Sheet1Value At Risk Models(Example from Anthony Saunders, "The VAR Approach",in Credit Risk Measurement, New York: John Wiley and Sons, 1999, pp. 37-57)2002P. LeBelValue at Risk (VAR) models measure the maximum loss (in value) on a given asset or liability over a giventime period at a given confidence level (e.g., 95%, 97.5%, 99%). Most VAR models, like those based onearnings at risk, rely on an underlying distribution of returns to generate explicit potential loss values.Our initial focus is on single asset VAR models.Example:What is the value at risk(VAR) of a1.00%percent loss in equity? First, derive the correspondingnormal standard deviation associated with a given probability. Then multiply the correspondingnormalized standard deviation times the standard deviation of an asset and subtract this productfrom the current asset value (e.g., current equity value).$80.00Current equity value1.00%Query Standard Probability (QSP)$10.00Standard Deviation2.33Query Standard Deviation (QSD)Adjusted Value =$56.74= ($80.00-(2.33)x($10.00)VAR =$23.26= ($80.00-$56.74)The lower the query standard probability, the higher will be the query standard deviation, and the higher the VAR.Table 1Query Standard Probability, Query Standard Deviation, and Value at RiskQSPQSDVAR1.00%2.33$23.262.00%2.05$20.543.00%1.88$18.814.00%1.75$17.515.00%1.64$16.456.00%1.55$15.557.00%1.48$14.768.00%1.41$14.059.00%1.34$13.4110.00%1.28$12.82QSP =Query Standard ProbabilityQSD =Query Standad DeviationVAR =Value at RiskAssessing VAR models:1.The current market value of a loan is not directly observable because most loans are not traded.Possible solution: create a secondary market in loans.2.If the current market value of a loan is not directly observable, there is no time series on whichto calculate the standard deviation of its value.3.As with other models, the normality assumption of returns creates potential bias in small samples.Loans tend to have highly truncated upside returns and long downside risks., i.e., they are asymmetric.Existing approaches involve compiling data on a borrower's credit rating (credit scoring), on theprobability that a rating will change in the next year (a rating transition probability matrix), recoveryrates on defaulted loans, and credit spreads and yields in the bond (or loan) market.The CreditMetrics Approach to Loan Valuation RevisionsCreditMetrics, a public research arm of J.P. Morgan, has developed an approach to VARin debt and equity markets. We consider here a debt example.Table 2Bond Categories, Interest Rates, and Ratings Transition Probabilitiesfor a Hypothetical Benchmark BBB BondRatingInterest RateTrans.Prob.AAA5.00%0.20%AA5.25%0.40%A5.50%5.00%BBB6.00%85.00%BB6.50%6.00%B7.00%1.50%CCC8.00%1.20%Default9.50%0.70%Total100.00%Source: Adapted from CreditMetrics-Technical Document, J.P. Morgan, April 2, 1997, p. 11Transition probabilities illustrate the likelihood of a benchmark bond moving from its base level to eitheran upgrade or a downgrade. These transition probabilities are derived by industry analysts based oncredit scoring systems, the level and volatility of earnings over time, and, where available, observedchanges in volatility of a firm's equity capital share prices. Changes in ratings translate into the required creditrisk spreads or premiums on a loan's remaining cash flows, and thus, on the implied market (or present)value of a loan. If a loan is downgraded, the required credit spread premium should rise, while an upgradeproduces the opposite effect.We now illustrate how bond ratings are used in conjunction with the term structure of interest rates toderive the corresponding VAR. Consider a relatively riskless asset such as a T-bond. In the absence ofrisk, for various time horizons, the yield curve portrays the underlying rate of discount. Usually the yieldcurve will be upward sloping, although inverted yield curves can occur in the presence of expected declinesin interest rates that are reflective of current and evolving economic conditions.Once one introduces risky assets such as bank loans and commercial bonds, one expects that for anygiven time horizon, the corresponding yield will be higher. For our present purposes, we will assumea monotonic relationship, I.e., for a given time horizon and a given bond category, the term interest ratewill be proportionately higher by the ratio of the corresponding T-bond rate to the period 0 rate.Consider the term structure of interest rates in Table 3. If T-Bonds have a current rate of 3 percent anda year-1 rate of 3.72 percent, we derive the year-1 rate for AAA bonds as 1.0891. We do this as follows:1.0891 =(1.03721.0000)x1.0500Of course, for actual gradesof bonds, the spread of yields will vary according to market conditions and perceived levels of risk.Table 3The Term Structure of Interest RatesRateCategory012340.0300T-Bonds1.00001.03721.04321.04931.05320.0500AAA Bonds1.00001.08911.09541.10181.10590.0525AA Bonds1.00001.09171.09801.10441.10850.0550A Bonds1.00001.09421.10061.10701.11110.0600BBB Bonds1.00001.09941.10581.11231.11640.0650BB Bonds1.00001.10461.11101.11751.12170.0700B Bonds1.00001.10981.11621.12281.12690.0800CCC Bonds1.00001.12021.12671.13321.13750.0950Default Bonds1.00001.13571.14231.14901.1533Let us now examine the effect of different bond ratings on the present value of a remaining loan.Suppose now that we have the following information on a benchmark bond whose current value, coupon rate,and interest rate are given below. The coupon payment is the same, whose stream and end-term principalwe discount using the term structure of interest rates as present worth factors.Example:B =$100.00Existing value of a loani =0.06Current interest rateC =$6.00Current coupon paymentt =5Remaining years on loanUsing the term structure of interest profile in Table 3 above, we derive the corresponding present value ofbond assets using the term structure of rates as present worth factors (PWF):Table 4Present Value of a T-Bond01234Coupon (+ end Principal)$6.00$6.00$6.00$6.00$106.00PWF1.00001.03721.04321.04931.0532Annual Present Values$6.00$5.78$5.75$5.72$100.65Asset Present Value (PV) =$123.90From Table 3, we now make the same calculations for each category of commercial debt.Table 5Present Values of Various Grades of Benchmark Commercial Debt01234PVT-Bonds$6.00$5.78$5.75$5.72$100.65$123.90AAA$6.00$5.51$5.48$5.45$95.85$118.29AA$6.00$5.50$5.46$5.43$95.63$118.02A$6.00$5.48$5.45$5.42$95.40$117.75BBB$6.00$5.46$5.43$5.39$94.95$117.23BB$6.00$5.43$5.40$5.37$94.50$116.70B$6.00$5.41$5.38$5.34$94.06$116.19CCC$6.00$5.36$5.33$5.29$93.19$115.17Default$6.00$5.28$5.25$5.22$91.91$113.67The present values (PV) in the right-hand column reflect the effect of differential interest rates of variousbond rating categories.Let us now integrate the role of interest rates and bond ratings within a VAR framework. To do so, wegenerate a probability distribution of present values, whose standard deviation is then used to calculatethe corresponding VAR.Table 6VAR Calculations for a Benchmark LoanABCDE= AxB=B-Mean PV=(D)^2RatingTrans.Prob.PV LoanProb.PVDifferenceDifference^2AAA0.20%$118.29$0.24$1.12$1.26AA0.40%$118.02$0.47$0.86$0.74A5.00%$117.75$5.89$0.59$0.35BBB85.00%$117.23$99.64$0.06$0.00BB6.00%$116.70$7.00-$0.46$0.21B1.50%$116.19$1.74-$0.97$0.95CCC1.20%$115.17$1.38-$1.99$3.98Default0.70%$113.67$0.80-$3.49$12.18Weighted Mean PV =$117.16Variance =$19.67St.Deviation =$4.44VAR:5.00%1.00%Standard Normal Distribution$7.30$10.32Actual Distribution$5.84a$8.09ba. Actual distribution 5% level approximated by 9.4%=6%+1.5%+1.2%+.7%b. Actual distribution 1% level by 3.4%=1.5%+1.2%+.70%$113.670.70%$115.171.20%$116.191.50%$116.706.00%$117.2385.00%$117.755.00%$118.020.40%$118.290.20%Differences in the transition probability distribution create a bias in terms of the actual VAR,as is shown in the above figure.What implications derive from the VAR calculations? First is that a VAR estimate for an assetprovides a benchmark standard for the level of required capital, or reserves. This may be quitedifferent from a standardized capital reserve requirement as set by a central bank or byan international standard such as the tier reserve requirements of the Basle Accords of 1988.While VAR models represent an important step from simple capital reserve models, it shouldbe kept in mind that VAR estimates depend on the intertemporal stability of transitionprobabilities as well as on the normality of the underlying probability distribution. VAR maynot capture the effects of asymmetric information arising the presence of moral hazard oradverse selection.

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Query Standard DeviationValue At RiskValue At Risk Under Alternative Levels of Risk

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T-BondsAAA BondsAA BondsA BondsBBB BondsBB BondsB BondsCCC BondsDefault BondsThe Term Structure of Interest Rates

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Distribution of Transition Probability Present Values

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