Dynamic Pricing for Personal Unsecured Loans

Price Sensitivity (0.1)
Towards 1:1 pricing

Modern/dynamic pricing approaches applied to personal unsecured loans (PUL).
Paul Golding, Dec 2018 @pgolding

Preface
• We want to predict a borrower’s sensitivity to price and use this information to
optimize prices to achieve maximum expected profit (or some other business
goal - TBD)

• This deck is a “think aloud” conversation to gain a common understanding of the
principles and possible solutions to the pricing sensitivity problem. It also poses
questions that need answers to move forward with building a pricing model.

• The deck contains a mostly mathematical tour of the principles because the goal
is to discover a computational method(s) that might achieve 1:1 pricing
segmentation via a fully automated system.

• We heavily lean upon the work of R. Phillips’ (Pricing Credit Products) and
referenced materials. A literature review suggests that his work is a definitive
baseline. In places, we have formulated the math in a more accessible fashion.

Credit Risk
1. Risk is a variable cost

2. Risk is correlated with price sensitivity: riskier customers tend to be less
price sensitive

3. Price influences risk: all else equal, raising prices leads to higher losses -
price-dependent risk.

Terms
• Exposure at Default (EAD) - remaining balance at default

• Loss Given Default (LGD) = EAD/Total Loan Amount

• Probability of Default (PD) = P(Default) | t<=loan_term

• EAD and LGD are random variables

• (Note: we largely stick to Phillips’ terminology to be consistent with his
theoretical foundations. Variation from more standard terminology is
noted.)

Expected Loss
We care about expected values of random vars
PD = pd
(t)
t=1
T
∑
EEAD =
pd
(t) × EADt
t=1
T
∑
PD
ELGD =
pd
(t) × EADt
t=1
T
∑ × LGDt
pd
(t) × EADt
t=1
T
∑
EL = pd
(t) × EADt
t=1
T
∑ × LGDt
∴ EL = PD × ELGD × EEAD
Probability of default at t=1 or t=2 or t=3 … or t=T
Expected loss is major determinant of
expected profitability of a loan and
thereby influences price.

Simplified Pricing Threshold
E π
( )= 1− PD
( ) p − pc
( )− PD × (1+ pc ) One-period supply capital rate pc
One-period price p
Expected profit for $1 1-period loan (to simplify understanding risk/price relationship):
profit loss
p >
PD 1+ pc
( )
1− PD
+ pc
cost of capital
Risk-adjusted premium
This only suggests a lower bound.

It does not suggest the price:

• Maximize profitability?

• Maximize bookings subject to profit constraint?

• Price sensitivity of customer
Extremely simplified: ignores dynamics of payment/
default in multi-period loan and that PD is not
independent of price at which loan offered. But sufficient
to show the role of risk.
A function of PD
pc =0.1
Risk cut-off equation:

Risk Prediction
s x
( )= ln
Pr{G | x}
Pr{B | x}
⎛
⎝
⎜
⎞
⎠
⎟
score of observables vector log odds (i.e. monotonic)
modeled using ML
(e.g. XGBoost)
Presumably decision-tree based for reasons of interpretability (actual adverse actions methodology: see D. Paulsen).

(Look ahead: can we get better sensitivity predictions if we don’t care about interpretability?)
P̂{B | x}

Risk Factors
P̂{B | x} • Risk prediction based upon borrower
observables

• But also the loan itself

• Amount

• Leverage (e.g. LTV, DTI)

• Period (longer loans riskier due to information
asymmetry, random shock etc.)

• And price-dependent risk
Not part of x
Borrower dependent risk
Product dependent risk

Price-dependent Risk
If for all values of
E π
( )= 1− PD
( ) p − pc
( )− PD × (1+ pc )
⇒ E π
( )= 1− PD p
( )
( ) p − pc
( )− PD p
( )× (1+ pc )
P ′
D p
( )> 0 i.e. positive slope, or risk increases with price
PD p
( )>
p − pc
p +1
p ≥ pc
then there is no price at which a loan is profitable

Price-dependent Risk
• Fraud - fraudulent borrowers are insensitive to price

• Adverse private information - e.g. unobservable knowledge of a
borrower’s forthcoming redundancy

• Competitive alternatives - less riskier customers have more options and
so loans at a certain (higher) price will attract riskier borrowers

• Behavioral factors - behavioral economics, mental accounting, time
discounting etc. (Dynamic contextual factors unrelated to loan
characteristics or borrower credit-worthiness attributes.)

• Aﬀordability - e.g. DTI (capacity to pay)

Incremental Profitability
• Optimal price imposes two questions:

• How will profitability of a loan change with price?

• How will demand for a loan change with price?

• Profitability for loans non-trivial calculation due to:

• Time value of money (discounted value of future costs/revenues)

• Uncertainty: default, prepayment, utilization (for credit line)

• We need a model for incremental profitability for PUL

• What are the variable costs? (Holds, channels?)

• What is the profit goal? (For whom?)

• If we sell/securitize loans, how does this aﬀect optimization?

• Longer loans are more profitable (via interest)

• Large-portfolio holders should act more risk-neutral (assuming risk of loans are statistically independent across
the portfolio) - see Phillips

• Also: should pricing model take LTV into account - e.g. cross-selling other products?  
(Segmentation question?)
Action: construct the incremental profitability model

Price Response Curve
• We want to know the price-
response function per product
per segment (vs. market-
demand curve per product) - i.e.
their Willingness To Pay (WTP)

• Price optimization uses this along
with expected incremental profit
calculation to price for maximum
expected total profitability (for a
loan portfolio)
d p
( )= DF p
( )
total applications

(i.e. “market size”)
take-up rate (i.e. WTP)
price-response

(i.e. “demand”)
Adapted from curve in this paper: Journal of Business Research
Loan APR
Loan APR
F p
( )= f w
( )dw
p
∞
∫
probability distribution of WTP

WTP is probability of a customer paying

at most p - i.e. note that it is the complementary

cum. dist. func. or 1 - F(p) (often seen in the literature)
D = d(0) - i.e. willing to buy at all

(source: DTU ME)
Additional source: BPO lecture notes.
Nominal demand curve
(closed form continuous)
In general, d(p) = D ( 1-F(p) ) = D(F(infinity)-F(p)).

Some derivations
d p
( )= DF p
( )⇒ ′
d p
( ) = −D ′
F p
( )= Df p
( )
F p
( )= f w
( )dw
p
∞
∫ ⇒ − ′
F p
( )= f p
( )
For a given WTP curve, the corresponding density
Slope -ve, hence introduce minus sign
In some cases, we are interested in absolute slope of the take-up rate, in which case:
′
F p
( )⇒ ′
d p
( ) / D = f p
( )
Note: the derivative can be interpreted as the percentage willing to pay exactly p:
′
F p
( )≈
F(p + h)− F(p)
F(p)
For small h

Measuring Sensitivity
Typical curve more like this
Note: the curve is actually time-dependent,

but this is not reflected in the baseline math.
There are many different ways in which product demand might
change in response to changing prices but all price-response
functions assume:

• non-negative (p≥0)

• continuous (no gaps in market response to prices)

• differentiable (smooth and with well-defined slope at every
point)

• downward sloping (raising prices decreases demand)

• notwithstanding “behavioral economics effects”
Additional source: DTU Management Engineering
Most common measures are:
• Slope - d’(p1) is the derivative of the price-response function at p1
• Hazard rate (slope/demand)
• Elasticity: %demand change from 1% change in price:
ε p1
( )= ph p1
( )= ′
d (p1)p1 / d p1
( )= pf p
( )/ F p
( )
h p1
( )=
′
d (p1)
d p1
( )
= f p
( )/ F p
( )
point elasticity
We want to estimate this curve
from any actual take-up data.
“Failure rate”
(See Yao)
has important underlying
properties when calculating
optimal pricing.

Sensitivity and risk
• As said, higher prices = higher default rates AND higher risk customers are less price sensitive (e.g. price
elasticity higher for higher FICO)

• But the two phenomenon are the same: if price sensitivity decreases with risk in a population, then the
populate will demonstrate price-dependent risk.
db p
( )= DbFb p
( )
D = Dg + Db
dg p
( )= DgFg p
( )
#applicants take-up rate
#goods #bads
DR = Db / Dg + Db
However, the rate is price dependent, so: DR(p) =
DbFb p
( )
DbFb p
( )+ DgFg p
( )
Diﬀerentiating wrt price, we get: D ′
R p
( )= (hg p
( )− hb p
( ))DR p
( )(1− DR p
( ))
failure rate diﬀerence between
goods and bads
hg p
( )= − ′
Fg p
( )/ Fg p
( )= fg p
( )/ Fg p
( )
hb p
( )= − ′
Fb p
( )/ Fb p
( )= fb p
( )/ Fb p
( )
via substitution, chain-rule

and product rule

Sensitivity and risk
D ′
R p
( )= z p
( )DR p
( )(1− DR p
( ))
z p
( )= hg p
( )− hb p
( ) Difference in hazard rate
Price-dependent risk rate (or PDR rate) - i.e. rate at which risk changes with price
commercial lending region
for constant
z(p) and
z(p)>0
In this part of the curve, price-dependent risk more salient for

higher-risk populations.

“As found in practice - i.e. effect of price on risk much stronger in sub-
prime than prime. Very little impact in super prime.”

What does it imply for pricing? Net interest income for a loan of value
1 (for simplicity) and loss-given default:
Π p
( )= (p − c)dg p
( )− ldb p
( )
0 < l ≤1
There is a positive number k such that z(p)>k for all p >=c in which case the lender cannot achieve a profit at any price.
Whether or not it is profitable to offer a loan depends not only on bads:goods but on their difference in price sensitivity as
reflected in the differential hazard rate z(p)

Pricing/Underwriting
Π p
( )= (p − c)dg p
( )− ldb p
( )
There is a positive number k such that z(p)>k for all p >=c in which
case the lender cannot achieve a profit at any price.
Whether or not it is profitable to offer a loan depends not only on
bads:goods but on their difference in price sensitivity as reflected in
the differential hazard rate z(p)
k = hb p
( )− hg p
( )

Estimating d(p)
• We assume a data-driven approach based upon random testing (of
prices), but we need to evaluate the current random-testing method/data

• Are we storing all endogenous variables? (if any)

• Do we have suﬃcient test data for each segment?

• (Note: we can model using available data and also estimate with
piecewise numerical estimator - see later.)

• Given the historical data, we now estimate a price-response model using
ML (GLMs or ANN?)

• Do we have endogeneity?

• What might cause it? e.g. agent intervention in the sale of the loan

Model
• Considerations

• Feature selection (for price sensitivity)

• Feature engineering overhead

• Monotonicity - The assumption of pricing optimization is that prices change monotonically

• Can we short-cut feature engineering stage with use of a novel DL method, like Deep Lattice
Networks?

• Pros: shortcut feature engineering, Tensorflow models, organizational learning

• Cons: lack of proven field experience (with DLN). Expertise better used elsewhere (portfolio
optimization?)
• Updating: frequency and method (to track temporal changes)

• To what extent can this be automated?

• Should we include other product co-joint analysis?  
(“Price to entice” as gateway to higher LTV.)

Pricing Segmentation
M Product dimensions
(e.g. amount, term)
N Customer dimensions
(e.g. scores, employment)

observable/explainable
Price - Price -
- Price Price -
- Price Price
- - - Price
NxM buckets
(upper bound)

logical limit 1:1 pricing

or.. NxM dimensional surface

whose geometry determines revenue/profit

and should be correlated to:

• incremental profitability

• price sensitivity
• Smaller (more targeted) buckets => profitability, but also
complexity. What is the trade-off? (We need to do work to
analyze the frontier of ROI trade-off curves.)

• All else being equal, charge lower prices to segments with:

• higher inc. profitability

• higher price sensitivity

• other business incentive e.g. promote certain
channels? (Again, what are the business constraints.)
• Customer dimensions:

• Risk, [age], loyalty, [geography] etc.

• Product dimensions:

• Loan amount, term

• Term - should we consider a higher resolution of term (than
just 3/5 year) to obtain better results? (And give a more
satisfactory UX.)?

Are there other ways to differentiate product besides price? e.g. “no-charge term rescheduling or repayment rewards”

i.e. customer segmentation achieved via product segmentation correlated to price sensitivity

WTP: Pricing
Demand
Price
Demand
Price
Demand
Price
p* p1* p2*
c c c
Single price p*

missed revenue = A,B
Two prices c<p1*<p2*

missed = A,B,C
1:1 pricing

missed = “0”
• We now see the importance of predicting WTP = revenue/profit

• However, WTP extends beyond risk-based customer variables e.g. to include things like financial
sophistication, brand aﬃnity, urgency, competitive environment etc.

• Note: are some of these observable, say via random experiments in design (or other means). Do we
capture all the necessary data? What other data sources can we use?

• i.e. pricing (sensitivity) is not a “backend” function only

• Can we gather more user data? (e.g. for DM clients can we aﬀord “second stage?”)

online tasks
Dynamic segmentation
Query $
Price
Dims
Optimization
Prices for all

segments
offline tasks
Optimization $
Offer Offer
Application Application
Analysts
reviewers
• d(p) can be continually computed (and “event driven”)

• New data resolution (higher freq)

• Market shifts (lower freq)

• What are the operational implications of moving to a 1:1
model?

• To what extent can the entire process be automated?

• Optimization model (inc. freq)

• Operational model (as an operationally acceptable rule-
based system - what are the constraints?)

• Can/do we capture all the data we need (e.g. UX state)?

• What is the limit at which other approaches should be
considered (due to behavioral economics factors)?

• The exact nature and implementation of the current offline
tasks need to be audited and automation ROI calculated
Compute power is cheap. If we can find an end-to-end computational solution, even highly complex (AI), it potentially pays!
What are the strategic (tech) imperatives in a commodity PUL market?

Optimizing Prices (Simple loan)
E π p
( )
( )= F p
( )[ 1− PD
( )× p − c
( )− PD × l]
Consider a single two-period loan (just makes math simpler initially) [And assume LGD independent of price, which it is not.]
l = LGD + c
[simplistic incremental profit]
price response
= F p
( )[ 1− PD
( )× p − c −
l
o
⎛
⎝
⎜
⎞
⎠
⎟] o = (1− PD) / PD odds that loan will not default
loss given default
cost of funds
⇒ o*
=
LGD + c
p − c
underwriting criterion for a price-taking lender - i.e. take any loans with odds > o*
But we are interested in price-setting lender. In which case we want to charge a price p* that would maximize

Here we omit the analytical solution for as there is no closed-form solution for any non-linear price response function.

Rather, we can see that if we can compute then we can calculate the maximum using numerical methods.
E π p
( )
( )
′
E π p
( )
( )
F p
( )

Price-dependent risk
Analytical solution for
h p*
( )=
1
p*
− c −
l
o
′
E π p
( )
( )= 0
But we have so far assumed that PD and loss are independent of price of loan, which is incorrect.

If we now assume that we have a population Dg of good and Db bad customers.
E π p
( )
( )= Fg p
( )Dg p − c
( )− Fb p
( )Dbl
′
E π p
( )
( ) ′
E =0
⎯ →
⎯⎯ hg p*
( )=
1
p*
− c −
fb p*
( )
fg p*
( )
⎛
⎝
⎜
⎞
⎠
⎟
l
o0
⎛
⎝
⎜
⎞
⎠
⎟
This holds for price-dependent risk (diﬀerent dist. for bads/goods)
′
E π p
( )
( ) ′
E =0
⎯ →
⎯⎯ hg p̂
( )=
1
p̂ − c −
l
o p̂
( )
⎛
⎝
⎜
⎞
⎠
⎟
If we assume the odds will not change (i.e. ignore price-dependent risk)
BUT: we do expect goods have higher failure rate => fb p
( )/ fg p
( )> Fb p
( )/ Fg p
( )
And therefore (plugging into above equations):

Which means the (amateur) lender who does not consider influence of price upon risk will set a price that is higher than the
expert lender, which means higher losses and lower profit. Note: this aﬀects competitors profitability - see Nomis white paper.
p*
< p̂
population odds

Adaptive Optimization
1. Iterate k pricing adjustments α1 > α2 > α3...> 0
2. Use 2-point price estimation (testing) to estimate ĥ pk
( )= 2 d1D2 − d2D1
( )/[δ d1D2 + d2D1
( )
( )]
3. Calculate price gap g(pk ) = ĥ pk
( )−1/(pk − c − l / o)
4. if − ε ≤ g(pk ) ≤ ε set p*
= pk and stop
5. if g(pk ) < −ε set pk+1 = pk +αk and go to step 2
6. if g(pk ) > ε set pk+1 = pk −αk and go to step 2
1/ p − c − l / o
( )
h p
( ) assumed unknown
c + l / o r*
ε
ε
p
Raise Hold Lower
inverse margin
What if we don’t know h(p)?

Optimization Over Segments
p*
= argp max DF p
( )π p
( )
( )
!
p*
= argp max DiFi pi
( )πi pi
( )
i=1
N
∑
⌣
"
p = argp max DiFi pi
( )Ri pi
( )
i=1
N
∑
!
p = argp max DiFi pi
( )πi pi
( )
i=1
N
∑ + DiFi pi
( )Ri pi
( )
i=1
N
∑
⎡
⎣
⎢
⎤
⎦
⎥
Or with constraints:
e.g Bounds, structural constraints, monotonicity of pricing, segment bounds, rate endings
!
p = argp max ΦC ( DiFi pi
( )θi pi
( )
i=1
N
∑ )
⎡
⎣
⎢
⎤
⎦
⎥
c=1
M
∑
⎡
⎣
⎢
⎤
⎦
⎥
Subject to feasibility whilst trying to avoid post-hoc pricing adjustments (“overlays”) due to “shortcomings” in the model.

And subject to idealized segmentation (that avoids mixed [separable] failure curves with implied lack of global maximum).

And subject to location of the portfolio on the eﬃcient frontier (next).

Where/what are the automation opportunities for constraint review and management?
(assume independently solvable)

Efficient Frontier
Return
Risk
• We can estimate the frontier and where our portfolio lies within it

• But we may have chosen unrealistic (or too many) constraints

• We might have chosen constraints outside of the model (i.e. tuned prices manually)

• Optimal portfolio selection is an area of R&D opportunity (reinforcement learning, genetic algorithms, etc.)

• Optimal segmentation is also an area for R&D opportunity
Can I have the highest revenue with greatest profit and lowest risk please? No.

Dynamic Pricing for Personal Unsecured Loans

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