the life-length of banknotes* - peter · pdf fileof n notes contains d defective notes, these...

The life-length of banknotes*

A case study

by P. KOEZE**

Abstract With the aid of the Bank's banknote sorting system the issue·and subsequent withdrawalof f 25-banknotes on three varieties of paper have been recorded for two-and-a-half years.

The aim was to measure the durability of the three paper varieties in circulation. The results of thissecond trial with f 25-banknotes confirm the statistical model developed previously for the rust trialwith f 100-banknotes. G RESHAM 's Law is equally not applicable, neither to f 25-banknotes nor tof 100-banknotes. A two-parameter gamma distribution fits the cumulative fraction of banknotes withdrawn reasonably well.

Key Words: gamma distribution, reliability estimation, failure model.

Symbols

Capitals are used for quantities describing all series of banknotes in circulation and lower

case for quantities describing test series. ISO-recommendations regarding SI-units are ad

hered to; in particular the normalized decimal comma is used throughout.

a Banknotes culled and rejected in weekly returns (week-I), fraction of original issueC number of banknotes in circulation

c banknotes in circulation, fraction of original issue

d number of defective banknotes per series

H, h hazard rate, ratio of unfit banknotes withdrawn per week to banknotes in circula

tion (week-I)cumulative fraction of banknotes withdrawn

N total number of banknotes issued

n number of new banknotes per series

0, 0 return rate, ratio of banknotes returned per week to banknotes in circulation

(week-I)

p fit banknotes in two-weeks' storage, fraction of original issue

Q, q withdrawal rate, ratio of unfit banknotes withdrawn per week to bartknotes returned

per weekr correlation coefficient

s fit banknotes in weekly returns (week-I), fraction of original issue

T historical time or date (week)

t circulation time (week)

* An earlier version of this paper was presented for the Banknote Printers Conference Dublin 1980.** De Nederlandsche Bank N.V., Amsterdam, The Netherlands.

Statistica Neerlandica 36 (1982), Uf. 4. 187

u time, dummy variable (week)

v unfit banknotes in weekly returns (week-I), fraction of original issuer gamma function

e banknotes seemingly in circulation, immobile fraction of original issue

1] shape parameter of the gamma distribution and possibly a measure of the paperquality

;\ scale parameter of the gamma distribution and possibly a measure of the circulationrate of banknotes (week-I)

J.L scale parameter of the lognormal distributionv series number

a shape parameter of the lognormal distribution

1 Introduction

Since the beginning of 1975 at De Nederlandsche Bank we have a banknote sorting system

in operation which enables us to assess the quality of banknotes experimentally. The Dutch

banknote sorting machines, developed under direction of the Bank, were built for the pur

pose of mechanically sorting circulated banknotes into fit notes suitable for recirculation

and unfit notes to be destroyed. Apart from sorting, the machines read the numbers of

all banknotes as they are processed and record these daily on magnetic tape. The tapes are

processed on a computer for counterfeit detection and for statistical purposes. Thus circu

lation trials are feasible issuing a certain batch of banknotes and observing their subse

quent withdrawal as a function of circulation time.

Earlier I have reported on our first circulation trial with f IOO-banknotes (K 0 EZ E

(1979)). Since the publication of that paper we have started a second trial with f 25-banknotes. Also we have continued the observations in the first trial and extended the statis

tical model. It is the aim of this paper to present comprehensively our knowledge gained

so far and to give the results of the f 25-trial. For comparable results from another coun

try, Canada, see GILLIESON (1977).

2 Issue and withdrawal of banknotes

In figure 1 a simple sketch illustrating the life of a batch of Dutch banknotes has been

drawn. A certain quantity of paper is made by our paper mill V AN HOUTUM & PALMand printed by our printing works J OH. ENSCHEDE EN ZONEN. The result is a batch

of numbered notes. ENSCHEDE'S take particular care to ensure that a banknote number

is used only once, and that the number of every defective note is known and reported tothe Bank. Series of n notes are kept together. A series is defined as a batch of all notes

bearing the same initial four digits called the series number v.

The ready notes are delivered to the Bank and stored at the Bank's vaults. If a series

of n notes contains d defective notes, these are taken away before issue and destroyed.

The remaining (n-d) notes are issued when required and added to the circulating notes

188 Statistica Neerlandica 36 (1982), nr. 4.

defect dcotton Paper-Printing

new n rejectadestruc-pulpchemicals Mill Works

Bank

unfit

vtion

ink 1 1coating

return

fitnewls+v+al

s{n -dl

circulation c

Fig. 1. The life of a batch of n Dutch banknotes of which d notes are defective. During sorting vnotes appear unfit and s notes fit, and a notes are culled. In circulation are c notes.

already in the hands of the public. After some time some notes are paid by the public into

branches of commercial banks or the Post Office and passed on in part to one of the

branches of the Bank. After some irregular delay batches of returned notes arrive at the

Head Office where they are run through banknote sorting machines and separated into vunfit notes and s tit notes according to some fit/unfit criterion. The unfit notes are de

stroyed, the fit notes are reissued after the customary two weeks' storage time. During

the process of mechanical sorting a notes are culled or rejected by the machines and de

stroyed together with the unfit notes.The Dutch banknote sorting machines have several tasks. The main task, as said above,

is to sort the notes returned daily to the Bank into fit notes and unfit notes by means of

an optical scanner. Unfit notes comprise soiled notes, torn notes, notes with holes, notes

with missing corners, notes with heavy writing, and notes mended with adhesive tape. Thesecond task of the machines is to read the banknote numbers with an OCR-B number

reading system. By means of this system the numbers of all notes are read as they are pro

cessed and recorded on magnetic tape. The tapes are processed on a computer for counter

feit detection and for statistical purposes. Every night a computer printout is produced

stating the total numbers of fit and unfit notes per series and the total numbers of fit and

unfit notes of all series together. These daily observations are accumulated per week and

used as raw data in our investigations. The unit of circulation time is set at the seven-day

week finishing Friday night to avoid spurious effects due to a weekly circulation pattern.

The date of issue is defined as the Friday night nearest to the middle of the time interval

during which the notes of the series of interest were issued.In figure 2 issue and withdrawal of one series have been sketched. In week 0 n notes

minus d defective notes are issued. With increasing circulation time t more and more unfit notes are withdrawn. The cumulative fraction of notes withdrawn i as a function of

circulation time t is called the withdrawal curve. The fraction of notes in circulation c is

Statistica Neerlandica 36 (1982), nr. 4. 189

1VI

~o~ 0,5c:ro.c

.•..oc:

~£

o

o circulation time _

Fig. 2. Withdrawal of banknotes after their issue at week O.

called the survival curve. Obviously, the sum of notes withdrawn i, notes in circulation c,

and fit notes in store at the Bank's vaults p must be equal to one at all times: thus i+p +c= 1.A small fraction of notes e will never be withdrawn due to several reasons:

(i) Some notes are culled or rejected by the sorting machines and, consequently, their

numbers are unknown. The notes are destroyed with the unfit notes.

(ii) Technical and organisational failures may regrettably render magnetic tapes illegible.

We have certain rescue procedures but still, sometimes, unfit notes have to be destroyedwhile their numbers remain unknown.

(iii) The general public may destroy, forget, or hoard some notes. For these reasons iwill

ultimately tend to (1 - e) instead of to 1, and c will tend to e instead of 0, e being a small

positive number. As mentioned, we know exactly which notes were defective and, therefore, were not issued. So these notes are not included in e.

The rate at which new series are issued is determined by the number of unfit notes to

be replaced and by the demand of the public. The number of notes to be replaced is de

termined in turn by the threshold of fitness below which notes are withdrawn, by the

banknote quality, and by the circulation rate. The demand of the public may be deter

mined by the inflation rate, the population, the circulation rate, the preferences and paying

habits of the public, and possibly by other causes. The main instrument by which the Bank

can influence the demand for new notes in the short run is changing the threshold of fit·

ness below which notes are withdrawn. In the long run the demand is influenced most

effectively through the banknote quality.Normally, series will be issued by the Bank in the order of their series number v. In the

case of a circulation trial this sequence may be abandoned. If the historical time or date

is denoted by T, then the issuing policy of the Bank is given by

190

v = v(T) (1)

Statistica Neerlandica 36 (1982), Ilf. 4.

and the number of banknotes issued up to time T by

N = N(D = L {n(v) - d(v)}.v

3 A numerical model of the Bank's sorting procedure

(2)

In the following, if some quantity x assumes a new value in the course of week t, the new

value of x is always assigned to x(t). At the beginning of week t x may have the value

x (t - 1) or perhaps even an earlier value if no notes were sorted for a long time. Thus x(t)

denotes the value of x on Friday night of week t.Assume a series of banknotes with series number v is issued at time T, and take T as

the origin of time axis t. The number of notes issued is (n - d). All further quantities con

cerning numbers of notes are expressed as fractions of (n - d). We define the followingfractions:

notes withdrawn up to and including week tunfit notes in weekly return tfit notes in weekly return tfit notes in storage at the end of week tnotes in circulation at the end of week t.

The storage time at the Bank's vaults is two weeks.

The following equations describe our sorting system:

i(t) + pet) + c(t) = 1

i(t) = i(t-l) + v(t)

p(t) = pet ~ 1) + s(t) - s(t-2)

c(t) = c(t -1) - vet) - set) + s(t -2).

Moreover the following boundary conditions hold:

i(t)

vet)

set)

pet)

c(t)

(3)

(4)

(5)

(6)

i(O) = 0

v(O) = 0

c(O) = 1

lim i(t) = 1 - et=oo

lim vet) = 0t=oo

p(O) = 0

s(O) = 0

lim pet) = 0t=oo

lim s(t) = 0t=oo

(7)

(8)

Hm c(t) = et=oo


with € a small positive number. The better the sorting system functions, the smaller € is.It follows that

ti(t) = ~ v(u)

u=!

p(t) = s(t-l) + set)

t

e(t) = 1 - ~ v(u) - s(t-l) - s(t).u=!

(9)

(10)

(11)

The circulation e(t) may be compared with the "reliability function" mentioned in litera

ture on quality control. Further we define the withdrawal rate ("quality") as

q(t) == vet) / {vet) + set)},

the return rate as

o(t) == {vet) + set)} / e(t-l),

and the hazard rate as

het) == vet) / e(t-l).

These three quantities are interdependent:

het) = o(t)· q(t).

(12)

(13)

(14)

(15)

In figure 3 the typical behaviour of q, 0, and h as a function of t has been sketched. It

follows easily from (8) that

lim o(t) = 0 and lim het) = O.t= 00 t= 00

If no fit notes were kept in storage, then according to BUR Y (1975)

t

i(t) = 1 - exp {- ~ h(u)}.u=!

(16)

(17)

Analogously, for the whole circulation comprising all series issued we may define the withdrawal rate as

~ {n(v) - d(v)} v(v; t)v

Q(t) == ~ {n(v) _ d(v)}{v(v; t) + s(v; t)} ,v

the return rate as

~ {n(v) - d(v)} {v(v; t) + s(v; t)}v

O(t) == ~ {n(v) -d(v)} e(v; t-1)v

(18)

(19)


1

o

o

h

circulation time _

Fig. 3. Typical behaviour of the withdrawal rate q, the return rate 0 and the hazard rate h as afunction of circulation time t.

and the hazard rate as

~ {n(v) - d(v)} v(v; t)v

H(t) =. ~ {n(v) _ d(v)} c(v; t-l)v

(20)

In formulas (18), (19) and (20) v, sand c are not only dependent on time t but also on

the series number v. Again it follows that

H(t) = O(t) • Q(t). (21)

Note that so far no assumptions whatsoever have been made regarding the nature of the

circulation. The formulas (1) to (21) present an objective mathematical description of

issue and withdrawal in our sorting procedure. They enable us to evaluate circulation

trials numerically.

4 A statistical model of the circulation

In statistical terms a circulation trial comes under order statistics (B U R Y (1975)). As usual

in such cases the data on life-length are generated as an ordered sample and are censored

to the right. During the trial varying numbers of notes are inspected at fixed time inter

vals which is known as multiple type I censoring or multiple time censoring. This censor

ing schedule is so uncommon that it is not dealt with in the literature.

In practice only two quantities can be measured at the Bank: s(t) and v(t). So the sta

tistical model of the circulation should not be too complex. In its simplest form the cir

culation may be thought of as five processes:


I Issue of new notes by the Bank to the public

2 Wear and tear of the notes in the hands of the public

3 Return of part of the notes by the public to the Bank

4 Sorting and withdrawal of unfit notes by the Bank

5 Reissue of fit notes after two-weeks' storage to the public.

Which of these is essentially of statistical nature? Necessarily, new notes are issued in un

certain numbers during several days. However, I assume that all notes are issued in one

single week (in both trials it was two weeks) and thus any statistical variation on this

account is ignored. Wear of the notes as well as their return are essentially statistical pro

cesses. During sorting some of the notes are culled or rejected, the machines are not work

ing ideally and identically from machine to machine and from day to day. So these are

statistical processes but we are trying continually to keep their variation as small as pos

sible. Therefore I will neglect them. Reissue of fit notes is a statistical process because it

may take some days before the notes reach the public again. However, I assume that the

notes are put into circulation immediately after storage. In conclusion, then, I recognize

the statistical nature in the case of the second and third processes only.

4.1 Wear and tear of banknotes in the hands of the public

What happens to a banknote when it is in the hands of the public? Figure 4 is an approxi

mate representation of what happens. After issue at time t = 0 a note starts circulating

which means that it changes hands frequently and is exposed to treatments such as fold

ing, soiling and tearing. We could call these the elementary events. Although we do not

know much about the paying habits of the public, it is not unreasonable to suppose that

the elementary events occur at a constant average rate A. which is the well-known Poisson

model. This process goes on till, by chance, the note is returned to the Bank after 171

elementary events.

If the note is fit after sorting, it is reissued. Immediately, it is subjected to further ele

mentary events with frequency of occurrence A. till after 172 events it is returned again to

the Bank. Now this goes on till, eventually, after 17 events the note becomes unfit and is

withdrawn. The time between issue and withdrawal can be described by a gamma distri

bution with parameters (A., 17). If we could be reasonably sure that the note is of such a

issue returnreissuereturnreissue return

stora ge

stora ge••••ithdra ••••al

I

!I I II!!I!!"I !II I!I I" 11I! !I •

1

23... q1q1+1 ~2~/1 ~ltime t

t=O

Fig. 4. Elementary events in the life of a banknote.


quality that it can sustain exactly 11 elementary events and that it is then promptly with

drawn, the gamma distribution would be an exact model of its life-length. But these con

ditions do not apply. It is not likely that all notes are withdrawn after the same number

of elementary events; even if all notes become unfit after 11 events, many will continue

to circulate because they do not happen to return to the Bank. I have found in Ko E Z E

(1979) that the withdrawal curve may be approximated by a two-parameter gamma distribution:

i(t; A, 11)

o , elsewhere.

(22)

Then the mean time to failure is given by BUR Y (1975)00

MTTF = J t v (t) dt = 11/A.o

(23)

Using the method of WILK et al. (1962a) the maximum likelihood parameters of the

gamma distribution can be estimated after some involved computation.

If two varieties of notes have parameter pairs (AI, 111) and (A2, 112), the switch from

variety 1 to variety 2 will lead to a reduction in the consumption of notes by

(24)

If the model given above is valid, the parameter A is determined solely by the circulation

rate and 11 by the banknote quality and the threshold of fitness below which notes are

withdrawn. A large A means a high circulation rate and a large 11 a strong paper quality

with respect to the fitness threshold. Both parameters are dependent on the denominationbecause the circulation rate varies with the nominal value of the denomination, and the

fitness threshold with the colour of the printing ink.

So far the model has been presented as if a single failure mode is responsible for the

withdrawal of banknotes. Our optical scanner, however, distinguishes between several

failure modes: soiled notes, notes with tears or holes, missing corners, dog-ears, adhesive

tape and heavy writing. Therefore the observed withdrawal curve is in principle composed

of several (statistically independent) withdrawal curves, each of which is due to a par

ticular failure mode. One could imagine that catastrophic failures like missing corners or

heavy writing give rise to exponential distributions and that degradation failures like soil

ing and perhaps tears (mended or not) give rise to gamma distributions. But as long as we

lack data on these competing failure modes, I have to confine myself to the present grossmodel.


4.2 Return of banknotes to the Bank

When does a banknote return to the Bank? This question has been asked long before, at

least for coins. As early as the 16th century SIR THOMAS GRESHAM, an English finan

cier, formulated an observation which has become known as GRESHAM'S Law. It states

that, when two varieties of coins are equal in face value but unequal in intrinsic value, the

one having the lesser intrinsic value tends to remain in circulation while the other is hoarded

or exported as bullion. Over the centuries the validity of GRESHAM'S Law has been sub

stantiated for coins, e.g. see VISSERING (1916). In more recent years GRESHAM'S Lawhas been extended in modified form to banknotes, in the belief that worn and torn notes

are discarded by the public more readily than fit notes. This implies that the return ratewould increase with deterioration of the notes. This, however, has been refuted duringour circulation trials as will be shown.

The total fraction of notes returned to the Bank is equal to c(t-l). o(t) = (fraction

of fit notes in circulation) x (return rate of fit notes) + (fraction of unfit notes in circula

tion) x (return rate of unfit notes). Of course, the fraction of notes in circulation is equalto the sum of the fractions of fit and unfit notes in circulation. So, it follows that (in

shorthand writing)

fit unfito(t) = ---- x return rate fit + ---- x return rate unfit =

fit + unfit fit + unfit

(unfit ) unfit1 - f f x return rate fit + f . x return rate unfit.It + un It It + unfit

With increasing circulation time the second quotient on the right-hand side will increase.

If unfit notes were discarded by the public more readily than fit ones, the return rate of

unfit notes would be larger than the return rate of fit notes. Both statements togethermean that the return rate 0 would increase with deterioration of the notes and circulation

time. On the other hand, if fit and unfit notes have the same return rate, the return rate 0will be independent of deterioration of the notes and circulation time. This is exactly

what I have found. The return rate 0 is independent of the withdrawal rate q and of the

time since issue t except for the very long survival times when the number of notes of a

given vintage becomes so small that their return is quite erratic. Every note, fit or unfit,

appears to have the same probability of return. This means that the process of wear and

the process of return are independent.

So far I have dealt with one series v only. There seems to be no reason why the general

public would treat a note of one series differently from a note of another as long as the

notes have the appearance to be from the same stock. So it does not seem unreasonable

to assume that the return rate per series has the same constant value irrespective of age,

deterioration or series number. A more formal argument for a similar question has been

given by HEMELRIJK (1978).

Unfortunately, this has not been found in our trials. In the beginning there was a dif

ference between the mean values of 0 and 0 of the order of 25 per cent with 0 exceeding


o. Since such a difference was found for the first time (K 0 E Z E (1979)), a number of meas

ures have been taken to improve the accuracy of the raw data: the disturbing effects ofrejects and cullings have been partly eliminated, the date of reporting C has been shifted

from Monday night to Friday night, and lately a better estimate of the number of notes

in storage at the Bank's vaults p has been made available. As a result, over the years the

correlation coefficient between o(t) and O(t) has gradually improved from 0,3 to 0,8 al

though the means are still quite different. It is likely that our next trial will show further

improvement.

Because the processes of wear and return are independent, we need a second statistical

model to represent the return of notes to the Bank. In 1978 the "Stichting Mathematisch

Centrum" (Foundation Mathematical Centre) at Amsterdam has conducted an investiga

tion at the request of the Bank and shown empirically that the daily return of all five

denominations may be represented fairly well by an autoregressive model based on normal

probability distributions.

Two further implications bear on the withdrawal rate. The withdrawal rate is equal to

q(t) = (unfit notes in circulation) x (return rate of unfit notes) -;-{(fit notes in circu

lation) x (return rate of fit notes) + (unfit notes in circulation) x (return rate of un

fit notes)}.

Because the return rate of fit notes is equal to the return rate of unfit notes, the expected

withdrawal rate at time t is equal to the relative number of unfit notes in circulation at

time t. Hence there is no time lag between the relative numbers of unfit notes returnedand in circulation. The withdrawal rate as measured at the Bank is an instantaneous meas

ure of the fraction of unfit notes in circulation, per series or for more series taken to

gether.

Before, it was thought at the Bank that one could sort notes on fitness by the series

number. Implicitly, it was assumed that the withdrawal rate would tend to 1 for large t.In that case a series could be withdrawn completely as soon as the withdrawal rate would

exceed a predetermined value, e.g. 0,9, thus dispensing with the costly development of a

high-quality fitness detector. Now it is clear that such a procedure is uneconomical or im

possible, at least for! 25-notes and! lOO-notes, because the withdrawal rate does not exceed a relatively low value. This has been observed in our trials and is in accordance with

my model. From (15) it follows that

q(t) = h (t) /o(t).

Further, for the gamma distribution it can be shown that (e.g. BURY (1975))

lim het) = A. = constantt=oo

and above I have shown that

E(o) = constant.

Statistica Neerlandica 36 (1982), nr. 4.

(25)

(26)

(27)

197

As both constants are independent and as the definition (12) indicates that q (t) ~ 1, gen

erally,

lim q(t) < 1.t=oo

(28)

5 Results of a circulation trial with f 25-banknotes

In 1975 we ordered a certain amount of banknote paper for f25-notes from another paper

mill than our own. The paper was produced in two varieties: with and without flax

(called hereafter "paper with flax" and "paper without flax"). The quality of these papers

was to be compared with paper containing flax from our own paper mill V AN HauTEM&PALM.

The three varieties of paper were tested in the laboratories of V AN HaUTEM &

PALM and ENSCHEDE. Within the attainable accuracy no differences were found. Each

variety was printed as usual by EN SCHEDE in the denomination off 25. In the period of

Monday 20 June 1977 till Friday 1 July 1977 six series comprising 595000 immaculate

notes of each variety were issued by the Bank's Head Office at Amsterdam. I take Friday24 June 1977 as the date of issue. Since then the withdrawal of the notes was recorded.

On 21 December 1979, the last day of the present trial, approximately 90 per cent of

each variety had been withdrawn after 130 weeks which means that this is our most com

plete circulation trial.

A computer programme has been written which performs the analysis of par. 3 and

par. 4 and gives the results in tables and plots. It is fully operational so that the work load

is minimal and all attention can be devoted to the interpretation of the results.

5.1 The return rate

In figure 5 the return rate of the three test series has been plotted and in figure 6 of the

entire circulation of the same denomination. Despite the scatter of the data points it is

clear that o(t) is almost equal for the three paper varieties. The correlation coefficients are,

accordingly, very high:

VHP with flax/paper with flax

VHP with flax/paper without flax

paper with flax/paper without flax

r = 0,9982

r = 0,9805

r = 0,9853.

In the early stage of the trial the mean return rate appears to be constant over time as is

assumed in my model. Using the method of least squares straight lines were fitted to the

data sets. Under the hypothesis that the slope of the lines is zero the t-statistic has been

calculated for a few selected circulation times (HawKER and LIEBERMAN (1972)).The absolute values are shown in figure 7 as well as the 0,05 and 0,01 significance levels of

the t-statistic. From the figure it is clear that for t ~ 90 week the hypothesis is not re

jected. Thus it may be concluded that the mean return rates are constant. Beyond t = 90


0,20

0,16

1 0,12c OM

0,04

o = VHP paper with flaxx = paper with flax+ = paper without flax* = coinciding points

10 20 30 40 50 60 70 80 90 100 110 120 week

0,20

0,16

t-Fig. 5. Return rate of the test series.

I 0,12

c 0,08

0,04

[ u[J[J[J [J [J [J [J

[J •••

OJ p[J [J U [J [J [JEO [J [J J [J

[J[J[t5ln [J [Jee"tJ ,.., [J~ [J [J

J

10 20 30 40 50 60 70 80 90 100 110 120 week

t_Fig. 6. Return rate of all f 25-series in circulation.

week the return rates diminish gradually with the return rate of the shortest-living variety

diminishing first. This is caused by cullings and rejects as explained in par. 2 and shown

in formula (16). At t = 90 week the circulation trial has evidently reached its natural end

and the results become unreliable thereafter. Hence, all results to be presented in the fol

lowing paragraphs are based on the first 90 weeks. The equations of the straight lines fitting the observed return rates are in that case

VHP with flax

paper with flax

paper without flax

o(t) = 0,0461 - 0,000055t week-1 (-0,6108)

o(t) = 0,0465 - 0,000063t week-1 (-0,6910)o(t) = 0,0472 - 0,000146t week-1 (-1,6798).

The numbers between brackets give the values of the t-statistic at t = 90 week.


10

u:;:

VI

~ 4••.• 1

n11!!ii

+11 *I1+

1\

+1\ +-l~1 \

\

"- +I

+,---+------"-

-----0,01--------------+---*-------- ----------0,05 +==1==*0

TX

+~* ft

lifT

10 20 30 40 50 60 70 BO 90 100 110 120 week

t_Fig. 7. Absolute value ofthe t-statistic ofthe return rate under the hypothesis that the return rate

is constant. The 0,01 and 0,05 significance levels are indicated.

Under the hypothesis that the three sets of data points are independent (approximately

normally) distributed random variables with equal means, the t-statistic at t = 90 weekhas been calculated:

VHP with flax/paper with flax

VHP with flax/paper without flax

paper with flax/paper without flax

It 1= 0,0034

I t 1=0,9495

I t 1= 0,9347.

On a 5 per cent significance level the hypothesis is not rejected. The overall mean returnrate of the three varieties during the first 90 weeks is then 0 = 0,0426 week-I.

The conclusion is that the mean return rates of the three paper varieties are equal and

constant. Weekly deviations from the mean are not due to differences in paper quality

but to statistical fluctuations in the circulation and in our sorting procedure, and to cul

lings and rejects. Accordingly, this trial gives a fair comparison of the deterioration of

the three varieties in the circulation. Faster withdrawal of one variety (see par. 5.4) is not

due to faster discarding of these notes by the public and thus to a higher circulation rate,

but it is a genuine measure of faster deterioration in circulation.

A stilI unexplained result has to be mentioned. During the first ten weeks comparatively

high return rates are observed. This effect was observed also by KOEZE (1979) during

our first trial with f lOO-notes. Now that the effect occurred again with a second denomi

nation, it appears likely that it is a systematic occurence at the beginning of every trial. It

could be caused by the geographical or economical situation at Amsterdam or by the com

mercial banks or by our sorting procedure.

As was shown in par. 4.2 the expected return rate of the test series should be equal to

the expected return rate of the circulation, but it is not. The mean return rate of the cir

culation during the first 90 weeks is 0 = 0,0597 week -I which is considerably larger thanthe mean of the test series. The correlation coefficient between VHP with flax and the

circulation, which also consists of VHP-paper, is r = 0,8030. Although much was improved

200 Statistica Neerlandica 36 (1982), Uf. 4.

since we set up the first trial, this is not yet satisfactory. We are taking further measuresto diminish differences in future trials.

5.2 The hazard rate

In figure 8 the hazard rates of the test series are shown and in figure 9 of the circulation.

The hazard rates appear to have an asymptote as time increases, at least to have a finite

maximum. This is in accordance with the model of par. 4.1 which was based upon the

gamma distribution, and with formula (26)(Bu Ry (I975),HAHN and SHAPIRO (I967)).In par. 3 it was stated that formula (17) would be valid if no notes were kept in storage

after sorting:

0.070

+

0.056

10'042.c: 0,028

0,014

10 20 30 40 50 60 70 80 90 100 110 120 wee k

0,040

0,032

r 0,024~ 0,016

0,008

t_Fig. 8. Hazard rate of the test series.

[][]nI[]- [][]

~[][] Cc []

,.~[]

[][][]U []pU[][][[ []C

19[]

PIIl ~[][][][] [] []ID [][] []J [][]~ [][][] [][] D [][] []~ n 0

[][]

[][]rP[]~ a51lQ[] []

[]u'b

r# ~ to[]

:o[][][][] [] P[][] [] [][] n []

[] Cl[] d ~

10 20 30 40 50 60 70 80 90 100 110 120 week

t_Fig. 9. Hazard rate of all f 25-series in circulation.


t

i(t) = 1 - exp {- ~ h(u)}.u=1

As less notes are kept in storage, this formula will be better satisfied. For f 25-notes it is

satisfied during the first 90 weeks within an accuracy of about 2 per cent.

90VHP with flax 1 - exp {- ~h(u)} = 0,8241

u=1

compare

i(90) = 0,8119

90

paper with flax 1 - exp {- ~h(u)} = 0,8268u=1

compare

i(90) = 0,8148

90

paper without flax1 - exp {- ~h(u)} = 0,8766

u=1

compare

i(90) = 0,8703.

5.3 The withdrawal rate

In figure 10 the withdrawal rate of the test series are shown in figure 11 of the circula

tion. The rates of the test series increase with circulation time. VHP with flax and paper

with flax have nearly the same durability in circulation whereas paper without flax is

withdrawn considerably faster.

It is important to notice that the withdrawal rate has an asymptote smaller than 1 in

accordance with formula (29). It means that even after a long time the notes returned to

the Bank are not withdrawn completely but only partly. This was also the case with f 100-

1.0 -n0,8 +++ +~ ot~+

t" + :::~1~~w~:#+ti~~I + i+'**~, ~o0,4

"" 0 I ** ix+ +~+0,2 ~*

~*

° -~° 10 20 30 40 50 60 70 80 90 100 110 120 week

t_Fig. 10. Withdrawal rate of the test series.


10 20 30 40 50 60 70 80 90 100 110 120 week

t_Fig. 11. Withdrawal rate of all f 25-series in circulation.

notes in Ko E Z E (1979). It is to be expected that the asymptote will be the smaller the

higher the nominal value of the denomination is.

The correlation coefficients of the withdrawal rate and the return rate per paper

variety are small:

VHP with flax

paper with flax

paper without flax

r = -0,1282

r = -0,1480r = -0,1757.

The hypothesis that the correlation coefficient is zero is not rejected at a 5 per cent sig

nificance level (BOWKER and LIEBERMAN (1972)).

So in accordance with the model of par. 4.2 the return rate is independent of the with

drawal rate and deterioration of the notes. In other words, a high proportion of unfit

notes in those returned does not necessarily entail that very many notes were returned.

5.4 The withdrawal curve

In figure 12 the withdrawal curves of the test series are shown using linear scales. From

the first week onwards paper without flax is withdrawn much faster than both paperswith flax. The median times to failure are

VHP with flax

paper with flax

paper without flax

to,s = 41,4 week

to,s = 42,0 week

to,s = 32,1 week.

In figure 13 the withdrawal curves have been replotted on gamma probability paper (WILK

et aJ. (1962b )). In accordance with the model of par. 4.1 the gamma distribution provides

a satisfactory fit, at least during the first 90 weeks. Using the method of WILK et aJ.


1,0

0,8

i 0,6>_ 0,4

0,2

o *o 10 20 30 40 50 60 70 80 90 100 110 120 week

4,0

3,53,0

r

2,5

V>

2,0~ :;:;c:'"::><r 1,5

1,00,5

t __

Fig. 12. Withdrawal curves of the test series, linear scales.

+

*+

xo

+

*+

*+

*+

I

*+

*+-

~*

+1** ++1;**t**itlt

10 20 30 40 50 60 70 80 90 100 110 120 week

t __

Fig. 13. Withdrawal curves of the test series, gamma probability plot with 17 = 1,8.

(1962a) the maximum likelihood parameters of the gamma distribution are estimated:

VHP with flax

paper with flax

paper without flax

A= 0,0345 week-1

A= 0,0361 week-1

A= 0,0438 week-1

1/ = 1,88

1/ = 1,96

1/ = 1,88.

The scale parameter A is approximately four times larger for f 25-notes than for f 100

notes. The shape parameter 1/ is approximately equal for both. However, it must be stressed


that the estimated values are probably unreliable. According to WILK et al. the accuracy

of their method may drop to less than one digit for the range of values of interest here.

For the mean times to failure it follows from formula (23)

VHP with flax

paper with flax

paper without flax

MITF = 1,88 / 0,0345 = 54,5 week

MITF = 1,96 / 0,0361 = 54,3 week

MITF = 1,88 / 0,0438 = 42,9 week.

Comparison of the gamma and the log-normal distributions indicate that under certain

circumstances the two are similar. In the present circulation trial, after some time, the

withdrawal curves may be represented by a 10 log-normal distribution with the following

parameters (graphically estimated from figure 14).

VHP with flax

paper with flax

paper without flax

J.l. = 1,61

J.l. = 1,61

J.l. = 1,51

a = 0,38

a = 0,37

a = 0,39.

These results are qualitatively in agreement with the Canadian results. GILLIESON (1977)

has found, likewise, that the withdrawal curves of three circulation trials could be repre

sented by log-normal distributions from a certain circulation time onwards

++j ++~*-

+'*'.••.-h*++if*++**...-*++1*

'f>+'++~*+ **..t+t*"'"(/++++-:11'+ I*,+ *+~2

+-"'"-~Sl+

~ Sl

Sl

(

0,95

0,900,800,600,40

I0,20

0,100,05

0,010,002

0,0001

1 4 8 10 20 40 B0100 week

t_Fig. 14. Withdrawal curves of the test series, log-normal probability plot.


6 Conclusions

The following general conclusions apply to f 25-banknotes as well as to f 100-banknotes.

Presumably, they are valid for all denominations of Du tch banknotes.

6.1 Return and wear of banknotes are two independent statistical processes.

6.2 The return rate as measured at the Bank is equal to the probability of a banknote incirculation to be returned.

6.3 The expected return rate is a constant irrespective of age, deterioration or banknote

quality. Accordingly, GRESHAM'S Law does not apply to banknotes.6.4 The withdrawal rate as measured at the Bank is an instantaneous measure of the frac

tion of unfit banknotes in circulation, calculated per series, or for several series, orover all series.

Table 1. Experimental values of Dutch circulation statistics

mean return rate, circulation (week-! )mean return rate, test series (week-! )

asymptote of withdrawal rateVHP with flaxVHP without flax

paper with flaxpaper without flax

median time to failure (week)VHP with flaxVHP without flax


mean time to failure (week)VHP with flaxVHP without flax


shape parameter of gamma distributionVHP with flaxVHP without flax


scale parameter of gamma distribution (week-!)VHP with flaxVHP without flax


asymptote of hazard rate (week-I)VHP with flaxVHP without flax


[25

0,05970,0426

0,51

0,520,63

41,4

42,032,1

54,5

54,342,9

1,88

1,961,88

0,0345

0,03610,0438

0,021

0,0220,024

flOO

0,03530,0299

0,270,18

121,5154,0

150196

1,642,27

0,01090,0116

0,00710,0054


6.5 The withdrawal rate does not tend to 1 but to a value smaller than 1.

6.6 The withdrawal of banknotes may be represented by a two:parameter gamma distri

bution, the scale parameter possibly being a measure of the circulation rate and the

shape parameter possibly being a measure of the paper quality. Thus the effects of

circulation rate and paper quality are separated and measured independently.

In table I the experimentally established statistics characterizing the two denomina

tions and the tested paper varieties are collected. The values for f 1OO-banknotes are based

on a circulation time of 125 weeks (K 0 E Z E (1979)).

Acknowledgement

I am indebted to prof. Or. l.S. CRAMER and Ors. M. ZIJLSTRA for their valuable remarks.

References

BOWKER, A.H. and G.J. LIEBERMAN (1972), Engineering Statistics, Prentice Hall, EnglewoodCliffs (N.J.).

BURY, K.V. (1975), Statistical models in applied science, Wiley, New York.GILLIESON, A.H. (1977), Research into the extension of the life of banknotes: Results of 1973,

1975 and 1976 field trials, Bank of Canada Technical Report 10.HAHN, G.J. and S.S. SHAPIRO (1967), Statistical models in engineering, Wiley, New York.HEMELRIJ K, J. (1978), Rules for building statistical models, Stat. Neerl. 32, 123-134.KOEZE, P. (1979), An accurate statistical estimation of the life-length of f 100-banknotes: A circula

tion trial with two qualities of currency paper, Int. Stat. Rev. 47, 283-297.VISSERING, G. (1916), The Netherlands East-Indies and the gold exchange standard, Martinus Nij

hoff, The Hague.WILK, M.B.., R. GNANADESIKAN and M.L HUYETT (1962b), Estimation of parameters of the

gamma distribution using order statistics, Biometrika 49, 525-545.WILK, M.B., R. GNANADESKAN and M.J. HUYETT (1962b), Probability plots for the gamma dis

tribution, Technometrics 4, 1-20.

Received June 1980, Revised April 1982.

Statistica Neer\andica 36 (1982), nr. 4. 207

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