Stochastic Programming in Finance and Business management

1. Stochastic Programming in Finance and Business management Lecture 7 Leonidas Sakalauskas Institute of Mathematics and Informatics Vilnius, Lithuania EURO Working Group on Continuous Optimization

2. Content • Introduction • Interbank payment management • Rational cash planning and investment management • Power production planning • Supply chain management • Network stochastic optimization

3. Introduction • The aim of stochastic programming is being applied in a wide variety of subjects ranging from agriculture to financial planning and from industrial engineering to computer networks and data mining. • Our prime goal is to help to develop an intuition on how to model uncertainty into mathematical problems, what uncertainty changes bring to the decision process, and what techniques help to manage uncertainty in solving the problems.

4. Interbank payment management

5. 0 6 1 3 0 0 1 8 1 8 4 3 1 0 1 4 3 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 U S D 0 6 5 2 0 0 3 5 6 4 2 5 0 2 0 0 1 0 1 3 0 CHECK NUMBER DRAWER ACCOUNT NUMBER DRAWEE BANK NUMBER PAYEE BANK NUMBER AMOUNT CURRENCY PAYEE ACCOUNT NUMBER DATE Only the information is sent to the clearing house Electronic Clearing The paper check is just a carrier of information. Electronic transmission is better. We dematerialize the check (remove the paper). MAKER (DRAWER) DATE PAYEE CHECK NUMBER AMOUNT DRAWEE BANK CURRENCY AUTHORIZED SIGNATURE OF MAKER’S AGENT DRAWEE BANK NUMBER

6. Interbank payment management

7. Interbank payment management • Active introduction of means of electronic data transfer in banking and concentration of interbank payments were related with creation of an automated system of clearing (ACH). • ACH should provide the principles of stability, efficiency, and security. The participants of a system must meet the requirements of liquidity and capital adequacy measures. • Sensitivity of the sector of interbank payments and settlements to changes makes the subject of investigation on simulation and optimization of interbank payments topical both in theory and in practice.

8. Automated Clearing House (ACH) • Nationwide wholesale electronic payments system • Transactions not processed individually • Banks send transactions to ACH operators • Batch processing store-and-forward • Sorted and retransmitted within hours • Banks • Originating Depository Financial Institutions (ODFIs) • Receiving Depository Financial Institutions (RDFIs) • Daily settlement by RTGS • Posted to receiver’s account within 1-2 business days • Typical cost: $0.02 per transaction; fee higher

9. Noncash Transactions Per Person Annual Annual Percentage Country Paper-based Electronic Electronic Payments Switzerland 2 65 97% Netherlands 19 128 87 Belgium 16 85 84 Denmark 24 100 81 Japan 9 31 78 Germany 36 103 74 Sweden 24 68 74 Finland 40 81 67 United Kingdom 7 58 50 France 86 71 45 Canada 76 53 41 Norway 58 40 41 Italy 23 6 20 United States 234 59 20

10. ACH Credit Transaction 1. BUYER SENDS AN ORDER TO BUYER’S BANK TO CREDIT $X TO SELLER’S ACCOUNT IN SELLER’S BANK BUYER SELLER 6. SELLER’S BANK CREDITS SELLER’S ACCOUNT WITH $X 2. BUYER’S BANK SENDS TRANSACTION TO AUTOMATED CLEARINGHOUSE BUYER’S BANK SELLER’S BANK 4. BUYER’S BANK PAYS $Y TO SETTLEMENT BANK SETTLEMENT BANK CLEARINGHOUSE 3. CLEARINGHOUSE DETERMINES THAT BUYER’S BANK OWES SELLER’S BANK $Y (ALL TRANSACTIONS ARE NETTED) 5. SETTLEMENT BANK PAYS $YTO SELLER’S BANK

11. Daylight Overdrafts • An overdraft that must be repaid by the close of business the same day • U.S Federal Reserve allows daylight overdrafts • Hong Kong does not U.S. Federal Reserve Daylight Overdraft History

12. Analisys Input of settlement flow data Calibration of transaction flow Simulation and Optimization Generation of transaction flow Simulation of settlement process Analysis of costs and liquidity Stochastic optimization of settlement system SYSTEM OF SIMULATION AND OPTIMIZATION OF INTERBANK SETTLEMENTS

13. Simulation of execution Data input Analysis Įvesties duomenų bazė Simulation parameters Output data Statistics • systems • agents • orders • balanses • reserves • siytems • agents • orders • balances • reserves Output analyzer Input module Imitavimo process Export module Settlement algorithms User’s choices Data editor CSV data of input CSV files Simulation of settlement process(BoF-PSS2 simulator)

14. BoF-PSS simulation parameters 1) 2) 3)

15. DATA FLOWS AND MODEL CALIBRATION INPUT DATA

16. Transaction data DATA FLOWS AND MODEL CALIBRATION • Transaction numberID; • Sender code a; • Recipient codeb; • Date and time of transaction t; • Transaction valueP.

17. DATA FLOWS AND MODEL CALIBRATION (CNS SYSTEM) ij–payment flow ithto jthagent; k –payment number Pij –payment from ithto jth agent; l – day of payment; C – indicator of settlement performance (C=1 – performed, C=0 – delayed, in CNS systems C=1) ; zij –number of payments from ithto jthagent.

18. Poisson-lognormal model of transaction flow DATA FLOWS AND MODEL CALIBRATION Transaction flow is Poisson with intensity ;, Transaction value is lognormallogN(,σ2). Intensities of interbank flows where: i –intensity of ithagent flow; pi –probability of ithagent payment; rij – conditional probability of transaction from ith to jthagent;  – average of logarithms transaction value; 2– variance of logarithms transaction value.

19. Estimation of parameters of Poisson lognormal model DATA FLOWS AND MODEL CALIBRATION where: z – number of transactions; J –number of agents; zi – number of transactions of ithagent; zij –number of transactions from ithto jthagent; tz – time of the session end.

20. Generation of transactions SIMULATION OF TRANSACTION FLOW ,  – random U[0,1]; – standard normal; t – transaction application time; Pij –transaction value; l – day of settlement, ; k –payment number; T –period. where:

21. Costs of ith agent COSTS OF PERIOD OF SETTLEMENTS Di – total costs of i-thagent; REi –premium of satisfaction of reserve requirements; Fi – penalty of violation of reserve requirements; Bi– costs of short time loans; TTi– losses due to freeze of finance; ACi–operacional costs.

22. COSTS OF PERIOD OF SETTLEMENTS Correspondent Account where: Kil-1 – correspondent account of ith agent at l-1 day ; i –day balance: Gi –deposit (or withdrawal).

23. Premium of satisfaction of reserve requirements COSTS OF PERIOD OF SETTLEMENTS LR – interest rate of refinansing; RRi – reserve requirements.

24. Penalty of violation of reserve requirements COSTS OF PERIOD OF SETTLEMENTS p –penalty percent points (usually 2.5) .

25. Costs of short time loans COSTS OF PERIOD OF SETTLEMENTS il– day balance; STL– overnight loan interest rate Xi– deposit (or withdrawal).

26. Losses due to freeze of finance COSTS OF PERIOD OF SETTLEMENTS IBR– interbank interest rate.

27. COSTS OF PERIOD OF SETTLEMENTS Operational costs – cost of performance of one transaction

28. Probability of system liquidity COSTS OF PERIOD OF SETTLEMENTS H– Heavyside function; T– period. Condition of unliquidity

29. Statement of optimization of settlement costs STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS Agent costs during period: Vector of day balances: Expected costs of agent: Objective function (total expected costs): where:

30. Analytical example (one agent, one day period) STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS LBR=5 %, IBR=9%, STL=10% 1) 2) .

31. Statistical simulation of expected costs function and its gradient STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS where: - statistical estimate of expected agent costs; -costs during period; - estimator of gradient of cost functionLi(Xi ); -generalized gradient of cost function; N– number of simulated periods;

32. Stochastic optimization STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS  - step multiplier, >0. Sakalauskas (2000) - Fisher -quantile with degrees of freedom A (.) – sampling covariance matrix

33. Statistical termination criterion STATISTICAL OPTIMIZATION OF SETTLEMENT COSTS Testing of optimality hypothesis with significance  according to Hotelling criteria: 1) Confidence interval of estimator of objective function: 2) - standart normal  quantile; - sampling standard deviation of the objective function.

34. COMPUTER MODELLING Initial data: - interest rate of short time loans; - interbank interest rate; - premium interest rate; - penalty percent points on violation of reserve requirements; - period length J=11 – number of agents.

35. COMPUTER MODELLING Dependencies of expected costs L(X)and gradient Q(X)on deposit (1 ir 10 agents)

36. COMPUTER MODELLING Total expected costs and deposit under number of iterations

37. COMPUTER MODELLING Total expected costs and deposited sum under number of iterations (1 ir 10 agents)

38. COMPUTER MODELLING

39. DISCUSSION AND CONCLUSIONS • The statistical Poisson-lognormal model of electronic settlement flows has been created as well as the methodology for calibrating the settlement flow model has been developed and adapted to the analysis of real time settlement data • The methodology of modeling interbank settlement flows by the Monte-Carlo estimator has been developed following to instructions of Central Bank • The algorithm of stochastic optimization of settlement costs, a view on settlement costs and liquidity risk has been created

40. Rational Cash Planningand Investment Management

41. Introduction Problems of cash management often arise in public, non-profit-making or business institutions Many companies are faced with choosing a source of short-term funds from a set of financing alternatives Various constraints are placed on the financial options. The objective is to minimize the short run financial costs of the financial alternatives plus expected penalty costs for shortages and surpluses in stage’s balances.

42. Financial instruments (1) Line of credit – is the maximum amount of credit that a financial institution can give out to a business firm. This alternative has two interest rates – one on taken part of the credit line and another on not taken. So, if the firm doesn’t have needs for additional financing, she pays only small interest rate on limit for credit line.

43. Financial instruments (2) Pledging of accounts receivable (factoring) – is the financial transaction whereby a firm may borrow by pledging its accounts receivable to a third part as security for loan. The bank will lend up 70-90% of the face value of pledged accounts receivable and will get remainder 30-10% accounts receivable part then debtor pay all his debt to the bank. For this option firm must pay interest rate for difference from borrowed and reminded parts of the accounts receivable.

44. Financial instruments (3) Stretching of accounts payable – the firms, at this option, may delay payments of accounts payable. The firm may stretch up to 80% of the payments due in the period. Term loan – the firm may take out a term loan from bank at the beginning of the initial period. Marketable securities – the firm may invest any cash in short term securities.

45. Stochastic linear model Thus, interpreting financial instruments described above the stochastic linear model for this task is: The objective is to minimize the short run financial costs of the financial alternatives plus expected penalty costs for shortages and surpluses in stage’s balances.

46. Model details (1) The formulation given below refers to a short term financial planning model based on Kallberg, White and Ziemba, 1982. Funds are received or disbursed at the beginning of periods. All quantities are in thousands of dollars. In this model xit – denote the amount obtained from option i in period t, t = 1,2.

47. Model details (2) Used financial alternatives: 1 – line of credit (x1t), 2 – pledging of accounts receivable (factoring) (x2t), 3 – stretching of accounts payable (x3t), 4 – term loan (x4), 5 – marketable securities (x5t). Another used options: ARj – accounts receivable, j = 0,1,2. (j – denote planning moments), APj – accounts payable, LR – liquidity reserve, L1 – contribution to liquidity reserve from credit line option,

48. Model details (3) x6j+, x6j- – surpluses and shortages respectively, j=0,1,2. r12, r11 – interest rate for taken/not taken part of the credit line, r2 – interest rate for pledging of accounts receivable, r3 – interest rate for stretching of accounts payable, r4 – interest rate for term loan, r5 – interest rate for marketable securities, rv – norm of cash which can be invested, β1 – upper bound of a limit of a credit line, β3 – upper bound of stretching of accounts payable, β4a,β4v – lower and upper bound of a term loan, β41 , β42 – upper bound for constraints on financing combinations,

49. Model details (4) First stage constraints: x11 + L1 ≤ β1 x4 ≤ β4v x21 ≥ 0.7 · AR0 x11 + x4 ≤ β41 x21 ≤ 0.9 · AR0 x21 + x4 ≤ β42 x31 ≤ 0.8 · AP0 x51 ≤ AR0 ·rv x4 ≥ β4a x51 + L1 ≥ LR Second stage constraints: x12 - L1 ≤ 0 x11 + x12 + x4 ≤ β41 x22 ≥ 0.7 · AR1 x21 + x22 + x4 ≤ β42 x22 ≤ 0.9 · AR1 x52 ≤ AR0 ·rv x32 ≤ 0.8 · AP1 x51 + x52 + L1 ≥ LR x31 + x32 ≤ β3

50. Model details (5) Initial balance: First stage balance: Second stage balance: Objective function: