**A Lagrangian Approach to Customer Relationship Management:
Variable-Rate Pricing Strategy**

Joseph George Caldwell, PhD

Lagrangian Solutions

Tel. (001)(864)439-2772

E-mail: jcaldwell9@yahoo.com

Website: http://www.foundationwebsite.org

© 2006 Joseph George Caldwell. All Rights Reserved

**Contents**

3. Mathematical Formulation of the Problem

This paper describes an optimization-based approach to
customer relationship management (CRM), and illustrates the methodology with an
application to the problem of determining a variable-rate pricing model for
banking loan products. The methodology
employed to solve the problem is

This paper consists of five additional sections. Section 2, Description of the Problem, describes the general problem to be solved – customer relationship management – with specific reference to the problem of determining an optimal pricing policy for bank loans. Section 3, Mathematical Formulation of the Problem, formulates the problem in mathematical terms. Section 4, Solution of the Problem, describes the GLM methodology for solving the mathematical problem. Section 5 describes the design of a computer-program model for implementing the methodology in the variable-rate pricing example.

“Customer relationship management” refers to the collection of policies and procedures that an organization adopts and applies in dealing with its customers. The term is used mainly by organizations having very large numbers of customers, such as a bank, credit-card company, insurance company or financial advisory service. The number of customers or potential customers in these applications is very large, often in the millions. Any organization dealing with such large numbers of customers cannot deal individually with each customer or potential customer, and needs automated procedures for conducting operations such as mass mailings to solicit business, extend offers to existing customers, or modify the terms of accounts. Because of the very large numbers involved, even small improvements in the policies or procedures used to deal with customers can result in significant increases in profit. For this reason, large banks and other organizations having large customer bases invest heavily in scientific research, such as in statistical models of customer behavior, statistical decision functions and optimization models.

The past couple of decades has seen the application of a variety of technical methodologies such as statistical models and artificial-intelligence models to describe customer behavior (or potential-customer behavior) and optimization models to determine good strategies for dealing with customers. These methodologies the use of logistic-regression models to estimate the probability that a customer or non-customer will respond favorably to a mass mailing offer, and the use of neural-networks to estimates the likelihood of loan default on the basis of observed customer characteristics such as credit score, income, age, or marital status.

At present, a massive amount of effort is invested by customer-relationship-management organizations in the conduct of statistical analysis of customer data to describe customer behavior as a function of customer and product / service characteristics. This includes not only efforts to develop sophisticated analytical models based on statistical experimental-design methodology, but also the use of “exploratory data analysis” or “data mining” to detect apparent relationships and thereby suggest hypotheses that may be investigated by methodologies oriented toward identification of causal relationships. In contrast, a much smaller amount of effort is invested in the development of optimization or control models to achieve specified objectives, such as maximization of income or profitability subject to constraints such as funds availability or return on investment; or to support risk management. This is partly due to the fact that the formulation and solution of optimization problems is often more difficult than the formulation and solution of statistical estimation or statistical decision problems, and partly due to the fact that the “statistical analysis industry” (represented by SAS, SPSS, CART, etc.) is much larger than the “optimization” industry, and the availability of appropriately skilled personnel to apply statistical-analysis software, which is highly automated, is much greater than the availability of appropriately skilled personnel to develop optimization models.

The problem to be addressed in this article is the problem of determining an appropriate action to be taken with respect to each customer (or non-customer) so as to maximize (or minimize) a specified quantity, subject to specified constraints. For example, it may be desired to conduct a mass mailing to non-customers to invite them to accept a bank credit card, subject to a fixed budget (number of letters to be mailed). A limited amount of information is known about each non-customer, such as credit score, marital status, location (e.g., urban/rural status), and number of credit cards already using. A “traditional” way of dealing with this problem is to use historical data (from earlier mailings) to estimate the probability (likelihood) that a potential mail recipient will accept the offer, using a logistic regression model. Then, all of the potential mail recipients are ranked in decreasing order of the estimated probability (i.e., from the most likely to the least likely), and as many addressees as desired are selected from the top of the list. The terms of the offer (interest rate, fees, credit limit) are typically indicated in the offer letter, and they are usually the same for all recipients (in a particular mailing).

This approach maximizes the number of favorable responses to the mailing. Since it maximizes a quantity, it is an optimization problem, but it is a rather trivial one. The bank’s fundamental objective is not to maximize the number of people responding (favorably) to the offer, but to maximize its income, or return on investment, or stockholder equity. It would be considerably more relevant to the bank’s ultimate objective to select mail recipients in such a way as to maximize one of these quantities, rather than to maximize the favorable-response rate. Using the traditional approach of sending to those who are most likely to accept the offer, it could happen that the least creditworthy of the target population responds, so that the bank obtains lots of new customers, but these customers will not generate much income for the bank.

In order to target a mass mailing in such a fashion as to
maximize bank income or profitability, it is necessary to develop a
mathematical model that expresses these quantities as functions of customer
characteristics and offer characteristics, and to tailor the mailing and offer
to customers in a way to maximize the quantity of interest. In mathematical terminology, the quantity to
be maximized, such as bank income or return on investment, is called the *objective function* or *payoff function*.

In general, the problem of customer relationship management involves allocating resources to a target population (customers or potential customers) in such a way as to maximize a specified objective function, subject to various constraints.

This article will illustrate the optimization-based approach to customer relationship management by means of an example in which it is desired to determine a pricing strategy for bank loan products (e.g., credit cards, installment loans, mortgages) that maximizes stockholder value added, subject to a constraint on the return on the capital allocated as a provision against loss. The “pricing” of the loan is specified by two quantities: (1) the decision to make the loan; and (2) the interest rate to be charged. The pricing strategy is determined by specifying the loan decision (i.e., to extend the loan or not extend the loan) and the interest rate for each customer. Since the interest rate may vary for each customer (or class of customers), this pricing policy is referred to as a “variable-rate pricing” strategy. (This example does not address the problem of determining a strategy for setting fees, which may be substantial for some loan products, such as credit cards – it deals only with determining an optimal strategy for maximizing interest income, not fee income.)

The motivation for adopting a variable-rate pricing (VRP) strategy is that bank profits may be increased by making credit decisions and interest rates sensitive to customer characteristics (i.e., to the risk). Credit may be extended to riskier customers at higher rates, which offset the increased losses associated with those customers, or it may be extended to premium customers at lower rates, to retain them or attract more of them. With this approach, the customer base may be increased, and higher-risk customers will pay higher rates, with the end result that total bank income may be increased. In order to produce higher income, however, it is necessary to determine rules for the credit decision and the interest rate in a satisfactory way. In order to maximize bank income, it is necessary to determine an optimal rule for making the credit decision and for setting the interest rate.

Under the so-called Basel II Accord (the *International Convergence of Capital Measurement
and Capital Standards - A Revised Framework*) of the Bank for International Settlements,
international banks are to relate the size of their reserve capital (provision
for loan defaults) to risk. Large
international banks take risk into account in pricing their products, and the
increased risk sensitivity is reflected in the willingness of these banks to
lend to higher-risk borrowers at higher prices.
An advantage of variable-rate pricing to higher-risk customers is that
some that were previously excluded from consideration have a chance to establish
a good credit history. (See Wikipedia
entry, *Basel II* at http://en.wikipedia.org/wiki/Basel_II
; also, *International Convergence of
Capital Measurement and Capital Standards, A Revised Framework*, Updated
November 2005 at http://www.bis.org/publ/bcbs118.pdf
.

We will
illustrate the problem of optimization-based customer-relationship management
by means of a specific example, viz., the determination of an optimal
variable-rate pricing strategy for loans.
In practice, a different pricing stragegy would be developed for each
different type of credit product, such as credit cards, installment loans, and
secured loans (e.g., home mortgages).

Let us denote by
the symbol x a pricing stragegy for a single customer, and by X the pricing
strategy for all customers. The pricing
strategy consists of a credit decision (i.e., whether to extend the loan to the
customer) and an interest rate for the loan if it is extended. If x_{i} denotes the pricing stragegy
for customer i, then X = (x_{1}, x_{2},…,x_{n}), where
n denotes the total number of customers.
Let h_{i}(x_{i}) denote the payoff resulting from using
strategy x_{i} with the i-th customer.
Let H denote the total payoff from all customers, and H(X) denote the
total payoff (e.g., net income, or net income after taxes) if strategy X is
used. Then

H(X) = Σ_{i}(h_{i}(x_{i})),

where Σ_{i} denotes summartion over the index i.

In the absence of
any constraints, the objective of the bank would be to determine the stratgegy
X so as to maximize the total payoff, H(X).
But, in this resource-constrained, risk-sensitive world, there are
always constraints. First, the amount of
funds available to any particular business application are always constrained
in some fashion. Second, a bank must pay
in some way for the funds that it loans, either by means of interest payments
to depositors, or a return to stockholders, or as a “discount rate” to to the
central (reserve) bank from which it borrows funds. What it pays for this capital (i.e., what
rate) is the “cost of capital.” For the
bank to continue to make loans of a certain type, it is necessary that the
return on the loans that it makes be greater than the cost of the funds that it
uses for the loans. To avoid the risk of
bank default, banking regulations require that a bank must set aside a “capital
allocation,” or “capital requirement,” for each type of loan (credit) product,
such that the risk of losing the entire capital allocation because of loan
defaults is very small. In recognition
of these conditions, we shall address the problem of determining X so as to
maximize H(X) subject to a constraint on the return on investment of the
capital allocation (the total capital set aside as provision for bad loans).

The amount of the
capital allocation varies by credit product type, depending on the level of
risk associated with that type. The size
of the capital allocation is determined by a risk analysis, such as a risk-adjusted
performance measurement analysis.

Let us denote the
capital allocation as CA, and the cost of capital as CC. Let NI denote the net income for the loans,
after deduction of all expenses except the charge for the capital
allocation. In this article, we shall
ignore taxes, and focus attention on pretax income. (Whether attention focuses on pretax income
or posttax income has no significant effect on methodology for determining the
pricing strategy.) We define two other
terms: the return on capital, or ROC, which is defined as

ROC = NI/CA

and the
shareholder value added, SVA, which is defined as

SVA = NI – CC CA .

The return on
assets, ROA, is defined as

ROA = NI/A

where A, assets,
is the sum of the loan account balances.
The ratio A/CA is called the financial leverage, FA, so that the
relationship of ROC to ROA is ROC = FL ROA.

The four
quantitites, net income (NI), return on capital (ROC), return on assets (ROA)
and stockholder value-added (SVA) will be referred to as performance
measures. Once the optimal pricing
strategy is determined, the value of these performance measures will be
determined for the optimal pricing strategy, the bank’s current pricing
strategy, and any other user-specified pricing strategy of interest.

Net income is
determined by a variety of income and expense components. In developing an optimization model to
maximize net income, it is necessary to know the relationship of net income to
customer / account characteristics and the pricing policy. That is, it is necessary to estimate net
income for a customer as a function of the customer’s characteristics (e.g.,
credit score) and the loan terms (interest rate, fee provisions). To do this it is necessary to estimate the
relationship of demand for loans and risk of default to customer
characteristics (such as credit score) and loan product characteristics (such
as interest rate). In this example, we
shall be concerned only with determining a policy for determining the credit
decision and the interest rate, and not with other loan terms, such as fee
schedules. (If the methodology were to
be extended to include determination of optimal fee schedules, then demand and
risk models would have to be available for fee income.) The following formula expresses net income,
NI, as a function of various components of income and expense:

NI = Net Interest
Income (interest income – cost of funds
– provision for credit loss)

+ Fee Income – Direct Operating
Expenses (cost of origination, cost of servicing)

- Indirect Operating Expenses (indirect operating expenses and overhead) .

In determining an
optimal policy for loan pricing (credit decision and interest rate), we are most
concerned with the components of NI that are affected by loan pricing. These include net interest income, cost of
servicing and fee income. The other
components of NI (indirect costs) are necessary to calculate the value of NI
correctly, but they are not affected by the pricing. (They do affect the optimal pricing strategy,
since they affect the value of ROC, and pricing policy is determined by a
constraint on ROC.)

The objective of
extending loans is to maximize net income, subject to whatever constraints are imposed
on the operation. The three principal
constraints to loan operations are (1) the total amount of capital available is
limited to a certain amount; (2) the (total) return on capital must exceed a
particular level; and (3) the risk of loss of the capital allocation must be
very small. In this illustrative
example, we shall assume that the size of the capital allocation has already
been determined in a fashion so that the risk of loss of the capital is very
low, and so no constraint will be included in this imposed to reflect that
concern (i.e., the probability of loss exceeding the capital allocation). Also, it will be assumed that, as long as the
return on capital meets or exceeds the prescribed level, there is not
constraint on the amount of capital available, i.e., capital is an “infinite
resource.” It is also assumed that there
is no requirement to utilize all of the capital allocated to the operation,
i.e., there is no penalty for failure to make loans up to the maximum level
permitted by the capital allocation.

The problem of
variable-rate pricing may be focused on determining the pricing strategy for a
single credit product or for a group of credit products. The pricing strategy for a particular loan
product will differ in these two cases, since in the first case the constraint
on ROC must be met by that particular product, while in the second case the
constraint on ROC must be met by the entire group of products (in which case
the ROC may be less on some products and higher on others, such that the ROC
for the total group of products meets the constraint).

In view of these
remarks, the problem to be addressed here is to maximize net income subject to
the single constraint that the return on allocated capital exceed a specified
level. If NI(X) denotes the net income
generated by pricing strategy X, and if ROC(X) denotes the return on capital
under pricing strategy X, then, in mathematical terminology, the problem is to
determine pricing strategy X so as to maximize NI(X), subject to the constraint
that ROC(X) >= CC (where CC denotes the cost of capital). Or, more compactly, determine the pricing
strategy, X*, such that

NI(X*) = max_{X} NI(X)

subject to

ROC(X*) >= CC ,

where max_{X}
NI(X) denotes the maximum value of NI(X) with respect to X.

The constraint
ROC(X*) >= CC may be written as

ROC(X*) = NI(X*) / CA(X*) >= CC ,

or as

SVA(X*) = NI(X*) – CC CA(X*) >= 0 .

This section will
present a solution to the optimization problem defined in the preceding
section, for the situation in which the loan pricing may be done independently
for each customer account. This
situation applies if the income (“payoff”) for a particular account depends
only on the pricing of that account and not on the pricing of other
accounts. In this situation, a very
powerful optimization methodology is available to determine the solution to the
problem (i.e., to determine the optimal pricing strategy). That methodology is the Generalized Lagrange
Multiplier (GLM) method developed by Hugh Everett III. The GLM methodology is a very powerful
technique for solving “cell-separable” constrained-optimization problems in
which the quantity (the *objective
function*, or *payoff function*) to
be maximized (or minimized) may be represented as a sum of a large number of
independent terms (such as income from customer accounts). It is much more flexible than most
constrained-optimization methodologies, since it can handle objective (payoff)
functions that are nonlinear, nonconvex, and discontinuous.

Under the GLM
approach, the solution to the constrained optimization problem posed above,
viz., determine strategy X* such that

NI(X*) = max_{X}(NI(X))

subject to

NI(X*) – CC CA(X*) >= 0 ,

is the same as
the solution to the following *unconstrained*
optimization problem:

Determine
strategy X* such that

NI (X*) – λ [CC CA(X*) – NI(X*)] = max_{X} [NI(X) –
λ {CC CA(X) – NI(X)}]

where λ denotes a lagrange multiplier (nonzero number), the value of
which is determined so that the constraint NI(X*) – CC CA(X*) >= 0 is satisfied. The tremendous power of the GLM method is that
it is much easier to solve the unconstrained extremization problem than the
original constrained extremization problem.

It is customary to simplify the preceding expression by combining
terms. The right-hand-side of the
expression may be written as

max_{X} [(1 – λ) NI(X) – λ CC
CA(X)] .

For a specified value of λ, the value of X that maximizes this
expression is the same as the value that maximizes

NI(X) – λ’ CA(X)

where λ’ = CC λ / (1 – λ) is chosen so that the constraint
NI(X) >= CC CA(X) is satisfied. For
simplicity, the constant λ’ is relabelled as λ. With this reformulation, the problem is to
determine strategy X* such that

NI(X*) – λ CA(X*) = max_{X} [NI(X) –
λ CA(X)]

= max_{X} [Σ_{i}(NI_{i}(x_{i}))
– λ Σ_{i}(CA_{i}(x_{i})) ] ,

where λ is chosen so that the constraint NI(X) >= CC CA(X) is
satisfied.

In order to solve this problem, the essential step is to take into account
the assumption that the income for the i-th account is a function of the
pricing for that account only, and not of the pricing of other accounts. Under this assumption, the problem has a
“cell-separable” payoff function, and my be solved by the GLM method. This assumption is reasonable for accounts
owned by different customers, but it may not be reasonable for accounts
associated with the same customer. For
example, if a customer is denied a home mortgage loan, or such a high price is charged for this mortgage that
he declines, he may decide to cancel all of his other accounts with the bank (e.g.,
his credit card accounts, or perhaps other mortgages). (In this case, the expected net income for
the mortgage loan would not be zero, but negative, if the decision were made
not to extend the loan.)

Under the assumption that the net income (the “payoff”) for the i-th
account is independent of the net income for other accounts, then the price
strategy x_{i} for each account may be determined independently for
every account. In this case, the
expression max_{X} [Σ_{i}(NI_{i}(x_{i})) –
λ Σ_{i}(CA_{i}(x_{i}))] may be maximized by
maximizing each term NI_{i}(x_{i}) – λ CA_{i}(x_{i})
independently, i.e.,

max_{X} [Σ_{i}(NI_{i}(x_{i}))
– λ Σ_{i}(CA_{i}(x_{i}))] = Σ_{i}(max_{xi}
[NI_{i}(x_{i}) – λ CA_{i}(x_{i})]) .

That is,

NI(X*) – λ CA(X*) = Σ_{i}(max_{xi}
[NI_{i}(x_{i}) – λ CA_{i}(x_{i})]) .

The optimization problem is hence solved by determining, *independently for each account*, the
price strategy x_{i} that maximizes the expression NI_{i}(x_{i})
– λ CA_{i}(x_{i}), which is called a *lagrangian function*. To do
this, all that is required is to determine λ so that the constraint NI(X)
>= CC CA(X) is satisfied. This is
done by using an interative numerical method (such as Newton’s method) to find
the minimum value of λ for which the constraint is satisfied. (There may be multiple values of λ for
which the constraint is satisfied. The
reason why we seek the minimum value of λ is that it is desired to
maximize net income, not just find a solution that satisfies the constraint.)

The original problem of determining a pricing strategy that maximizes total
net income subject to the constraint on return on capital allocation has now
been reduced to the simpler problem of determining, independently for each
account, a pricing strategy that maximizes the account lagrangian
function. To solve this problem, it is
necessary to know the relationship of net income (NI_{i}) and capital allocation (CA_{i}) to
the account pricing and account characteristics. The value of CA_{i} is known – it is
either zero if no loan is extended or a fixed percentage of the loan value (the
percentage factor being the same for all accounts of a given loan type). It remains to specify the relationship of NI_{i}
to account pricing and account characteristics.

The net income from a customer account depends on a number of factors,
including the decision to extend the loan, the interest rate, the event of a default
if the loan is extended, and the amount of the loss in the event of
default. The decision to extend the loan
and the interest rate are under the control of the bank, and will be made
according to the strategy determined by the solution to the optimization problem
described above (which strategy will take into account customer
characteristics). The other two
variables (the event of a default and the loss in the event of a default) are
random variables. Hence the relationship
of net income to pricing strategy and customer characteristics will be a stochastic
(probabilistic, statistical) one, not a deterministic one. This relationship will typically be expressed
as a table or formula that specifies the expected (mean, average) net income as
a function of pricing strategy (loan decision, interest rate) and customer
characteristics (e.g., credit score).

The customer (or potential customer) characteristics of interest in this
application include all available data (known facts) about the customer that
may affect loan income or expense. For a
simple model, or for a potential customer, this information might be a single
variable – the individual’s credit score.
Whatever customer (or potential customer) data are available, it is
necessary to know the relationship of net income to those variables. These relationships would typically be
expressed as statistical tables or statistical regression functions that
specify the expected value of net income as a function of the available
explanatory variables. These relationships
would be derived from a statistical analysis of available data, or from expert
judgment for situations in which no historical data were available (e.g.,
estimation of net income for an interest rate / credit score combination that
had not been used before).

The pricing strategy, x_{i}, consists of two components – the credit
decision and the interest rate. We shall
represent the credit decision by a numerical variable, d_{i}, whose
value is 0 if the decision is made not to extend the loan and 1 if the decision
is made to extend the loan. We denote
the interest rate as r_{i}. In
terms of d_{i} and r_{i}, we shall denote the pricing strategy
as x_{i} = (d_{i}, r_{i}). Let us denote the collection of customer (or
potential customer) variables as y_{i} (a vector, y_{i} = (y_{i1},
y_{i2},…,y_{in})). For
simplicity in this illustrative example, we shall assume that there is a single
customer characteristic that is to be taken into account in determining pricing
strategy, namely, the customer’s credit score, which we shall denote as cs_{i}. That is, y_{i} in this example will
have but a single component, y_{i1} = cs_{i}.

In the formulation of the optimization problem presented above, net income
was represented as a deterministic variable, and the variables over which the
account lagrangian was to be maximized were not explicitly shown. Representing net income as an expected value,
the problem is to determine the pricing strategy x_{i} = (d_{i},
r_{i}) so as to maximize the lagrangian function

E(NI_{i}(x_{i},y_{i})) –
λ E(CA_{i}(x_{i},y_{i}) = E(NI_{i}(d_{i},r_{i},cs_{i}))
– λ E(CA_{i}(d_{i},r_{i},cs_{i})) ,

where the symbol E(.) represents the expectation (expected value) operator. (Note:
The formula for determining net income and capital allocation might
differ from customer to customer, and for this reason the index i is appended
to each symbol, viz., NI_{i} and CA_{i }. In general, however, the functional form will
usually be the same for all accounts of a particular loan type. It is conceivable, however, that different
functions might be used if different variables are known for some customers and
not others (e.g., the only variable known for potential customers might be
credit score, but many additional variables, such as income and account age,
would be known for customers, and so a different formula could used to
calculate net income in these two cases.)

In this example, we shall assume that the capital allocation is the same
rate for all customers of a given loan type, and not dependent on the
individual customer’s credit score or the interest rate to be charged. In this case, CA_{i}(d_{i},r_{i},cs_{i})
is a deterministic quantity, and so E(CA_{i}(d_{i},r_{i},cs_{i}))
= CA_{i}(d_{i},r_{i},cs_{i}). If d_{i} = 0 (i.e., the decision is
made to not make the loan) then CA_{i} = 0. If d_{i} = 1, then CA_{i} =
CC times (amount of loan to the i-th customer).

We shall now present a formula for determining the value of the expected
net income, E(NI_{i}(x_{i},y_{i})).

Note that if a decision is made to not make a loan, then it does not matter
what the interest rate is – the value of the interest rate matters only if the
loan is made. Hence, the two components
of the loan pricing – the credit decision and the interest rate – may be
considered to be independent. If a
decision is made to not make the loan, i.e., d_{i} = 0, then the net
income is zero (no matter what the interest rate). The lagrangian function is hence maximized,
as a function of both d_{i} and r_{i}, by maximizing it as a
function of r_{i} in the case d_{i} = 1 (the loan is made),
comparing the resulting value to zero (corresponding to d_{i} = 0, the
loan is not made), and then choosing the combination of (d_{i}, r_{i})
corresponding to the larger of these two quantities. For this reason, we shall present in the
discussion that follows only the expression for expected net income conditional
on making the loan (i.e., d_{i} = 1).

In the case of some loans (e.g., installment loans, mortgages), the amount
of the loan is known – initially it is the full amount of the loan, and
subsequently it is the balance owed. For
credit cards, the amount of the loan is not known when the offer is made to the
customer. All that is known is the
maximum amount (credit limit) that will be extended, and the balance, which may
fluctuate from day to day. For loans
that are not yet made, the analysis will focus on the expected size of the loan
to be made. For loans that are already
made, the analysis will focus on the loan balance (current market value of the
loan). This quantity will be denoted by VAL_{i}
for the i-th customer.

The net income for an account equals the sum of all of the revenues
associated with the account minus the sum of all expenses associated with the
account. The account revenues include
net interest income, fee income, insurance revenue, and capital earnings. The account expenses include provision for
credit loss, capital cost, loan acquisition costs, insurance, and
overhead. In order to be able to
calculate ROC, *all* components of
revenue and expense must be included in the calculation, not just the direct
ones, or the ones that are affected by the loan pricing. For simplicity of presentation, we shall
combine all of the income other than net interest income (NII) into a single
quantity, “other income,” denoted by OI_{i} for the i-th customer, and all
of the expenses that are not affected by the loan pricing into a single
quantity, “other expenses,” denoted by OE_{i} for the i-th customer. Each of these two quantities must be
disaggregated into two components – one that is dependent on pricing and
account characteristics, and the other that is independent of these variables.

As mentioned, net income is a random variable that depends on a number of
stochastic variables, including the probability that the customer accepts the
loan, the probability of loan default, and the expected size of the loss in the
event of default. We shall denote the
probability that the customer accepts the loan as pacc_{i}. We denote the probability of loan default as
pdef_{i}. We shall represent the
expected size of the loss as the expected proportion of the loan that is lost,
plos_{i}. These quantities are
estimated from historical data or from expert opinion (e.g., a panel of loan
officers may be able to provide insight on demand that may not be available
from historical data). They may be
expressed as tables or as equations, and they may be as simple or as complex as
desired (depending on the availability of historical data and analytical
resources). In a simple application of
this methodology, it may be, for example, that the probability that the
customer accepts the loan is a function of the interest rate, that the
probability of default is a function of the customer’s credit score, and that
the expected proportion of loss in event of default is also a function of the
credit score.

What characteristics are taken into account in determining the models for
demand (probability of acceptance) and default depends on what data are
available about the customer or potential customer. The data elements that are available for
potential customers or new customers may differ from those that are available
for existing customers. For example, all
that may be known for potential customers in a mass mailing may be the
customer’s credit score, whereas for existing customers the available customer
/ account characteristics may include many additional variables, such as annual
income, age, marital status and occupation.

Using the preceding notation, a general expression for the expected net interest
income from the i-th account, if the loan is offered at interest rate r_{i}
and is accepted, is:

E(NII_{i}(d_{i}=1, r_{i}))
= VAL_{i} pacc(r_{i})
[(1 – pdef_{i}) r_{i} – pdef_{i} plos_{i} ]

where, to summarize the notation,

E(.) denotes the
expectation (expected value) operator

index i refers to the i-th
customer

d_{i} = loan decision
(1 corresponding to extending the loan, 0 to denying the loan)

r_{i} = interest
rate

NII_{i}(d_{i}, r_{i}) = net
interest income (a function of d_{i} and r_{i}, and whatever
other customer characteristics are included on the right-hand-side of the
equation, such as credit score, cs_{i})

VAL_{i} = loan
balance

pacc(r_{i}) =
probability that the customer accepts the loan (a function of r_{i},
and possibly of customer characteristics as well, such as credit score)

pdef_{i} =
probability of loan default (may be a function of customer characteristics,
such as credit score)

plos_{i} =
expected proportion of loan lost in event of default (may be a function of
customer characteristics, such as credit score) .

The expression for expected net income, E(NI_{i}), is expected net
interest income plus other income minus other expenses:

E(NI_{i}(d_{i}=1, r_{i})) =
VAL_{i} pacc(r_{i}) [(1
– pdef_{i}) r_{i} – pdef_{i} plos_{i} ] + OIi –
OEi .

If the decision is made not to extend the loan (i.e., d_{i}=0), the
expected net income is zero.

Once the functions pacc, pdef and plos have been determined (by analysis of
historical data), it is possible to calculate the expected net income for each
account. For a specified value of the
lagrange multiplier, λ, the credit decision (value of d_{i}) and
the interest rate, r_{i}, may be determined so as to maximize the value
of the lagrangian function

E(NI_{i}(x_{i},y_{i}))
– λ E(CA_{i}(x_{i},y_{i}) .

The solution to the constrained optimization problem is obtained by
adjusting the value of λ so that the constraint on ROC is satisfied. For problems involving a constraint on a
single resource (e.g., capital allocation, as in this example), algorithms for
adjusting λ so as to satisfy the constraint are easy to develop, and they
converge very quickly to a solution.

Note that under the preceding formulation of the problem of determining a
variable-rate pricing strategy, a different credit decision and interest rate
are determined for each customer, as a function of the customer’s exact credit
score. It may be that this strategy is
much more “detailed” than what is actually desired, particularly when
conducting “exploratory” analysis to investigate a variety of pricing
strategies. If it is desired to
determine a pricing strategy that determines a credit decision and interest
rate as a function of *categories* of
credit scores (“credit bands”), then all that is required to be done is to
solve the problem using models of pacc, pdef and plos that depend on the
customer’s credit category, and not on the customer’s exact credit score. In this case, it is not necessary to iterate
over the entire customer account set in determining the optimal solution, but
simply over a frequency table of loan-size by credit-score categories (in which
case the value of total net income and total return on capital are determined by
weighting the net income and captial allocation values for each loan-size by
credit-score category by the number of accounts in each category).

Such a simplification greatly increases the speed of the numerical
algorithm for determining the optimal solution, since the optimization is
performed over a small number of loan-size by credit-score categories, rather
than over the set of all customers or potential customers. (For a mass credit-card mailing to
non-customers, the same initial loan amount may be used for all customers, such
as $5,000, so that there is but a single loan amount. In this case, the optimization is done simply
over the total number of credit bands, e.g., 20.) Also, it may be desired to restrict the
interest rate to a small number of discrete values (e.g., 3 or 5), rather than
allow a continuum of values. Such a
restriction is readily accommodated by the GLM methodology, simply by
determining the maximum of the account lagrangian function over the restricted
set of values. With such
simplifications, the result of the optimization could be a simple table that
shows the optimal credit decision and interest rate as a function of credit
score band.

If the optimization is performed over a frequency table of customers rather
than over a sample of customer records in order to determine results quickly,
the optimization may be speeded up even more by restricting the interest rates
to a small number such as three or five.
If the optimization is performed over a sample of records in order to
determine a pricing strategy that is “fine-tuned” to each customer’s characteristics,
it would likely be desired to allow for a finer selection of interest rates,
such as ten.

It may be desired to investigate the relationship of the optimal pricing
strategy (or any other pricing strategy) with respect to customer
characteristics, such as demographic characteristics (referred to as
“segmentation variables” in the banking industry). When working with a customer sample, it is
easy to determine the relationship of features of the strategy (such as the
credit decision) with respect to any known customer characteristic, such as
age, race, sex, marital status, income, or location. If the optimal pricing strategy is determined
by optimizing over a frequency table of customer characteristics (rather than
over a sample of customer records), then determination of strategy
characteristics with respect to customer characteristics requires two steps –
the first step is to determine the optimal strategy, and the second is to scan
through a sample of customer records to talley customer characteristics with
respect to the strategy characteristic of interest (e.g., credit decision).

Under the preceding GLM solution of the variable-rate pricing problem, the
credit decision is not necessarily monotonic with respect to credit score,
i.e., it is possible that credit could be extended for a particular score, but
not for every higher score. This is
unlikely to happen, but is possible if the distribution of customers by credit
score is “unusual” in that historical data indicate a significant proportion of
customers having a higher default rate than customers with lower credit
scores. To avoid this anomaly, it is
desirable to modify the solution so that it produces a credit-decision “cutoff”
point, such that credit is extended to all customers having a credit score
above the cutoff point, and no credit is extended to customers having credit
scores lower than the cutoff point.

The GLM methodology is implemented very easily through the use of numerical
algorithms programmed on a digital computer, using a programming language or
application development system such as Microsoft Visual Basic or a database
development program such as Microsoft Access.
The program can be set up to accept user input parameters, such as the
value of the constraint on the total capital allocation.

At a minimum, the program would be set up to determine the optimal pricing
strategy for a single credit product type and specified customer (or potential
customer) data set. A more comprehensive
model might include all of the bank’s credit products.

The model input data include the customer characteristics and
specifications of the demand and default functions (pacc, pdef, plos). As explained previously, the customer data
may be specified in the form of a file containing a record for each customer
(or potential customer), or in the form of a table that presents the frequency
distribution of customers by known characteristics, such as credit score. The demand and default functions may,
therefore, be specified as tables or as continuous functions. These functions enable the calculation of expected
net interest income for each customer (or “cell” in the frequency distribution
table). In the simple model described
above, pacc would be specified as a function (table) of credit score and
interest rate, and pdef and plos would be specified as a function of credit
score. The probability of default would
be specified as a function of time (e.g. probability of default in 60 or 90 days,
as a function of credit score). In
addition, the model input data include specification of all income and expense
components other than interest, including the portion that depends on pricing
and account characteristics (e.g., account service cost) and the portion that
does not (e.g., overhead).

In summary, the model input data include:

- Whether
the optimization is to be done using a sample of records for individual
customers or using an account probability distribution over credit
scorebands.

- The demand
and default functions, pacc, pdef, and plos, as functions of credit score
(or scoreband).

- Specification
of all income and expense components other than interest.

As output data, the model should display, in addition to all input
parameters, the optimal pricing strategy (credit decision, interest rate), net
income, return on capital allocation, return on assets and stockholder value
added. The output should include a
presentation of the frequency distribution of assets by customer
characteristics (such as credit score). For
an application in which the only customer characteristic on which these
functions depend is credit score band, the optimal pricing strategy (credit
decision, interest rate) depend only on credit score band, and the optimal
pricing strategy may be displayed very easily, on a single screen.

The following are examples of model output that may be produced, under the
assumption that the various model functions depend on a single customer
characteristic, viz., credit score (this output would be presented for each
loan type). For each output described
below, the user should have the option of specifying, (1) for the optimal
pricing strategy, the minimum acceptable ROC, and whether the model should
determine the optimal credit decision independently for each credit scoreband or
as a credit-score cutoff; and (2) for a user-specified pricing strategy: (a)
the interest rate (either a specified constant interest rate or the optimal
constant interest rate); and (b) the credit decision as a cutoff score, either
as a specified cutoff or as an optimally determined cutoff.

In summary, the computer model should produce all of the following output,
in the case in which the optimization is done over a frequency table of
accounts by credit score:

- A table displaying
all model input parameters that depend on credit score, by credit score
(e.g., the table row headings are credit score band (interval) and the
table column headings are parameter name, and the table entry is the
parameter value).

- A table displaying
all model input parameters that do not depend on credit score.

- A graph
displaying each of the following model parameters as a function of credit
score: (1) 60-day default probability; (2) 90-day default probability; (3)
proportion of loss in event of default; (4) probability of acceptance (of
the loan offer by the customer); (5) expenses (loan service / collection costs)
that are dependent on customer attributes and pricing strategy parameters

- Asset
distribution (frequency or proportion of assets) by credit score, for the
following pricing strategies: (1) current pricing strategy; (2) current
pricing strategy, but no credit-decision cutoff; (3) current pricing
strategy, but no credit-decision cutoff and all customers accept the loan
offer; (4) optimal pricing strategy; (5) user-specified pricing strategy
(other than current pricing strategy).
(All graphics specified in items 1-4 may be displayed on the same
output page, by providing the user options by option buttons (“radio” buttons).

- On a
single page, the following: At the top of the page, a three-dimensional
graph showing the distribution of expected net income, by credit score,
for a range of interest rates (i.e., the same interest rate for all credit
bands). At the bottom of the page,
a graph showing the interest rate by credit score, for any selection of
(1) optimal pricing strategy; (2) current pricing strategy; (3)
user-specified strategy.

- On a
single page, the following: At the top of the page, a graph showing the
expected return on capital (NI/CA) if credit is extended. At the bottom of the page, a graph
showing the optimal credit decision (acceptance rule), for any selection
of (1) optimal pricing strategy; (2) current pricing strategy; (3)
user-specified strategy.

- On a
single page, the following: The distribution, by credit score, of any
selected performance measure: (1) net income (NI); (2) return on capital
(ROC); (3) return on assets (ROA); or (4) shareholder value added (SVA).

- On a
single page, the following: For any selected performance measure (NI, ROC,
ROA, SVA), the net income by pricing strategy (optimal, current, or
user-specified) and product type.
(Product on one axis, performance measure by strategy on the
other.)

- On a
single page, the following: For any selected performance measure (NI, ROC,
ROA, SVA), the net income by pricing strategy (optimal, current, or
user-specified) (summed over all product types).

- A single table
summarizing all numerical outputs by product type. A different table may be presented for
each selection of pricing strategy (optimal, current, or user-specified)
and output measurement scale (dollars or basis points).

If the optimization is done over a sample of customer records, then the
model output should include, in addition to the above, the following:

- The credit
decision (extend the loan or not) and the interest rate, under the optimal
pricing strategy, the bank’s current pricing strategy, and a
user-specified pricing strategy.

- Graphs of
various dependent variables, such as mean optimal interest rate, mean optimal
credit decision, and frequency of occurrence as a function of various
dependent variables (account characteristics) such as credit score, race,
and account age. For example, it
may be desired to know the proportion of Asians (or Whites, or Blacks, or
Hispanics) who are denied credit under the bank’s current pricing strategy
vs. the optimal pricing strategy, and the mean interest rate charged to
them under the bank’s current pricing strategy vs. the optimal pricing
strategy.

Joseph George Caldwell, PhD

Consultant in Statistics, Operations Research
and Information Technology

**Professional
Profile:**

Career in management
consulting, research, and teaching.
Directed projects in strategic planning, policy analysis, program
evaluation, economics, public finance, statistics, operations research /
systems analysis, optimization (

Experience includes
monitoring and evaluation of national health, education and welfare programs in
the

2001- Management
Consultant (US Agency for International Development,

1999-2001 Director
of Management Systems, Bank of

1991-1998 Management
Consultant (Wachovia Bank, Charlotte, NC; US Agency for International
Development, Egypt, Malawi, Ghana; Asian Development Bank, Bangladesh; TD Canada
Trust Bank, Toronto, Canada)

1989-1991
President,
Vista Research Corporation,

1982-1991
Director
of Research and Development and Principal Scientist, US Army Electronic Proving
Ground’s Electromagnetic Environmental Test Facility / Bell Technical
Operations Corporation and Combustion Engineering; Adjunct Professor of Statistics,
University of Arizona; Principal Engineer, Singer Systems and Software
Engineering; Arizona

1964-1982
Consultant
or employee to firms in

**Education:**

PhD, Statistics,
University of North Carolina at Chapel Hill, Chapel Hill, NC, 1966

BS, Mathematics,

Graduate of

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