Since 2008, businesses and banks must manage and track more risk than ever before. Financial risk management helps companies and banks decrease the risk of investment and trade. Additionally, financial risk management gives a guide on how to forecast and manage the risk efficiently. More specifically, the  three major risks are market risk, credit risk, and operational risk. This report will focus on the credit risk: introducing the definition of credit risk, single factor model, the relationship between coefficient and default probability, and the relationship of m coefficient and default probability. Using the single factor model, we will extend the definition and application to the double factor model. Furthermore, the coding will be provided

Chen, Siwen

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Undergraduate Student Research Internship Report:Analysis of Credit Risk and Single / Two Factor ModelBySiwen (Reeves) Chen 250888852Supervisor: Lars StentoftAugust 15, 2022ABSTRACTSince 2008, businesses and banks must manage and track more risk than ever before. Financial riskmanagement helps companies and banks decrease the risk of investment and trade. Additionally, financialrisk management gives a guide on how to forecast and manage the risk efficiently. More specifically, thethree major risks are market risk, credit risk, and operational risk. This report will focus on the credit risk:introducing the definition of credit risk, single factor model, the relationship between coefficient and βdefault probability, and the relationship of m coefficient and default probability. Using the single factormodel, we will extend the definition and application to the double factor model. Furthermore, the codingwill be provided.2TABLE OF CONTENTSABSTRACT 2CREDIT RISK 4The situation may increase the credit risk 4Credit Risky Securities 5Economic Balance Sheet: Asset, Equity, and Debt 6Definition of Default and its Application 6SINGLE FACTOR MODELS 8Conditional and Unconditional Default Probability 9Relationship between Beta Coefficient(β) and Default Probability 10Conclusion 12TWO FACTOR MODEL 13Introduction 13Conclusion 15REFERENCES 173CREDIT RISKIn Financial Risk Management, credit is defined as an economic obligation. For example, we would liketo purchase a good. However, we do not have enough money to pay for that good. In this case, the creditcan help us to obtain the good in advance and we are bound to pay back the money later. Under thedefinition of credit, we are also curious about the outcome when borrowers fail to repay. We consider thissituation as a default. Furthermore, we consider the credit risk as economic loss from default or changesin ratings. From a quantitative perspective, for any individual and financial institution, if their creditpoints decrease, banks will re-evaluate their credit risk, and always, their default risk would rise. Ingeneral, the low credit points will result in bad consequences for them, such that they can not get a loanfrom the bank.Credit risk strictly sticks with market risk, since every risk is not independent, for example, if the societysuffers an economic recession, the unemployment rate has dramatically increased, then individuals andcompanies may not be able to pay back loans, then their credit points will be lower than before and thedirect result will be they can not get loans again.The situation may increase the credit riskIndividuals and financial institutions may suffer bankruptcy or lose an incredible amount of money.Hence, they will fail to pay back the loan to the banks. Secondly, many debtors’ collateral value is notworth that much money, hence, they might default, this also may increase the credit risk. Thirdly, due toinflation, a country may increase the interest rate, in this case, most people are not able to afford theincreased interest, which may lead them to fail to repay the loan. For example, in July 2022, the Bank ofCanada raised the interest rate by 100 basis points to solve the high inflation issue. Since most people'ssalaries did not improve, their default risk may increase. Another example is in 2008, during the global4financial crisis, in the United States of America, the Delinquency rate on mortgages was almost 12%, thismeans every 100 households that paid the mortgage will result in 12 people paid late or unpaid.Credit Risky SecuritiesLet us introduce several credit risky securities, they are corporate debt securities, sovereign debt, creditderivatives and structured credit products.1. Corporate debt securities: it is usually described as bonds and bank loans; they all belong to fixedincome.2. Sovereign debt: It is a tool to measure the currency and is usually issued by local government andgovernment-controlled companies.3. Credit derivatives: it is a contract, and this value depends on the creditworthiness.4. Structured credit products are individual and commercial mortgages and loans. For example,student loans for school, credit cards etc.5In conclusion, these credit risky securities have a common feature: use credit to measure its value.Nevertheless, credit risky securities may suffer a big problem in the economic recession due to people’sincome and society’s employment rate.Economic Balance Sheet: Asset, Equity, and DebtBefore we go through the credit risk, we need to be familiar with the several definitions of the economicbalance sheet, they have two categories which are Assets and Liabilities. Now from the graph, we knowthat the value of the firm(At ) belongs to the Assets category, Equity( Et ) and Debt(Dt ) belong toLiabilities. We also have the balance sheet formula = +𝐴𝑡𝐸𝑡𝐷𝑡Asset LiabilityAsset ( )𝐴𝑡Equity( )𝐸𝑡Debt( )𝐷𝑡In the assets, A t is an asset of the firm, Et is capital invested by the firm’s shareholders, and Dt is acontract to pay a fixed amount of money. The equity ratio is equal to while we denote the as the𝐸𝑡𝐴𝑡𝐴𝑡𝐸𝑡leverage ratio.Definition of Default and its ApplicationIn Malz, A. M. (2011), we refer to default as the failure to pay a financial obligation. The probability ofdefault is denoted by PD. Credit exposure means the lenders may suffer a loss with default situations.Loss Given Default (LGD), is the amount the creditor loses in the event of a default. We can expressexposure as “exposure = recovery + LGD” where the recovery amount is part of the money owed that the6creditors receive in the event of bankruptcy. We usually denote recovery as R located in [0,1]. In addition,we can also know that R= = = 1- (Note that we always treat the LGD as a𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒−𝐿𝐺𝐷𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒𝐿𝐺𝐷𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒known parameter).Expected loss is the expected value of the credit loss. The formula presented asEL= PD×(1-R)×exposure = PD× LGD. Note that LGD and recovery rate are conditional expectations,that isE[loss|default]= = LGD, with this formula, we can conclude several points:𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑙𝑜𝑠𝑠𝑃𝐷1.  In this formula, PD can not be zero.2. As default probability becomes very low, the LGD can be very large3. LGD is conditional expectation7SINGLE FACTOR MODELSSingle factor model can be considered as a type of structural model, which is related to the risk of creditloss. We set the logarithmic asset return is .Note we have T= t + d, d is denoted as time𝐴𝑇= 𝑙𝑜𝑔 (𝐴𝑇−𝐴𝑡𝐴𝑡)interval.Now we can state the single factor model, that a firm’s asset return could be represented byIn this formula , we indicate that the β correlation factor; m is the correlation𝑎𝑇=  β𝑚 + 1 − β2 ε𝑖.between default and the economic performance; is an idiosyncratic risk. We can consider that m and εε𝑖are standard normal, these are and ,i,i.d.𝑚~𝑁 0, 1( ) ε~𝑁(0, 1)And we can conclude  that ,𝐸 𝑎𝑇[ ] = β𝐸[𝑚] + 1 − β2𝐸[ε𝑖] = β · 0 + 1 − β2 · 0 = 0. And since the ~N(0,1) is standard normal distributed,𝑉𝑎𝑟 𝐴𝑇[ ] = β2𝑣𝑎𝑟[𝑚] + (1 − β2)𝑣𝑎𝑟[ε𝑖] = 1 𝑎𝑇we can apply Z-value tables, and the calculation is presented as we can say that𝑍 = 𝑋−µσ  , where PD is default probability, is the quantile function of𝑃𝐷 = 𝑃 𝑎𝑇≤𝑘[ ]⟺𝑘 = ϕ−1 𝑃𝐷( ) ϕ−1 *( ) standard normal distribution and is PD-th quantile of . Moreover, we can denote the k asϕ−1 𝑃𝐷( ) 𝑎𝑇default threshold.The graph of and k ‘s relationship- Strictly follow the Z-value table, in the example , we only select𝑃𝐷0.1, 0.05,0.01 and 0.001:8π k0.1 -1.280.05 -1.650.01 -2.330.001 -3.09Conditional and Unconditional Default ProbabilityConditional default probability: , however, we would like to find out the𝑃𝐷(𝑚) = ϕ( 𝑘−β𝑚1−β2)Unconditional Default Probability,  these are the steps to find how to make it happen:1. Consider the loss level as a random variable X=x.Also, we consider x .∈ [0, 1]2. We set the PD = loss level , loss level and market factor returnx(m) = p(m) = ϕ( 𝑘−β𝑚1−β2)Now we can find m by giving a specific loss level of x, that is,(x) = ⇒ (x) = k-ϕ−1 𝑘−β𝑚1−β2ϕ−1 1 − β2 β𝑚=k- (x) ⇒ m=β𝑚 ϕ−1 1 − β2 𝑘− ϕ−1(𝑥) 1−β2β93. Now we can consider P[loss level] is equal to the P[m], we also know that the market factor is astandard normal, that is ,P[X x]=≤ ϕ(𝑚) = ϕ( 𝑘− ϕ−1(𝑥) 1−β2β )4. We re-do step 1-3 to obtain the distribution of x .Relationship between Beta Coefficient(β) and Default ProbabilityIn Single factor model , we have found that where i=1,2,… Now we introduce𝑎𝑖=  β𝑖𝑚 + 1 − β𝑖2 ε𝑖a new notation, k, default threshold and we denote the probability default as π, then the . Nowπ =  ϕ 𝑘( )we are trying to figure out the Beta(β) default probability ‘s relationship, we have several steps,1. Get default threshold k(numeric value)2. Get value , remain this fixed𝑚3. State this formula, p(m) , i= 1,2…,note this is the conditional cumulative default= ϕ𝑘𝑖−β𝑖𝑚1−β𝑖2( )probability function , also this satisfies the normal distribution4. Coding presentation in R:beta <- seq(-0.99999, 0.99999, by = .00001)# we can not set up -1 or 1 since dominator can not be zerom<-0.9 # make this value fixedpm <- dnorm(beta, mean = -beta*m, sd = sqrt(1-beta^2))# pm means conditional cumulativedefault probability10plot(beta,pm, main = "Normal Distribution", col = "blue")From the graph above, we are pretty sure that the beta coefficient with conditional probability isbell-shaped, due to m being fixed the highest conditional probability is less than 0.4, still, we need to becareful that beta can not be -1 or 1, since the denominator part can not be zero.115. How to use the probability of default to estimate the beta, firstly we have the formula, let’s define the p(m) as the conditional probability of default, CPD,𝑝 𝑚( ) = ϕ𝑘𝑖−β𝑖 𝑚1−β𝑖2( )6. In this case, we easily apply the Beta coefficient in the programming,  the basic idea to apply inthe coding is to remain k fixed to get the conditional default probability then to get the Betacoefficient. The exact derivation is: denote the conditional probability of default as CPD, then weknow thatCPD= ϕ( 𝑘−β𝑚1−β2) ⇒  𝑍 = ϕ−1(𝐶𝑃𝐷) = 𝑘−β𝑚1−β2⇒ 𝑍 = 𝑘−β𝑚1−β2⇒ (𝑘 − β𝑚)2 = 𝑍2(1 − β2)by using quadratic⇒ 𝑘2 + 𝑘2𝑚2 − 2𝑘β𝑚 = 𝑍2 − 𝑍2β2 ⇒ (𝑚2 + 𝑍2)β2 − 2𝑘𝑚β + 𝑘2 − 𝑍2formula: , let insert the a,b,c value in to−𝑏± 𝑏2−4𝑎𝑐2𝑎 𝑎 = (𝑚2 + 𝑍2),  𝑏 =− 2𝑘𝑚,  𝑐 = 𝑘2 − 𝑍2 ,  the quadratic formula: = , in addition to this we know that2𝑘𝑚+ 4𝑘2𝑚2−4(𝑚2+𝑍2)(𝑘2−𝑍2)2(𝑚2+𝑍2)β, we can also substitute (*) into quadratic formula:𝑘 = ϕ−1(𝑃𝐷) (*) ,  where Z= , m and z can not be2ϕ−1(𝑃𝐷)𝑚+ 4[ϕ−1(𝑃𝐷)]2𝑚2−4(𝑚2+𝑍2)([ϕ−1(𝑃𝐷)]2−𝑍2)2(𝑚2+𝑍2)ϕ−1(𝐶𝑃𝐷)zero.ConclusionAs for the single factor model, we only have one factor that can not be realistic, in fact, the more factorswe have , the more accurate we can achieve. In the financial risk management, as we used to talk the risk,the major risks never exists independently, we rely on each other, and so does the single factor model, weneed to add more possible factors in it, such as more assets and more companies, after figured out their12correlation, wc can calculate the Beta more accurate.We will also extend the double factor model andshow how it exact works.TWO FACTOR MODELIntroductionSince we already have the single- factor model, we can now extend the idea to two factors. Suppose youhave the two companies, i and j, then they are the pairs of companies, the market factor is and ,β1 β2m~N(0,1) and ~N(0,1) , base on single factor model theory, we can extend as two- factor models, weε𝑖can set up them as = ,  where , and are N(0,1) and iid.𝑚𝑗 β1𝑚𝑖+ 1 − β12 ε𝑖𝑚𝑖𝑚𝑗ε𝑖In these two assets, there is a relationship between them , , ) .Here are someβ β =  𝑐𝑜𝑟𝑟(𝑚𝑖𝑚𝑗notations:1.  m is a common factor and its standard normal distributed N(0,1)2. is an idiosyncratic factor and only affects a company and it’s also a standard normally distributed ε𝑖N(0,1).3. is asset correlation for i and j.β4. Note that the firm's values and defaults are independent.Recall that we have the single factor model, then we can create the two company (A and B)  with andβ1β213And we have the two companies asset formulas:+𝑎𝐴𝑖 = β1𝑚𝑖1 − β21 ε𝑖+ where , and are N(0,1) iid.𝑎𝐵𝑗 = β2𝑚𝑗1 − β22 ε𝑗𝑚𝑖𝑚𝑗ε𝑗As we talked in single factor model, we have developed the conditional default probability as :CPD(m) , so we can see the two companies assets= ϕ𝑘𝑖−β𝑖𝑚1−β𝑖2( )as ( 1)𝐶𝑃𝐷𝐴(𝑚𝑖) = ϕ(𝑘1−𝑚𝑖β11−β12)(2)𝐶𝑃𝐷𝐵(𝑚𝑗) = ϕ(𝑘2−𝑚𝑗β21−β22)With (1) and (2), we can also derive them in the same way to deal with CPD and correlation ’sβrelationship, firstly, we have the three formulas:(A)𝐶𝑃𝐷𝐴(𝑚𝑖) = ϕ(𝑘1−𝑚𝑖β11−β12)(B)𝐶𝑃𝐷𝐵(𝑚𝑗) = ϕ(𝑘2−𝑚𝑗β21−β22)= (C)𝑚𝑗 β1𝑚𝑖+ 1 − β12 ε𝑖Now we can derive A and B, to find a beta coefficient and conditional probability(CPD)’s relationship,14as we used that in the single factor model, in equation A, we have the in this𝐶𝑃𝐷𝐴(𝑚𝑖) = ϕ(𝑘1−𝑚𝑖β11−β12)case, we can get the ϕ−1(𝐶𝑃𝐷𝐴) =𝑘1−𝑚𝑖β11−β12⇒  𝑠𝑒𝑡 𝑍𝐴= ϕ−1(𝐶𝑃𝐷𝐴),  𝑡ℎ𝑒𝑛 𝑍 =𝑘1−𝑚𝑖β11−β12 ⇒,(𝑘1− 𝑚𝑖β1)2 = 𝑍2(1 − β12 ) ⇒ (𝑚𝑖2 + 𝑍2)β12 − 2𝑘1𝑚𝑖β1+ 𝑘12− 𝑍2 = 0  , again, with quadratic formula we have the final equation of𝑎 = 𝑚𝑖2 + 𝑍2,  𝑏 =− 2𝑘1𝑚𝑖, 𝑐 =  𝑘12 − 𝑍2, that is , we get the value, where Z= , special noteβ1β1=2𝑘1𝑚1± 4𝑘12𝑚𝑖2−4(𝑚𝑖2+𝑍2)(𝑘12−𝑍2 )2(𝑚𝑖2+𝑍2)β1 ϕ−1(𝐶𝑃𝐷𝐴)that , since the denominator can not be zero.𝑚𝑖2 + 𝑍2 ≠ 0Move to Equation (B) and equation (C), let’s find out the relationship between and , in this case,β2𝐶𝑃𝐷𝐵we need to use equation (C), in the equation (B), similar to the ’s derivation, we can get the equation,β1β2that is, , where Z= and can not be zero.What ifβ2=2𝑘2𝑚𝑗± 4𝑘22𝑚𝑗2−4(𝑚𝑗2+𝑍2)(𝑘𝑗2−𝑍2 )2(𝑚𝑗2+𝑍2) ϕ−1(𝐶𝑃𝐷𝐵) 𝑚𝑗2 + 𝑍2we don’t have the ’s value , we can calculate that with equation (C ) , that is𝑚𝑗= .𝑚𝑗 β1𝑚𝑖+ 1 − β12 ε𝑖ConclusionTwo-Factor Models indicate that in a market, two companies with two assets have a correlation betweeneach other. We can find out the beta coefficient and CPD relationship more accurately than in a single15factor model, however, it’s not as accurate and powerful as the Multi-Factor Model, we will introduce thatin the future report.16REFERENCESMalz, A. M. (2011). Financial Risk Management: Models, history, and institution models, history,and Institution. Wiley.S. R. | O. M. (2016, February 4). A recipe for the 2008 financial crisis. Turing Finance. RetrievedAugust 18, 2022, from http://www.turingfinance.com/recipe-for-the-financial-crisis/Céspedes,J. C.G. and Martín D. G.(2002,January).THE TWO-FACTOR MODEL FOR CREDITRISK: A comparison with the BIS II one-factor model17

Analysis of Credit Risk and Single / Two Factor Model

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Analysis of Credit Risk and Single / Two Factor Model

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