moody’s analytics risk practitioner conference 2014 · csun scenario analysis specific shock...
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Moody’s Analytics Risk Practitioner Conference ‐ 2014
Quantitative Modeling Approaches for Mid‐size Institutions
Thomas L ThomasQuantitative Portfolio Manager
City National Bank
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Development of Commercial Stress Testing Process
1/20/2015 2
CNB Idiosyncratic Scenarios – Under Development
CRE PPR Model
Investor Properties – tested
Owner Occupied Properties –Under Development
Land & Construction –Under Development
Macroeconomic Scenario – “Fed Test "Bottom Up Industry Segment Macro Scenario – Moody’s
Top Down Macro Scenario CSUN
Scenario Analysis Specific Shock Testing
“Breakage Points” ‐General Sensitivity Analysis (Shock Testing) in CreditManager
Migration Analysis Spreadsheet +/‐ PD % (TARP Stress Testing)
Credit Manager Worst Case Transition Matrix We chose to take an incremental approach:
1. There are many stress testing techniques including:
• Shock Testing• Proxy Analysis• Scenario Analysis• Econometric Modeling
2. Staring with simpler models first and then moving to more complex econometric models
• Helps understand your data and loss behavior
• Sets parameters (boundaries' for later modeling)
• Acts as a challenge and reasonableness check
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Important to Map DFAST Process Flow
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As the graphic above illustrates, the credit capital stress testing modeling process consists of the following three main steps: • Data extraction and preparation• Applying stress test models• Export results to Fed reporting templates Helps understand potential data issues and explain the process and data flow to both regulators as well as senior management.
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Econometric Approach ‐ Regression Analysis
• Econometrics is a branch of economics that uses the “science and art of building models which utilizes a set of quantitative tools to construct and test mathematical representations of the real world.”
• More specifically the branch attempts to unify economics with math and statistics to make forecasts and empirically test economic theory.
• While the analyst may employ several different mathematical and statistical techniques to build his or her econometric model, regression is often the starting point.
• The regression is represented through a modification of the formula for a straight line by adding an error term (e): = a+B1X1+ e
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Regression ApproachesTwo general regression approaches: Introspective Counterfactual Queries and Prospective Counterfactual Queries.
• Introspective Counterfactual Queries ‐ The goal in this case is to determine the values that are driving the dependent variable or imply casual relationships.
• Here statistical methods are used to estimate the strength of the causal relationships. • The independent variables are said to be efficient when exhibiting good fit via statistical
measures like the adjusted R‐squared and statistically significant “F” and “t” tests.
• Prospective Counterfactual Queries ‐ Take the form what value would the dependent variable take if the independent variable X were set to a specific value. Such models are simply conditional expectation models that are more in the line of traditional curve fitting techniques.
• If the question we are trying to answer is one of conditional expectations, and not causal, then we do not have to be as concerned whether other causal factors are present.
• Measurements of goodness of fit, like R‐squared, Adjusted R‐squared, F and t statistics are sufficient to estimate the best fit line given a specific value of “X.
• Due to feedback issues in the independent variables – we believe a Prospective Counterfactual or “curve fitting” approach is best.
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Issues to Consider• Multicollinearity –
• In causal models is an implicit assumption in many regression models that the relationship between the independent (exogenous regression variables) are assumed to be independent.
• In addition, the relationship between the independent variables and errors between predicted values and the actual values are also assumed to be independent.
• These can be checked by looking at the sings in the regression coefficients and through test like variance inflation factor test (VIF) to diagnose potential multicollinear variables.
• Stationarity –• That is the independent variables do not change dramatically over time thereby
skewing the regression forecast. • Plotting trends and using histograms to view the independent variable’s
distribution help identify if the data is stationary. • If the data is not stationary – utilizing transposition techniques like “Tuckey’s
Ladder” can be used – the simplest is using the period‐over‐period change in the data for your regression.
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Choosing the Appropriate Horizon
It is important to choose the correct time horizon to model the behavior you are interested in examining.
The model above illustrates a regression analysis of predicted charge‐off rates using Federal Reserve data to proxy the impact to LGD during a stress event. Note the first analysis includes a long‐time horizon incorporating an expansionary
period between 2000 and 2005. This pulls the regression downward as a result the actual charge‐off rates exceed the regressions upper 95% confidence level.
When we only model the stressed recessionary period – the actual charge off rates and forecast now fit within our confidence level and mimic the correct expected behavior.
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Picking Independent VariablesComm.
Delinquency Nominal
GDP Growth
Real Disposable
Income Growth
Unemployment Rate
CPI Inflation
Rate
3-Month Treasury
Yield10-Year Treasury
YieldBBB Corporate
Yield
Dow Jones Total Stock
Market Index
Market Volatility
Index (VIX)House Price
Index
Commercial Real Estate Price Index
Ln 3-Month Treasury
Yield
Ln 10-Year Treasury
Yield
Ln BBB Corporate
YieldComm. Delinquency 1Nominal GDP Growth -0.3433 1Real Disposable Income Growth -0.3306 0.4913 1Unemployment Rate 0.9555 -0.19 -0.3141 1CPI Inflation Rate -0.2171 0.5125 0.3789 -0.1835 13-Month Treasury Yield -0.6818 0.2487 0.2423 -0.7253 0.1623 110-Year Treasury Yield -0.7184 0.1802 0.3042 -0.7955 0.1258 0.6355 1BBB Corporate Yield 0.0195 -0.7384 -0.3239 -0.145 -0.465 -0.0165 0.3292 1
Dow Jones Total Stock Market Index -0.0962 0.3358 0.2813 -0.1503 0.411 0.342 0.0198 -0.4909 1Market Volatility Index (VIX) 0.4638 -0.8034 -0.5257 0.3412 -0.5521 -0.4333 -0.3286 0.6681 -0.5194 1House Price Index -0.4465 0.2791 0.2096 -0.4167 0.2098 0.4702 0.1687 -0.2929 0.6338 -0.5115 1
Commercial Real Estate Price Index -0.0423 -0.1383 0.0235 -0.1451 0.1392 0.235 -0.1253 -0.0576 0.7393 -0.0585 0.7206 1Ln 3-Month Treasury Yield -0.8974 0.219 0.313 -0.9431 0.1868 0.8337 0.8313 0.1412 0.1587 -0.4135 0.4425 0.1316 1Ln 10-Year Treasury Yield -0.7054 0.1626 0.3002 -0.7822 0.1224 0.6156 0.9901 0.3322 0.0125 -0.3351 0.2018 -0.1078 0.8318 1Ln BBB Corporate Yield -0.0207 -0.695 -0.2822 -0.1893 -0.4156 0.0247 0.389 0.9952 -0.4733 0.6251 -0.2745 -0.0554 0.1946 0.3929 1
Correlation Matrix
• We chose to utilize only the Fed Scenario Variables – This kept it more manageable and reduced the need for additional forecasts.
• Use common sense when picking variables. For example if you are modeling commercial loan defaults, one would expect the house price index to add little value. Indeed, the housing index exhibited a low correlation 45% compared to 10‐year treasury which exhibited 72% correlation.
• Use a correlation matrix (something easily created in Microsoft Excel) to help choose relevant variables and confirm which variables to exclude.
Keep in mid the signs to make sure the are institutively correct with expected economic behavior.
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Example Stepwise RegressionCommercial Factors R2
3‐Month Treasury 68.39%10‐Year Treasury 58.24%VIX Index Only 26.78%Nominal GDP Growth 14.66%CPI Inflation Rate 10.24%Dow Jones Total Index Only 4.89%BBB Corporate Yield 3.35%
Consumer Factors R2
Unemployment Only 93.56%Real Disposable Income Only 14.90%
Multi‐Factor Regression R2 ChangeChange to Base
3‐Month, 10‐Year Treasury (BASE) 70.70%3‐Month, 10‐Year Treasury, VIX 72.28% 1.58% 1.58%3‐Month, 10‐Year Treasury, VIX,GDP 71.87% ‐0.40% 1.18%3‐Month, 10‐Year Treasury, VIX,GDP, DowJones 74.70% 2.82% 4.00%3‐Month, 10‐Year Treasury, VIX,GDP, DowJones,BBB 74.77% 0.07% 4.07%3‐Month, 10‐Year Treasury, VIX,GDP, DowJones,BBB, Comml RE 79.38% 4.62% 8.68%3‐Month, 10‐Year Treasury, VIX, Unemployment 96.33% 16.95% 25.64%3‐Month, 10‐Year Treasury, VIX, Unemployment, Dow Jones 97.27% 0.93% 26.57%10‐Year Treasury, VIX, Unemployment, Dow Jones 97.34% 0.07% 26.64%All Factors 97.97% 1.64% 27.27%
This is an example of stepwise regression utilizing Federal Reserve delinquency data as a proxy for PD behavior• Note in this process we first begin by regressing each of the individual “Fed”
scenario factors against delinquencies.• We then go through a process of combining various factors to obtain the “Best
Fitting Regression Line.”
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Testing the Results Scenario Change
020,000,00040,000,00060,000,00080,000,000100,000,000120,000,000140,000,000160,000,000180,000,000200,000,000
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9
Moody's Test Portfolio
2012Stress 2013Severe 2014Severe
18%
‐28%
‐18%
• When picking a vendor model – make sure you understand the impact from changing variables. • The analysis above shows the expected loss results from changes to the Fed Variables for the severely
adverse case.
• The graph above highlights the results from Moody’s Test Case Portfolio for each of the Severely Adverse Scenarios (Note in 2012 there was only a base case and stress case scenario provided by the Federal Reserve).
• The credit losses estimated by the RiskCalc Plus model were 18% lower for 2014 stress test compared to the 2013 stress test and 28% lower when compared to the 2012 stress case.
• The results suggest that in the 2014 Severely Adverse Scenario the economic factors are similar to the economic factors in the 2013 Severely Adverse stress test, but recover much more quickly over the last 5 quarters of the stress test. 10
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Macro Factor Comparison
• When we compare the macroeconomic variables utilized by RiskCalc Plus, we observe a overall improvement between the 2013 and 2014 Severely Adverse Scenarios.
• One would expect then to see some reduction in forecast losses. • For example, the Unemployment rate is substantially lower in the 2014 stress test
suggesting fewer firms are expected to default thereby reducing the number of people unemployed.
• A similar reasoning can be used for all of the above factors.
88.59
9.510
10.511
11.512
12.5
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
Unemployment
2013 2014
0
0.5
1
1.5
2
2.5
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
10Y Treasury
2013 2014
3
3.5
4
4.5
5
5.5
6
6.5
7
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
BBB Corporate
2013 2014
6000
8000
10000
12000
14000
16000
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
DowJones Index
2013 2014
0102030405060708090
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
VIX
2013 2014
3.00
3.50
4.00
4.50
5.00
5.50
6.00
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
BBB 10Yr Treasury Spread
2013 2014
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Understand the impact to internal variables ‐ CNB Balance Sheet Mix Example
9/30/2008 9/30/2009 9/30/2010 9/30/2013LOAN
OUTSTANDING TOTALS
LOAN OUTSTANDING
TOTALS
LOAN OUTSTANDING
TOTALS
LOAN OUTSTANDING
TOTALS
1 34,275.5 42,304.1 38,961.4 90,022.4 47,718.32 726,686.5 785,059.3 708,929.8 1,794,998.4 1,009,939.13 893,349.7 1,019,598.1 806,631.2 1,360,747.4 341,149.44 1,982,097.6 1,536,847.1 1,242,879.8 2,698,935.4 1,162,088.25 2,462,053.8 1,785,202.8 1,773,974.3 3,269,869.0 1,484,666.26 1,262,103.9 1,106,293.5 1,277,892.1 1,406,674.5 300,381.17 392,855.2 437,795.9 255,081.8 109,041.8 (328,754.1)8 389,012.7 1,023,939.5 849,355.2 342,189.2 (681,750.3)9 81,364.7 101,476.4 25,743.5 2,732.1 (98,744.3)PG 4,007,607.6 4,288,463.1 4,392,936.0 5,265,791.3 977,328.2
Not rated** 47,011.8 41,510.5 46,157.1 225,131.3 183,620.8Total $12,278,419.1 $12,168,490.2 $11,418,542.4 $16,566,132.8 $4,397,642.5
9/30/2008 9/30/2009 9/30/2010 9/30/2013
LOAN % TOTALS LOAN % TOTALS LOAN % TOTALS LOAN % TOTALS
1 0.28% 0.35% 0.34% 0.54% 0.20%2 5.92% 6.45% 6.21% 10.84% 4.38%3 7.28% 8.38% 7.06% 8.21% -0.16%4 16.14% 12.63% 10.88% 16.29% 3.66%5 20.05% 14.67% 15.54% 19.74% 5.07%6 10.28% 9.09% 11.19% 8.49% -0.60%7 3.20% 3.60% 2.23% 0.66% -2.94%8 3.17% 8.41% 7.44% 2.07% -6.35%9 0.66% 0.83% 0.23% 0.02% -0.82%
PG 32.64% 35.24% 38.47% 31.79% -3.46%Not rated** 0.38% 0.34% 0.40% 1.36% 1.02%
Total 100.00% 100.00% 100.00% 100.00% 0.00%
Difference between Peak
Loss Exposure 2009 and 2013
Difference between Peak
Loss Exposure 2009 and 2013
Risk Rating
Risk Rating
1 0.35% 2
6.45%
3 8.38%
4 12.63%
5 14.67%
6 9.09%
7 3.60%
8 8.41%9
0.83%
PG35.24%
Not rated**0.34%
Outstanding Balances by Risk GradeSeptember 2009
1 0.54%
2 10.84%
3 8.21%
4 16.29%
5 19.74%
6 8.49%
7 0.66%
8 2.07%
9 0.02%
PG31.79%
Not rated**1.36%
Outstanding Balances by Risk GradeSeptember 2013
• The Severely Adverse stress test loss rates (2.95% vs. 2.40%) are approximately 19% lower that what CNB experienced over the recent economic downturn.
• CNB’s current loan portfolio balance mix is a major contributing factor to the expected decline in credit losses.
• The above tables and pie charts compare CNB’s book balances between our worst year 2009 and the most recent quarter‐end used in our stress Test (September 2013).
• Note that CNB has experienced a significant improvement in credit quality in grades 6,7,8,9 compared to the period just preceding 2009‐2010.
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Use External Comparisons for Reasonable Testing ‐ Loss Scalars Example
First Lien Mortgages 2.35% 6.05% 2.58 0.12% 0.30% 2.48 ELC and Lien Mortgages 3.48% 9.40% 2.70 0.81% 3.88% 4.82 Commercial & Industrial Loans 3.29% 6.50% 1.98 3.53% 3.37% 0.95 Commercial Real Estate Loans 5.94% 7.85% 1.32 5.33% 2.80% 0.53 Credit Cards 13.27% 15.85% 1.19 4.22% 8.78% 2.08 Other Consumer 3.65% 4.10% 1.12 3.78% 4.94% 1.31
Average 1.82 Average 2.03
Loss Scalar Analysis ‐ Actual Losses Vs. Stress Test Loss Rates
CNB 2014 Severly Adverse Case
Loss Scalar
ASSET CLASSIFICATION
Median 2‐Year Worst Case Loss CCAR Banks 2009‐2010
2013 CCAR Median Average Loss Rates
Loss Scalar
CNB Loss Rate 2009‐2010
• One way to measure the reasonableness of our loss projection is to compare our loss rates to those of the CCAR Stress Test.
• In this comparison we would expect our loss rates to be similar in scale to those experienced by the CCAR banks that have been performing macro economic stress testing over the last several years and have much more modeling experience.
• Note on average the CCAR Stress Test Loss rates are generally higher than what the CCAR banks experienced over the stress periodbetween 2009 – 2010. This suggest the CCAR stress scenarios are more severe than what was experience in the recent economic downturn.
• Our Severely Adverse Stress Test scenario suggest we may experience losses that are on average 2 times greater than what CNB experienced between 2009‐2010 compared to 1.8 times for the CCAR banks.
• The reduced loss scalar for CRE is primarily attributable to the reduced concentration of ADC loans
• This result suggest our Severely Adverse credit loss forecast is reasonably scaled up for such a severe economic event.
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Impact Analysis
• One way to estimate the impact of the balance sheet mix and the changing scenarios is to run the period‐end balance sheet through the Moody’s model for the same historical time period and same stress scenario and measure the differences.
• This acts like a controlled experiment where we change one variable – in this case balances or scenario, hold everything else, constant and then measure the impact.
• To keep as many variables constant as possible, we utilized the same PDs and LGDs in each test. • In addition, we utilized the 2013 Severely Adverse Scenario because we began this test before the Fed
released the 2014 scenarios.
• We also tested at a total portfolio level since GL and Fed Class code roll‐ups were different between the two periods.
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Total 2 Yr Loss Rate Delta2009 Historical (2009 Balances) 44,022 37,418 33,179 33,023 29,833 26,027 22,802 21,601 23,917 271,822 2.23%2009 Historical (2013 Balances) 33,517 28,476 25,482 26,065 23,839 20,949 18,533 17,871 20,261 214,992 1.77% ‐20.91%
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Total 2 Yr Loss Rate DeltaCCARSevere2013 (2009 Balances) 50,136 60,849 73,421 80,705 85,746 83,042 75,955 67,066 57,403 634,324 5.21%CCARSevere2013 (2013 Balances) 40,597 54,791 68,400 80,178 83,892 78,333 69,069 59,101 49,522 583,883 1.77% ‐7.95%
Balance Issue Risk Rating
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Challenger Model
• Another way to examine the reasonableness of the stress test results is to compare the Moody’s results against our CSUN Challenger Model.
• The CSUN Challenger Model develops PD scalars base on Federal Reserve delinquency data. 1
• While the quarterly results are somewhat different since the two models employ different modeling variables, the forecast credit losses and loss rate are generally within the same overall range.
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Graphical Comparison Severely Adverse Scenario
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
9.00%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
Unemployment
StressPDUnEmp+1STD BaseStress StressPDUnEmp‐1STD
3.00%3.50%4.00%4.50%5.00%5.50%6.00%6.50%7.00%7.50%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
10‐Year Treasury
StressPD10Y+1STD BaseStress StressPD10Y‐1STD
3.00%3.50%4.00%4.50%5.00%5.50%6.00%6.50%7.00%7.50%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
BBB
StressPDBBB+1STD BaseStress StressPDBBB‐1STD
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
DJX
StressPDDJX+1STD BaseStress StressPDDJX‐1STD
2.50%
3.50%
4.50%
5.50%
6.50%
7.50%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
Vixhigh
StressPDVixhigh+1STD BaseStress StressPDVixhigh‐1STD
• The graphs above illustrated the results to a +/‐ one standard deviation change in each individual macro variable. • Unlike the Base Case which exhibits generally linear downward trends, as one would expect the resulting PDs
rapidly increase in the first part of the stress scenario, peak around the mid part, and then decline during the remaining forecast quarters.
• All factors exhibit a doubling in absolute PD rates as the stress is increased; however, the volatility of the spread to a one standard deviation change is greater for some of the factors, such as unemployment and DJX.
• The model also exhibits a more stable relationship for the 10‐year Treasury / BBB Credit Spread the VIX index in that the volatility around these measures narrows slightly compared to the base case.
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Sensitivity Analysis• We also conducted a Sensitivity Analysis in order to analyze the model’s results
and determine if the model results were directionally consistent with our expectation for a given change in a macroeconomic variable input.
• Like the impact analysis above we conducted a controlled experiment where we changed one macro variable one standard deviation up or down while holding all the other independent variable in the model constant.
• The independent variables we tested were the independent macro variables employed in the vendor model. They include:
o Unemployment Rate
o BBB Bond Rate
o 10 – Year Treasury Rate
o VIX Index (VIXhigh)
o Dow Jones Index (DJX)
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• Although the BBB bond rate is a macroeconomic variable employed in the Moody’s model, the Moody’s model actually measures the “credit spread” between the BBB bond rate and the 10‐year treasury rate. 2
• If the bond rate increases relative to 10‐Year Treasury, the credit spread widens suggesting the market is pricing for perceivedincreased credit risk. Hence we would expect an increase in PDs.
• Conversely, if BBB decreases relative to 10‐Year Treasury, the perceived risks are lower and PDs decline. • As one would expect a one standard deviation increase in the BBB rate results in an increased PD forecast by the Moody’s
model. • However, the results between a one standard deviation change upward Vs. a one standard deviation downward in the BBB rate
are asymmetrical – The model exhibits a larger change in PDs for a one standard deviation decrease that by a one standard deviation increase by roughly 0.59 times.
• This suggests that in the Moody’s Model, BBB market rates are slightly more sensitive to downward default rates and re‐pricing the credit spread then when defaults increase.
1.50%
2.00%
2.50%
3.00%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
BBB
StressPDBBB+1STD BaseStress StressPDBBB‐1STD
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4StressPDBBB+1STD 2.90% 2.78% 2.63% 2.51% 2.38% 2.25% 2.15% 2.05% 1.94%BaseStress 2.79% 2.62% 2.47% 2.35% 2.23% 2.10% 2.00% 1.91% 1.81%StressPDBBB‐1STD 2.62% 2.36% 2.21% 2.10% 1.97% 1.83% 1.74% 1.66% 1.55%
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Sensitivity Analysis ‐ BBB Bond Rate Example
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Graphical Comparison Severely Adverse Scenario
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
9.00%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
Unemployment
StressPDUnEmp+1STD BaseStress StressPDUnEmp‐1STD
3.00%3.50%4.00%4.50%5.00%5.50%6.00%6.50%7.00%7.50%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
10‐Year Treasury
StressPD10Y+1STD BaseStress StressPD10Y‐1STD
3.00%3.50%4.00%4.50%5.00%5.50%6.00%6.50%7.00%7.50%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
BBB
StressPDBBB+1STD BaseStress StressPDBBB‐1STD
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
DJX
StressPDDJX+1STD BaseStress StressPDDJX‐1STD
2.50%
3.50%
4.50%
5.50%
6.50%
7.50%
2012Q4 2013Q1 2013Q2 2013Q3 2013Q4 2014Q1 2014Q2 2014Q3 2014Q4
Vixhigh
StressPDVixhigh+1STD BaseStress StressPDVixhigh‐1STD
• The graphs above illustrated the results to a +/‐ one standard deviation change in each individual macro variable. • Unlike the Base Case which exhibits generally linear downward trends, as one would expect the resulting PDs
rapidly increase in the first part of the stress scenario, peak around the mid part, and then decline during the remaining forecast quarters.
• All factors exhibit a doubling in absolute PD rates as the stress is increased; however, the volatility of the spread to a one standard deviation change is greater for some of the factors, such as unemployment and DJX.
• The model also exhibits a more stable relationship for the 10‐year Treasury / BBB Credit Spread the VIX index in that the volatility around these measures narrows slightly compared to the base case.
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