An outline of the mathematics and technicalities of predicting personalized loan pricing -- i.e. the next step beyond pure risk-based pricing. Based largely on the work of Robert Phillips.
Several ways to calculate option probability are outlined, including the derivation that relies on terms from the Black-Scholes (Merton) formula. Programming formulas are provided for Excel. Delta is discussed, as a proxy for option probability and the differences in various volatility measures are described.
This blog is a place by Apoorva javadekar to discuss various aspects of life. But in specific, it discusses Economics, Mathematics, Politics and sports events.
I like to write about recent economic policies, Socio - political issues, reviews about new books, cricket and fitness. This blog is a good place to take a look at least once a week.
Several ways to calculate option probability are outlined, including the derivation that relies on terms from the Black-Scholes (Merton) formula. Programming formulas are provided for Excel. Delta is discussed, as a proxy for option probability and the differences in various volatility measures are described.
This blog is a place by Apoorva javadekar to discuss various aspects of life. But in specific, it discusses Economics, Mathematics, Politics and sports events.
I like to write about recent economic policies, Socio - political issues, reviews about new books, cricket and fitness. This blog is a good place to take a look at least once a week.
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www.truthinoptions.net
olagues@gmail.com
504-875-4825
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Alpha Index Options Explained. These can be used to efficiently convert conce...Truth in Options
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Alpha Index Options Explained. These can be used to efficiently convert concentrated employee stock or options positions to a diversified portfolio by
Jacob Sagi and Robert Whaley
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www.truthinoptions.net
olagues@gmail.com
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http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470471921.html
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A C++ based program which prices the fair value of a participating life insurance whereby the underlying follows a Kou process and the insurer's default occurs only at contract's maturity.
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ELASTICITY OF DEMAND
PharmaKhabar is an online platform that provides entire pharma related information, news, and articles at one place.
https://www.pharmakhabar.com/
1Valuation ConceptsThe valuation of a financial asset is b.docxeugeniadean34240
Ā
1
Valuation Concepts
The valuation of a financial asset is based on determining the present value of future cash flows. Thus we need to know the value of future cash flows and the discount rate to be applied to the future cash flows to determine the current value.
The market-determined required rate of return, which is the discount rate, depends on the marketās perceived level of risk associated with the individual security. Also important is the idea that required rates of return are competitively determined among the many companies seeking financial capital. For example ExxonMobil, due to its low financial risk, relatively high return, and strong market position, is likely to raise debt capital at a significantly lower cost than can United Airlines, a financially troubled firm. This implies that investors are willing to accept low return for low risk, and vice versa. The market allocates capital to companies based on risk, efficiency, and expected returnsāwhich are based to a large degree on past performance. The reward to the financial manager for efficient use of capital in the past is a lower required return for investors than that of competing companies that did not manage their financial resources as well.
Throughout this course, we apply concepts of valuation to corporate bonds, preferred stock, and common stock. For that purpose we have to be aware of the basic characteristics of each form of security as part of the valuation process. We have to consider the following:
Ā· The valuation of a financial asset is based on the present value of future cash flows.
Ā· The required rate of return in valuing an asset is based on the risk involved.
Ā· Bond valuation is based on the process of determining the present value of interest payments plus the present value of the principal payment at maturity.
Ā· Preferred stock valuation is based on the dividend paid.
Ā· Stock valuation is based on determining the present value of the future benefits of equity ownership.
List of terms:
required rate of return
That rate of return that investors demand from an investment to compensate them for the amount of risk involved.
yield to maturity
The required rate of return on a bond issue. It is the discount rate used in present-valuing future interest payments and the principal payment at maturity. The term is used interchangeably with market rate of interest.
real rate of return
The rate of return that an investor demands for giving up the current use of his or her funds on a noninflation-adjusted basis. It is payment for forgoing current consumption. Historically, the real rate of return demanded by investors has been of the magnitude of 2 to 3 percent.
inflation premium
A premium to compensate the investor for the eroding effect of inflation on the value of the dollar.
risk-free rate of return
Rate of return on an asset that carries no risk. U.S. Treasury bills are often used to represent this measure, although longer-term government securities have al.
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GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
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The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
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Welocme to ViralQR, your best QR code generator.ViralQR
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Welcome to ViralQR, your best QR code generator available on the market!
At ViralQR, we design static and dynamic QR codes. Our mission is to make business operations easier and customer engagement more powerful through the use of QR technology. Be it a small-scale business or a huge enterprise, our easy-to-use platform provides multiple choices that can be tailored according to your company's branding and marketing strategies.
Our Vision
We are here to make the process of creating QR codes easy and smooth, thus enhancing customer interaction and making business more fluid. We very strongly believe in the ability of QR codes to change the world for businesses in their interaction with customers and are set on making that technology accessible and usable far and wide.
Our Achievements
Ever since its inception, we have successfully served many clients by offering QR codes in their marketing, service delivery, and collection of feedback across various industries. Our platform has been recognized for its ease of use and amazing features, which helped a business to make QR codes.
Our Services
At ViralQR, here is a comprehensive suite of services that caters to your very needs:
Static QR Codes: Create free static QR codes. These QR codes are able to store significant information such as URLs, vCards, plain text, emails and SMS, Wi-Fi credentials, and Bitcoin addresses.
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Pricing and Packages
Additionally, there is a 14-day free offer to ViralQR, which is an exceptional opportunity for new users to take a feel of this platform. One can easily subscribe from there and experience the full dynamic of using QR codes. The subscription plans are not only meant for business; they are priced very flexibly so that literally every business could afford to benefit from our service.
Why choose us?
ViralQR will provide services for marketing, advertising, catering, retail, and the like. The QR codes can be posted on fliers, packaging, merchandise, and banners, as well as to substitute for cash and cards in a restaurant or coffee shop. With QR codes integrated into your business, improve customer engagement and streamline operations.
Comprehensive Analytics
Subscribers of ViralQR receive detailed analytics and tracking tools in light of having a view of the core values of QR code performance. Our analytics dashboard shows aggregate views and unique views, as well as detailed information about each impression, including time, device, browser, and estimated location by city and country.
So, thank you for choosing ViralQR; we have an offer of nothing but the best in terms of QR code services to meet business diversity!
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In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
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Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
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Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
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Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. Whatās changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
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
ā¢ We want to predict a borrowerās sensitivity to price and use this information to
optimize prices to achieve maximum expected profit (or some other business
goal - TBD)
ā¢ This deck is a āthink aloudā 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.
ā¢ 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.
ā¢ We heavily lean upon the work of R. Phillipsā (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
ā¢ Exposure at Default (EAD) - remaining balance at default
ā¢ Loss Given Default (LGD) = EAD/Total Loan Amount
ā¢ Probability of Default (PD) = P(Default) | t<=loan_term
ā¢ EAD and LGD are random variables
ā¢ (Note: we largely stick to Phillipsā 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
ā
EEAD =
pd
(t) Ć EADt
t=1
T
ā
PD
ELGD =
pd
(t) Ć EADt
t=1
T
ā Ć LGDt
pd
(t) Ć EADt
t=1
T
ā
EL = pd
(t) Ć EADt
t=1
T
ā Ć LGDt
ā“ EL = PD Ć ELGD Ć EEAD
Probability of default at t=1 or t=2 or t=3 ā¦ or t=T
Expected loss is major determinant of
expected profitability of a loan and
thereby influences price.
6. Simplified Pricing Threshold
E Ļ
( )= 1ā PD
( ) p ā pc
( )ā PD Ć (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
( )
1ā PD
+ pc
cost of capital
Risk-adjusted premium
This only suggests a lower bound.
It does not suggest the price:
ā¢ Maximize profitability?
ā¢ Maximize bookings subject to profit constraint?
ā¢ 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 oļ¬ered. But suļ¬cient
to show the role of risk.
A function of PD
pc =0.1
Risk cut-oļ¬ equation:
7. Risk Prediction
s x
( )= ln
Pr{G | x}
Pr{B | x}
ā
ā
ā
ā
ā
ā
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 donāt care about interpretability?)
PĢ{B | x}
8. Risk Factors
PĢ{B | x} ā¢ Risk prediction based upon borrower
observables
ā¢ But also the loan itself
ā¢ Amount
ā¢ Leverage (e.g. LTV, DTI)
ā¢ Period (longer loans riskier due to information
asymmetry, random shock etc.)
ā¢ And price-dependent risk
Not part of x
Borrower dependent risk
Product dependent risk
9. Price-dependent Risk
If for all values of
E Ļ
( )= 1ā PD
( ) p ā pc
( )ā PD Ć (1+ pc )
ā E Ļ
( )= 1ā PD p
( )
( ) p ā pc
( )ā PD p
( )Ć (1+ pc )
P ā²
D p
( )> 0 i.e. positive slope, or risk increases with price
PD p
( )>
p ā pc
p +1
p ā„ pc
then there is no price at which a loan is profitable
10. Price-dependent Risk
ā¢ Fraud - fraudulent borrowers are insensitive to price
ā¢ Adverse private information - e.g. unobservable knowledge of a
borrowerās forthcoming redundancy
ā¢ Competitive alternatives - less riskier customers have more options and
so loans at a certain (higher) price will attract riskier borrowers
ā¢ Behavioral factors - behavioral economics, mental accounting, time
discounting etc. (Dynamic contextual factors unrelated to loan
characteristics or borrower credit-worthiness attributes.)
ā¢ Aļ¬ordability - e.g. DTI (capacity to pay)
11. Incremental Profitability
ā¢ Optimal price imposes two questions:
ā¢ How will profitability of a loan change with price?
ā¢ How will demand for a loan change with price?
ā¢ Profitability for loans non-trivial calculation due to:
ā¢ Time value of money (discounted value of future costs/revenues)
ā¢ Uncertainty: default, prepayment, utilization (for credit line)
ā¢ We need a model for incremental profitability for PUL
ā¢ What are the variable costs? (Holds, channels?)
ā¢ What is the profit goal? (For whom?)
ā¢ If we sell/securitize loans, how does this aļ¬ect optimization?
ā¢ Longer loans are more profitable (via interest)
ā¢ Large-portfolio holders should act more risk-neutral (assuming risk of loans are statistically independent across
the portfolio) - see Phillips
ā¢ Also: should pricing model take LTV into account - e.g. cross-selling other products? āØ
(Segmentation question?)
Action: construct the incremental profitability model
12. Price Response Curve
ā¢ 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)
ā¢ 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. āmarket sizeā)
take-up rate (i.e. WTP)
price-response
(i.e. ādemandā)
Adapted from curve in this paper: Journal of Business Research
Loan APR
Loan APR
F p
( )= f w
( )dw
p
ā
ā«
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
( )ā ā²
d p
( ) = āD ā²
F p
( )= Df p
( )
F p
( )= f w
( )dw
p
ā
ā« ā ā ā²
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:
ā²
F p
( )ā ā²
d p
( ) / D = f p
( )
Note: the derivative can be interpreted as the percentage willing to pay exactly p:
ā²
F p
( )ā
F(p + h)ā 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 diļ¬erent ways in which product demand might
change in response to changing prices but all price-response
functions assume:
ā¢ non-negative (pā„0)
ā¢ continuous (no gaps in market response to prices)
ā¢ diļ¬erentiable (smooth and with well-defined slope at every
point)
ā¢ downward sloping (raising prices decreases demand)
ā¢ notwithstanding ābehavioral economics eļ¬ectsā
Additional source: DTU Management Engineering
Most common measures are:
ā¢ Slope - dā(p1) is the derivative of the price-response function at p1
ā¢ Hazard rate (slope/demand)
ā¢ Elasticity: %demand change from 1% change in price:
Īµ p1
( )= ph p1
( )= ā²
d (p1)p1 / d p1
( )= pf p
( )/ F p
( )
h p1
( )=
ā²
d (p1)
d p1
( )
= f p
( )/ F p
( )
point elasticity
We want to estimate this curve
from any actual take-up data.
āFailure rateā
(See Yao)
has important underlying
properties when calculating
optimal pricing.
15. Sensitivity and risk
ā¢ As said, higher prices = higher default rates AND higher risk customers are less price sensitive (e.g. price
elasticity higher for higher FICO)
ā¢ 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
( )
Diļ¬erentiating wrt price, we get: D ā²
R p
( )= (hg p
( )ā hb p
( ))DR p
( )(1ā DR p
( ))
failure rate diļ¬erence between
goods and bads
hg p
( )= ā ā²
Fg p
( )/ Fg p
( )= fg p
( )/ Fg p
( )
hb p
( )= ā ā²
Fb p
( )/ Fb p
( )= fb p
( )/ Fb p
( )
via substitution, chain-rule
and product rule
16. Sensitivity and risk
D ā²
R p
( )= z p
( )DR p
( )(1ā DR p
( ))
z p
( )= hg p
( )ā hb p
( ) Diļ¬erence 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.
āAs found in practice - i.e. eļ¬ect of price on risk much stronger in sub-
prime than prime. Very little impact in super prime.ā
What does it imply for pricing? Net interest income for a loan of value
1 (for simplicity) and loss-given default:
Ī p
( )= (p ā c)dg p
( )ā ldb p
( )
0 < l ā¤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 oļ¬er a loan depends not only on bads:goods but on their diļ¬erence in price sensitivity as
reflected in the diļ¬erential hazard rate z(p)
17. Pricing/Underwriting
Ī p
( )= (p ā c)dg p
( )ā 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 oļ¬er a loan depends not only on
bads:goods but on their diļ¬erence in price sensitivity as reflected in
the diļ¬erential hazard rate z(p)
k = hb p
( )ā hg p
( )
18. Estimating d(p)
ā¢ We assume a data-driven approach based upon random testing (of
prices), but we need to evaluate the current random-testing method/data
ā¢ Are we storing all endogenous variables? (if any)
ā¢ Do we have suļ¬cient test data for each segment?
ā¢ (Note: we can model using available data and also estimate with
piecewise numerical estimator - see later.)
ā¢ Given the historical data, we now estimate a price-response model using
ML (GLMs or ANN?)
ā¢ Do we have endogeneity?
ā¢ What might cause it? e.g. agent intervention in the sale of the loan
19. Model
ā¢ Considerations
ā¢ Feature selection (for price sensitivity)
ā¢ Feature engineering overhead
ā¢ Monotonicity - The assumption of pricing optimization is that prices change monotonically
ā¢ Can we short-cut feature engineering stage with use of a novel DL method, like Deep Lattice
Networks?
ā¢ Pros: shortcut feature engineering, Tensorflow models, organizational learning
ā¢ Cons: lack of proven field experience (with DLN). Expertise better used elsewhere (portfolio
optimization?)
ā¢ Updating: frequency and method (to track temporal changes)
ā¢ To what extent can this be automated?
ā¢ Should we include other product co-joint analysis? āØ
(āPrice to enticeā 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:
ā¢ incremental profitability
ā¢ price sensitivity
ā¢ Smaller (more targeted) buckets => profitability, but also
complexity. What is the trade-oļ¬? (We need to do work to
analyze the frontier of ROI trade-oļ¬ curves.)
ā¢ All else being equal, charge lower prices to segments with:
ā¢ higher inc. profitability
ā¢ higher price sensitivity
ā¢ other business incentive e.g. promote certain
channels? (Again, what are the business constraints.)
ā¢ Customer dimensions:
ā¢ Risk, [age], loyalty, [geography] etc.
ā¢ Product dimensions:
ā¢ Loan amount, term
ā¢ 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 diļ¬erentiate product besides price? e.g. āno-charge term rescheduling or repayment rewardsā
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 = ā0ā
ā¢ We now see the importance of predicting WTP = revenue/profit
ā¢ However, WTP extends beyond risk-based customer variables e.g. to include things like financial
sophistication, brand aļ¬nity, urgency, competitive environment etc.
ā¢ 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?
ā¢ i.e. pricing (sensitivity) is not a ābackendā function only
ā¢ Can we gather more user data? (e.g. for DM clients can we aļ¬ord āsecond stage?ā)
22. online tasks
Dynamic segmentation
Query $
Price
Dims
Optimization
Prices for all
segments
offline tasks
Optimization $
Oļ¬er Oļ¬er
Application Application
Analysts
reviewers
ā¢ d(p) can be continually computed (and āevent drivenā)
ā¢ New data resolution (higher freq)
ā¢ Market shifts (lower freq)
ā¢ What are the operational implications of moving to a 1:1
model?
ā¢ To what extent can the entire process be automated?
ā¢ Optimization model (inc. freq)
ā¢ Operational model (as an operationally acceptable rule-
based system - what are the constraints?)
ā¢ Can/do we capture all the data we need (e.g. UX state)?
ā¢ What is the limit at which other approaches should be
considered (due to behavioral economics factors)?
ā¢ The exact nature and implementation of the current oļ¬ine
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 Ļ p
( )
( )= F p
( )[ 1ā PD
( )Ć p ā c
( )ā PD Ć 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
( )[ 1ā PD
( )Ć p ā c ā
l
o
ā
ā
ā
ā
ā
ā] o = (1ā PD) / PD odds that loan will not default
loss given default
cost of funds
ā o*
=
LGD + c
p ā 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 Ļ p
( )
( )
ā²
E Ļ p
( )
( )
F p
( )
24. Price-dependent risk
Analytical solution for
h p*
( )=
1
p*
ā c ā
l
o
ā²
E Ļ 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 Ļ p
( )
( )= Fg p
( )Dg p ā c
( )ā Fb p
( )Dbl
ā²
E Ļ p
( )
( ) ā²
E =0
āÆ ā
āÆāÆ hg p*
( )=
1
p*
ā c ā
fb p*
( )
fg p*
( )
ā
ā
ā
ā
ā
ā
l
o0
ā
ā
ā
ā
ā
ā
This holds for price-dependent risk (diļ¬erent dist. for bads/goods)
ā²
E Ļ p
( )
( ) ā²
E =0
āÆ ā
āÆāÆ hg pĢ
( )=
1
pĢ ā c ā
l
o pĢ
( )
ā
ā
ā
ā
ā
ā
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 aļ¬ects competitors profitability - see Nomis white paper.
p*
< pĢ
population odds
25. Adaptive Optimization
1. Iterate k pricing adjustments Ī±1 > Ī±2 > Ī±3...> 0
2. Use 2-point price estimation (testing) to estimate hĢ pk
( )= 2 d1D2 ā d2D1
( )/[Ī“ d1D2 + d2D1
( )
( )]
3. Calculate price gap g(pk ) = hĢ pk
( )ā1/(pk ā c ā l / o)
4. if ā Īµ ā¤ g(pk ) ā¤ Īµ set p*
= pk and stop
5. if g(pk ) < āĪµ set pk+1 = pk +Ī±k and go to step 2
6. if g(pk ) > Īµ set pk+1 = pk āĪ±k and go to step 2
1/ p ā c ā l / o
( )
h p
( ) assumed unknown
c + l / o r*
Īµ
Īµ
p
Raise Hold Lower
inverse margin
What if we donāt know h(p)?
26. Optimization Over Segments
p*
= argp max DF p
( )Ļ p
( )
( )
!
p*
= argp max DiFi pi
( )Ļi pi
( )
i=1
N
ā
ā£
"
p = argp max DiFi pi
( )Ri pi
( )
i=1
N
ā
!
p = argp max DiFi pi
( )Ļi pi
( )
i=1
N
ā + DiFi pi
( )Ri pi
( )
i=1
N
ā
ā”
ā£
ā¢
ā¤
ā¦
ā„
Or with constraints:
e.g Bounds, structural constraints, monotonicity of pricing, segment bounds, rate endings
!
p = argp max Ī¦C ( DiFi pi
( )Īøi pi
( )
i=1
N
ā )
ā”
ā£
ā¢
ā¤
ā¦
ā„
c=1
M
ā
ā”
ā£
ā¢
ā¤
ā¦
ā„
Subject to feasibility whilst trying to avoid post-hoc pricing adjustments (āoverlaysā) due to āshortcomingsā 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 eļ¬cient frontier (next).
Where/what are the automation opportunities for constraint review and management?
(assume independently solvable)
27. Efficient Frontier
Return
Risk
ā¢ We can estimate the frontier and where our portfolio lies within it
ā¢ But we may have chosen unrealistic (or too many) constraints
ā¢ We might have chosen constraints outside of the model (i.e. tuned prices manually)
ā¢ Optimal portfolio selection is an area of R&D opportunity (reinforcement learning, genetic algorithms, etc.)
ā¢ Optimal segmentation is also an area for R&D opportunity
Can I have the highest revenue with greatest profit and lowest risk please? No.