In this section, we outline two methods of assessing the life of banknotes following Rush (
2015): the ‘traditional’ steady-state or turnover method, and the Feige (
1989) steady-state method, as well as recent innovations applying survival analysis methods. The traditional/steady-state methods are tools that central bankers and suppliers of currency may use to estimate roughly the duration of banknote circulation. Survival analysis methods are more statistically rigorous, in that the probability of a banknote returning for disposal is predicted as a survival function driven by macromonetary factors (Aves
2019; Rush
2015).
Steady-State Methods
As the name suggests, the traditional steady-state or turnover method (Rush
2015) simply tracks the stock of banknotes on issue (averaged over a 12-month period) relative to the total number of banknotes destroyed (in the same 12-month period). This would translate to the ratio of issued to disposed from the data presented in Figs.
3,
4,
5. This is essentially a measure of turnover rate that computes the average number of years that an issued banknote is circulated until it returns for disposal (likely on account of being deemed unfit as per the fitness rules discussed in earlier sections).
As Rush (
2015) states, a key limitation of the traditional steady-state methodology is that the measure varies widely when there are disproportionate issues relative to disposals of a banknote in a given period. As a partial solution, Feige’s (
1989) ‘banknote life’ is an alternate measure that makes a distinction about the number of times a banknote is used in a transaction over its assumed lifetime of 12 months. This is a key assumption made by this method which drives the estimation of the life of a banknote. Thus, the Feige method for banknote life (for the
\({D}^{th}\) denomination) uses the formula:
$${F}_{D}=\frac{Mean stock of banknotes on issue over 12 months}{(Annual disposal+Annual issuances) / 2}.$$
(1)
This is a marginal improvement over the traditional-steady-state method as it does not assume a constant hazard or destruction rate.
One of the key limitations of these methods is that they assume that the probability of a banknote being deemed unfit is assumed to be independent of their age (i.e., the year that they were issued). They are also typically unable to take into account changes in currency management, such as changes in quality of banknote production or other innovations in their supply. One of the key changes that the RBI does regularly undertake in this regard is new issuances with newer security features and new inks. Furthermore, as Rush (
2015) states, these methods do not account for exogenous changes in demand for currency, such as the global financial crisis (Cusbert and Rohling
2013) or similar macroeconomic events. In India, there might also be changes in the network of issuance and disposal of banknotes (such as the Cash Distribution and Exchange Scheme, or CDES) that may affect banknote life as measured by these methods.
Survival Analysis
Survival models are gaining traction in the literature on currency management and life of banknotes (Aves
2019; Deinhammer and Ladi
2017; Rojas et al.
2020; Rush
2015). In order to further examine life of banknotes across various denominations, recent studies (Rush
2015; Aves
2019) have made use of survival models that are common in the medical literature in tandem with statistical optimization algorithms to fit and model survival curves across different series of a currency denomination over time. Although more commonly used in the biological sciences, survival analysis is useful in explaining the ‘death’ of a banknote as one that is removed from circulation after a particular point in time. Unlike a typical application, however, central banks do not maintain data on each banknote in circulation once it is disposed, and data available is more aggregate in nature.
6 Further, by using such survival models, we are able to relax the assumption of the constant hazard rate and use standard probability distributions to estimate models of the survival function for banknotes. The actual number of fit banknotes is defined in these models as the total banknotes ever issued less the total number of destructions up to that point in time. Each issuance (for each denomination) therefore will be characterized by a potentially unique survival function—a likelihood that defines the fraction of fit banknotes based on a set of parameters)—and can be used to compute the expected number of aggregate fit banknotes.
We therefore follow Rush (
2015) and Aves (
2019) in proposing a joint survival function using aggregate issuances and disposals data from the RBI. Thus, the total quantity of surviving banknotes at any given date and denomination is a sum of all surviving banknotes from each issuance. This is a tenable description since at the time of every issuance, individual banknote pieces are identical and do not differ in quality or other characteristics. In the sections that follow, we describe our survival and death functions, and the destruction and hazard rates.
The Model
For a time of destruction
T, the survival function is defined as
\(S\left(t\right)=P(T> t)\), where the function
\(P (T)\) describes the probability that a banknote will survive (not be disposed) at time period
T. The lifetime distribution function (which is a cumulative density function for the probability that a banknote will be destroyed at time
t) is given by
\(L\left(t\right)=P\left(T\le t\right)=1-S(t)\). Scaling the first derivative of the lifetime distribution function (
\(\dot{L}(t)\), which is the destruction rate) by the proportion of surviving banknotes yields the hazard rate (Aves
2019), given by:
\(\dot{L}(t)/S\left(t\right).\) This quantity is essentially the instantaneous failure rate at a given time, and may also be interpreted as the negation of the logarithmic derivative of the survival function S(t). For instance, if S(t) corresponds to an exponential distribution, which is the canonical distribution for a survival function, then the hazard rate is constant, reflecting the memoryless nature of the distribution. For other distributions the magnitude of this quantity may indicate other properties of the distribution, such as a higher rate of failure for older individuals (a tendency to wear out over time), or the opposite phenomenon (a tendency towards early failure rather than later).
As Rush (
2015) notes, the expected number of ‘fit’ banknotes at any given time can be defined as the sum of new issuances since banknote production commenced (in this case, since data on banknote production is available from the RBI), multiplied by the issuances’ survival function. Based on aggregate data, this is given by
$$E\left({F}_{t}|\boldsymbol{\alpha }\right)= \sum_{n=1}^{t}S\left({t}_{n}, \boldsymbol{\alpha }\right).{I}_{n},$$
(2)
where
\({F}_{t}\) is a measure of outstanding banknotes which is derived from aggregate RBI currency management data as the sum of total issuances less the disposals for that particular denomination (over the total number of time periods,
t) and
\({I}_{n}\) is the number of issued banknotes at time
n.
\(\boldsymbol{\alpha }\) is a vector of parameters that governs the shape of the survival function
S. As described elsewhere, this vector is likely to factor in determinants of banknote survival that cannot be fully captured in the data (e.g., changes in handling of banknotes, preferences for cash payments, among others). In our survival model, the functional form of the survival function determines what is contained within
\(\boldsymbol{\alpha }\). We also define an issuance function in this manner for each banknote denomination (INR 10, 20, 50, 100, 500, and 1000). The key addition that we are able to make to the literature on currency management in using this approach relates to the use of non-linear regression analyses with this survival function applied to the case of India. Thus, in contrast to the steady-state approaches, we are able to explain or predict the probability of banknote disposal (and fitness) using currency management policies. A notable shortcoming remains, as Rush (
2015) acknowledges, which is that there is very little in terms of demand-side preferences or tastes that one can account for in this analysis (e.g., preferences for cash). Finally, our non-linear regression specification that estimates the joint survival function is defined as
$${t}_{n}={e}^{{X}_{n}{\prime}\beta }+{t}_{n-1}.$$
(3)
In line with Aves (
2019), time is defined as ‘activity time’
7 (i.e., the duration for which banknotes are assumed to undergo wear and tear as a result of circulation in the economy). Thus,
t represents activity time, and
X is a vector of explanatory variables (dummy variables for changes in banknote series, global financial crisis, demonetization, and the velocity of cash circulation), and
\(\beta\) is the set of parameters to be estimated.
In general, in work that uses survival analysis, either the Weibull or the generalized Gamma distribution are preferred due to their flexibility in approximating other distributional forms. These distributions are determined by two or three parameters, which determine the scale, shape and location of the function. For a detailed discussion on the choice of probability function (i.e., the functional form), we refer the reader to Rush (
2015) and Aves (
2019). Since there are no prior studies that suggest appropriate values for these parameters, a key challenge is in calibrating the parameter values so that the model converges. There is little to no statistical guidance on what appropriate values are as these could differ by the nature of the data as well as the number of explanatory variables and observations under consideration.
Selecting Explanatory Variables
Literature on currency management is small and sporadic and is largely restricted to countries where central banks maintain monthly data on currency notes issued and destroyed. In the case of India, the RBI makes available only annual data on currency issuances and disposals (by denomination), and data on banknote quality changes or changes in issuance processes is scarce outside the annual reports. To help explain the probability of survival (and disposal) of a banknote within a denomination class, we start with indicator variables that prior work suggests will play a role in the life of banknotes. First, we create two indicator variables for macroeconomic conditions that have been shown to influence the circulation and velocity of currency used in transactions: the global financial crisis (GFC, henceforth), and the demonetization and remonetisation period spanning December 2016 to April 2017. The GFC variable for India takes a value of 1 for the period between August 2008 and December 2010 (Dua and Tuteja
2016). Similarly, the demonetization variable takes a value of 1 for the period between December 2016 and April 2017, and zero otherwise.
8
Furthermore, our model considers the process by which the circulation of notes change—i.e., the velocity of transactions using cash that could increase the frequency of usage thus shortening the time to becoming ‘unfit’. To proxy for this, we use the ratio of ATM withdrawals to currency in circulation (Aves
2019). This also helps us overcome the shortcomings of the steady-state models presented previously, since we are able to account for the role of mechanical defects and inkwear in the life of a banknote. Thus, we can do away with the assumption implicit in turnover models that a banknote is assumed to remain in circulation until it is returned to the RBI for disposal.
As such, Eq. (
3) for the ‘activity time’ is estimated as
$${t}_{n}={e}^{\beta {GFC}_{n}+\beta {CIC}_{n}+\beta {Velocity}_{n}+\beta {Velocity}_{n}*t}+{t}_{n-1},$$
(4)
where
GFC is an indicator variable equal to one between October 2008 and August 2009 as well as between December 2016 to April 2017 that account for any precautionary demand for banknotes during the global financial crisis and the demonetization and remonetisation periods;
CIC is the total currency in circulation by value;
Velocity is the ratio of monthly ATM withdrawals, by value, to circulating banknotes, by denomination as a proxy for the velocity of cash; and
Velocity*
t is the Velocity variable times a time trend.
However, banknotes (especially of higher value) might be used as a store of value and hoarded for a long time until changes in the macroeconomy might induce changes in the composition of notes circulating.
9 To account for such variations in cash use by denomination, the choice of probability distribution is critical. Prior work in this domain has suggested the use of the Weibull (Rush
2015) and the generalized gamma distribution (Aves
2019). The generalized gamma (GG, henceforth) allows for a wide variety of possibilities in the behaviour of the survival function and is commonly used in various survival applications.
The Weibull distribution is the probability distribution admitting the density:
$$f\left(t\right)=\frac{k}{\lambda }{\left(\frac{t}{\lambda }\right)}^{\left\{\left(k-1\right)\right\}{e}^{\left\{-{\left(\frac{t}{\lambda }\right)}^{k}\right\}}},$$
(5)
for
\(t \ge 0\), and
\(f(t)=0\) for
\(t<0\). Here
k and
λ are positive parameters, called the
shape and
scale parameters, respectively. Note that a random variable with a Weibull distribution is positive with probability 1. Our objective is to estimate
\(f(t)\) as a survival function with explanatory variables and estimate the probability of survival using the functional form specified in Rush (
2015). We adapt the GG function from Aves (
2019) to take a similar form.
The generalized gamma distribution is the probability distribution admitting the density:
$$f\left(t\right)=\Gamma {\left(\frac{d}{p}\right)}^{\left\{-1\right\}\left(\frac{p}{{a}^{d}}\right){{\text{t}}}^{\left\{\left(d-1\right)\right\}{e}^{\left\{-{\left(\frac{t}{a}\right)}^{p}\right\}}}},$$
(6)
for t ≥ 0, and f(t) = 0 for t < 0. Here
a, d, and
p are positive parameters which determine the properties of the distribution. Note that a random variable with a generalized gamma distribution is positive with probability 1, which is appropriate in light of our interpretation of
t as time until failure. Appropriate choices for the parameters
a, d, and
p can yield simpler important distributions, such as, the exponential, Weibull, and gamma distributions, but the generalized gamma distribution is more flexible than these and as a result is popular in survival analysis (Cox et al. 2007; Aves
2019).
One noteworthy feature of the generalized gamma is that the failure rate, which is defined as
\(\Lambda \left(t\right)=\frac{f\left(t\right)}{1-F(t)}\) with
F(t) the cumulative distribution function, takes a complicated form that allows for a variety of different behaviours of a banknote. In particular, the failure rate for a generalized gamma need not be monotone in
t. This contrasts with the Weibull distribution, whose failure rate is assumed to be monotonically increasing.
10 In what follows, we discuss results of estimation from both models, acknowledging caveats associated with using annual data, which is lower frequency than monthly data typically used in modelling banknote life using survival analyses. For optimization, we use the non-linear least squares (NLS) function in R, which provides a set of summary statistics following iteration through model parameters. Using NLS poses some challenges and requires key assumptions for the model to converge, especially with a small number of observations. The optimization technique is sensitive to the initial values, the number of variables specified, and potentially any measurement error. We attempt to address these concerns by iterating the model using a limited number of explanatory variables, restricting optimization to non-negative values, setting assumptions on the initial values (see discussion above), and restricting our analysis to a denomination that has a median lifespan of 4–5 years.