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1、Chapter TenMarket RiskChapter OutlineIntroductionMarket Risk MeasurementCalculating Market Risk ExposureThe RiskMetrics Model The Market Risk of Fixed-Income Securities Foreign Exchange Equities Portfolio AggregationHistoric or Back Simulation The Historic (Back Simulation) Model versus RiskMetrics
2、The Monte Carlo Simulation ApproachRegulatory Models: The BIS Standardized Framework Fixed Income Foreign Exchange EquitiesThe BIS Regulations and Large Bank Internal ModelsSummarySolutions for End-of-Chapter Questions and Problems: Chapter Ten1.What is meant by market risk?Market risk is the uncert
3、ainty 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 marke
4、t risk important to the manager of a financial institution?Measurement of market risk can help an FI manager in the following ways:a. Provide information on the risk positions taken by individual traders.b. Establish limit positions on each trader based on the market risk of their portfolios.c.Help
5、allocate resources to departments with lower market risks and appropriate returns.d.Evaluate performance based on risks undertaken by traders in determining optimal bonuses.e.Help develop more efficient internal models so as to avoid using standardized regulatory models.3.What is meant by daily earn
6、ings at risk (DEAR)? What are the three measurable components? What is the price volatility component?DEAR or Daily Earnings at Risk is defined as the estimated potential loss of a portfolios value over a one-day unwind period as a result of adverse moves in market conditions, such as changes in int
7、erest 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
8、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 $1,402,552. The bond is trading at a yield to maturity of 7.00 percent. The historical mean change in daily yields is 0.0 percent, and the standard devia
9、tion is 12 basis points.a.What is the modified duration of the bond?MD = 5 (1.07) = 4.6729 yearsb.What is the maximum adverse daily yield move given that we desire no more than a 5 percent chance that yield changes will be greater than this maximum?Potential adverse move in yield at 5 percent = 1.65
10、s = 1.65 x 0.0012 = .001980c.What is the price volatility of this bond?Price volatility = -MD x potential adverse move in yield = -4.6729 x .00198 = -0.009252 or -0.9252 percentd.What is the daily earnings at risk for this bond?DEAR = ($ value of position) x (price volatility) = $1,000,000 x 0.00925
11、2 = $9,2525.What is meant by value at risk (VAR)? How is VAR related to DEAR in J.P. Morgans RiskMetrics model? What would be the VAR for the bond in problem (4) for a 10-day period? With what statistical assumption is our analysis taking liberties? Could this treatment be critical?Value at Risk or
12、VAR is the cumulative DEARs over a specified period of time and is given by the formula VAR = DEAR x N. VAR is a more realistic measure if it requires a longer period to unwind a position, that is, if markets are less liquid. The value for VAR in problem four above is $9,252 x 3.1623 = $29,257.39.Th
13、e relationship according to the above formula assumes that the yield changes are independent. This means that losses incurred on one day are not related to the losses incurred the next day. However, recent studies have indicated that this is not the case, but that shocks are autocorrelated in many m
14、arkets over long periods of time.6.The DEAR for a bank is $8,500. 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 = 8,500 x 10 = 8,500 x 3.1623 = $26,879.36 For the 20-day period: VAR =
15、8,500 x 20 = 8,500 x 4.4721 = $38,013.16 The reason that VAR20 (2 x VAR10) is because 20 (2 x 10). 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 five
16、 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 90 percent confidence limit is required; that is, adverse moves will not occur more than one day in 20?If yi
17、eld changes are normally distributed, 90 percent of the area of a normal distribution will be 1.65 standard deviations (1.65s) from the mean for a one-tailed distribution. In this example, it means 1.65 x 15 = 24.75 bp. Thus, the maximum adverse yield change expected for this zero-coupon bond is an
18、increase of 24.75 basis points in interest rates.b.What is the highest yield change expected if a 95 percent confidence limit is required?If a 95 percent confidence limit is required, then 95 percent of the area will be 1.96 standard deviations (1.96s) from the mean. Thus, the maximum adverse yield
19、change expected for this zero-coupon bond is an increase of 29.40 basis points (1.96 x 15) in interest rates.8.In what sense is duration a measure of market risk?The market risk calculations typically are based on the trading portion of an FIs fixed-rate asset portfolio because these assets must ref
20、lect 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 wi
21、th a face value of $400 million. The bonds currently are yielding 9.5% in the over-the-counter market.a.What is the modified duration of these bonds?Modified duration = (MD) = D/(1 + r) = 15/(1.095) = -13.6986.b.What is the price volatility if the potential adverse move in yields is 25 basis points?
22、Price volatility = (-MD) x (potential adverse move in yield)= (-13.6986) x (.0025) = -0.03425 or -3.425 percent.c.What is the DEAR?Daily earnings at risk (DEAR) = ($ Value of position) x (Price volatility)Dollar value of position = 400/(1 + 0.095)15 = $102.5293million. Therefore,DEAR = $102.5293499
23、million x -0.03425 = -$3.5116 million, or -$3,511,630. d.If the price volatility is based on a 90 percent confidence limit and a mean historical change in daily yields of 0.0 percent, what is the implied standard deviation of daily yield changes?The potential adverse move in yields (PAMY) = confiden
24、ce limit value x standard deviation value. Therefore, 25 basis points = 1.65 x s, and s = .0025/1.65 = .001515 or 15.15 basis points.10.Bank Two has a portfolio of bonds with a market value of $200 million. The bonds have an estimated price volatility of 0.95 percent. What are the DEAR and the 10-da
25、y VAR for these bonds?Daily earnings at risk (DEAR)= ($ Value of position) x (Price volatility) = $200 million x .0095 = $1.9million, or $1,900,000Value at risk (VAR) = DEAR x N = $1,900,000 x 10= $1,900,000 x 3.1623 = $6,008,327.5511.Bank of Southern Vermont has determined that its inventory of 20
26、million euros () and 25 million British pounds () is subject to market risk. The spot exchange rates are $0.40/ and $1.28/, respectively. The ss of the spot exchange rates of the and , based on the daily changes of spot rates over the past six months, are 65 bp and 45 bp, respectively. Determine the
27、 banks 10-day VAR for both currencies. Use adverse rate changes in the 95th percentile.FX position of = 20m x 0.40 = $8 millionFX position of = 25m x 1.28 = $32 millionFX volatility = 1.65 x 65bp = 107.25, or 1.0725%FX volatility = 1.65 x 45bp = 74.25, or 0.7425%DEAR = ($ Value of position) x (Price
28、 volatility)DEAR of = $8m x .010725 = $0.0860m, or $85,800DEAR of = $32m x .007425 = $0.2376m, or $237,600VAR of = $138,000 x 10 = $85,800 x 3.1623 = $271,323.42VAR of = $237,600 x 10 = $237,600 x 3.1623 = $751,357.1712.Bank of Alaskas stock portfolio has a market value of $10,000,000. The beta of t
29、he portfolio approximates the market portfolio, whose standard deviation (sm) has been estimated at 1.5 percent. What is the 5-day VAR of this portfolio, using adverse rate changes in the 99th percentile?DEAR= ($ Value of portfolio) x (2.33 x sm ) = $10m x (2.33 x .015)= $10m x .03495 = $0.3495m or
30、$349,500VAR= $349,500 x 5 = $349,500 x 2.2361 = $781,505.7613. Jeff Resnick, vice president of operations of Choice Bank, is estimating the aggregate DEAR of the banks portfolio of assets consisting of loans (L), foreign currencies (FX), and common stock (EQ). The individual DEARs are $300,700, $274
31、,000, and $126,700 respectively. If the correlation coefficients rij 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.EstimatedAss
32、ets DEAR rS,FXrS,BrFX,BStocks (S)$300,000-0.100.750.20Foreign Exchange (FX)$200,000Bonds (B)$250,000What is the amount of risk reduction resulting from the lack of perfect positive correlation between the various assets groups?The DEAR for a portfolio with perfect correlation would be $750,000. Ther
33、efore 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 pro
34、cess, (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
35、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 five percent worst case provides an estimate with 95 percent confidence that the value of the portfolio will not fall mor
36、e than this amount.16.Export Bank has a trading position in Japanese Yen and Swiss Francs. At the close of business on February 4, 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 U.S. Dollar at the Close of Business 2/4 2/
37、3 2/2 2/1 1/29 1/28Japanese Yen112.13112.84112.14115.05116.35116.32Swiss Francs1.41401.41751.41331.42171.41571.4123a.What is the foreign exchange (FX) position in dollar equivalents using the FX rates on February 4?Japanese Yen:300,000,000/112.13 = $2,675,465.98Swiss Francs:Swf10,000,000/Swf1.414 =
38、$7,072,135.78b.What is the definition of delta as it relates to the FX position?Delta measures the change in the dollar value of each FX position if the foreign currency depreciates by 1 percent against the dollar.c.What is the sensitivity of each FX position; that is, what is the value of delta for
39、 each currency on February 4?Japanese Yen:1.01 x current exchange rate = 1.01 x 112.13 = 113.2513/$Revalued position in $s= 300,000,000/113.2513 = $2,648,976.21Delta of $ position to Yen= $2,648,976.21 - $2,675,465.98 = -$26,489.77Swiss Francs:1.01 x current exchange rate = 1.01 x Swf1.414 = Swf1.42
40、814Revalued position in $s= Swf10,000,000/1.42814 = $7,002,114.64Delta of $ position to Swf= $7,002,114.64 - $7,072,135.78= -$70,021.14d.What is the daily percentage change in exchange rates for each currency over the five-day period?DayJapanese Yen:Swiss Franc2/4-0.62921%-0.24691% Change = (Ratet/R
41、atet-1) - 1 * 1002/30.62422%0.29718%2/2-2.52934%-0.59084%2/1-1.11732%0.42382%1/290.02579%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 FrancsTotalDayDelta% Rate DRiskDelta% Rate DRiskRisk2/4-$26,489.77-0.6292
42、%$166.68-$70,021.14-0.2469%$172.88$339.562/3-$26,489.770.6242%-$165.35-$70,021.140.2972%-$208.10-$373.452/2-$26,489.77-2.5293%$670.01-$70,021.14-0.5908%$413.68$1,083.692/1-$26,489.77-1.1173%$295.97-$70,021.140.4238%-$296.75-$0.781/29-$26,489.770.0258%-$6.83-$70,021.140.2407%-$168.54-$175.37The worst
43、-case day is February 3, and the best-case day is February 2.f.Assume that you have data for the 500 trading days preceding February 4. Explain how you would identify the worst-case scenario with a 95 percent degree of confidence?The appropriate procedure would be to repeat the process illustrated i
44、n part (e) above 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 five percent value at risk (VAR) position would be interprete
45、d for business on February 5.Management would expect with a confidence level of 95 percent that the total risk on February 5 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
46、 at the end of the day on February 5? 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 the 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 affect does the inclusion of more observation days have as a re