Cash Flow Entropy

0.24 MB·English

by  Ulrich Kirchner

Cash Flow Entropy
Ulrich Kirchner
3 PO Box 72277, Parkview, 2122, South Africa
1 [email protected]
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a Simon Moolman
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3 Department of Physics
2 University of the Witwatersrand

Johannesburg
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January 24, 2013
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[ Abstract
groceries, rent, monthly bills, etc). By considering

1 thesources,destinationsandrelativesizesofthepay-
v In this paper we aim to find a measure for the diver- ments, it is possible to write down the probability of
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sity of cash flows between agents in an economy. We where a dollar of an agent’s income came from and
0
argue that cash flows can be linked to probabilities whereitisgoing. Withthiswecandescribethediffer-
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5 of finding a currency unit in a given cash flow. We encebetweenthelargeinflowsanddispersedoutflows
. then use the information entropy as a natural mea- in terms of the entropy of these probabilities.
1
sureofdiversity. Thisleadstoahirarchicalinequality
0 Let us assume an economy in which no money is
3 measure, which includes the well-known Theil index
created or destroyed and in which, over a period,
1 as a special case, and a constraint on agent entropy
cashflowsonlyoccurbetweenasetofdiscreteagents.
: differentials, i.e., the difference between cash inflow
v Suchaneconomycanclearlynotexistifeveryagent’s
i and outflow entropies. The last result is particularly
X cash flow situation is as above — few large inflows
intriguing as it formalises the fact that an economy
andmanysmalloutflows. Theremustbeagentswho
r must contain both, cash flow aggregating and cash
a collect small cash flows and pay out large sums. Ex-
flow diversifying agents.
ampleswhichcometomindaregenerallycompanies,
for example selling a mass produced product (small
cash flows) and buying supplies and services in bulk
1 Introduction
(large payments).
This hints at the idea that the entropies of the
Most people, when inspecting their bank account
inflow and outflow probabilities of all the agents in
statement, could make the following observation:
the economy must be related in some way.
while inflows tend to come in a few large payments
(salaryetc),outflowsaremuchmoredispersed(daily Thispaperwillbestructuredasfollows: InSection
1
2 we formalize the cash flow-probability relationship An intriguing property of the information entropy
and introduce information entropy to measure diver- is how it respects the grouping of possible outcomes.
sity. In Section 3 we examine the results for the Lets say we group the N possible outcomes into n<
simpler case of a steady-state economy. Section 4 N groups. The overall entropy can then be written
contains examples for the two and three agent cases. as
(cid:88)
ThisisfollowedbytheConclusioninSection5. Inthe H =H + p H , (5)
g j j
appendix we derive the information entropy subdivi- j
sion property for the case of 2/3 possible outcomes.
where p is the probability of the outcome being in
j
group j, H is the “internal entropy” of group j (en-
j
tropyoftheprobabilitiesconditionalonthefactthat
2 Cash flows and probabilities
the outcome was in group j), and
Let us assume that in an economy cash flows occur (cid:88)
H d=ef− p log p (6)
over a period of time between N agents. Let cj rep- g j 2 j
k j
resent the cash flow from agent j to agent k. This
canbeconvinientlyrepresentedinacashflowmatrix, is the information entropy for the aggregate groups.
whereeachrowcorrespondstoasourceagent(upper Byapplyingaboveidentityrecursivelytotheprob-
index)andeachcolumntoadestinationagent(lower abilities corresponding to all possible cash flows one
index) obtains a hirarchical set of inequality/diversity mea-
 c1 ··· c1  sures.
1 N
 ... ...  (1) Asafirststepletusputallsavingscii inonegroup,
andallremainingcashflowsinanothergroup. Using
cN ··· cN
1 N above identity we find
Withcashflowcj wecanthenassociateaprobability
k H =H +p H +p H , (7)
{sc} {s} {s} {c} {c}
pjk d=ef (cid:80)cjk cj. (2) gwahteereprpo{bs}abd=ielfit(cid:80)yiopfiiaacnudrrpe{nc}cy=u1n−itpb{esi}ngarientahe“saagvgirneg-
k,j k
cash flow” and inter-agent cash flow respectively,
which represents the likelihood that a random cur-
rency unit in the economy would have been part H d=ef−p log (p )−p log (p ) (8)
{sc} {s} 2 {s} {c} 2 {s}
of cash flow cj (which assumes identifiable currency
k
units). Note that “saving” would correspond to a is a measure of diverisification between savings and
cashflow cj from an agent j to himself. spending,
j
To measure the diversity of all cash flows in the H d=efK(c1,...,cN) (9)
{s} 1 N
economy we will use the information entropy
is a measure of the saving diversity in the economy
(cid:88) (it is highest when all agents have the same saving),
H d=ef− pj log (pj), (3)
k 2 k and
j,k H d=efK(c1,...,cn ) (10)
{c} 2 n−1
which is measured in bits (due to the choice of 2 as isameasureofdiversityforallinter-agentcashflows.
the base of the logarithm). To simplify the notation Cash flows can now be grouped by their origin
letusextendthisdefinitionto“un-normalized”prob- or destination. We define the aggregated cash flows
abilities by defining a new function from agent j and to agent k excluding savings as
(cid:18) (cid:19) (cid:88) (cid:88)
K(x1,...,xn)d=efH (cid:80)x1x ,...,(cid:80)xnx . (4) cj d=ef cjk ck d=ef cjk. (11)
i i i i k,k(cid:54)=j j,j(cid:54)=k
2
(cid:88)
Theprobabilitiesoffindingacurrencyunit(which H d=ef− pj log (pj ). (20)
k (k) 2 (k)
is not a saving) in the cash-out/in flows of an agent j
are
Taking the sum and difference of the two expres-
cj c
pj = (cid:80) ci pk = (cid:80)kc (12) sions for H{c} we find
i l l
We can now define probabilities similar to above, H(cid:48)+H(cid:48)(cid:48) 1(cid:88)
H = + (p H +pjHj) (21)
but within one of the cash flow groups to or from an {c} 2 2 j j
agent for j (cid:54)=k j
cj cj H(cid:48)(cid:48)−H(cid:48) (cid:88)
p(kj) = ckj(1−δkj) pj(k) = ck(1−δkj). (13) 0= 2 + (pjHj −pjHj) (22)
k j
This is the probability to find an arbitrary currency
The last equation is particularly interesting as it re-
unit leaving/going to agent j in a cash flow to/from
lates the difference of overall income and spending
agent k. Evidently
inequality to individual agents cash flow diversifica-
(cid:88)p(j) =1 (cid:88)pj =1 (14) tion.
k (k)
k j
Bygroupingthecashflowbytheoriginatingagent 3 The Steady State Economy
(upper index) we find
Cash flows to and from agents over the period are
(cid:88)
H{c} =H(cid:48)+ pjHj, (15) related by
j
 
where p(1) ··· p(1)
H(cid:48) d=ef−(cid:88)pjlog2(pj) (16) [c1···cN] 1... N... =[c1···cN] (23)
 
j p(N) ··· p(N)
is the entropy of the aggregate agent cash out-flow 1 N
probabilities pj and
For a stationary wealth distribution we need c =
j
Hj d=ef−(cid:88)p(j)log (p(j)) (17) cj and hence
k 2 k p =pj. (24)
k j
is the entropy of each agents cash outflows, i.e., the Assuming stationarity and dividing above equation
entropy of p(j) for fixed j. It is maximized if all cash by the sum of all cash flows gives
k
flows from the agent are equal, and minimized (in
 
whichcaseitiszero)ifthereisonlyasinglecashflow p(11) ··· p(N1)
from agent j. Hence this is a measure of spending [p1···pN] ... ... =[p1···pN] (25)
diversity for agent j.  
p(N) ··· p(N)
Similarly this can be done for cash in-flows 1 N
(cid:88)
H =H(cid:48)(cid:48)+ p H , (18) In the case of stationarity we have H¯ d=ef H(cid:48) = H(cid:48)(cid:48)
{c} k k
k and expression 21 becomes
where H(cid:48)(cid:48) is the entropy of the aggregate agent cash
in-flow probabilities pk and Hk is the entropy of the H =H¯ + 12(cid:88)pj(Hj +Hj), (26)
pj for fixed k. Here similarly to above j
(k)
H(cid:48)(cid:48) d=ef−(cid:88)p log (p ) (19) which shows how the overall entropy splits into an
k 2 k inter-agentcomponentH¯ andaninternalcomponent
k
3
for each agent Hj+Hj. We suggest that H¯ is a mea- thewholeincomecomesintheformofasinglesalary
2
sureofincomeinequalityasitmeasuresthediversity payment) while the latter could be a store or service
of incomes. However, H¯ includes contributions from provider (many customers pay for goods while the
all agents, including non-human agents like corpora- payments will go to few suppliers).
tions and government. Using the same identity as
above one can express H¯ in terms of agent groups,
4 Examples
for example
H¯ =H∗+p∗H∗ +p∗H∗ +p∗H∗, (27) 4.1 The two-agent economy
G G C C P P
where H∗ measures the income distribution between Thisisthesimplestpossible,butsomewhatrestricted
P
case. Let us define the matrix of cash flows
human individuals and can be recognized as related
to the Theil index of income inequality [2]. H∗ and (cid:20) c1 c1 (cid:21)
H∗ are similar measures for government and cGorpo- C d=ef c12 c22 (29)
C 1 2
rates. H∗ then measures the inequality between the
sectors in the economy. It is well know that these Note that there is the freedom to rescale the cash
properties of the information entropy make it a suit- flowsbyaconstantfactor. Forstationaritywerequire
able inequality measure [3].
c1 =c2. (30)
Similarly (22) simplifies to 2 1
With
0=(cid:88)pj(H −Hj), (28) (cid:20) 1 (cid:21)
j (cid:126)ud=ef (31)
j 1
which says that the probability weighted sum of the we have (cid:20) c1 (cid:21)
difference of agent inflow and outflow entropy van- =C(cid:126)u (32)
c2
ishes. This equation is interesting as it shows how
the entropy differentials between agent in and out (cid:20) p1 (cid:21) C(cid:126)u
= (33)
flows have to balance in the economy. p2 (cid:126)u(cid:48)C(cid:126)u
For the agent in the economy labelled j, the en-
tropy H represents the uncertainty of which source (cid:104) p(i) p(i) (cid:105)=(cid:2) ci ci (cid:3)/(ci +ci) (34)
j 1 2 1 2 1 2
a dollar of j’s came from. Similarly, the entropy Hj
Fromthesealltheentropiesofinterest(H(cid:48),H and
is the uncertainty of where a dollar of j’s payments j
Hj) can be calculated.
will go. The difference (H −Hj) is, therefore, the
j Let us scale the cashflows such that c1 = c2 = 1,
difference of uncertainty between the inflow and out- 2 1
i.e., the savings are measured relative to the inter-
flow probabilities. If the difference is positive, then
agent cashflows. The cashflow matrix then takes the
an agent will be more uncertain about where an in-
form
flowdollarcamefromthanwhereanoutflowdollaris (cid:20) (cid:21)
a 1
going. If the difference is negative then the opposite C d=ef , (35)
1 b
will be true.
Sincetheprobabilitiespj in(28)areallpositive,it where a and b are non-negative numbers. We then
isnotpossibleforallagentstohaveapositiveentropy have
differenceoralltohaveanegativeentropydifference. 2
p = p =1−p (36)
Someagentsmustbemorecertainaboutwheretheir {c} 2+a+b {s} {c}
nextdollarwillcomefromrelativetowhereitisgoing
H =K(a,b) H =1 (37)
and other agents must be more certain about where {s} {c}
their next dollar is going than where it came from. Figures1to4illustrateprobabilitiesandentropies
Typically, the former could be an individual (often for this case.
4
0 Hsc 1
0 ps 0.9
0.7 0.5 0.8
0.5 0.6 0.7
1
1 0.5 0.6
b1.5 0.5
0.4
b1.5 0.4
2
0.3 0.3
2
0.2 2.5 0.2
2.5 0.1
0.1 3 0 0.5 1 1.5 2 2.5 3 0
3 0 0.5 1 1.5 2 2.5 3 0 a
a
Figure 3: Heat-map of H for the 2-agent case.
{sc}
Figure 1: Heat-map of p for the 2-agent case.
{s}
H
0 2
1.9
0.5 1.8
0 Hs 1 1.7
1
0.9 1.6
0.5 0.8 b1.5 1.5
1 0.7 1.4
0.6 2 1.3
b1.5 0.5 2.5 1.2
0.4 1.1
2
0.3 3 0 0.5 1 1.5 2 2.5 3 1
2.5 0.2 a
0.1
3 0 0.5 1 1.5 2 2.5 3 0
a
Figure 4: Heat-map of the overall cash-flow entropy
H for the 2-agent case. The previous plots are a
Figure 2: Heat-map of H{s} for the 2-agent case. decompositionofthisentropy. WehaveH =H{sc}+
pH +p H .
{s} {c} {c}
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4.2 The three-agent economy
Letusassumestationarityandnosavings. Thecash-
flow matrix is then
 0 c1 c1 
2 3 Total cashflow Entropy H for k=0.3
 c21 0 c23  (38) 0
c3 c3 0 2.4
1 2
0.2 2.2
There are two constraints resulting from the station- 2
0.4
arity requirement b 1.8
0.6 1.6
c1 =c c2 =c (39)
1 2 1.4
0.8
(thisimpliesc3 =c )andonefreedomto“choosethe 1.2
3
cashdenomination”(allowingustore-scaleallvalues 1 0 0.2 0.4 0.6 0.8 1 1
a
in the cash-flow matrix).
Let us choose the normalization such that the av-
erage agent income/spending is 1, i.e.,
c1+c2+c3 =3. (40)
Furthermore,letuschoosethefollowingthreeparam-
eters for the cash flow matrix Figure 5: Heat-map of the overall cash-flow entropy
H for the 3-agent case with k =0.3.
c1 c2 c3
2 =a 3 =b 1 =k. (41)
c1 c2 c3
Then the definition of cj implies
c1 c2 c3
3 =1−a 1 =1−b 2 =1−k. (42)
c1 c2 c3
Total cashflow Entropy H for k=0.5
Writing down the definitions for c1, c2 and the mean 0
constraint as a linear system in the cj gives 2.4
0.2 2.2
 c1   0 1−b k  c1  2
0.4
 c2 = a 0 1−k  c2  (43) b 1.8
3 1 1 1 c3 0.6 1.6
1.4
and hence 0.8
 c1   0  1.2
 c2 =B−1 0 , (44) 1 0 0.2 0.4 a 0.6 0.8 1 1
c3 3
where
 
−1 1−b k
B = a −1 1−k  (45)
1 1 1
Theoverallentropy(whichisequaltoH because
{c} Figure 6: Heat-map of the overall cash-flow entropy
we excluded savings here) for different values of k is
H for the 3-agent case with k =0.5.
presented in the heat plots 5 and 6.
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To illustrate the identity (28) let us assume a spe- inequality measures between all possible groupings
cific case, by fixing the parameters for the cash flow of agents and internal inequality measures for each
matrix to group.
Another interesting result is a kind of overall en-
a=0.1 b=0.3 k =0.7. (46)
tropy balance expressed in equation (22) and (28).
This essentially says that there must be a balance
This corresponds to the cash flow matrix
between entropy increasers (receiving few large cash
 0 0.1235 1.1118  flow, making many small payments) and decreasers
 0.3516 0 0.1507  (47) (receiving many small payments and making a few
0.8837 0.3787 0 large payments). Whether this can be linked to an
“economic energy” is an interesting question which
The inflow entropies (entropies in each column) are wehavenotyetaddressed. However,onemightspec-
then ulate that entropy increasers (consumers) tend to
paytheprofitmarginsreceivedbyentropydecreasers
H =0.8617; H =0.8048; H =0.5275 (48) (companies).
1 2 3
The formalism presented here does not allow for
while the outflow entropies (entropies in each row)
credit, which is obviously an important factor in the
are then
real world. We believe that credit can be incorpo-
ratedintheformalismandthiswillbethesubjectof
H1 =0.4690; H2 =0.8813; H3 =0.8813. (49)
further research.
Let us define the entropy differentials (inflow minus Another possible extension is an inclusion of time
outflow entropy) D d=efH −Hj so that entropy cannot just be measured over one
j j
time step, but also over a number of time steps. One
D =0.3927; D =−0.0765; D =−0.3538 way of doing this is to consider discrete times and to
1 2 3
(50) treat the same agent at different times as ’different’.
Weseethatagent1isanentropyreducer,whileagent This should lead to entropies for payment profiles
3 is an entropy increaser. Agent 2 is almost entropy over time, for example for loans repayed in install-
neutral. From the cash flow matrix we find the p as ments. We hope to be able to explore this more in a
j
seperate paper.
p =0.4118; p =0.1674; p =0.4208. (51)
1 2 3
Nowitiseasytoverifythattheidentity(28)issatis-
fied. This means that the weighted entropy differen- A Subdivision Identity Exam-
tials for any two agents imply the weighted entropy
ple
differential for the remaining agent.
Let us assume that we have 2 possible outcomes la-
5 Conclusion
beledby1and2withprobabilitiesp andp =1−p .
1 2 1
The information entropy is then
We presented a formalism linking cash flows to an
entropy measure. The measure naturally adopts to
H =−p log(p )−p log(p ). (52)
sub-goupings of the cash flows and as such produces 1 1 2 2
a number of inequality measures, one of them being
the Theil Index. Nowletoutcome2bethecombinationoftwopossible
While the Theil Index is well know, we believe the outcomes 21 and 22 with probabilities p and p
21 22
over all framework we presented is new. It produces such that p =p +p . We then find for the over-
2 21 22
7
all entropy
H = −p log(p )−p log(p )−p log(p )
1 1 21 21 22 22
= −p log(p )
1 1
(cid:18) (cid:19) (cid:18) (cid:19)
p p
−p log p 21 −p log p 22
21 2 p 22 2 p
2 2
= −p log(p )−p log(p )
1 1 2 2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
p p p p
+p − 21 log 21 − 22 log 22
2 p p p p
2 2 2 2
The first two terms are the information entropy for
outcome 1 and the aggregate outcome 2. The second
term then adds the internal information entropy re-
sulting from the sub-division of outcome 2, which is
theweightedinformationentropyoftheprobabilities
conditional on the knowledge that outcome 2 is true.
Above derivation can be generalized in an obvious
way to arrive at (5).
References
[1] Jaynes E T, Bretthorst L G (2003), Probability
Theory: The Logic of Science, Cambridge Uni-
versity Press, Cambridge
[2] TheilH(1967),Economics and Information The-
ory, North Holland Publishing Company, Ams-
terdam
[3] Shorrocks A F (1980), ‘The Class of Additive
Decomposable Inequality Measures’, Economet-
rica Vol 48 No 3, 613-25
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