the standard deviation associated with level I risk factor, $30 500). The square of the standard deviation (the variance) is then listed in column 5 (i.e. $930 250 000 for level I). Summing the squared standard deviations gives the portfolio variance ($9 170 086 250) and taking the square root of this gives the portfolio standard deviation ($95 761). Finally, this portfolio standard deviation is multiplied by a scalar (currently set at 4) to achieve the overall expected shortfall for the portfolio ($383 042).

Level Equity Risk Net Risk Position (EUR) Standard Deviation (i.e. shift of risk factor) Standard Deviation of Net Risk Position Square the Standard Deviation of the Net Risk Position (i.e. variance)

I Worldwide equity index $762 500 4% $30,500 $930,250,000 4% $30 500 $930 250 000

II Industry equity index $762 500 6% $45 750 $2 093 062 500

III BHP Billiton share price $865 000 9% $77 850 $6 060 622 500

Woolworths share price –$102 500 –9% $9 225 $85 100 625

N-h* Woolworths share price –$102 500 1% $1 025 $1 050 625

Portfolio Sum the squared standard deviations (portfolio variance) €$9 170 086 250

Portfolio Take the square root (portfolio standard deviation) $95 761

Portfolio Multiply by scalar to obtain expected shortfall $383 042

27. Suppose an FI’s portfolio VaR for the previous 60 days was $3 million and stressed VaR for the previous 60 days was $8 million using the 1 per cent worst case (or 99th percentile). Calculate the minimum capital charge for market risk for this FI. Capital charge = ($3 million × √10 × 3) + ($8 million × √10 x 3) = $104.355 million Web question 28 Go to the websites of each of the four largest Australian banks. From their latest annual report, find their approaches to market risk measurement and management. Compare the methodologies. The answer will depend on the date of the assignment. For example, at the Westpac Bank website: www.westpac.com.au , go to ‘Westpac info’, and then click on ‘Investor Centre’, where you will find references to the latest annual report. Click on the full annual report and look up interest rate sensitivity report. Then go to each of the other major banks’ websites: National Australia Bank: www.nab.com.au; Commonwealth Bank: www.cba.com.au; and ANZ Bank: www.anz.com.au; and follow similar steps to those set out for Westpac. 1. Value at Risk (VaR): • VaR is a widely used measure to assess the potential loss in value of a portfolio due to adverse market movements over a specified time horizon and confidence level. • Banks calculate VaR by estimating the potential loss in the value of their trading portfolios based on historical market data and statistical models. 2. Stress Testing: • Stress testing involves assessing the impact of extreme and adverse scenarios on a bank's portfolio, including changes in interest rates, exchange rates, and asset prices. • Banks conduct stress tests to evaluate their resilience to severe market shocks and identify potential vulnerabilities in their risk exposure. 3. Scenario Analysis: • Scenario analysis involves simulating various hypothetical scenarios to assess the impact on a bank's portfolio and earnings under different market conditions. • Banks analyze the sensitivity of their portfolios to changes in key risk factors such as interest rates, credit spreads, and market volatility. 4. Value Adjustments (XVA): • XVA represents a family of adjustments made to derivative contracts and other financial instruments to account for counterparty credit risk, funding costs, and capital requirements. • Banks calculate XVAs to measure the potential impact of market risk factors on their derivative portfolios and assess the overall risk-adjusted profitability of their trading activities. 5. Risk Limits and Controls: • Banks establish risk limits and controls to manage and mitigate market risk exposure effectively. • Risk limits define the maximum allowable exposure to specific market risk factors, such as interest rate risk, foreign exchange risk, and commodity price risk. 6. Portfolio Diversification: • Banks diversify their trading portfolios across different asset classes, geographic regions, and industries to reduce concentration risk and enhance risk-adjusted returns. • Portfolio diversification helps mitigate the impact of adverse market movements on the bank's overall performance. While these are general methodologies commonly used by banks for market risk measurement and management, the specific approaches adopted by each of the four largest Australian banks may vary based on their risk appetite, business model, and regulatory requirements. For the latest and most accurate information on their approaches to market risk measurement and management, it's recommended to refer to the annual reports and disclosures of each bank. Integrated mini case: calculating DEAR on an FI’s trading portfolio An FI wants to obtain the DEAR on its trading portfolio. The portfolio consists of the following securities. Fixed-income securities: The FI has a $1 million position in a six-year zero-coupon bonds with a face value of $1 543 302. The bond is trading at a yield to maturity of 7.50 per cent. The historical mean change in daily yields is 0.0 per cent, and the standard deviation is 22 basis points. The FI also holds a 12-year zero bond with a face value of $1 000 000. The bond is trading at a yield to maturity of 6.75 per cent. The price volatility of the potential adverse move in yields is 65 basis points. Foreign exchange contracts: The FI has a €2.0 million long trading position in spot euros at the close of business on a particular day. The exchange rate is €0.80/$1, or $1.25/€, at the daily close. Looking back at the daily changes in the exchange rate of the euro to dollars for the past year, the FI finds that the volatility or standard deviation (σ) of the spot exchange rate was 55.5 basis points (bp). Equities: The FI holds a $2.5 million trading position in stocks that reflect the Australian stock market index (e.g. All Ordinaries). The β = 1. Over the last year, the standard deviation of the stock market index was 175 basis points. Correlations (ρij) among Assets Six-yr zero-coupon 12-yr zero-coupon €/$ Stock market index Six year, zero-coupon - 0.75 –0.2 0.40 12-year, zero-coupon - - –0.3 0.45 €/$ - - - 0.25 Stock Index - - - - Calculate the DEAR of this trading portfolio. Solution: Fixed-income securities: 1. MD = D/(1 + R) = 6/(1.075) = 5.581395 => Potential adverse move in yield at 5 per cent = 1.65 = 1.65 × 0.0022 = .00363 => Price volatility = MD x potential adverse move in yield = 5.581395 × 0.00363 = 0.02026 or 2.026 per cent and the daily earnings at risk for this bond is: DEAR = ($ value of position) × (price volatility) = $1 000 000 × 0.02026 = $20 260 2. Dollar value of position = $1m./(1 + 0.0675)12 = $456 652. The modified duration of these bonds is: MD = D/(1 + R) = 12/(1.0675) = 11.24122 => Price volatility = (MD) × (potential adverse move in yield) = (11.24122) × (0.0065) = 0.073068 or 7.3068 per cent. => DEAR = $456 652 × 0.073068 = $33 367 Foreign exchange contracts: Dollar equivalent value of € position = FX position × ($/€ spot exchange rate) = €3.5 million × $ per unit of foreign currency Dollar value of € position = €2.0 million × $1.25/€ = $2 500 000 FX volatility = 1.65 × 55.5 bp = 91.575 bp or 0.91575% => DEAR = Dollar value of DM position × FX volatility = $2 500 000 × 0.0091575 = $22 894 Equities: Stock market return volatility = 1.65 σm = 1.65 × 175 bp = 0.28875 = 2.8875% => DEAR = Dollar market value of position x Stock market return volatility = $2 500 000 × 0.28875 = $72 187 Portfolio DEAR: Using the correlation matrix along with the individual asset DEARs, the risk (or standard deviation) of the whole (four asset) trading portfolio is: DEAR portfolio = [(DEARz6)2 + (DEARz12)2 + (DEAR€)2 + (DEARUS)2 + (2 × ρz6,z12 × DEARz6 × DEARz12) + (2 × ρz6€ × DEARz6 × DEAR€) + (2 × ρz6,US × DEARz6 × DEARUS) + (2 × ρz12€ × DEARz12 × DEAR€) + (2 × ρz12,US × DEARz12 × DEARUS) + (2 × ρ€,US × DEAR€ × DEARUS)]2 = [(20 260)2 + (33 367)2 + (22 894)2 + (72 187)2 + 2(0.75)(20 260)(33 367) + 2(–0.2)(20 260)(22 894) + 2(0.4)(20 260)(72 187) + 2(–0.3)(33 367)(22 894) + 2(0.45)(33 367)(72 187) + 2(0.25)(22 894)(72 187)]2 = $108 597 Solution Manual for Financial Institutions Management Anthony Saunders, Marcia Cornett, Patricia McGraw 9780070979796, 9780071051590

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