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MAT1856S/ APM466S Mathematical Theory of Finance Final Exam - April 2003 Problem 1. Consider a call option on a stock. The stock value today is S, the option value when the stock is worth S at time t is f (S, t) and the strike price for the option is K. Determine whether the following statements are true or false. Justify your answer. S f (S, t) > 0. K f (S, t) > 0. limS f (S, t) = 0. limK f (S, t) = 0. Problem 2. If the yield curve today is given by r(T ) = 0.05 T +1 , T + 10 calculate the forward curve today. Problem 3. Assume a certain stock is valued at $1 today, and over a period of one year it will be worth either $2 or $0.5. Determine the valid range of interest rates so that there are no arbitrage opportunities. Problem 4. Consider a stock with price St , for 0 t 1 and S0 = 1. A convertible bond BS is a bond that, at time 1, will be worth max (1, ST ) . Calculate the value of BC now, when S is the stock in problem 3. If ST is normally distributed,with mean 1 and standard deviation 0.1, determine which of the following three portfolios has lower VaR: BC , BC + S or BC S? Calculate the VaR of BC . Problem 5. Consider a convertible bond like above, on a stock like the one in problem 3. Assume the bond is issued by the company that owns the stock, which may default (and S = 0 at that point). Determine which of the following portfolios has lower Credit VaR: BC , BC + S or BC S? PortfolioCreditRisk Luis Seco University of Toronto seco@math.utoronto.ca TheGoodrich-Rabobankswap:1983 Belgiandentists U.S.SavingsBanks LIBOR+0.5%(Semi) 11%annual B.F.Goodrich Rabobank BBB-rated AAArated 5.5 million (11% fixed) Once a year Swap (LIBOR - x) % Semiannual Morgan Guarantee Trust 5.5 million Once a year Swap (LIBOR - y) % Semiannual Reviewofbasicconcepts Cashflowvaluation Creditpremium The discounted value of cash flows, when there is probability of default, is given by qi denotes the probability that the counter-party is solvent at time ti Thelargerthedefaultrisk(q small),thesmalleritsvalue. Thehigherthecreditrisk(q small),thehigherthe payments,topreservethe samepresentvalue Thecreditspread Since , we can write the loan is now valued as Default-prone interest rate increases. Firstmodel:twocreditstates What is the credit spread? Assume only 2 possible credit states: solvency and default Assume the probability of solvency in a fixed period (one year, for example), conditional on solvency at the beginning of the period, is given by a fixed amount: q According to this model, we have which gives rise to a constant credit spread: ThegeneralMarkovmodel In other words, when the default process follows a Markov chain, Solvency Default Solvency q 1-q Default 0 1 the credit spread is constant, and equals Goodrich-Morganswap The fixed rate loan G-RBCreditMetricsanalysis:setup The leg to consider for Credit Risk is the one between JPMorgan and BF Goodrich Cashflows of the leg (in million USD): 0.125 upfront 5.5 per yr, during 8 years Assume: onstant spread h = 180 bpi c state transition probabilities matrix 2 G-RBCreditMetrics:expectedcashflows Since Expected[cashflows] = ($cashflows) * Prob{non_default} Then E[cashflows] = .125 + Sum( 5.5 * P{nondefault @ each year}) But at the same time E[cashflow] = G-RBCreditMetrics:probabilityofdefault Under our assumptions: P {non-default} = exp(-h) = exp(-.018) = .98216 onstant for each year c The 2 state matrix: BBB D BBB .9822 .0178 D 0 1 G-RBCreditMetrics:computecashflows Inputs {default of BBB corp.} = 1.8%; P 1-exp(0.018)=0.9822 he gvmnt zero curve for August 1983 was T r = (.08850,.09297,.09656,.0987855,.10550, .104355,.11770,.118676) for years (1,2,3,4,5,6,7,8) G-RBCreditMetrics:cashflows E[cashflows] Risk-less Cashflows (cont) G-RBCreditMetrics:Expectedlosses Therefore E[loss] = 1 - ( E[cashflows] / Non-Risk Cashflow) = .065776 i.e. the proportional expected loss is around 6.58% of USD 24.67581 million Or roughly E[loss] = 1.623 (USD million) Non-constantspreads Adefaulto-defaultmodel (suchasCreditRisk+) leadstoconstantspreads, unlessprobabilitiesvarywith time Inordertofitnon-constantspreads, andbeabletofitthemodeltomarket observations,oneneedstoassume either: Time-varyingdefaultprobabilities Multi-ratingsystems(suchascredit metrics) MarkovProcesses Transition Probabilities Constant in time 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 t=0 1 7 7 t=1 t=2 Transitionprobabilities Conditional probabilities, which give rise to a matrix with n credit states, ... ... ... : Pij = cond prob of changing from state i to state j : : : ... ... Creditratingagencies There are corporations whose business is to rate the credit quality of corporations, governments, and also of specific debt issues. The main ones are: Moody's Investors Service, Standard & Poor's, Fitch IBCA, Duff and Phelps Credit Rating Co StandardandPoor'sMarkovmodel AAA AA A BBB BB B CCC D AAA 0.9081 0.0833 0.0068 0.0006 0.0012 0.0000 0.0000 0.0000 AA 0.0070 0.9065 0.0779 0.0064 0.0006 0.0014 0.0002 0.0000 A 0.0009 0.0227 0.9105 0.0552 0.0074 0.0026 0.0001 0.0006 BBB 0.0002 0.0033 0.0595 0.8693 0.0530 0.0117 0.0012 0.0018 BB 0.0003 0.0014 0.0067 0.0773 0.8053 0.0884 0.0100 0.0106 B 0.0000 0.0011 0.0024 0.0043 0.0648 0.8346 0.0407 0.0520 CCC 0.0022 0.0000 0.0022 0.0130 0.0238 0.1124 0.6486 0.1979 0 0 0 0 0 0 0 1 D Longtermtransitionprobabilities Transition probability between state i and state j, in two time steps, is given by In other word, if we denote by A the one-step conditional probability matrix, the two-step transition probability matrix is given by Transitionprobabilitiesingeneral If A denotes the transition probability matrix at one step (one year, for example), the transition probability after n steps (30 is specially meaningful for credit risk) is given by For the same reason, the quarterly transition probability matrix should be given by This gives rise to a number of important practical issues. CreditLoss CreditExposure Exposure(99%) Exposure(95%) It is the maximum loss that a portfolio can experience at any time in the future, taken with a certain confidence level. Evolutionofthemark-to-marketofa20-monthswap RecoveryRate-LossGivenDefault When default occurs, a portion of the value of the portfolio can usually be recovered. Because of this, a recovery rate is always considered when evaluating credit losses. It represents the percentage value which we expect to recover, given default. Loss-given-default is the percentage we expect to lose when default occurs: Defaultprobability(frequency) Each counterparty has a certain probability of defaulting on their obligations. Some models include a random variable which indicates whether the counterparty is solvent or not. Other models use a random variable which measures the credit quality of the counterparty. For the moment, we will denote by b the random variable which is 1 when the counterparty defaults, and 0 when it does not. The modeling of how it changes from 0 to 1 will be dealt with later Measuringthedistributionofcreditlosses For an instrument or portfolio with only one counterparty, we define: Credit Loss = b x Credit Exposure x LGD Randomvariable: Number: Number: Dependsonthecreditquality ofthecounterparty Dependsonthemarketrisk oftheintrumentorportfolio Usually,thisnumberisa universalconstant(55%),but morerefinedmodelsrelateitto themarketandthecounterparty Measuringthedistributionofcreditlosses(2) For a portfolio with several counter-parties, we define: Credit Loss = (b i x Credit Exposure x LGD) i i i Randomvariable: Number: Number: Normallydifferentfor differentcounterparties Normallydifferentfor differentportfolios,samefor thesameportfolios Usually,thisnumberisa universalconstant(55%),but morerefinedmodelsrelateitto themarketandthecounterparty NetReplacementValue The traditional approach to measuring credit risk is to consider only the net replacement value NRV = (Credit Exposures) i i This is a rough statistic, which measures the amount that would be lost if all counter-parties default at the same time, and at the time when all portfolios are worth most, and with no recovery rate. Creditlossdistribution Thecreditlossdistributionisoftenverycomplex. Unexpectedloss AswithMarkowitztheory,wetrytosummarizeits statisticswithtwonumbers:itsexpectedvalue,andits standarddeviation. Inthiscontext,thisgivesustwovalues: Theexpectedloss Theunexpectedloss Expectedloss CreditVaR/WorstCreditLoss Worst Credit Loss represents the credit loss which will not be exceeded with some level of confidence, over a certain time horizon. A 95%-WCL of $5M on a certain portfolio means that the probability of losing more than $5M in that particular portfolio is exactly 5%. CVaR represents the credit loss which will not be exceeded in excess of the expected credit loss, with some level of confidence over a certain time horizon: A daily CVaR of $5M on a certain portfolio, with 95% means that the probability of losing more than the expected loss plus $5M in one day in that particular portfolio is exactly 5%. Usingcreditriskmeasurementsintrading Marginal contribution to risk When considering a new instrument to be traded as part of a certain book, one needs to take into account the impact of the new deal in the credit risk profile at the time the deal is considered. An increase of risk exposure should lead to a higher premium or to a deal not being authorized. A decrease in risk exposure could lead to a more competitive price for the deal. Remuneration of capital Imagine a deal with an Expected Loss of $1M, and an unexpected loss of $5M. The bank may impose a credit reserve equal to $5M, to make up for potential losses due to default; this capital which is immobilized will require remuneration; because of this, the price of any creditprone contract should equal Price = Expected Loss + (portion) Unexpected loss Netting When two counterparties enter into multiple contracts, the cashflows over all the contracts can be, by agreement, merged into one cashflow. This practice, called netting, is equivalent to assuming that when a party defaults on one contract it defaults in all the contracts simultaneously. Netting may affect the credit-risk premium of particular contracts. Assuming that the default probability of a party is independent from the size of exposures it accumulates with a particular counter-party, the expected loss over several contracts is always less or equal than the sum of the expected losses of each contract. The same result holds for the variance of the losses (i.e. the variance of losses in the cumulative portfolio of contracts is less or equal to the sum of the variances of the individual contracts). Equality is achieved when contracts are either identical or the underlying processes are independent. ExpectedCreditLoss:Generalframework In the general framework, the expected credit loss is given by Expectationusingthe jointprobability distribution Jointprobabilitydensity forallthreerandom variables: efaultstatus(b) d reditExposure C ossgivendefault L ExpectedCreditLoss:Specialcase Because calculating the joint probability distribution of all relevant variables is hard, most often one assumes that their distributions are independent. In that case, the ECL formula simplifies to: Probabilityof default Expected Credit Exposure Expected Severity Example Consider a commercial mortgage, with a shopping mall as collateral. Assume the exposure of the deal is $100M, an expected probability of default of 20% (std of 10%), and an expected recovery of 50% (std of 10%). Calculate the expected loss in two ways: Assuming independence of recovery and default (call it x) Assuming a -50% correlation between the default probability and the recovery rate (call it y). What is your best guess as to the numbers x and y. 1. x=$10M, y=$10M. 2. x=$10M, y=$20M. 3. x=$10M, y=$5M. 4. x=$10M, y=$10.5M. Example Consider a commercial mortgage, with a shopping mall as collateral. Assume the exposure of the deal is $100M, an expected probability of default of 20% (std of 10%), and an expected recovery of 50% (std of 10%). Calculate the expected loss in two ways: Assuming independence of recovery and default (call it x) Assuming a -50% correlation between the default probability and the recovery rate (call it y). What is your best guess as to the numbers x and y. 1. x=$10M, y=$10M. 2. x=$10M, y=$20M. Cannotbe:x 3. x=$10M, y=$5M. hastobe smallerthany 4. x=$10M, y=$10.5M. Tree-basedmodel 1 1 0 0 -1 -1 Correlatingdefaultandrecovery Assume two equally likely future credit states, given by default probabilities of 30% and 10%. Assume two equally likely future recovery rates, given by 60% and 40%. With a -50% correlation between them, the expected loss is EL = $100M x (0.375x0.6x0.3 + 0.375x0.4x0.1 + 0.125x0.4x0.3 + 0.125x0.6x0.1) = $100M x (0.0825 + 0.0225) = $10.5M Probabilitiescalibratedtostated correlations Goodrich-RabobankExample Consider the swap between Goodrich and MGT. Assume a total exposure averaging $10M (50% std), a default rate averaging 10% (3% std), fixed recovery (50%). Calculate the expected loss in two ways Assuming independence of exposure and default (call it x) Assuming a -50% correlation between the default probability and the exposure (call it y). What is your best guess as to the numbers x and y. 1. x=$500,000, y=$460,000. 2. x=$500,000, y=$1M. 3. x=$500,000, y=$500,000. 4. x=$500,000, y=$250,000. Goodrich-RabobankExample Consider the swap between Goodrich and MGT. Assume a total exposure averaging $10M (50% std), a default rate averaging 10% (3% std), fixed recovery (50%). Calculate the expected loss in two says Assuming independence of exposure and default (call it x) Assuming a -50% correlation between the default probability and the exposure (call it y). What is your best guess as to the numbers x and y. 1. x=$500,000, y=$450,000. Cannotbe:x 2. x=$500,000, y=$1M. hastobelarger thany 3. x=$500,000, y=$500,000. 4. x=$500,000, y=$250,000. Correlatingdefaultandexposure Assume two equally likely future credit states, given by default probabilities of 13% and 7%. Assume two equally likely exposures, given by $15M and $5M. With a -50% correlation between them, the expected loss is EL = 0.5 x (0.125x$15Mx0.13 + 0.125x$5Mx0.07 + 0.375x$15Mx0.07 + 0.375x$5Mx0.13) = 0.5 x ($0.24M + $0.04 + $0.40M + $0.24M) = $460,000 Example23-2:FRMExam1998Question39 \"Calculate the 1 yr expected loss of a $100M portfolio comprising 10 B-rated issuers. Assume that the 1-year probability of default of each issuer is 6% and the recovery rate for each issuer in the event of default is 40%.\" Example23-2:FRMExam1998Question39 \"Calculate the 1 yr expected loss of a $100M portfolio comprising 10 B-rated issuers. Assume that the 1-year probability of default of each issuer is 6% and the recovery rate for each issuer in the event of default is 40%.\" 0.06 x $100M x 0.6 = $3.6M Variationofexample23-2. \"Calculate the 1 yr unexpected loss of a $100M portfolio comprising 10 B-rated issuers. Assume that the 1-year probability of default of each issuer is 6% and the recovery rate for each issuer in the event of default is 40%. Assume, also, that the correlation between the issuers is 1. 100% (i.e., they are all the same issuer) 2. 50% (they are in the same sector) 3. 0% (they are independent, perhaps because they are in different sectors)\" Solution 1. The loss distribution is a random variable with two states: default (loss of $60M, after recovery), and no default (loss of 0). The expectation is $3.6M. The variance is 0.06 * ($60M-$3.6M)2 + 0.94 * (0-$3.6M)2 = 200($M)2 The unexpected loss is therefore sqrt(200) = $14M. Solution 2. The loss distribution is a sum of 10 random variables Xi, each with two states: default (loss of $6M, after recovery), and no default (loss of 0). The expectation of each of them is $0.36M. The variance of each is (as before) 2. The variance of their sum is Solution 3. The loss distribution is a sum of 10 random variable, each with two states: default (loss of $6M, after recovery), and no default (loss of 0). The expectation of each of them is $0.36M. The standard deviation of each is (as before) $1.4M. The standard deviation of their sum is sqrt(10) * $1.4M = $5M Note: the number of defaults is given by a Poisson distribution. This will be of relevance later when we study the CreditRisk+ methodology. Example23-3:FRMexam1999 \"Which loan is more risky? Assume that the obligors are rated the same, are from the same industry, and have more or less the same sized idiosyncratic risk: A loan of 1. $1M with 50% recovery rate. 2. $1M with no collateral. 3. $4M with a 40% recovery rate. 4. $4M with a 60% recovery rate.\" Example23-3:FRMexam1999 \"Which loan is more risky? Assume that the obligors are rated the same, are from the same industry, and have more or less the same sized idiosyncratic risk: A loan of 1. $1M with 50% recovery rate. 2. $1M with no collateral. 3. $4M with a 40% recovery rate. 4. $4M with a 60% recovery rate.\" The expected exposures times expected LGD are: 1. $500,000 2. $1M 3. $2.4M. Riskiest. 4. $1.6M Example23-4:FRMExam1999. \"Which of the following conditions results in a higher probability of default? 1. The maturity of the transaction is longer 2. The counterparty is more creditworthy 3. The price of the bond, or underlying security in the case of a derivative, is less volatile. 4. Both 1 and 2.\" Example23-4:FRMExam1999. \"Which of the following conditions results in a higher probability of default? 1. The maturity of the transaction is longer 2. The counterparty is more creditworthy 3. The price of the bond, or underlying security in the case of a derivative, is less volatile. 4. Both 1 and 2.\" Answer 1. True 2. False, it should be \"less\