Optimal Cash Management with Dynamic Programming

MotivationThe Results

Summary

Optimal Cash Management withDynamic Programming

Performance Measure on the Example ofHungarian Post Co. Ltd

Dániel Havran ∗

∗Department of FinanceCorvinus University of Budapest

CAMEF, Budapest Workshop2008

Daniel Havran Optimal Cash Management with Dynamic Programming


Summary

Outline

1 MotivationProblems on the Postal Cash Management SystemModel of Cash-System: A Short Introduction

2 The ResultsComparison of Optimal and Observed StrategiesImplementation



Summary

Problems on the Postal Cash Management SystemModel of Cash-System: A Short Introduction

Outline





Summary


Problems on the Postal Cash Management System

How can we describe and model the working ofHungarian postal cash management system?We found that:

the aggregate cash level (demand and surplus) is dependon the local decisions of post offices andthe offices are almost independent from the centraladministration.

Consequence 1: Nobody knows the efficiency of theircash-stocking strategies.

Consequence 2 : The uncoordinated decisions of postoffices are important risk factor of the system’s liquidity .

How could be described and measured the activites ofpost offices?



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Outline





Summary


Model of Cash-System: Short Description

Actors:Customers: generate random cash-flowscheque (+), pension (-)Post offices: do services day-to-day and keep theirliquidityCash-handling centre: provides needed cash for postofficesCentral Bank : provides extra notes and coins for thesystem in case of aggregate shortage



Summary


Working Rules for Post Offices

Let:

St : Stock (cash level) at the start of the day

It : Ordered cash from cash-handling centres (In)

Ct : Cheque: positive cash-flows

Pt : Pensions that offices have to pay: negative cash-flows

Ot : Extra cash sent back to the centres (Out)

Then ∀ discrete t :

St + It + Ct = St+1 + Ot + Pt

St ≤ k

St + It ≥ Pt



Summary


Working Rules for the Whole System

Let:

CHCIt : Cash-handling centre cash income

CHCOt : Cash-handling centre cash outcome

Zt : Cash level at cash-handling centre

CBIt : Central bank cash income (from ChC)

CBOt : Central bank cash outcome (to ChC)

Then:

Zt + CHCIt = Zt+1 + CHCOt

CHCIt =∑

Ot + CBOt

CHCOt =∑

It + CBIt+1

(CBOt − CBIt+1)− (Zt+1 − Zt) = (∑

It+1 −∑

Ot)



Summary


Time Series of the Aggregate Cash-Flows



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Comparison of Optimal and Observed StrategiesImplementation

Outline





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Method of Comparison

Concentrate on only one post office. We seek the distance ofthe office leader’s real strategy from the optimal one!Properties of optimal strategy:

The officer is rational.

The officer observes future cash-flows deterministicly andknows all kinds of costs.

And the reality:

Irrationality: the officer apply heuristic strategy.

The cash-flows are stochastic and costs aren’t well-known.

Measure of distance:

The total cost.



Summary


Working Rules for Post Offices

Illustration: A typical day at the post offices

The post offices decide on:It at time t − 1Ot at time t



Summary


Definition of Optimal Strategy

An {(Os, Is)}s optimal strategy on s ∈ [0, S] if satisfies:

S∑s=1

βs−1 [TC] → min{Os,Is}

TC=̇c (Os) + c (Is) + (Us −Os + Is) rUs+1 = Us + Is + Cs − Ps −Os

Us + Is −Os ≥ Ps

Us −Os ≤ kSs, Is, Os ≥ 0

where:Us=̇Ss−1 + Is−1 + Cs−1 − Ps−1 ≡ Ss + Os

c (y) : transfer cost function (step function)r : interest rateβ : discount factor =̇1/ (1 + r)



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On Optimal Strategy

The shape of TC:



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Outline





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Steps of Solution

Implementation:

We use discrete state dynamic programming.

We solve the problem with Policy Iteration Algorithm.

The Bellman equation:

Vs (Os, Is) = minOs,Is

[XTCs (Os, Is) + βVs+1]

with VS = 0where XTC is an extended total cost function.Simulation:The optimal cash-management strategy (the trajectories) andthe costs are easily computable from the optimal policyfunction.



Summary


Policy Iteration Algorithm

Let:N : the number of state (here: levels of cash)x ∈ R2×NS

+ : policy functionf ∈ RNS×NS : reward functionv ∈ R2×NS : value function

Thus the shape of the Bellman equation eg. for S = 3:



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Steps of Policy Iteration Algorithm

Steps of PIM:1 Given the current value approximant v, update the policy x:

x = arg maxx {f (x) + βv}and update the value by settingv = (I − βI)−1 f (x)

2 If ∆v = 0 then stop, otherwise return to step 1.



Summary


Example: Results of a Simulation



Summary

Summary

We built up a formal way to describe working postal cashmanagement system.

Effectivity comparison seems to be reachable andcountable.

On some tests we get that post office strategies areseemingly close to the theoretical optimum!

OutlookRethinking the problem in a stochastic environment.To analyse that the two rule (level-limit and liquidity-rule) inwhich way influence the officer leaders strategies.


Appendix Some Helpful Books

Some Helpful Books I

M. J. Miranda and P. L. Fackler.Applied Computational Economics and Finance.MIT Press, 2002.

S. Bertsekas.Dynamic Programming and Optimal Control: 2nd edition,Vol I. II.Athena Scientific, 2001.


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Optimal Cash Management with Dynamic Programming

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