Statistics for Retail Finance 1
Overview > We now consider regulatory requirements for managing risk on a portfolio of consumer loans. Regulators have two key duties: 1. Protect consumers in the financial market place. 2. Provide the means for a stable and efficient financial market. Lenders must ensure they have sufficient capital to cover possible losses on a portfolio of loans. This is their capital requirement. 2
How much is a sufficient capital requirement? Well, the banks could cover every 1 lent with 1 of cash. o However, that would be an expensive and impractical solution. The usual solution is to cover some percentage of the loan value on the grounds that only a small percentage of the loans will go bad. o A usual figure in the past has been to have 8% of loan value as capital reserve. o This is a risk, however, since it is possible to under-estimate the number of bad loans. o Now, however, capital requirement is calculated based on measuring the risk of a portfolio of loans. 3
Risk on a portfolio of loans > We consider the loss on a portfolio (collection) of loans as a means to assess risk. Suppose we have loans. Then, Let { } denote default for observation ; Let denote loss given default (LGD) for observation ; Let denote exposure at default (EAD) for observation. LGD is the proportion of the loan that is recovered, whilst EAD is the amount exposed at default. Then individual loss is given as the product and total loss is 4
To understand risk, we need to develop a distribution for an estimate of total loss. This is not easy since separate models for each of the three components are required along with an understanding of their correlations: Default models are well-understood. However, models of LGD and EAD are still being developed. Although there is some evidence of a correlation between LGD and default, the size of the correlation is still not well-understood. 5
Expected Loss with fixed EAD and LGD > For this reason, often the loss is simplified by taking EAD and LGD as constant for all loans. Therefore, for all, we assume, for some fixed values. Define the default rate as Then, total loss is given by and expected loss (EL) on an individual is ( ) where ( ) is the probability of default (PD) for observation. 6
In the credit risk literature, this is usually written as Expected Loss = PD LGD EAD EL on a portfolio (collection) of loans is then given by where ( ) ( ) ( ) ( ) Therefore, EL can be computed using just a model of default, and PDs can be taken from a calibrated scorecard (eg using log-odds scores). 7
Variance of Default Rate > The EL does not tell us about the risk. Traditionally, in portfolio management, it is the variance that tells us about risk. We can compute variance on Default Rate: ( ) (( ) ) ( ( )) ( ) ( ) since. [ ( ) ( ( ) )] 8
Therefore ( ) [ ( ) ] where ( ). If we assume no covariance between observations, then this allows us to easily compute variance. However, this is not a reasonable assumption since defaults are likely to be correlated with each other. So we need to be able to estimate the covariance structure. 9
Value at Risk > Although variance gives a measure of risk, what is of most interest to lenders and regulators is that lenders can cover a large proportion of possible losses and in particular the extreme losses that might be experienced during a recession. For this reason, in credit risk management the focus is on estimating an upper quantile of the loss distribution. In Finance, this is known as the Value at Risk (VaR). Let be a cumulative distribution function on loss. Then, for a percentage ( ), ( ) ( ) Typical VaR are estimated at 99% or 99.9% levels. 10
So, VaR(99%) gives us an upper bound on 99% of possible losses. Therefore, it will only underestimate loss in 1% of cases. Please do not confuse VaR with Variance! 11
Example 8.1 This graph illustrates the loss distribution. It shows Expected Loss (EL) and VaR (99% level). VaR can be much larger than EL, especially when the loss distribution has a long right tail (this is usual). 0 5 10 15 20 Loss (in million) Loss distribution Expected Loss VaR(99%) 12
Economic and Regulatory Capital > Economic capital is the capital a bank deems necessary to run its business. It will cover the Expected Loss, whilst allowing for unusually high losses. This is a matter for internal bank policy. Regulatory capital (RC) is the capital that regulators believe the banks need to keep against unexpected losses. There may be a cross-over with economic capital, but the calculation of RC is imposed by financial regulators. Each country has its own regulatory authority (eg Federal Reserve in the USA and Financial Services Authority in the UK). However, most countries now follow the Basel III rules set out by the Bank for International Settlements (BIS) in Switzerland. 13
Basel III > The Basel III Accords is an international agreement on bank regulation. It is composed of three Pillars : Pillar I: Calculation of Regulatory Capital (RC). Pillar II: Supervisory procedure; ie the powers regulators should have in working with banks. Pillar III: Financial market discipline; this ensures banks disclose critical information to enable an efficient financial market. Most of Basel is about Pillar I, and that is also what we are interested in. 14
Basel III Regulatory Capital (Pillar I) > Basel III allows banks to calculate RC in one of three ways. Standard approach; Internal-ratings based (IRB) approach: foundational; Internal-ratings based (IRB) approach: advanced. The standard approach follows the traditional method of setting RC as a percentage of loan amount (Basel I). However, it is not quite so simplistic since different types of loans require a different percentage to be set aside based on risk levels of the loan type. Both IRB approaches allow banks to use their own risk models to determine RC. This is attractive since it allows knowledge about a specific credit product to inform the RC computation. Since models of default are so well-established in the industry, both approaches require banks to provide 15
estimates of PD. However, since LGD is less well-understood, the foundational approach allows banks to use a fixed value of LGD provided by the regulator. Additionally both approaches allow banks to use an estimated value for EAD. To summarize: RC approach Bank estimate required? PD LGD EAD Standard No No No IRB foundational Yes No May be used IRB advanced Yes Yes Yes The major banks have chosen to use one of the IRB approaches. 16
Basel IRB Regulatory Capital calculation > In Basel III, RC is simply calculated as the difference between the VaR and EL. It therefore covers all unexpected losses ; ie losses that could occur in extreme conditions. Therefore, ( ) ( ) Note that ( ) is expected to be covered by provisioning and pricing of the credit product by the lender. We can easily compute ( ), but a loss distribution is required to compute VaR. o As we have seen, this is a problem since we need to estimate the correlation structure between default events. 17
One Factor Model > We cannot realistically model all possible correlations between accounts. Instead, we assume portfolio invariance; ie that there is one factor that governs the correlation between default events. Additionally, we use Merton s (1974) assumption: A borrower will default if the borrower s assets fall below the value of debts. Suppose we want to measure risk of default for an individual over a period of time. For each borrower, Let be the fixed debt over that period. Let be fixed assets at the start of the period. Let period. be a random variable representing assets at the end of the 18
Assume that the difference in asset values is normally distributed: ( ) Then model this difference using a one factor model where ( ) is a common systematic factor, scaled by and ( ) is an idiosyncratic factor scaled by, such that and are independent and the s are independent, and. This is known as a Merton-type model. 19
Notice that this model assumes that changing assets is associated with a combination of A common systematic factor ; and An idiosyncratic factor which depends on the individual. The common factor represents changing economic or social conditions which are universal to everyone in the loan portfolio. 20
Asset Correlation > We are interested in the correlation coefficient between the assets for two different borrowers. This is given by ( ) It turns out this correlation coefficient is a constant. This is known as the asset correlation. Theorem 8.1. For all observations,. Proof. ( ) ( ) ( ) ( ) since and are independent. 21
Theorem 8.2. For all observations and such that, Proof. ( ) (( )( )) ( ) ( ) ( ) ( ) since ( ), ( ) and and are independent. Therefore, ( ), since each and are independent. The result then follows using Theorem 1. 22
PD in Merton s Model > Remember Merton s assumption: A borrower will default if assets fall below the value of debts. We can now use this to estimate PD. It is the probability that assets at the end of the period are less than the debt: ( ) ( ) ( ) ( ) where is the cumulative standard normal distribution, since ( ). 23
Conditional PD > PD is then derived conditional on the value of factor : ( ) ( ) ( ) ( ) ( ) ( ( ) ) Notice this is a latent variable model, since values of needed to extract PD. are not actually 24
Loss Distribution on PD > The loss distribution on PD is then constructed on the random variable ( ). All that is needed to get a specific loss distribution is 1. An estimate of usual PD; We can use a standard scorecard model to do that, and 2. An estimate of the asset correlation. In Basel III, VaR is computed at the 99.9% level (so RC is underestimated with only a 1/1000 chance). We adjust to account for that. Larger is linked to higher assets, therefore we take the lowest 0.1 percentile. Since ( ), this is ( ) ( ). 25
Therefore, the VaR on PD at 99.9% is ( ( )) ( ) ( ( ) ( ) ) Basel II also specifies asset correlations. For instance, =0.04 for revolving credit; =0.15 for residential loans. These asset correlations have been estimated based on historic data. By considering loss at the individual level, if we fix LGD and EAD, RC is then given by ( ) ( ) ( ( ) ) 26
RC (per unit of EAD) Statistics for Retail Finance Example 8.2 The following graph shows the shape of RC, per unit of EAD, with LGD=0.75. 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 PD Revolving credit Residential loans So, for instance, if a residential loan has PD=0.02, then RC= 0.117 for every 1 of the loan. 27
Criticism of Merton-type models for consumer credit > There have been several criticisms of the Merton model as used in consumer credit. 1. The Merton model was originally designed for corporate loans. In that context it makes sense. It was used for consumer loans because no satisfactory alternative models were available for consumer credit. 2. The Merton assumption may not apply in consumer credit. Do all borrowers wait until asset value falls below debt before defaulting? In particular, many defaulters are those that simply won t pay, even if they have assets. Indeed, evidence suggests that for individuals (rather than corporation) cash-flow affects default, rather than overall assets. 3. Estimates of asset correlation have been criticized. It is not clear how they have been estimated, and it is suggested they have been set too high. 28
Exercise 8.1 In the usual one factor model for consumer credit, difference in assets a borrower is modelled by for where ( ) is a common factor scaled by, ( ) is an idiosyncratic factor scaled by, and and are independent and the s are all independent. a) Show that the correlation coefficient between difference in assets and for two borrowers and is for all and such that, where. Suppose the one factor model is modified so that there is a fixed correlation between the idiosyncratic factors. That is, the correlation coefficient between and for all and such that is given by. 29
b) Show that with this modification, the correlation coefficient between difference in assets and for two borrowers and is for all and such that. c) Express the probability of default (PD) conditional on the realization of, ( ), in terms of the unconditional PD,, and. d) Fixing values of, and, describe the effect of different values of [ ] on ( ). In particular, what is ( ) at the extremes of the range: and? 30
Stress Testing > In Pillar II, regulators also have powers to stress test banks, to determine how they will respond to extreme recessionary circumstances. Regulators expect banks to be able to survive extreme, but plausible, economic conditions. There are three general approaches to stress testing: 1. Analytic: Use a parametric loss distribution to compute extreme risk; 2. Scenarios: Either historic scenarios or designed by economists; 3. Simulation: Based on a dynamic model of default. 31
The 2 nd and 3 rd approach both rely on models that allow inclusion of extreme conditions to determine how this changes Expected Loss. For example, a survival model of default could be built with macroeconomic variables as time varying covariates (TVCs). Then risk of default is then a function of macroeconomic conditions. The scenario approach (2 nd ) produces one point estimate of effect on Expected Loss for each scenario. The simulation approach (3 rd ) will use Monte Carlo simulation to generate a distribution of Expected Loss based on a multivariate distribution of plausible economic scenarios. The scenario approach has the advantage that it reflects realistic historical conditions. The simulation approach, however, is able to take account of plausible but not historic conditions. Something to think about: Is one economic crisis just like another???? 32
If a distribution-based approach is used (either 1 st or 3 rd methods above) then extreme loss is considered as the loss beyond VaR. This is called expected shortfall (ES) and defined as ( ) ( ( )) That is, ES is the expected value of loss in the most extreme highestpercentile of values. 33
This graph shows the loss distribution with scenarios and ES also included. 0 5 10 15 20 Loss (in million) Loss distribution Expected Loss VaR(99%) ES(99%) region ES(99%) Scenarios 34
Example 8.3 The FSA has published an anchor scenario which can be used for stress testing. It specifies a series of economic conditions over a period of years of a recession. Therefore, a bank would need to determine how their portfolio of loans would behave over those years. Unemployment GDP CRE HPI FTSE All-Share Year 1 8.9-2.1-21.7-10.1-18.7 Year 2 10.3-3.0-18.5-11.3-7.0 Year 3 11.3-0.1-3.8-7.1 3.8 Year 4 11.0 1.4-0.8-0.7 3.7 Year 5 10.3 1.8-4.3 0.4-5.9 CRE = Commercial Real Estate and HPI = House Price Index Note: Values for GDP, CRE, HPI and FTSE represent annual change. Source: www.fsa.gov.uk. These figures are correct as of January 2013. 35
Review of Chapter 8 > We have covered the following topics on regulation and capital requirements. Capital Requirements Loss Distribution Value at Risk Basel III Accord One Factor / Merton Model Stress Testing 36