Table of Contents
Abstract
Acknowledgements
1. Introduction to Microfinance
1.1 Background
1.2 Concept
3. Life Time Analysis
3.1 Duration Data
3.1.1 Type I Censoring
3.1.2 Type II Censoring
3.1.3 Right-, Left- and Intervalcensoring
3.2 Duration Models
3.2.1 Hazard Function
3.2.2 Parametric Hazard Rate Models
3.2.3 Nonparametric Hazard Rate Models
4. Parameter Estimation
4.1 Maximum Likelihood
4.1.1 Likelihood function
4.1.2 Information matrix
4.2 Properties of Maximum Likelihood Estimators
4.3 Likelihood Ratio Test
5. Spandana Microfinance Model
5.1 Spandana Microfinance Institute
5.2 Evolution of the Spandana Model
5.1.1 Grameen Model
5.1.2 SHG Model
5.1.3 The Spandana Model
5.3Spandana Products and Services
5.4.1 Loans
5.4.2 Savings
5.4.2 Insurances
6. Life Time Analysis of Spandana Loans
6.1 Data
6.2 Model Selection
6.3 Regression Variables
6.4 Definitions and Assumptions
6.4.1 Discretization
6.4.2 Default, Prepayment and Non Default
6.5 Derivation of Likelihood Function
6.5.1 Time Independent Hazard
6.5.2 Time Dependent Hazard
6.6 Model Fitting
7. Implementations
7.1Data Collection
7.2 Maximization
7.2.1 Iterative Method
7.2.2 Linear Dependence of Binary Variables
7.3 Model Fitting
8. Results
7.1 Time Independent Hazard
7.2 Time Dependent Hazard
8. Conclusions
9. References
Abstract
Spandana Microfinance Institute has been issuing loans to poor women in the district of Guntur since 1998. The aim of this thesis is to determine how different attributes of a loan affect the probability of default.
By assuming a proportional hazards model, a maximum likelihood estimate of the regression parameters was found. The variances corresponding to the estimates were determined by construction of the observed Fisher-information matrix. Finally, the significance of the each regression parameter was established with a likelihood ratio test.
Due to problems such as linear dependence or near linear dependence in the data matrix and linear dependence of binary regression variables, the initial model had to be modified in various ways in order to attain accurate and reliable results. Modifications included both elimination and merging of regression variables.
Acknowledgements
1. Introduction to Microfinance
1.1 Background
[SAR03] The concept of micro finance is to provide financial services for the poor. Today, nearly 25% of India’s population falls into the category of poor or underprivileged. Their access to financial services from banks and other formal institutions is practically non-existing, and the lack of adequate credit facilities further limits the income generating potential of this low income group.
There are several reasons why banks do not consider the poor as potential clients and why the poor remain hesitant in approaching them. Firstly, the banks continue to rely on asset collaterals as a risk mitigation tool, which automatically excludes the poor from their target clientele. Secondly, the operational procedures for accessing credit are often complicated and tedious which the semi – illiterate/illiterate clients naturally find difficult to cope with. Finally, many of the banks are inconveniently located in the commercial centers of larger towns, complicating the accessibility for the low income rural and urban slum inhabitants. e group.
Shut out from the formal banking sector, the poor are left with informal money lenders for meeting their credit needs. This informal market is characterized by high interest rates and tough loan conditions, which often leads to dept traps and bankruptcy. A small part of this market is now being catered by Micro Finance Institutes (MFI:s), who realized that provision of access to credit plays an important role in reducing poverty.
The experiences from this sector have been very encouraging in terms of increasing demand and excellent repayment rates. The micro finance industry, which began in 1976 with the establishment of Grameen Bank in Bangladesh, is now a worldwide movement comprising thousands of specialist banks, credit unions, cooperatives, village credit societies and charities spanning both the richest and the poorest countries.
1.2 Concept
The common purpose of MFI:s is to extend the outreach of banking services, especially business credit, to those who do not qualify for normal bank loans. Having been established to meet the credit needs for the poor, MFI:s were faced with the task of developing systems and procedures for delivery of financial services to a low income clientele.
The risk involved in lending money without collateral was overcome by introducing the concept of group liability. By letting the borrowers form groups where each member is a co-guarantor of the other group members, a peer pressure is created within the group, which exerts similar risk mitigating effects as traditional asset collaterals. Effective risk reduction is also achieved by creating incentives for repayment through a promise of continued credit.
To suit client cash flows, repayments are usually made in small installments and at frequent intervals. This also enables the service providers to maintain regular contact with their borrowers for monitoring purposes. Furthermore, since the MFI:s are located near their customers, they are able to reduce administrative costs. These basic principles together with realistic loan pricing, enables them to cover their costs and still keep interest rates much lower than those charged in the traditional informal money lending sector.
3. Life Time Analysis
3.1 Duration data
[ELS96] In the analysis of time until failure, the variable of interest is the time that elapses from the beginning of some event until its end. This could for example be the time until failure of an electric component or the survival time after an operation. Observation data will typically consist of a sequence of durations . The process observed may have begun at different points in time. For example bank loans are often disbursed on different dates.
Lifetime data can often be incomplete in the sense that some observations are not known exactly. A common case is when the time measurement precedes termination of the event of interest, i.e. the measurement is made while the process is ongoing. This phenomenon is called censoring and is a common problem in the analysis of duration data. The two most frequent types of censoring are type I and type II censoring.
3.1.1 Type I censoring
Suppose units are placed under test for a period of time . Failure times of units are recorded. The test is terminated at with surviving units. The number of failures , is a random variable that depends on the duration of the test and the applied stress level.
The time at which the test is terminated is referred to as the test censoring time, and this type of censoring is referred to as type I censoring. A typical example can be drawn from medical research. Consider analyzing the survival times of heart transplant patients - at the time of measurement, observations on any individual still alive, is type I censored.
3.1.2 Type II censoring
Suppose units are placed under test and the failure times of failed units are recorded. The test continues to run until exactly failures occur, which implies that the test duration time will be a random variable. Since failures are specified in advance, the exact amount of data obtained from the test will also be known in advance.
In this type of censoring the censoring parameter is the number of failures and the censoring is referred to as type II censoring.
3.1.3 Right- Left- and Intervalcensoring
Both type I and type II censored data can be right-, left- or interval-censored. When a test on a number of units is stopped before all units have failed, observations of units surviving the length of the test, having , are referred to as right-censored. It is also possible that units may have failed prior to some initial time, having . This is called left-censoring. Units which fail at some unknown point between two observation times give rise to interval-censored observations. In practice, data which are classified as uncensored are in fact interval censored due to the inaccuracy of nearly all measurements.
3.2 Duration Models
3.2.1 Hazard Function
[WOL99] A question of interest in failure time analysis is how prone an item is to failure, having survived to time. The answer to the question is given by the hazard function, derived below.
Let be a non-negative, continuous random variable representing the duration time. Suppose further that has a continuous probability distribution, where is a realization of . The distribution function of will be given by (3.1)
The probability that the duration time is at least , is given by the survival function(3.2)
Consider the failure rate in a small interval of time . The unconditional probability that a unit fails in this interval is . For very small , this is approximately . Now let A be the event “surviving beyond ” and B the event “failing in ”. The probability that the unit fails in , given that it has not failed in is
The function , defined as (3.3)
, is called the hazard function and describes how likely the unit is to failure after a certain length of time.
3.2.2 Parametric Hazard Rate Models
[ELS96] The hazard function may take a variety of forms:
(i)Constant. The basic case in many analyses is a hazard function that does not vary with time. This is characteristic of a process that has no memory, since the conditional probability of failure in a given interval is the same regardless of when the observation is made. Assuming a constant hazard function , a simple differential equation is obtained from the earlier definitions (3.4)
With the terminal condition , the solution will be (3.5)
This survival function can be recognized as the exponential distribution, which is frequently used to model the time until failure of electronic components, due to its property of having no memory. An exponential life time distribution corresponds to “no ageing”.
(ii)Increasing. When the hazard function is an increasing function of t, the unit is subject to ageing through for example wear or accumulated damage. In practice, this is one of the most common cases.
(iii)Decreasing. This less common case may for example occur in manufacturing processes, where low-quality components are likely to fail early.
(iv)Bathtub-shaped. The most general case, where an initially decreasing hazard function (the burn-in phase) is followed by a fairly constant period and a finalwear-out phase, where the hazard rate increases. Often the different phases can be treated separately.
A distribution whose hazard function slopes upward is said to have positive duration dependence. For such distributions, the likelihood of failure at time t, conditional upon duration up to time t, is increasing in t. The opposite case, a decreasing hazard function, is called negative duration dependence. It might be uncertain whether the data in the analysis to be carried out, can be characterized by positive or negative duration dependence. In such cases, it is counterproductive to assume a distribution of that displays positive or negative duration dependence over the entire range of .
3.2.3 Nonparametric Hazard Rate Models
[ELS96] Nonparametric hazard rate models are used when time to failure is be dependent on external random variables – covariates. The covariates,, may be continuous or discrete and make the survival function conditional on the vector :
The nonparametric Proportional Hazards Model (PHM) was introduced by Cox in 1972.The model is said to be “distribution-free”, since no assumptions need to be made about the failure time distribution. The only condition imposed by the model is that the hazard functions for different devices when tested under different stress levels must be proportional to one another.
The proportional hazards model is given by (3.6)
, where
= the hazard rate at time for a device under test with regression variables
= regression variables (or covariates)
= regression coefficients
= the base-line hazard function
Common to nonparametric hazard models is the appearance of the base-line hazard function, which corresponds to the lifetime behavior for some standard or initializing condition. The regression variables account for the effects of external stresses on the hazard function. In the case of electronic components the regression variables may for example be temperature, humidity and voltage.
4. Parameter Estimation
[GRE00] The accuracy of an estimate of a set of parameters depends on both the sample size and the method used for estimating the parameters. In choosing between different inference methods, two aspects should be considered: its fitness for the problem of interest and the properties of the resulting estimate.
A good estimator should have the following properties:
- Unbiased. The estimator is an unbiased estimator for a parameter if and only if .
- Consistent. The estimator is said to be a consistent estimator if the probability of making errors tends to zero as the number of observations tends to infinity
as
- Efficient. An efficient estimator is a consistent estimator whose standard deviation is smaller than the standard deviation of any other estimator forparameter.
- Sufficient. A sufficient estimator is an estimator that utilizes all information about the parameter that the sample possesses.
4.1 Maximum Likelihood
4.1.1 Likelihood Function
[WOL99] Consider a random sample of observations from a distribution with density, where is the parameter to be estimated and indicates the sample data from observation. Since the observations are independent, their joint density is the product of the individual densities. This joint density function is called the likelihood function and is given by (4.1)
The likelihood function is proportional to the probability of getting the observed data under the model assumed. Since the likelihood function is globally concave,the value of the parameter for which the likelihood function assumes its maximum is obtained bysetting the derivativeof the likelihood function equal to zero (4.2)
The maximizing value, , is referred to as the maximum likelihood estimator (MLE). For simplicity reasons, one often prefers to maximize the logarithm of the likelihood function. Since the logarithm is a monotonic function, the value that maximizes is the same that maximizes. Thus, solving for below (4.3)
will result in the maximum likelihood estimates as well.
4.1.2 Information Matrix
[WEL96] The logarithm of the likelihood function can be used to construct the so-called Fischer information matrix. The inverse of this matrix is the covariance matrix of the maximum likelihood estimator.The information matrix , is constructed as follows (4.5)
When the expectation above is hard to find, the observed information matrixcan be used as well(4.6)
4.2 Properties of Maximum Likelihood Estimators
[GRE00] Maximum likelihood estimators are most attractive because of their asymptotic properties.Under regularity, the MLEs hold the following properties:
P1.Consistency
P2.Asymptotic normality:.
P3.Asymptotic efficiency
P4.Invariance: the maximum likelihood estimator of is .
4.3 Likelihood Ratio Test
[GRE00] Consider maximum likelihood estimation of a parameter and a test of the hypothesis . A commonly used test procedure for this kind of restriction is the likelihood ratio test, which isconstructed as follows:
Let be a vector of parameters to be estimated and specify some sort of restriction on these parameters. Let be the maximum likelihood estimator of without regard to the constraints, and let be the constrained maximum likelihood estimator. If and are the likelihood functions evaluated at these two estimates the likelihood ratio is (4.7)
Since a restricted optimum is never superior to an unrestricted one, lies between zero and one. The closer it is to one the more probable is .
The large sample distribution of gives the Likelihood ratio tests statistics: (4.8)
, where the degrees of freedom is equal to the number of restrictions imposed. The null hypothesis is rejected if this value is smaller than the critical value of the - distribution.
Matrix operties of Estimate5. Spandana Microfinance Model
5.1 Spandana Microfinance Institute
[SAR03] Spandana is a microfinance institute providing micro finance products to poor women in the district of Guntur, Andhra Pradesh, India. It was the first organization to start working in the slums of Guntur town in 1997-1998. Availability of credit at 10% interest rate without any collateral was very attractive for poor women in Guntur, where the prevailing interest rates varied between 60% and 300%. After five years of operation, Spandana is presently one of the fastest growing micro finance institutions (MFI) in India. With around 45 000 borrowers, it s not only among the largest MFI:s in the country, it is also one of the most efficient - a fact that enables the provision of low interest micro finance loans on the market.
Starting out in 1997 the organization included only two branches. With a continuous expansion since the start, approximately tripling the number of disbursed loans every year, the organization now includes seventeen branches covering a major part of the Guntur district.
5.2 Evolution of the Spandana Model
The evolution of the Spandanamicrofinance model is based on the Grameen model and the SHG model as described below.
5.2.1 Grameen Model
Some characteristic features of the Grameen bank credit delivery system are:
- Exclusive focus on the poorest of the poor.
- Organization of borrowers into homogeneous groups of five members.
- Very small collateral-free loans, repayable in weekly installments over one year.
- Eligibility for subsequent loans depends upon repayment of former loans.
- Transparency in all bank transactions, most of which take place at center meetings.
The group formation process takes place in stages. In the first stage only two of the five group members receive a loan. The group is observed for a month to see if the members are conforming to the rules of the bank. Only if so, do the other members of the group become eligible for a loan.
5.2.2 SHG Model
In India, the microfinance bank programs are dominated by Self-Help Groups (SHG:s). Such a group consists of five to twenty rural poor entrepreneurs, having homogeneous social and economic background. Members of an SHG have voluntarily come together to:
- Save small amounts regularly.
- Contribute to a common fund to meet their emergency needs.
- Pursue collective decision making
- Take collateral free loans at market driven rates.
5.2.3 The Spandana Model
Spandana’s model for credit delivery and recovery includes features from both the Grameen model and the SHG model. Initially, Spandana adopted the Grameen model in some branches and the SHG model in some branches. By drawing upon positive features in both of these models and incorporating various changes, the institute evolved a model of its own – the Spandana Model.