Kelly Criterion for Optimal Credit Allocation
Abstract
:1. Introduction
2. Conceptual Framework
2.1. Kelly Criterion
- (i)
- there is a continuing existence of the source of the probability signals during the allocation period;
- (ii)
- the information about the probability of default remains private at the time allocation decisions are made;
- (iii)
- capital can be divided into infinite amounts and reallocated; and
- (iv)
- the lender focuses on the long-term and keeps allocating capital, even when there are successive credit losses.
- (v)
- the credit allocation system allows fractional credit allocation by multiple lenders, with each providing a fraction of the full lending amount.2
2.2. Performance Measurement
2.3. Computational Implementation
3. Data
4. Empirical Results
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
1 | This is derived from the fact that equals , where is the compounding growth rate of capital after periods. The equations also exploit the facts that and where . |
2 | Essentially, we believe that given current advances in cost effective distributed computing technologies, building a lending platform that concurrently coordinates the issuance of small fractional loans from different lenders to the same borrowers is feasible from a technological perspective. |
3 | See Tran et al. (2021) for a discussion of the application of category theory to credit risk modeling. |
4 | In the bucketing process, PDs are put into different groups, or buckets, that establish how much a borrower could borrow and the credit charge. |
5 | Whilst the analytical process appears to take each borrower as being independent, the classification and prediction models, especially the machine learning models, are trained to recognize similarities and differences between borrowers in terms of their credit profile based on the set of features presented in the dataset. Thus, correlation and other similarity, if apparent, will be capitalized on by the training models when evaluating the probability of default, classify borrowers and make sound prediction as to the expected repayment amount. |
6 | By comparing the various approaches, the quality of the predictor can be checked. Results showed that the predicted results are close enough to the actual values across the models, warranting realistic and meaningful model comparison. |
7 | The exception is when rounding errors in the calculation of the Kelly fraction led to slight differences in the set of accepted borrowers, as in the case of the random forest and decision tree models. This happened because one scenario accepted borrowers that have a “non-default” classification status, while the other only accepted borrowers with a Kelly fraction greater than zero. When a borrower had a PD slightly greater than 0.5 but small enough to generate a zero Kelly fraction due to rounding, the borrower was present in one modeling scenario but not in the other. |
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Base | Kelly Strategy | System | |
---|---|---|---|
Data Process | Two samples were randomly extracted from the original dataset. One of the samples was used to train the models, the other for testing. This process was repeated 100 times. | The same datasets were used to train and test the model. | The testing datasets were used only. No classifier is trained in this approach. |
Model Selection | Nine standard classifiers available in the Mathematica software package: gradient boosted trees, logistic regression, Markov, nearest neighbors, random forest, support vector machine, neural network, decision tree and naive Bayes. | Identical classifiers were used. | No classifiers were used, with the results calculated directly from the actual data of funded amount and total payment. |
Accepted Borrowers/Allocation Strategy | All borrowers in the test sample received full funding. | In the testing sample, only borrowers with difference between non-default probability and default probability of at least 5% are accepted. An adaptive Kelly strategy was used for determining the fraction of credit allocated to these borrowers. | All borrowers in the test sample received full funding. Actual credit allocation decisions are as reflected in the test data. |
Performance Criteria | SCAP, ARC, AECE, AEVE and SEVE | SCAP, ARC, AECE and SEVE. | SCAP and ARC. |
Significant Tests | Yes, to compare SCAP and ARC to systems with Kelly strategy. H1: The Kelly strategy is no better than other strategies. H2: The Kelly strategy is better than other strategies. | Yes, to compare SCAP and ARC to systems without Kelly strategy. | Yes, to compare SCAP and ARC to systems with Kelly strategy. |
Payment Prediction | Yes, a predictive model was used to estimate loan payment. | Yes, a predictive model was used to estimate loan payment. | Yes, a predictive model was used to estimate loan payment for one approach (system). The other approach (system actual payment) used actual payment data to gauge how well the predictor worked. |
Sample Period | 2007–2018 |
Original Sample Size | 1,343,380 |
Filtered Samples | 1,074,704 |
Filtering Criteria | Reduction size, current loans, loans in grace period, and sample with one of the training features having no or abnormal value. |
Final Sample Size | 268,676 |
Default Sample | 53,638 |
Non-Default Sample | 215,038 |
Classes | “Default” and “Non-default” |
Number of Original Features | 115 |
Number of Final Features | 24 |
acc_now_delinq | Number of accounts on which the borrower is now delinquent. |
annual_inc | Self-reported annual income provided by the borrower during registration. |
chargeoff _within12mths | Number of charge-offs within 12 months. |
delinq_2yrs | The number of 30+ days past-due incidences of delinquency in the borrower’s credit file for the past 2 years. |
delinq_amnt | Past-due amount owed for the accounts on which the borrower is now delinquent. |
Dti | A ratio calculated using the borrower’s total monthly debt payments on the total debt obligations, excluding mortgage and the requested LC loan, divided by the borrower’s self-reported monthly income. |
home_ownership | Home ownership status provided by the borrower during registration or obtained from the credit report. |
inq_last_6mths | The number of inquiries in past 6 months (excluding auto and mortgage inquiries). |
loan_amnt | Listed amount of the loan applied for by the borrowers. |
funded_amnt | Actual amount received by the borrowers. |
open_acc | Number of open credit lines in the borrower’s credit file. |
pub_rec | Number of derogatory public records. |
pub_rec_bankruptcies | Number of public record bankruptcies. |
purpose | Category provided by the borrower for the loan request. |
revol_util | Revolving line utilization rate, or the amount of credit the borrower is using relative to all available revolving credit. |
term | Number of payments on the loan. Values are in months and can be either 36 or 60. |
tax_liens | Number of tax liens. |
total_acc | Total number of credit lines currently in the borrower’s credit file. |
verication_status | Indicates if income was verified (1) by LC or not verified (0). |
loan_status | Final status of loan has labelled outcome. “Non-default” for fully paid loans and “Default” for charged-off loans. |
emp_length | Number of years in employment represented by continues variable going from 0 to 10. |
earliest_cr_line | Number of years since the first credit line has been opened. |
fico_range_low | The low FICO value. |
fico_rangee_high | The high FICO value. |
Measures/Models | Logistic Regression | Nearest Neighbors | Random Forest | Gradient Boosted Trees | Decision Tree | Support Vector Machine | Markov | Naive Bayes | Neural Network |
---|---|---|---|---|---|---|---|---|---|
SCAP—System Actual Payment | 2.90% | 2.90% | 2.90% | 2.90% | 2.90% | 2.90% | 2.90% | 2.90% | 2.90% |
SCAP—System | 3.10% | 3.10% | 3.10% | 3.10% | 3.10% | 3.10% | 3.10% | 3.10% | 3.10% |
SCAP—Base | 3.60% | 3.20% | 3.40% | 3.40% | 3.80% | 3.20% | 3.30% | 4.20% | 3.60% |
SCAP—Kelly Strategy | 14.30% | 11.30% | 16.10% | 14.30% | 6.70% | 5.50% | 13.50% | 13.70% | 14.30% |
ARC—System Actual Payment | 91.643 | 91.643 | 91.643 | 91.643 | 91.643 | 91.643 | 91.643 | 91.643 | 91.643 |
ARC—System | 119.488 | 119.488 | 119.488 | 119.488 | 119.488 | 119.488 | 119.488 | 119.488 | 119.488 |
ARC—Base | 128.802 | 120.142 | 123.003 | 123.27 | 113.923 | 120.484 | 120.391 | 124.565 | 127.714 |
ARC—Kelly Strategy | 3763.89 | 650.865 | 1787 | 4821.15 | 209.044 | 207.813 | 1427.38 | 3216.45 | 3642.58 |
AEC—Base | 1715.76 | 1775.4 | 2302.67 | 1734.33 | 430.265 | 853.052 | 1705.24 | 991.763 | 1606.6 |
AEC—Kelly Strategy | 1715.76 | 1775.4 | 2295.6 | 1734.33 | 411.466 | 853.052 | 1705.24 | 991.763 | 1606.6 |
AEV—Base | 0.658 | 0.661 | 0.875 | 0.659 | 0.197 | 0.318 | 0.648 | 0.449 | 0.623 |
AEV—Kelly Strategy | 0.658 | 0.661 | 0.874 | 0.659 | 0.19 | 0.318 | 0.648 | 0.449 | 0.623 |
Measures/Models | Logistic Regression | Nearest Neighbors | Random Forest | Gradient Boosted Trees | Decision Tree | Support Vector Machine | Markov | Naive Bayes | Neural Network |
---|---|---|---|---|---|---|---|---|---|
SCAP—System Actual Payment | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 |
SCAP—System | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 |
SCAP—Base Models | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 |
ARC—System Actual Payment | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 |
ARC—System | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 |
ARC—Base Models | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 | <0.01 |
Measures/Models | Logistic Regression | Nearest Neighbors | Random Forest | Gradient Boosted Trees | Decision Tree | Support Vector Machine | Markov | Naive Bayes | Neural Network |
---|---|---|---|---|---|---|---|---|---|
SEVE—No Kelly Strategy | 0.055 | 0.048 | 0.039 | 0.052 | 0.196 | 0.103 | 0.051 | 0.095 | 0.058 |
SEVE—Kelly Strategy | 0.219 | 0.171 | 0.184 | 0.216 | 0.353 | 0.174 | 0.209 | 0.304 | 0.228 |
AECE—No Kelly Strategy | 0.075 | 0.068 | 0.053 | 0.072 | 0.267 | 0.145 | 0.071 | 0.128 | 0.080 |
AECE- Kelly Strategy | 2.165 | 0.369 | 0.776 | 2.769 | 0.510 | 0.248 | 0.833 | 3.163 | 2.222 |
Significance Test (p-value)—SEVE | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 |
Significance Test (p-value)—AECE | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 |
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Tran, S.; Verhoeven, P. Kelly Criterion for Optimal Credit Allocation. J. Risk Financial Manag. 2021, 14, 434. https://doi.org/10.3390/jrfm14090434
Tran S, Verhoeven P. Kelly Criterion for Optimal Credit Allocation. Journal of Risk and Financial Management. 2021; 14(9):434. https://doi.org/10.3390/jrfm14090434
Chicago/Turabian StyleTran, Son, and Peter Verhoeven. 2021. "Kelly Criterion for Optimal Credit Allocation" Journal of Risk and Financial Management 14, no. 9: 434. https://doi.org/10.3390/jrfm14090434