Statistics in Retail Finance
Chapter 7: Profit estimation
Statistics in Retail Finance
Chapter 7: Profit estimation
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Statistics in Retail Finance
Chapter 7: Profit estimation
Overview
>
In this chapter we cover various methods to estimate profits at both the
account and aggregate level based on the dynamic behavioural models
introduced in the previous chapter: Dynamic profit estimates using Markov transition models.
 Lifetime profit estimation using survival models.
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Statistics in Retail Finance
Chapter 7: Profit estimation
Introduction
>
 So far our interest has been in models of default. But why model
default, when ultimately our interest in in profit?
 There is now great interest in moving to profit modelling in retail banks.
 A default model is a risk model.
 A profit model is a model of risk and reward.
 When producing profit models, we can give account-level and/or
aggregate estimates (ie across a portfolio of loans).
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Statistics in Retail Finance
Chapter 7: Profit estimation
Profit calculation using Markov transition model
>
In this section, we consider using underlying Markov transition models as
the basis of profit calculations to rectify this problem.
This approach is particularly suited to revolving credit where there is no
fixed term of loan repayments.
We need to define some terms:
 Suppose we model states for loans.
 ( ) is the probability of moving from state to state
as estimated by a first-order Markov transition model.

is the state at period
: assume known.
 ( ) is profit associated with being in state .
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in time period ,
Statistics in Retail Finance
Chapter 7: Profit estimation
In this setting, we define profit recursively:

( ) is the expected value of profit at period
given an initial state of
at period
after the last
periods,
.
Therefore
()
{ ()
(
∑
)
()
 Notice that this calculation requires summing recursively over the
probability transitions from period
to .
 Expected profit across the whole period to
 It is computed as a recursive function.
5
is given by
( ).
Statistics in Retail Finance
Chapter 7: Profit estimation
Example 7.1
Suppose
(
,
, ( )
, ( )
and a stationary transition matrix
).
Compute expected profit up to and including time period .
Solution
( )
( )
[ (
[
[
)
(
( )
)
(
( )
)
(
]
( )]
)
( )]
[
6
[
(
]
)
( )
(
)
( )]
Statistics in Retail Finance
Chapter 7: Profit estimation
Introducing the default event
>
Define an indicator of default and a default cost:


{
is default cost.
Then extend the expected profit calculation to
()
{ ()
∑
(
)[
()
(
)]
The principle here is that default is a one-off cost. It is only included in the
)
model if the previous state was not a default. This is what the (
indicator does.
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Statistics in Retail Finance
Chapter 7: Profit estimation
Example 7.2
Suppose we have two risk grades (1,2) and a default state (3). The
transition matrix between states is represented (for all time periods) by
(
)
The profits associated with risk grades 1 and 2 are 10 and 8 respectively.
The cost of default is 200. Time periods are in months.
Calculations of expected profit for different values are given in this table:6
12
18
24
( ) 45.2 79.5 109.6 136.9
( ) 27.2 56.4 83.9 109.3
Therefore, if we suppose an initial state 2, then expected profits after 24
months are 109.3.
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Statistics in Retail Finance
Chapter 7: Profit estimation
Aggregate estimates
>
Aggregate expected profit can be computed by a weighted sum over
expected profits computed for each state.
So if the number of accounts with initial state is given by
expected value of aggregate profit at period is given by
()
∑
9
then the
Statistics in Retail Finance
Chapter 7: Profit estimation
Exercise 7.1
Consider a two state stationary Markov transition model with transition
matrix
(
)
as a model of “movers”, accounts that are less likely to stay on one state
than to move into another: more precisely,
and
.
Let ( )
and ( )
where
; ie state 2 incurs a loss.
1. Show that
2. If
( )
(
)(
)
.
, show that expected profit is always positive:
and states
and
.
Do not consider default in the profit calculations.
10
()
for all
Statistics in Retail Finance
Chapter 7: Profit estimation
Lifetime profit calculation
>
 Profit can be calculated more systematically through the lifetime of a
loan.
 Survival models can then be used to estimate risks within the context of
lifetime profit calculations.
 We begin by considering profit calculations without default.
 Then, later default events are introduced and expected profit
calculations are made.
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Statistics in Retail Finance
Chapter 7: Profit estimation
Some accounting terms
>
We introduce some new definitions for fixed term loans.





is number of periods of the loan (typically in months).
is interest rate per period.
(
)
o Note
o If
is annual interest rate and period is monthly then
(
) ⁄
.
is the risk-free rate per period.
o This is the return that the lender could have got with no risk to
capital, on the same loan amount (eg US government bonds), so
“raw” profits should be adjusted by this rate.
{
}.
is loan value at period for
o Since the loan is fully paid back by period ,
.
is repayment amount at time .
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Statistics in Retail Finance
Chapter 7: Profit estimation
Calculating profit without default
>
Profits are calculated either as “raw” monetary profit, , or as net present
value (NPV),
, when profit is adjusted by the risk-free rate:
∑
and
∑
(
)
The development of the loan in each period is expressed as
(
)
Therefore
(
)
∑
(
.
)
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Statistics in Retail Finance
Chapter 7: Profit estimation
Suppose we choose a fixed repayment amount
{
}.
Then
(
)
And, in particular, for
∑
(
)
(
,
(
)
per term, so
(
)
(
(
)
)
Then
[
(
]
)
and
[
(
(
14
)
)
]
)
for all
Statistics in Retail Finance
Chapter 7: Profit estimation
Introducing the default event
Let
(
>
) be the probability that payments have been made up to period t.
Then profitability calculation can be expanded to include this probability to
get an expected value of profit:
∑
(
)
Additionally, the possible amount recovered needs to be factored in.
 Expected recovery = PD (1-LGD) EAD.
 Then probability of default (PD) specifically in period is the
probability that payments have been made up to that period times
hazard of default at that period: ( ) ( ).

is loss given default (LGD).
 Exposure at default (EAD) is given by
.
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Statistics in Retail Finance
Chapter 7: Profit estimation
This gives the expected value of profit
( )
∑[
(
)
(
Take a fixed repayment amount
( )
(
∑[
(
)
(
[
(
)
(
∑[
(
)
)
∑[
(
(
)
)
(
)(
]
)
and substitute the formula for
(
(
)(
)(
)
(
:
)
)]
)
)(
(
)(
)
(
)
)
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(
(
(
)(
)
)(
)
(
(
)
)
)]
)]
]
Statistics in Retail Finance
Chapter 7: Profit estimation
The expected value of NPV can also be derived in a similar way:
(
)
[
(
)
∑[
(
)
(
) (
(
17
)(
)(
)
(
)
)
]
]
Statistics in Retail Finance
Chapter 7: Profit estimation
Profit calculation using the survival model
>
Including survival probability (for default) in lifetime profit model.
 Use lifetime profit calculation with default.

is time within which to measure defaults over missed payments.
o Eg if default is defined as three months consecutive missed
payments, then
.
(
), given a default survival model .
 Then ( )
 The hazard function can be the usual instantaneous hazard or, since the
period is not an instant, use the hazard probability given by
( )
(
)
( ).
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Statistics in Retail Finance
Chapter 7: Profit estimation
Example 7.3
A Cox PH model is run on a training data set of personal loans with predictor
variables: age, employment status, tenure and months at address.
Profits over a 12 month term can be calculated for any individual based on
the model’s survival probability.
One individual is a 42 years old, employed home owner with 3 months in
current address.
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Statistics in Retail Finance
Chapter 7: Profit estimation
Survival probabilities over are shown in the following graph for this
individual:
Based on this survival function, an expected profit rate can be computed,
given an interest rate and LGD.
(
)
0.1 0.8 -0.0079
0.2 0.8 0.0384
0.3 0.8 0.0823
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Statistics in Retail Finance
Chapter 7: Profit estimation
Extending the profit formulae
>
 The profit formulae expressed here are given specifically for fixed term
loans and where repayment amounts are fixed. However, it is feasible to
extend them to unfixed terms, revolving credit or other repayment
regimes (eg exponentially increasing amounts).
 In this model we are assuming LGD ( ) is fixed, but this may not be true
and we may have a model for LGD too. In particular, LGD may well be
correlated with PD. However, an LGD model could easily be incorporated
in this specification.
 We have assumed fixed account details ( ). However, these could easily
be treated as time-varying by indexing by time period (ie ). This would
fit in neatly with the use of time-varying covariates in survival modelling.
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Statistics in Retail Finance
Chapter 7: Profit estimation
o In particular, this would enable the use of macroeconomic variables
which would allow adjustment of profit forecasts based on forecasts
of changes in the economy.
 Survival models of attrition and prepayment can also be incorporated in a
similar way to the default model.
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Statistics in Retail Finance
Chapter 7: Profit estimation
Exercise 7.2
Specify the profit formula based on using two survival functions: one which
models default and another which models early account closure.
Note that early account closure implies full prepayment of the outstanding
loan.
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Statistics in Retail Finance
Chapter 7: Profit estimation
Review of Chapter 7
>
We covered profit estimation based on the following methods: Dynamic profit estimates using Markov transition models.
 Lifetime profit estimation using survival models.
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