MANG 2043 – Analytics for Marketing
MAT012 – Credit Risk Scoring
Lecture 4a
Copyright By PowCoder代写 加微信 powcoder
This Lecture’s Learning Contents
Behavioural scoring, profit scoring and risk based pricing
Survival analysis
Statistical methods for behavioural scoring
This is about how to manage existing customers, using information on their activities and personal characteristics. Former almost always dominate.
Information: data on customers’ activities in performance period such as payments, purchases, monthly credit bureau reports;
Usage: account management such as marketing campaigns, increase of credit limit pre-authorization etc.
Construction: the same methods as credit scoring
Alternative methods: Markov chain models
Extra performance information used by behavioural scoring
Balance of account (e.g. various weighted averages, trends, variances)
Total(number, amount, max) debit payments
total (or max) credit payments
time since overdrawn
number of “reminder” letters sent
no general economic data used
credit bureau information updated monthly
US bureaus have more info than UK who have more info than European ones
Current approaches
Score the customer for:
Default over all products (customer default)
Profit over one product (product profit)
Profit over all products (customer profit)
Customer default scoring is regularly done. Banks find information on current accounts very useful for estimating default on loans. It uses standard behavioural scoring methods.
Product profit scoring has been introduced. Typically, the lenders take a matrix of behavioural score (risk) and average balance or credit (return) as well as use rules based on these.
Developments needed to introduce customer profit scoring
Surely going from default scoring to profit scoring is just a change in definition of “good”?
Accounting problems
quantify profit on each transaction
Estimate LGD ( loss given default)
Information storage problems
customer’s transactions at customer level not account level
marketing/credit/operational databases connected
Data Protection legislation
Modelling problems
suitable time horizon ( longer than default)
economic conditions
Model attrition
Developing profit scoring models
Need for dynamic consumer credit risk models
Application scoring :
snapshot of applicant connected to snapshot of status at fixed time in future
Behavioural scoring :
video clip of customer’s recent behaviour connected to snapshot of his status at fixed time in future ( and video clip translated into snap shot of statistics0
Profit scoring :
video clip ( or snapshot) of customer’s behaviour/characteristics connected to video clip of future performance
Two ways of introducing dynamics (both allow for economic effects)
Markov chain models
Survival analysis approaches
Also allow for risk based pricing to increase profit
Existing Markov chain models of customer behaviour
Markov chains are not widely used in behavioural scoring even though first suggested in 1962.
Traditionally used in conjunction with behavioural scoring to forecast:
number of cases in arrears to manage resources
amount in arrears for debt provisioning
State: simplest example would be number of payments in arrears (0,1,2,3,4+).
Transition probabilities (roll rates) got from past data)
Second point: if gi or bi = 0 then expression for IV includes log0, which is infinite. Replace 0 by ½.
Parameter estimates
Given sample of R consumers history (s0r , s1r ,s2r ,…..sTr )
nt(i) is number of times str is i. n(i) = t=0T-1 nt(i)
nt(i,j) is number of times str is i and st+1r is j. n(i,j) = t=0T-1 nt(i,j)
Similar definitions for nt(i,j,k ) and n(i,j,k).
1. Estimate of
2. Test if the chain is Markov
is estimate of probability p(i,j,k) of going i->j->k.
Markovity corresponds to hypothesis p(1,j,k)=p(2,j,k)=…=p(m,j,k)
Can check this by using 2 test: are the populations 1,j,?; 2,j,?,..m,j,? the same ?
If n*(i,j) =t=0T-2 nt(i,j), then
States are number of months overdue – 0,1,2,2+
Number of transitions are as follows
What is the Markov chain and its transition matrix? If initially distribution of consumers in January is = ( 0.9,0.1,0,0). What is distribution in February and March?
From\To 0 1 2 2+
0 900 100 0 0
1 60 10 30 0
2 10 5 5 30
2+ 0 0 0 80
Markov chain to profit scoring
Markov chain profit scoring model won prize for best OR application in US in 2002 for Bank One.
Used Markov decision process approach
States, S: describes situation of customer
Had 6 dimensions , 2-4 intervals each dimension (recency , frequency of payments, purchases, etc) – 200 states in total
Actions, A: change in credit limit, interest rate, cross sell,
Affects immediate profit r(a,s) and transition probability p(a,s,s’) of moving from state s to state s’.- 50 actions in total
Found optimal policy by solving dynamic programming optimality equation for vn (s) optimal profit over next n periods on customer in state s.
vn (s) = maxa {r(a,s) + ,s’ p(a,s,s’) vn-1 (s’)}
Orthodox profit-based Markov chain model
States describe customers account, u= (b,n,i) –
b is balance
n is no. periods since last payment
i refers other info.
We need to set credit limit L. Estimate p(L,u,u’),prob. account goes from u to u’ under limit L and r(L,u) is the reward in period.
Estimation:
t(L,u,a), the prob. account in state u with credit limit L repays a next period;
q(L,u,o), the prob account in state u with credit limit L orders o next period;
w(L,u,i’), the prob in state u with credit limit L change information to i’
and defining transition probabilities by
p (L;b,n,i; b+o-a,0,i’) = t(L,u,a)q(L,u,o) w(L.u,i’), if b+o-a L, and a >0.
p (L;b,n,i; b-a,0,i’)=t(L,u,a). w(L,u,i’).( q(L,u,0)+ q(L,u,o): if o>.L-b+a;a >0.
p (L;b,n,i; b+o,n+1,i’) = t(L,u,0).q(L,u,o).w(L,u,i’), if b+o L.
p (L;b,n,i; b-a,n+1,i’) = t(L,u,0) w(L,u,i’)( q(L,u,0)+ q(L,u,o): if o>L-b,
f % of purchase that is profit. Company writes off bad debt after N periods
r(L;b,n,i) = f o q(L,s,o) – b.t(L,s,0) (n-(N-1))
V(n,u) expected profit over n periods given account in state u satisfies
V(n,u) = max L { r(L,u) + p(L;u,u’)V(n-1,u’) }
Mover-stayer models
Assumes the population consists of ‘stayers’ who never leave initial state and movers who make transitions
Example: Frydman et al ( Mgmt Sc.,1985, pp1203)
P: paid up state; C: “current” i.e owe but up to date with payment; D:overdue;
p(i,j) – probability mover moves from state i to state j;
s(i) proportion of stayers in state i (s(0)=1-s( P)-s(C) –s(D)):proportion of movers)
q(j,j)= s(j) + s(0)p(j,j); p(i,j) = s(0)p(i,j)
To get estimators using large number of periods N
n(i,j) no. of transitions from i to j
n(i) total number of visits to i;
s(i) number of sample who stay in state through whole of period
n*(i) number in sample initially in i
Stationary Markov Chain estimate p(i,j) =n(i,j)/n(i)
Mover –stayer estimates:
s(i) = s(i)/n*(i);
p(i,i) =(n(i,i)-Ns(i))/(n(i)-Ns(i));
Risk based pricing: Take-up modelling
Move from default based ( risk ) assessments to profit based assessments
Profit depends on much more than accept/reject decision of default scoring
Ease of acquisition/ response to mailings
Version of loan product offered and at what price (interest rate)
cannibalisation
How account is managed
Cross selling
Up selling
Attrition/prepayment
Key points in developing pricing models
Lost quote data is valuable
Find out who did not take offer ( and if possible why)
Regulations will set constraints on minimum and maximum prices
Could say take everyone but the price for some is so high no one will accept – but human behaviour always produces surprises (and regulations will tend to protect some whose behaviour is questionable).
Market changes much faster than economic changes
“response scorecards” need to be rebuilt faster than risk ones
Utilization of product is important for profitability
Adverse selection
Offer at interest rate 6% does not get normal population mix , but more of those who could not get better offer than 6%
Take Probability q(p,r)
Profitability depends vitally on take probability
Take probability is function of
risk probability p, (prob. of being good) of borrower
Rate offered r
Take probability can also depend on other features
Need to estimate this probability
Take scorecards need to be developed
Cannot estimate without considering adverse selection (i.e. does depend on p and more so than you may estimate)
Common risk-free response rate functions q(r)
q(r) – fraction who will take loan at rate r
w(r)- density function of maximum willingness to pay
Linear response function
Logistic response function
Linear response rate function with rL=0.04 and b=2.5
Logistic response rate function with a=4, b=32
Optimal price when response function depends only on rate charged
Example with logistic response
sresponse =4-32r, rF =0.05, lD =0.5
Probability of being Good, p Optimal interest rate r as % Take probability q(r ) as %
0.5 63.1 0.000009
0.6 44.8 0.003
0.7 31.7 0.2
0.8 22.0 4.5
0.9 15.5 28.0
0.94 13.7 40.2
0.96 13.0 45.7
0.98 12.4 50.5
0.99 12.2 52.7
1.00 11.9 54.7
Risk response rates q(r,p)
Same principle but now have to worry about
Adverse selection
Affordability
is probability of borrower being Good if interest rate charged is r, if p is probability of being Good at benchmark interest rate
Differentiate and set derivative to zero to find optimal rate
Example with logistic response
sresponse=4-32r-50p, rF =0.05, lD =0.5
Risk based pricing
Risk based pricing needs much more careful modelling and parameter estimation
Adverse selection
Cannibalisation
Other features might affect response rate not just “price” (interest rate charged)
Dynamic Price modelling will come. In the form of yield management, it has proved successful in airlines, hotels, and car rentals
Now appearing in China where limits on amounts banks can loan in any one year
Past Behaviour —> forecast——–> Future Behaviour
Performance
(12 months)
Observation
Outcome Period
£500-£2500
Beh. Score
£12,500 £15,000
Beh. Score
£2,000 £4,000 £10,000
Beh. Score
£500 £1,000
Overdraft Limit
Balance £500-£2500
Balance > £2500
Beh. Score >500
Beh. Score 300-500
Beh. Score <300 No overdraft 0 1 2 3 4+ 0 .985 .015 1 .25 .4 .35 2 .05 .05 .3 .6 3 .04 .02 .07 .12 .75 IV.2.1 Decisions and Data flows for Loans RejectReject AcceptAccept Bureau Data Management Not TakeNot Take PerformancePerformance Offer TermsOffer Terms to Borrowerto Borrower Offer TermsOffer Terms To BorrowerTo Borrower Response Data and Final Offer Terms DecisionsDecisions Figure IV.2.2 Price Tiers and RISK-BASED PRICEFigure IV.2.2 Price Tiers and RISK-BASED PRICE Probability of a Good Reject Tier 1 Tier 2 Fig. IV.2.3 OFFER AND ACCEPTANCE SCORES sCReject Accept Fig. IV.2.4(a) ADDING RESPONSE SCORE TO INFLUENCE DIAGRAM (without Adverse Selection)(without Adverse Selection) !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type ResponseResponse Score Score Response Response Take/No TakeTake/No Take Fig. IV.2.4(b) ADDING RESPONSE SCORE TO INFLUENCE DIAGRAM (with Adverse Selection)(with Adverse Selection) !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type ResponseResponse Score Score Response Response Take/No TakeTake/No Take IV.2.5 SEQUENTIAL DECISIONS WITH TWO-TIERSIV.2.5 SEQUENTIAL DECISIONS WITH TWO-TIERS High Rate Offer Low Rate Offer r=q2/q1 ModelTWO TIER Fig. IV.2.6(a)Expected Account Profit: Single TierFig. IV.2.6(a)Expected Account Profit: Single Tier Reject Accept (r + fD )(p − p1 ) q(r + fD )(p − p1 ) E[PA | r, p, fD ] Reject Accept q1(r1 + fD )(p − p1 ) q2 (r2 + fD )(p − p2 ) Tier 1Tier 1 Fig. IV.2.6(b) Expected Account Profit: Two TiersFig. IV.2.6(b) Expected Account Profit: Two Tiers E[PA | rk , p,qk , fD ] Tier 2Tier 2 IV.2.7 OPTIMAL ACCEPT AND OFFER POLICIES 0.760.76 0.840.84 0.920.92 1.001.00 qq2 2 / / qq11 Tier 1Tier 1 Tier 2 EXPECTED PROFIT E[P | p(s), q] and E[P | p(s)]EXPECTED PROFIT E[P | p(s), q] and E[P | p(s)] (g+l)(p-p)(g+l)(p-p) q(g+l)(p-p)q(g+l)(p-p) Reject Accept SEQUENTIAL DECISIONSSEQUENTIAL DECISIONS High Rate Offer Low Rate Offer r=q2/q1 Model EXPECTED PROFIT EXPECTED PROFIT !!E[P | p(s)] FOR OFFERS 1 AND 2E[P | p(s)] FOR OFFERS 1 AND 2 (l+g)(p-p)(l+g)(p-p) q(l+g)(p-p)q(l+g)(p-p) Reject Offer 1 Offer 2 Fig 2: Comparing Expected Account Profit of a Constant APRFig 2: Comparing Expected Account Profit of a Constant APR (Single Tier) with Risk Neutral Lender(Single Tier) with Risk Neutral Lender Probability of a Good (non-default) Risk Neutral Lender Slope: fD + r !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type Response Response Take/No TakeTake/No Take INFLUENCE DIAGRAMINFLUENCE DIAGRAM RISK BASED PRICING MODELRISK BASED PRICING MODEL ADDITION OF RESPONSE SCOREADDITION OF RESPONSE SCORE !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type ResponseResponse Score Score Response Response Take/No TakeTake/No Take !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type Response Response Take/No TakeTake/No Take PERFECT INFORMATION ON RISK PERFORMANCEPERFECT INFORMATION ON RISK PERFORMANCE IV.2.1 Decisions and Data flows for Loans RejectReject AcceptAccept Bureau Data Management Not TakeNot Take PerformancePerformance Offer TermsOffer Terms to Borrowerto Borrower Offer TermsOffer Terms To BorrowerTo Borrower Response Data and Final Offer Terms DecisionsDecisions Figure IV.2.2 Price Tiers and RISK-BASED PRICEFigure IV.2.2 Price Tiers and RISK-BASED PRICE Probability of a Good Reject Tier 1 Tier 2 Fig. IV.2.3 OFFER AND ACCEPTANCE SCORES sCReject Accept Fig. IV.2.4(a) ADDING RESPONSE SCORE TO INFLUENCE DIAGRAM (without Adverse Selection)(without Adverse Selection) !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type ResponseResponse Score Score Response Response Take/No TakeTake/No Take Fig. IV.2.4(b) ADDING RESPONSE SCORE TO INFLUENCE DIAGRAM (with Adverse Selection)(with Adverse Selection) !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type ResponseResponse Score Score Response Response Take/No TakeTake/No Take IV.2.5 SEQUENTIAL DECISIONS WITH TWO-TIERSIV.2.5 SEQUENTIAL DECISIONS WITH TWO-TIERS High Rate Offer Low Rate Offer r=q2/q1 ModelTWO TIER Fig. IV.2.6(a)Expected Account Profit: Single TierFig. IV.2.6(a)Expected Account Profit: Single Tier Reject Accept (r + fD )(p − p1 ) q(r + fD )(p − p1 ) E[PA | r, p, fD ] Reject Accept q1(r1 + fD )(p − p1 ) q2 (r2 + fD )(p − p2 ) Tier 1Tier 1 Fig. IV.2.6(b) Expected Account Profit: Two TiersFig. IV.2.6(b) Expected Account Profit: Two Tiers E[PA | rk , p,qk , fD ] Tier 2Tier 2 IV.2.7 OPTIMAL ACCEPT AND OFFER POLICIES 0.760.76 0.840.84 0.920.92 1.001.00 qq2 2 / / qq11 Tier 1Tier 1 Tier 2 EXPECTED PROFIT E[P | p(s), q] and E[P | p(s)]EXPECTED PROFIT E[P | p(s), q] and E[P | p(s)] (g+l)(p-p)(g+l)(p-p) q(g+l)(p-p)q(g+l)(p-p) Reject Accept SEQUENTIAL DECISIONSSEQUENTIAL DECISIONS High Rate Offer Low Rate Offer r=q2/q1 Model EXPECTED PROFIT EXPECTED PROFIT !!E[P | p(s)] FOR OFFERS 1 AND 2E[P | p(s)] FOR OFFERS 1 AND 2 (l+g)(p-p)(l+g)(p-p) q(l+g)(p-p)q(l+g)(p-p) Reject Offer 1 Offer 2 Fig 2: Comparing Expected Account Profit of a Constant APRFig 2: Comparing Expected Account Profit of a Constant APR (Single Tier) with Risk Neutral Lender(Single Tier) with Risk Neutral Lender Probability of a Good (non-default) Risk Neutral Lender Slope: fD + r !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type Response Response Take/No TakeTake/No Take INFLUENCE DIAGRAMINFLUENCE DIAGRAM RISK BASED PRICING MODELRISK BASED PRICING MODEL ADDITION OF RESPONSE SCOREADDITION OF RESPONSE SCORE !!DecisionDecision Accept/RejectAccept/Reject PerformancePerformance Score Score PerformancePerformance Good/Bad Good/Bad ProfitProfit DecisionDecision Offer TypeOffer Type ResponseResponse Score Score Response Response Take/No TakeTake/ 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com