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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.

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Dynamic Pricing for Personal Unsecured Loans

  • 1. Price Sensitivity (0.1) Towards 1:1 pricing Modern/dynamic pricing approaches applied to personal unsecured loans (PUL). Paul Golding, Dec 2018 @pgolding
  • 2. 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.
  • 3. 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.
  • 4. 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.)
  • 5. 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.
  • 6. 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:
  • 7. 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}
  • 8. 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
  • 9. 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
  • 10. 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)
  • 11. 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
  • 12. 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)).
  • 13. 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
  • 14. 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.
  • 15. 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
  • 16. 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)
  • 17. 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 ( )
  • 18. 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
  • 19. 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.)
  • 20. 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
  • 21. 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?”)
  • 22. 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?
  • 23. 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 ( )
  • 24. 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
  • 25. 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)?
  • 26. 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)
  • 27. 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.