Vanna-Volga, Option Pricing, Normal Bachelier model, Option Greeks, Delta, Vega, Vanna, Volga, Volatility Smiles, Volatility Frowns, SABR model, Interest Rates
Vanna-Volga is a popular method for the interpolation/extrapolation of volatility smiles. The technique is widely used in the FX markets context, due to its ability to consistently construct the entire Lognormal smile using only three Lognormal market quotes.
However, the derivation of the Vanna-Volga method itself is free of distributional assumptions. With this is mind, it is surprising there have been no attempts to apply the method to Normal volatilities (the current standard for interest rate markets).
We show how the method can be modified to build Normal volatility smiles. As it turns out, only minor modifications are required compared to the Lognormal case. Moreover, as the inversion of Normal volatilities from option prices is easier in the Normal case, the smile construction can occur at a machine-precision level using analytical formulae, making the approximations via Taylor-series unnecessary.
Apart from being based on practical and intuitive hedging arguments, the Vanna-Volga has further important advantages. In comparison to the Normal SABR model, the Vanna-Volga can easily fit both classical convex and atypical concave smiles (‘frowns’). Concave smile patterns are sometimes observed around ATM strikes in the interest rate markets, particularly in the situations of anticipated jumps (with an unclear outcome) in interest rates. Besides, concavity is often observed towards the lower/left end of the Normal volatility smiles of interest rates. At least in these situations, the Vanna-Volga can be expected to interpolate/extrapolate better than SABR.
Multi-year, Lifetime, Probability of Default, PD, Default Rates, Rating Transition Matrices, IFRS 9, Expected Credit Losses, ECL, Through-the-Cycle, TTC, Point-in-Time, PIT, Macroeconomic Adjustments, Time Series, Autoregression, Accounting, Financial Instruments, Maximum Likelihood Estimation
In banking practice, rating transition matrices have become the standard approach of deriving multi-year probabilities of default (PDs) from one-year PDs, the latter normally being available from Basel ratings. Rating transition matrices have gained in importance with the newly adopted IFRS 9 accounting standard. Here, the multi-year PDs can be used to calculate the so-called expected credit losses (ECL) over the entire lifetime of relevant credit assets.
A typical approach for estimating the rating transition matrices relies on calculating empirical rating migration counts and frequencies from rating history data. For small portfolios, however, this approach often leads to zero counts and high count volatility, which makes the estimations unreliable and unstable, and can also produce counter-intuitive prediction patterns such as non-parallel/crossing forward PD patterns.
This paper proposes a structural model which overcomes these problems. We make a plausible assumption of an underlying autoregressive mean-reverting ability-to-pay process. With only three parameters, this sparse process can well describe an entire typical rating transition matrix, provided the one-year PDs of the rating classes are specified (e.g. by the rating master scale).
The transition probabilities produced by the structural approach are well-behaved by design. The approach significantly reduces the statistical degrees of freedom of the estimated transition probabilities, which makes the rating transition matrix more reliable for small portfolios. The approach can be applied to data with as few as 50 observed rating transitions. Moreover, the approach can be efficiently applied to data consisting of continuous PDs (prior to rating discretization).
In the IFRS 9 context, the approach offers an additional merit: it can easily account for the macroeconomic adjustments, which are required by the IFRS 9 accounting standard.
Prediction, Probability of Default, PD, Default Rates, Through-the-Cycle, TtC, Point-in-Time, PiT, Credit Portfolio Model, Systematic Factor, Macroeconomic Factor, Time Series, Autoregression, Bayesian Analysis, IFRS 9, Accounting, Financial Instruments, Lifetime, Expected Credit Losses
This paper proposes a simple technical approach for the analytical derivation of Point-in-Time PD (probability of default) forecasts, with minimal data requirements. The inputs required are the current and future Through-the-Cycle PDs of the obligors, their last known default rates, and a measurement of the systematic dependence of the obligors. Technically, the forecasts are made from within a classical asset-based credit portfolio model, with the additional assumption of a simple (first/second order) autoregressive process for the systematic factor. This paper elaborates in detail on the practical issues of implementation, especially on the parametrization alternatives.
We also show how the approach can be naturally extended to low-default portfolios with volatile default rates, using Bayesian methodology. Furthermore, expert judgments on the current macroeconomic state, although not necessary for the forecasts, can be embedded into the model using the Bayesian technique.
The resulting PD forecasts can be used for the derivation of expected lifetime credit losses as required by the newly adopted accounting standard IFRS 9. In doing so, the presented approach is endogenous, as it does not require any exogenous macroeconomic forecasts, which are notoriously unreliable and often subjective. Also, it does not require any dependency modeling between PDs and macroeconomic variables, which often proves to be cumbersome and unstable.
Default Prediction, Financial Ratios, Liquidity, Solvency, Limited Liability Companies
In this study, we propose corporate default prediction models based on aggregated financial ratios and compare them with traditional models that draw on a pool of classical accounting ratios. The basis of the comparison is 22 selected balance sheet and income statement items, from which 47 traditional liquidity and solvency ratios are built. The aggregated and traditional ratios are calculated and analyzed for 6,174 financial statements of unlisted German companies with limited liability (GmbH), with reporting years spanning from 1997 to 2002. Of the companies included, 442 filed insolvency petitions in the period from 1999 to 2004. The aggregated financial ratios are built via simple weighting and division of accounting items. These ratios are optimized using maximum-likelihood methodology and are compared using a classical logit stepwise regression with traditional financial ratios. It becomes apparent that aggregated financial ratios reduce the problem of overfitting and improve the interpretation of estimated coefficients. In addition, the predictive out-of-sample performance measured by the standard accuracy ratio can be slightly improved when contrasted with comparable studies.
Default Prediction, Bankruptcy Prediction, Reduced Models, Non-Structural Models, Accounting Ratios, Market Ratios, Non-Classical Ratios, Missing Values, Logistic Regression, Variable Selection, Stepwise Selection, Lasso, L1 Regularization
This study pursues two issues in the context of reduced-type prediction of corporate bankruptcies. Firstly, we investigate the marginal usefulness of some non-classical ratios and indicators. We find that the performance of classical accounting and market ratios can be significantly improved by their squared terms and technical ratios built from itemized accounts. Also, missing values of many accounting items are proven to be informative.
Secondly, as the non-classical ratios dramatically increase the number of candidate variables, the variable selection becomes particularly important. We review various traditional selection methods and investigate the innovative lasso regularization technique. We show that lasso performs for these purposes better than the popular stepwise selection, and requires considerably less computational time. Furthermore, lasso proves to have a few specific advantages in the investigated context, e.g. being able to efficiently identify the non-classical ratios which complement (rather than replace) classical ratios.
Overall, the non-classical information used in this study increased the out-of-sample accuracy ratio by 1.9%. When applying the lasso instead of stepwise, this advantage increased to 2.3%.
Insolvency Prediction, Bankruptcy Prediction, Credit Ratings, Regression, Non-Linear, Lasso, Regularization, Variable Selection, Germany, USA, Ukraine, Financial Statements, Financial Ratios, Compustat
This PhD thesis has two main emphases. First, the author constructs a default prediction model for Ukrainian corporations. No previous advanced studies had previously investigated this issue, although the necessary data are publicly available, at least for Ukrainian Open-Joint-Stock companies. For this modeling, the author used the standard default-prediction methodology. However, some specific issues had to be accounted for. To better reflect Ukrainian specifics, some non-traditional ratios and indicators were investigated (and found to be useful). Also, the market information had to be ignored as it proved unreliable here.
The second emphasis in the PhD project at hand was made on the variable selection problem in the context of default prediction modeling for developed market economies. The modeling methodology being quite established and advanced here, still, no universal method for selecting explanatory variables (ratios, indicators) exists. The author proposes and investigates two methods for handling this problem. The first method is motivated economically and consists in aggregating the information from numerous accounting ratios into one single ratio. This ratio is intuitively built as weighted assets and revenues divided by weighted liabilities and expenses. This specification captures information contained in many traditional ratios from the categories ‘liquidity’ and ‘solvency’. The aggregated ratio was shown to be less susceptible to the problem of data overfitting than the traditional stepwise. The method can be extended to other categories, such as profitability.
The second selection technique proposed by the author is rather statistical, and consists in applying the innovative lasso regularization methodology. The author uses the method to select ratios from a large universe of explanatory ratios (built from Compustat accounting and market data). The method was shown to perform better than the traditional stepwise selection in terms of cross-validated prediction accuracy. Besides, lasso can be easily tuned to implement a hierarchy of explanatory variables. This feature was exploited to identify non-traditional ratios which complement (rather than displace) the traditional ratios.
All presented models achieved high degrees of prediction power. They can be used for real-life default prediction for Ukrainian and US corporations and German GmbH’s. All in all, the results of this PhD thesis indicate that there are still open areas in the field of single-obligor default modeling. In particular, the models may have to be adjusted considerably when extended to new emerging markets. Also, even for developed economies, the issue of optimal selection of explanatory variables is not yet resolved. Just two particular solutions (out of possibly many) consist in aggregating the candidate ratios into a single ratio, or in applying new innovative statistical selection methods such as lasso.
Insolvency Prediction, Bankruptcy Prediction, Discriminant Analysis, Conditional Probability, Logistic Regression, Survival Analysis, CUSUM Charts, Artificial Neural Networks
1. Introduction
2. Bankruptcy Prediction as a Classification Problem
2.1 Bankruptcy Prediction Models
2.2 Structural vs. Reduced Models and Explanatory Variables
2.3 Collinearity Issues
2.4 Sampling Considerations
2.5 Misclassification Costs
2.6 Measures for Model Performance
2.6.1 General
2.6.2 Performance Measures for Models with Probabilistic Output
2.6.3 Rank Correlation Measures
3. Discriminant Analysis
3.1 Discriminant Analysis as a Classification Technique
3.2 Bayesian Approach
3.2.1 Class membership as posterior probability
3.2.2 Assumption of multivariate normality
3.2.3 Distributions other than multivariate normal
3.3 Discriminant Functions Approach
3.3.1 General
3.3.2 Measures for importance of explanatory variables
3.3.3 Testing for mean differences
3.3.4 Importance and significance of discrimination functions
3.4 Stepwise Variable Selection
3.5 Sampling Considerations
3.6 Misclassifications Costs
3.7 Strengths vs. Weaknesses
3.8 Applications for Bankruptcy Prediction
4. Conditional Probability Models
4.1 General
4.2 Microeconomic Derivation
4.3 Model Estimation
4.4 Link to Discriminant Analysis
4.5 Significance Testing
4.5.1 Nested models and hypothesis testing
4.5.2 Test for overall significance of logit coefficients (omnibus test)
4.5.3 Wald test for linear restrictions and t-tests
4.5.4 Lagrange multiplier test
4.5.5 Raftery test
4.5.6 Confidence intervals
4.6 Goodness-of-Fit (GOF)
4.6.1 Mean loglikelihood
4.6.2 Saturated model and deviance
4.6.3 Categorical independents: Pearson and Deviance GOF
4.6.4 Hosmer-Lemeshow GOF test
4.6.5 Box-Tidwell nonlinearity test
4.6.6 Quasi-R²goodness-of-fit measures
4.6.7 Overdispersion
4.7 Variable Selection
4.8 Sampling Considerations
4.9 Extensions
4.9.1 Interaction and quadratic terms
4.9.2 Multinomial models
4.9.3 Mixed logit model
4.10 Collinearity Issues
4.11 Misclassification Costs
4.12 Strengths vs. Weaknesses
4.13 Applications for Bankruptcy Prediction
5. Survival Analysis
6. CUSUM Charts
7. Artificial Neural Networks
8. Some Other Techniques
9. Bankruptcy Prediction Models in Germany
10. Bankruptcy Prediction in Ukraine
11. Summary and Conclusions
Ukrainian Accounting Standards, UAS, IAS, IFRS, Financial Accounting, Financial Reporting
Die Entwicklung international anerkannter Rechnungslegungsstandards ist ein wesentlicher Bestandteil des ukrainischen Transformationsprozesses. Bei der Entwicklung ukrainischer Rechnungslegungsstandards (Ukrainian Accounting Standards, UAS) dienen bzw. dienten die IAS/IFRS als Vorbild. Dieser Beitrag bietet eine Einführung in die für kleine und mittlere Unternehmen relevanten UAS.
Der erste Abschnitt „Historische Entwicklung der Rechnungslegung in der Ukraine“ gibt zunächst einen kurzen Überblick über die Entwicklung der ukrainischen Rechnungslegung. Im folgenden Abschnitt „Grundlagen der handelsrechtlichen Rechnungslegung in der Ukraine“ wird auf die institutionellen Aspekte der handelsrechtlichen Rechnungslegung eingegangen. Der nächste Abschnitt vermittelt einen Überblick über die einzelnen Bestandteile eines Jahresabschlusses nach UAS. Die Ansatz- und Bewertungsvorschriften nach UAS werden schliesslich im letzten Abschnitt erläutert.