Banking
Risk Management Examples with Simulation, Optimisation, and Prediction
Chapter 2: Managing Credit Risk in Bank Loan Portfolio
In Book 2: Six Sigma Improvements for Basel III and Solvency II in Financial Risk Management
ISBN: 9781522572800
Author: Vojo Bubevski (Independent Researcher)
Abstract
In this chapter, the Six Sigma DMAIC approach is applied to improve Credit Risk Management in Banking Loan Portfolio selection. The objective is to select the optimal loan portfolio which achieves the banks’ investment objectives with acceptable credit risk according to their predefined limits. Stochastic Optimisation constructs the Efficient Frontier of optimal loan portfolios in banking with maximal profit minimising Loan Losses, i.e., credit risk. Simulation stochastically calculates and measures Mean Gross Profit, Loan Losses, Variance, Standard Deviation, and Sharpe Ratio. Six Sigma capability metrics determine if the loan portfolio complies with the bank’s limits regarding the Gross Profit; Loan Losses, which quantifies the Credit Risk; and Sharpe Ratio, i.e., a risk-adjusted measure. Also, the bank regulation limits are applied based on the bank’s capital to control the maximum loan amount per loan investment grade. The analysis allows for selecting the best Efficient Frontier loan portfolio with a maximum Sharpe Ratio. The new concept of What-If Analysis is applied to enhance the risk assessment.
Keywords: Financial Risk Management, Loan Portfolio, Credit Risk, Basel III, Six Sigma DMAIC, Stochastic Optimisation, Monte Carlo Simulation, What-If Analysis.
The Results
Evaluation of the two efficient Frontier loan portfolios is based on the comparison of the associated portfolio attributes. Table 3 presents Loan Portfolio Gross Profit Mean (LPGP – µ), Standard Deviation (σ), Total Loan Losses (TLL), and Sharpe Ratio (SR).
Table 3: Values of Efficient Frontier Portfolios’ Attributes
|
Portfolio |
LPGP – µ |
σ |
TLL |
SR |
|
1 |
$24.421MM |
0.7248 |
$3.199MM |
0.6577 |
|
2 |
$24.727MM |
0.5552 |
$2.346MM |
1.1003 |
Table 4: Probability of Attributes’ Compliance with Bank’s Target Limits
|
Portfolio |
LPGP – µ |
TLL |
SR |
|
1 |
91.2% |
61.8% |
34.6% |
|
2 |
99.2% |
62.4% |
62.4% |
Table 4 shows the probability that Loan Portfolio Gross Profit Mean (LPGP – µ), Total Loan Losses (TLL), and Sharpe Ratio (SR) would be within the bank’s specified target limits; that is the probability of compliance with the bank’s limits.
The evaluation presented in the following sections is based on the comparison of the attributes presented in Table 3 and Table 4.
The Gross Profit (LPGP)-Mean of Portfolio 2 is $24.727MM, which is $0.306MM more than the profit of Portfolio 1 (Table 3). Also, the probability of compliance with the bank’s target limits of Gross Profit (LPGP)-Mean is 99.2%, which is 8.0% more than the probability of profit compliance of Portfolio 1 (Table 4). Thus, considering the profit, Portfolio 2 is better than Portfolio 1.
The Total Loan Losses (TLL) of Portfolio 2 is $2.346MM, which is $0.853MM less than the losses of Portfolio 1 (Table 3). In addition, the probability of compliance with the bank’s target limits of Total Loan Losses (TLL) is 62.4%, which is 0.60% more than the probability of loss compliance of Portfolio 1 (Table 4). So, considering the losses, Portfolio 2 is better than Portfolio 1.
Chapter 6: Analysing Credit Losses for Banks
In Book 4: Risk Management for Businesses with Stochastic Six Sigma DMAIC Method
ISBN: 978-620-2-67095-1
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter analyses the credit losses of 200 bank customers. The bank has classified the customers into eight credit rating categories, starting with 1 – Normal to 8 – Default, with associated default probabilities, which are empirically calculated as average values from historical data collected. Every customer begins a year in a certain credit rating category, with a certain amount of credit exposure at default. By the end of the year, each customer has either defaulted or not. In case of default, the percentage that can be recovered is uncertain. A stochastic model is applied to calculate the Total Loss amount from those customers and the Percent Lost, which is the Total Loss percentage of the Total Amount of Exposure at Default. Also, it applies functions at several confidence levels to find the amounts of reserve required to be confident in covering the losses.
Keywords: Business Risk Management, Risk Assessment, Finance, Bank Credit Risk, Stochastic Model, Six Sigma DMAIC, Monte Carlo Simulation
The Results
The calculated simulation Outputs of Total Loss and Percent Lost are presented in Table 6 below.
Table 6: Total Loss and Percent Lost Results
|
Simulation Output |
Value |
|
Total Loss |
$4,833,664.47 |
|
Total Loss Confidence Level at 90% |
$10,258,442.42 |
|
Total Loss Confidence Level at 95% |
$12,028,788.54 |
|
Total Loss Confidence Level at 99% |
$15,173,954.52 |
|
Percent Lost |
0.9152% |
The Total Loss of Confidence Levels is interpreted as follows.
Total Loss Confidence Level at 90%: there is a 90% probability that the Total Loss will be equal to, or below $10,258,442.42. Total Loss Confidence Level at 95%: there is a 95% probability that the Total Loss will be equal to, or below $12,028,788.54. Total Loss Confidence Level at 99%: there is a 99% probability that the Total Loss will be equal to, or below $15,173,954.52.
Chapter 8: Projecting Loan Interest Rates and Payments
In Book 4: Risk Management for Businesses with Stochastic Six Sigma DMAIC Method
ISBN: 978-620-2-67095-1
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates two stochastic models for projecting variable interest rates and loan payment schedules for 21 years. In the Independent Rates Model (IRM), the yearly interest rates are generated independently. The rates are normally distributed with a given mean and standard deviation. In the Random Walk Model (RWM), only the first interest rate is normally distributed with a given mean and standard deviation, but each succeeding interest rate is normally distributed with a mean equal to the actual previous rate and the given standard deviation. The method that the interest rates are generated makes a substantial difference in the distribution of the total payment. Specifically, interest rates can vary much more in the Random Walk model, which causes a larger variation in the total payment. Compared to most typical loan payment schedules where all payments are equal, the schedules illustrated in this chapter pay a constant principal each year but have different yearly payments. Five simulations are run, each with a different mean and standard deviation to calculate the interest rates.
Keywords: Business Risk Management, Risk Assessment, Banking, Loan Interest Rate, Loan Payment Schedule, Stochastic Model, Six Sigma, DMAIC, Monte Carlo simulation
The Results
The five Total Net Payment simulation results of the two models, IRM and RWM, are presented below. The attained values for Total IRM Net Payment, Total IRM Net Payment Standard Deviation, Total RWM Net Payment, and Total RWM Net Payment Standard Deviation are given in Table 2.
Table 2: Five Simulations of IRM and RWM Results
|
Data |
Sim #1 |
Sim #2 |
Sim #3 |
Sim #4 |
Sim #5 |
|
Total IRM Net Payment |
$919,998 |
$972,498 |
$1,024,998 |
$1,077,497 |
$1,129,997 |
|
Total IRM Net Pay. SD |
$11,122 |
$12,512 |
$13,903 |
$15,293 |
$16,683 |
|
Total RWM Net Payment |
$919,998 |
$972,498 |
$1,024,998 |
$1,077,497 |
$1,129,997 |
|
Total RWM Net Pay. SD |
$101,699 |
$107,503 |
$113,306 |
$119,109 |
$124,913 |
Chapter 2: Analysis of Bank Loan Portfolio Credit Risk
In Book 7: Risk Analysis and Prediction in Finance and Insurance
ISBN: 978-620-3-02745-7
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates an analysis of banking loan portfolio credit risk. The objective is to select the optimal loan portfolio which achieves the bank’s investment objectives with an acceptable credit risk according to their predefined limits. Stochastic optimisation constructs an efficient frontier of optimal loan portfolios in banking with maximal profit and minimising loan losses, i.e. the credit risk. Simulation stochastically calculates and measures the gross profit and the objective profit. Also, the bank regulation limits are applied based on the bank’s capital to control the maximum loan amount per loan investment grade. This analysis allows for selecting the best Efficient Frontier loan portfolio which gains the maximum profit.
Keywords: Risk Analysis, Loan Portfolio, Credit Risk, What-If Analysis; Monte Carlo simulation; Optimisation; Stochastic model.
The Results
The optimisation and simulation are run and the results are presented below.
Efficient Frontier Portfolios are presented in Table 2.
Table 2: Efficient Frontier Portfolios’ Results
|
Portfolio |
Gross Profit Mean |
Gross Profit Standard Deviation |
Objective Profit Mean |
Objective Profit Standard Deviation |
|
1 |
$22.891 MM |
$0.8227 MM |
$20.527 MM |
$1.6450 MM |
|
2 |
$23.291 MM |
$0.5028 MM |
$21.263 MM |
$1.0056 MM |
|
3 |
$24.419 MM |
$0.7316 MM |
$23.557 MM |
$1.4224 MM |
|
4 |
$24.727 MM |
$0.5572 MM |
$24.101 MM |
$1.0600 MM |
The results presented above show that the best Efficient Frontier portfolio is Portfolio No 4 with a Gross Profit Mean of $24.727 MM. The Objective Profit Mean is $24.101 MM.
Chapter 4: Credit Risk Analysis in Banking
In Book 7: Risk Analysis and Prediction in Finance and Insurance
ISBN: 978-620-3-02745-7
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter discusses credit risk analysis in banking. A factory plans to accomplish a project and applies for credit from a bank to finance the project. The bank considers a loan to finance the factory project and assesses the credit risk. The chapter presents the analysis and measurement of different aspects of credit risk in order to answer how much should be lent to the factory project and for how long considering the risk inherent in the transaction. Credit risk is assessed considering: 1) Cash flow projection; 2) Count of negative cash flow; 3) Maximum negative cash flow; 4) Net Present Value (NPV) based on dividends; 5) Internal Rate of Return (IRR) based on dividends; 6) Capital asset NPV and IRR; 7) Solvency loan; 8) Risk of bankruptcy; 9) Financial Analysis Measures such as Gross Margin, Interest Coverage, Financial Coverage, Return on Investment, Return on Assets and Net Worth.
Keywords: Risk Analysis, Banking, Credit Risk, What-If Analysis; Monte Carlo simulation; Stochastic model.
The Results
The Cash Flow Projection for 20 years, based on the probability distribution, is as follows. The graph shows Cash Flow Mean, Cash Flow Mean from 25% to 75%, Cash Flow Mean from 5% to 95%, and Simulation Path. The Minimum Cash Flow Mean is in Year 5, which is negative; and the Maximum Cash Flow Mean is in Year 16, which is positive. The Cash Flow Mean is negative from Year 4 to Year 6.
The Cash Flow Projection for 20 years, based on calculation without probability distribution, is given below.
The Minimum Cash Flow Mean is in Year 4, which is negative; and the Maximum Cash Flow Mean is in Year 20, which is positive. Cash Flow Mean is negative for Years 4 – 6 and for Years 10 – 12. The calculated values are shown on the graph. It should be noted that the calculated Cash Flows are different from probability Cash Flows.
Chapter 6: Predicting Loan Applicants’ Timely Payments
In Book 12: Operations Research Applications for Decision Analysis and Prediction
ISBN: 978-620-5-49772-2
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates a prediction of the loan applicants’ timely payments with optimisation. A Neural Networks tool is used to predict unknown values of categorical dependent variables from known values of numeric and categorical independent variables. In this model, a neural net learns to predict whether an auto loan applicant will make timely payments, late payments, or default on the loan. The data contains information on applicants who took car loans in the past. The input data of five new applicants are also given. It is supposed that the bank executives want to allocate a certain amount of money in loans to the five applicants to minimise the probability of a default occurring. Therefore, Neural Networks and optimisation tools are used to predict the optimal values for the new applicants.
Keywords: Prediction, Risk Analysis, Banking, Loan Payments, Neural Networks; Optimisation.
The Results
Prediction Model results are presented in Table 1.
Table 1: Initial Prediction Results
|
Loan Applicant |
Timely Payments Probability |
Payment Status |
Loan Amount |
|
1 |
83.44% |
timely payments |
$8,000 |
|
2 |
93.25% |
timely payments |
$23,200 |
|
3 |
96.83% |
timely payments |
$24,500 |
|
4 |
99.52% |
timely payments |
$23,700 |
|
5 |
88.71% |
late payments |
$17,100 |
|
Probability of Default: |
0.0015 |
Total: |
$96,500 |
Table 2: Optimisation with Prediction Results
|
Loan Applicant |
Timely Payments Probability |
Late Payments Probability |
Default Payments Probability |
No Default Payments Probability |
Payments Status |
Loan Amount |
|
1 |
99.15% |
0.85% |
0.0% |
100% |
timely |
$316 |
|
2 |
58.11% |
41.89% |
0.0% |
100% |
timely |
$23,597 |
|
3 |
95.62% |
4.38% |
0.0% |
100% |
timely |
$30,000 |
|
4 |
99.63% |
0.37% |
0.0% |
100% |
timely |
$30,000 |
|
5 |
54.41% |
45.59% |
0.0% |
100% |
timely |
$12,587 |
|
Probability of Default: |
1.2554E-005 (0.0000126) |
Total: |
$96,500 |
|||
The prediction results are as follows. The Timely Payments Probability is 83.44% for Loan Applicant 1; 93.25% for Loan Applicant 2; 96.83% for Loan Applicant 3; 99.52% for Loan Applicant 4; 88.71% for Loan Applicant 5. The Payment Status is “timely payments” for Loan Applicants 1, 2, 3, and 4; “late payments” for Loan Applicants 5. The Loan Amount is $8,000 for Loan Applicant 1; $23,200 for Loan Applicant 2; $24,500 for Loan Applicant 3; $23,700 for Loan Applicant 4; $17,100 for Loan Applicant 5. The Probability of Default is 0.0015, i.e., 0.15%, which is a very low risk.
The results of the Optimisation Model with Prediction are presented in Table 2.
Table 2: Optimisation with Prediction Results
|
Loan Applicant |
Timely Payments Probability |
Late Payments Probability |
Default Payments Probability |
No Default Payments Probability |
Payments Status |
Loan Amount |
|
1 |
99.15% |
0.85% |
0.0% |
100% |
timely |
$316 |
|
2 |
58.11% |
41.89% |
0.0% |
100% |
timely |
$23,597 |
|
3 |
95.62% |
4.38% |
0.0% |
100% |
timely |
$30,000 |
|
4 |
99.63% |
0.37% |
0.0% |
100% |
timely |
$30,000 |
|
5 |
54.41% |
45.59% |
0.0% |
100% |
timely |
$12,587 |
|
Probability of Default: |
1.2554E-005 (0.0000126) |
Total: |
$96,500 |
|||
The Timely Payments Probability is 99.15% for Loan Applicant 1; 58.11% for Loan Applicant 2; 95.62% for Loan Applicant 3; 99.63% for Loan Applicant 4; 54.41% for Loan Applicant 5. The Payment Status is “timely payments” for all Loan Applicants. The Loan Amount is $316 for Loan Applicant 1; $23,597 for Loan Applicant 2; $30,000 for Loan Applicant 3; $30,000 for Loan Applicant 4; $12,587 for Loan Applicant 5. The Probability of Default is 1.2554E-005 which is insignificant (i.e., nearly zero) compared to 0.0015, i.e., 0.15%. This is a significant risk improvement.
Chapter 5: Bank Loan Portfolio Credit Risk Analysis
In Book 13: Financial Risk Assessment and Management with Six Sigma DMAIC Methods
ISBN: 978-620-5-49700-5
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates an analysis of banking loan portfolio credit risk. The objective is to select the optimal loan portfolio which achieves the bank’s investment objectives with an acceptable credit risk according to their predefined limits. Stochastic optimisation constructs an efficient frontier of optimal loan portfolios in banking with maximal profit and minimising loan losses, i.e., the credit risk. Simulation stochastically calculates and measures the gross profit and the objective profit. Also, the bank regulation limits are applied based on the bank’s capital to control the maximum loan amount per loan investment grade. This analysis allows for selecting the best Efficient Frontier loan portfolio which gains the maximum profit.
Keywords: Risk Analysis, Loan Portfolio, Credit Risk, What-If Analysis; Monte Carlo simulation; Optimisation; Stochastic model.
The Results
It was established that Efficient Frontier Portfolio No 4 is technically the best. Therefore, this portfolio is considered below. The distribution of Optimal Portfolio No 4 is shown below.
The Gross Profit Mean (Figure 7) is $24.7274 MM with a Standard Deviation of $0.5572 MM, i.e., 2.253%, which is a very low risk. The Six Sigma target parameters are: TV = $24.4056 MM; LSL = $23.1853 MM; and USL = $25.6259 MM. The Six Sigma process capability metrics are: Cp = 0.7300; Cpk = 0.5375; and σ-L = 2.0131. There is a 5.0% probability that the Mean will be below $23.902 MM; a 90.0% probability that it will be in the range of $23.902 MM – $25.587 MM; and a 5.0% probability that it will be above $25.587 MM.
Chapter 6: Estimating Loan Interest Rates and Payments
In Book 13: Financial Risk Assessment and Management with Six Sigma DMAIC Methods
ISBN: 978-620-5-49700-5
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates two stochastic models for estimating variable interest rates and loan payment schedules for 21 years. In the Independent Rates Model (IRM), the yearly interest rates are generated independently of one another. The rates are normally distributed with a given mean and standard deviation. In the Random Walk Model (RWM), only the first interest rate is normally distributed with a given mean and standard deviation, but each succeeding interest rate is normally distributed with a mean equal to the actual previous rate and the given standard deviation. The method that the interest rates are generated makes a substantial difference in the distribution of the total payment. Specifically, interest rates can vary much more in the Random Walk model, which causes a larger variation in the total payment. Compared to most typical loan payment schedules where all payments are equal, the schedules illustrated in this chapter pay a constant principal each year but have different yearly payments. To provide for a comprehensive analysis, five simulations are run, each with a different mean and standard deviation to calculate the interest rates.
Keywords: Risk Management, Banking, Loans Interest Rates & Payments, Stochastic Model, Six Sigma DMAIC, Monte Carlo Simulation, Basel III.
The Results
The five Total Net Payment simulation results of the two models, IRM and RWM, are presented below. The attained values for Total IRM Net Payment, Total IRM Net Payment Standard Deviation, Total RWM Net Payment, and Total RWM Net Payment Standard Deviation are given in Table 2.
Table 2: Five Simulations of IRM and RWM Results
|
Data |
Sim #1 |
Sim #2 |
Sim #3 |
Sim #4 |
Sim #5 |
|
Total IRM Net Payment |
$919,998 |
$972,498 |
$1,024,998 |
$1,077,497 |
$1,129,997 |
|
Total IRM Net Pay. SD |
$11,122 |
$12,512 |
$13,903 |
$15,293 |
$16,683 |
|
Total RWM Net Payment |
$919,998 |
$972,498 |
$1,024,998 |
$1,077,497 |
$1,129,997 |
|
Total RWM Net Pay. SD |
$101,699 |
$107,503 |
$113,306 |
$119,109 |
$124,913 |
Figure 1 graphically presented the five Total Net Payment simulation results of the two models.
The Total Net Payments of IRM and RWM are the same (i.e., they overlap in the graph). Total Net Payment starts at $919,998 in Sim #1 and then increases to $1,129,997 in Sim #5 with linear approximation.
However, the Total Net Payment Standard Deviations of the IRM and RWM models are quite different. The RWM Standard Deviation is 9.144 times greater than IRM Standard Deviation in Sim # 1, 8.592 times greater in Sim #2, 8.150 times greater in Sim #3, 7.788 times greater in Sim #4, and 7.487 times greater in Sim #5. Total Net Payment Standard Deviations start at $11,122 for IRM and $101,699 for RWM in Sim #1 and then increase to $16,683 and $124,913 respectively in Sim #5 with a linear approximation.