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Chapter Sixteen

Quantitative finance

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Learning objectives

1. Create a framework for modelling.

2. Explain and measure concentration risk

3. Define expected losses

4. Define and measure probability of default

5. Define and measure Loss Given Default

6. Define and measure prepayment risk

7. Identify the problems with quantitative modelling

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Introduction

• Lending is now driven by two important factors:

• Increasingly technologically driven statistical methodologies in lending decisions.

• The requirements of Basel Capital requirements and, in particular, the underlying assumptions of capital adequacy when using advanced methods

• It is important to understand the underlying assumptions.

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• There are two underlying assumptions for advanced methods in capital adequacy (and thus lending models):

• Granularity: Basel assumes perfect granularity meaning completely diverse lending portfolios. If this condition is not held, then advanced methods may be biased. Concentration risk will violate granularity

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• Asymptotic Single Risk Factor (ASRF) model: the model is defined below

but suggests that a lending portfolio would be affected by a single factor,

with the most obvious being interest rates. This condition is normally

violated by lenders with a large geographical spread.

• The model is as follow:

• V is the value of the assets of lender i at time t

• M is the systematic risk

• Z is the unsystematic risk

• ρ is the risk attached to the systematic risk

ititiit ZMV 21

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Concentration risk

• There are two main types of concentration risk:

• Name risk: This is where there is a large exposure to one ‘name’ such as a company group. This violates granularity.

• Sector risk: This is where lending may be concentrated in sectors or geographical jurisdictions. This violates ASRF assumptions.

• Due to the violations, models should indicate capital that should be put aside.

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• Most models indicate concentration but not the amount of capital required. The most common model is Hefindahl-Hirschman Index (HHI) which is defined as:

 

Where:

• H is the HHI

• s is the proportion of each firm’s loans to the overall portfolio

• n is the number of loans

n

iisH

1

2

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• There have been a number of attempts to modify HHI to allocate capital but as yet, none have been rigorous.

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Expected losses

• Expected losses are those losses that can be estimated. This is

defined as:

EL = PD x LGD x EAD

Where

• EL is Expected losses

• PD is probability of default

• LGD is loss given default

• EAD is exposure at default

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• Traditionally, this has been estimated from historical data. However, given that capital adequacy will require forecasts, more effort is focused on:

• Probability of default

• Loss given default

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Probability of default

• There are many functional forms for probability of default (PD). For

default to occur, either there is no liquidity or equity becomes

negative (e.g. KMV).

• Functional this is:

P(d) = f(lf, ne)

Where:

• P(d) is the probability of default

• Lf is liquidity failure

• Ne is negative equity

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• These models need to expanded to take into account not just borrower characteristics but external factors that affect the ability to repay.

• Thus the model will be as follows:

P(d) = f(micro-factors, macro-factors)

• Micro-factors would be the original model.

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• The task is to develop models that are rigorous. However, as most regression methods will give results that are bounded by infinity. Usually, most models are logistic to have bounded by 0 and 1.

logp1-p= iβimicro-factors+jβjmacro-factors

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Loss given default

• Loss given default is difficult to model since the distribution is rarely normal.

Therefore, modeling is often focused on technique rather than variables.

• Variables that are normally used are:

• Type of loan

• Seniority

• Collateral

• Term

• Seniority of mortgage

• Characteristic of company’s liquidity

• Level of interest rates

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• Most approaches to LGD is to create transformed distributions that can be used such as:

• Fractional response regressions.

• Inverse Gaussian models

• Decision trees/neural approaches

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Prepayment risk

• Prepayment risk is the risk created by fixed rate loan being prepaid early.

• Traditionally, models focus on interest rates. However, it has been found that interest rates are not the dominant factor.

• This can be explained in Australia by loans having a shorter interest rate maturity than the actual maturity (meaning they pay out early for a various number of reasons).

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• The normal approach is to model around the following:

• Monthly prepayment rate = (Refinance incentive) x (Season multiplier) x (Month multiplier) x (Burnout)• The interpretation of this formula is:

• Refinance incentive is the current level of interest rates with respect to interest rates in the portfolio. If current interest rates are higher then there will little incentive to refinance or prepay loans.

• Season multipliers recognise there are times there are higher than normal prepayments.

• Month multiplier is similar to the season multiplier

• Burnout recognise that the longer loans exist the more likely they are to be prepaid.