Using R for Analyzing Loans, Portfolios and Risk: From Academic Theory to Financial Practice

Using R in Academic Finance Sanjiv R. Das Professor, Santa Clara University Department of Finance h?p://algo.scu.edu/~sanjivdas/

Outline •  High‐performance compuGng for Finance •  Modeling the opGmal modiﬁcaGon of home loans using R •  IdenGfying systemically risky ﬁnancial insGtuGons using R network models. •  Goal‐based porOolio opGmizaGon with R •  Using R to deliver funcGons/models on the web, and for pedagogical purposes. R works well with Python and C.

h?p://www.rinﬁnance.com/RinFinance2010/agenda/ h?p://cran.r‐project.org/web/views/Finance.html

Calling C from R: An Example of Tax‐opGmized PorOolio Rebalancing

R CMD SHLIB tax.c

Topic 1 MODIFYING HOME LOANS WITH R MODELS THE PRINCIPAL PRINCIPLE: OpGmal ModiﬁcaGon of Distressed Home Loans (Why Lenders should Forgive, not Foresake Mortgages) STRATEGIC LOAN MODIFICATION: An OpGons based response to strategic default (joint work with Ray Meadows)

Game theoreGc problem: Lender determines the loan modiﬁcaGon that maximizes value of loan given that the borrower will act strategically in his best interest.

Model Home value HJM CorrelaGon h?p://algo.scu.edu/~sanjivdas/ 14

“Iso‐Service” Surface choose Loan balance = $300,000 Home value = $250,000 Remaining maturity = 25 years A = $1,933 per month Amax = $20,000 per year ($1,667 per month) 2/27/12 h?p://algo.scu.edu/~sanjivdas/ 15

Some R

Values of Iso‐Service Loans h?p://algo.scu.edu/~sanjivdas/ 17

Default Put Exercise Region L=225,000 L=250,000 h?p://algo.scu.edu/~sanjivdas/ 18

Cure risk and Re‐default Risk The risk of unnecessary relief, i.e., the Providing fuGle relief, leading to borrower would not have ulGmately ulGmate default anyway. defaulted. Value of loan accounGng A: borrower income available for housing for willingness to pay service, with mean μ and std. dev σ. h?p://algo.scu.edu/~sanjivdas/ 19

h?p://algo.scu.edu/~sanjivdas/ 20

Logit: Explaining Re‐default h?p://algo.scu.edu/~sanjivdas/

Reduced‐Form Analysis of SAMs Home values Normalize iniGal home value to 1. The opGon to default is ITM when (H > L). There is a home value D at which the borrower will default. D is a “default level” or default exercise barrier. D is a funcGon of the lender share θ, we write it as D(L, θ). D increases in L and in θ. Foreclosure recovery as a fracGon of H is ϕ. 22 h?p://algo.scu.edu/~sanjivdas/

Default Barrier and Lender Share 23 h?p://algo.scu.edu/~sanjivdas/

Barrier Model IntuiGon No default Payoﬀ=L Region of no default H0 = 1 and gains to SAM Default Payoff=фD D=L exp[‐γ(1‐θ)] Region of default 24 h?p://algo.scu.edu/~sanjivdas/

A Barrier OpGon DecomposiGon Non‐default component Default component Shared AppreciaGon component 25 PDE h?p://algo.scu.edu/~sanjivdas/

The Closed‐Form SoluGon 26 h?p://algo.scu.edu/~sanjivdas/

SAM or not? 27 h?p://algo.scu.edu/~sanjivdas/

Topic 2 MANAGING SYSTEMIC RISK BY ANALYZING NETWORKS USING R THE MIDAS PROJECT @IBM Paper: “Unleashing the Power of Public Data for Financial Risk Measurement, RegulaGon, and Governance” (with Mauricio A. Hernandez, Howard Ho, Georgia Koutrika, Rajasekar Krishnamurthy, Lucian Popa, Ioana R. Stanoi, Shivakumar Vaithyanathan) IBM Almaden)

Midas Financial Insights Proxy Statement Insider TransactionAnnual Report Loan Agreement Raw Unstructured Data Raw Unstructured Data Extract IntegrateRelated Companies Data for Analysis Exposure by subsidiary Loan Exposure … … 29

Systemic Analysis Systemic Analysis •  DeﬁniGon: the measurement and analysis of relaGonships across enGGes with a view to understanding the impact of these relaGonships on the system as a whole. •  Challenge: requires most or all of the data in the system; therefore, high‐ quality informaGon extracGon and integraGon is criGcal. Systemic Risk •  Current approaches: use stock return correlaGons (indirect). [Acharya, et al 2010; Adrian and Brunnermeier 2009; Billio, Getmansky, Lo 2010; Kritzman, Li, Page, Rigobon 2010] •  Midas: uses semi‐structured archival data from SEC and FDIC to construct a co‐lending network; network analysis is then used to determine which banks pose the greatest risk to the system. 30

Co‐lending Network •  DeﬁniGon: a network based on links between banks that lend together. •  Loans used are not overnight loans. We look at longer‐term lending relaGonships. •  Lending adjacency matrix: •  Undirected graph, i.e., symmetric •  Total lending impact for each bank: 31

Centrality •  Inﬂuence relaGons are circular: •  Pre‐mulGply by scalar to get an eigensystem: •  Principal eigenvector of this system gives the centrality score for a bank. •  This score is a measure of the systemic risk of a bank. 32

Data •  Five years: 2005—2009. •  Loans between FIs only. •  Filings made with the SEC. •  No overnight loans. •  Example: 364‐day bridge loans, longer‐term credit arrangement, Libor notes, etc. •  Remove all edge weights < 2 to remove banks that are minimally acGve. Remove all nodes with no edges. (This is a choice for the regulator.) 33

2005 CiGgroup Inc. J.P. Morgan Chase Bank of America Corp.

2006 2007 2008 2009 37

Network Fragility •  DeﬁniGon: how quickly will the failure of any one bank trigger failures across the network? •  Metric: expected degree of neighboring nodes R!2 averaged across all nodes. •  Neighborhoods are expected to expand when •  Metric: diameter of the network. 38

Top 25 banks by systemic risk 39

Topic 3 PORTFOLIO OPTIMIZATION USING R The research papers for this work are on my web page – just google it. h?p://algo.scu.edu/~sanjivdas/research.htm/ 1.  Das, Markowitz, Scheid, and Statman (JFQA 2010), “PorOolio OpGmizaGon with Mental Accounts” 2. Das & Statman (2008), “Beyond Mean‐Variance: PorOolio with Structured Products and non‐Gaussian returns.”

Standard OpGmizaGon Problem Risk aversion Mean Covariance matrix Portfolio weightsSOLUTION: See D, Markowitz, Scheid, Statman (JFQA 2010) Sanjiv Das 41

SoluGon Math Sanjiv Das 42

Final soluGon Sanjiv Das 43

Example: ConstrucGon of PorOolios: Available securiGes Expected returns Standard deviationsBond 5% 5%Low-risk stock 10% 20%High-risk stock 25% 50% The correlation between the two stocks is 0.2. Other correlations are zero. Sanjiv Das 44

Investor goals (sub‐porOolios) 45

Sub‐porOolios and overall porOolio The expected return of the overall porOolio is the weighted average of the expected returns of the sub‐porOolios. The risk of the overall porOolio is not the weighted average of the risk of the sub‐porOolios. Sanjiv Das 46

Mean‐variance eﬃcient fronGer Sanjiv Das 47

Real porOolios versus virtual porOolios Sanjiv Das 48

An alternate problem For normal returns Solve for γ Sanjiv Das 49

Risk as probability of losses Mean-variance problem: Minimize Risk (variance) subject to minimum level of Expected Return. Behavioral portfolio theory: Maximize Return subject to a maximum probability of falling below a threshold. Sanjiv Das 52

Eﬃcient FronGers in the BPT (Mental Account) World Sanjiv Das 53

Short‐Selling Constraints Linear program with non‐linear constraints. This is not a standard quadraGc programming problem (QP) like the Markowitz model. MVT uses a standard QP: quadra6c objecGve funcGon with linear constraints. 54

Modiﬁed Problem Standard QP Standard QP problem with linear constraintsAmenable to industrial opGmizers; we use the R system with the quadprog package and minpack.lm library. San Diego, 12‐Nov‐2007 55

MV FronGer with Short‐selling Sanjiv Das 56

DeviaGng from Normality with Copulas Sanjiv Das 57

Gaussian Copula Sanjiv Das 58

Extended OpGmizaGon Problem Sanjiv Das 59

SoluGon Sanjiv Das 61

Non‐Linear Products U is a set of states over r(u) is a n-vector of random returnsn assets Sanjiv Das 62 Compute p[r(u)]

Restatement of the problem This is a quadratic optimization with linear constraints.Not a quadratic optimization with non-linear constraints. Sanjiv Das 63

Introducing Structured Products Can we improve the risk-adjusted returns in a portfolio by using puts and calls? Derivatives are very risky. Sanjiv Das And so …. 64

Are puts opGmal? No, they add very little value to the portfolio. But … Sanjiv Das 65

Puts are needed when the threshold return is high For high thresholds the investor cannot get an acceptable portfolio without puts. Sanjiv Das 66

Should investors use calls? Calls are risky too.But have attractive and high mean returns! Sanjiv Das 67

Calls give be?er porOolios Improvement is greater than 60 bps ! Sanjiv Das 68

Structured Product: The Barrier‐M‐note Sanjiv Das 69

Barrier‐M Note Sanjiv Das 70

Truncated Straddle Barrier-M-noteReturn pick-up greater than 250 bps! Sanjiv Das 71

Equity‐Indexed Product Sanjiv Das 72

Conclusion •  Investors ﬁnd it easier to think in terms of mental accounts or sub‐porOolios when trying to reach their separate ﬁnancial goals. •  Behavioral porOolio theory deals with maximizing return subject to managing the risk of loss. This problem has a mathemaGcal mapping into mean‐variance opGmizaGon, yet is much more general. •  Even with short‐selling prohibited, the loss from sub‐porOolio opGmizaGon is smaller than the loss from misesGmaGng investor preferences. •  ReporGng performance by sub‐porOolio enables investors to track their goals be?er. •  Goal‐based opGmizaGon enables choosing porOolios even when normality is not assumed. •  Goal‐based opGmizaGon provides a framework for including structured products in investor porOolios. The research papers for this work are on my web page – just google it. h?p://algo.scu.edu/~sanjivdas/research.htm/ 1.  Das, Markowitz, Scheid, and Statman (JFQA 2010), “PorOolio OpGmizaGon with Mental Accounts” 2. Das & Statman (2008), “Beyond Mean‐Variance: PorOolio with Sanjiv Das 73 Structured Products and non‐Gaussian returns.”

Topic 4 PEDAGOGICAL USES FOR R USING THE WEB h?p://sanjivdas.wordpress.com/

Use the Rcgi package from David Firth: h?p://www.omegahat.org/CGIwithR/ You need two program files to get everything working. (a) The html file that is the web form for input data. (b) The R file, with special tags for use with the CGIwithR package.

R Code called from CGI

h?p://algo.scu.edu/~sanjivdas/Rcgi/mortgage_calc.html

High-performance computing (parallelR) Network modeling Q? OptimizationWeb functions Calling C from R Lattice dynamic optimization High-dimensional distributions with copulas

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Using R for Analyzing Loans, Portfolios and Risk: From Academic Theory to Financial Practice

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