## Introduction

## Currency Management in India

### Printing and Issuance

^{1}are via subsidiary companies of the RBI and the Union Government of India, the Bhartiya Reserve Bank Note Mudran Private Limited (BRBNMPL 2020) and the Security Printing and Minting Corporation of India Limited (SPMCIL). BRBNMPL operates two printing presses in India, both set up in 1996, one in Mysore (Karnataka) and another in Salboni (West Bengal). BRBNMPL has the capacity for both printing presses stands at 16 billion paper banknotes a year, which is well above the average of 0.25 million banknotes on average that have been supplied and issued since 2018–19 (Reserve Bank of India 2021). Subbarao (2010) provides a useful overview and history of the currency management in India.

^{2}in 31 cities that supply paper banknotes and 3519 small coin depots in charge of coinage. Notably, the number of currency chests has actually declined over time, from 4422 in 2002 to 4247 in 2010, and 4075 just prior to demonetization in 2016. As of 2020, the RBI is considering redesigning the model through which banknotes are distributed, where larger currency chests redistribute paper banknotes to smaller currency chests in a specific geographical location (Reserve Bank of India 2020a).

### Inventory Management and Disposal Rules

^{3}Further, a note is unfit if “is not suitable for recycling because of its physical condition or belongs to a series that has been phased out by Reserve Bank of India (Reserve Bank of India 2010).”

### Cost of Printing Banknotes

^{4}It may therefore be more cost effective to have steady replacement of currency notes over time.

### Issuance and Disposal of Banknotes

^{5}

## Methodology

### Steady-State Methods

### Survival Analysis

^{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.

#### The Model

^{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.

### Selecting Explanatory Variables

^{8}

^{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.

^{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.

## Results

### Traditional/Steady-State Results

Denomination | Mean | Median | SD |
---|---|---|---|

INR10 | 4.92 | 5.31 | 2.48 |

INR20 | 5.69 | 4.99 | 4.01 |

INR50 | 2.94 | 2.50 | 1.87 |

INR100 | 3.77 | 3.06 | 1.92 |

INR500 | 7.90 | 5.73 | 4.43 |

INR1000 | 42.49 | 19.03 | 46.16 |

Denomination | Mean | Median | SD |
---|---|---|---|

INR10 | 4.25 | 4.04 | 2.48 |

INR20 | 3.69 | 3.44 | 0.84 |

INR50 | 2.83 | 2.76 | 0.63 |

INR100 | 3.41 | 3.19 | 0.93 |

INR500 | 4.06 | 3.99 | 1.44 |

INR1000 | 5.10 | 4.59 | 2.78 |

^{11}This is particularly the case for the highest value banknote (INR 1000), for which we provide a separate graph (Fig. 7). The results suggest that there have been notable increases in banknote life for the lowest value banknote (INR10), whereas other denominations show a fairly stable duration of circulation over the period for which our data is available (2003–2016). Note that the spikes at the end of the graphs are likely on account of substantial changes to RBI focus on managing withdrawal of older banknotes and issuing new banknotes during the demonetization event.

### Survival Analysis Results

^{12}In Fig. 8 below, the graphs show the expected quantity of surviving banknotes by denomination for the INR 10, 20, 50, and 100 banknotes. The red line represents the actual quantity of surviving banknotes, whereas the solid black line presents the estimated surviving banknotes from the GG model. The grey lines indicate the continuous disposal of surviving banknotes. This graph illustrates whether the fit of the model improves over time, which we could argue for INR 10 and 20 to be the case, but we see little to no information from the estimates for medium-value banknotes (INR 50 and 100) from Fig. 8. The survival function here is based entirely on the data of net issuances for the period of analysis. Note that Fig. 9 is not necessarily indicating that the number of surviving INR 20 banknotes is increasing over time, instead it shows that the fit of the survival model (and the generalized gamma function) is improving over time for the INR 20 banknote. Thus, although the INR 10- and 20-rupee note are different denominations and therefore have differing issuance, use, and disposal patterns, our model is unable to fully account for these variations.

Weibull | INR10 | INR20 |
---|---|---|

Estimate | Estimate | |

Probability of survival | 0.36 | 0.36 |

lambda | 5.97 | 5.97 |

k | 0.29 | 0.29 |

GFC | 0.17 | 0.17 |

Currency in circulation | 0.00 | 0.00 |

Velocity of circulation | − 2.13 | − 2.13 |

Velocity × t | 0.15 | 0.15 |

MAPE | 536.40 | 666 |

RMSE | 16,044.66 | 16,045 |

Generalized gamma | INR10 | INR20 |
---|---|---|

Estimate | Estimate | |

Location | 3.48 | 17.91*** |

lambda | − 0.01 | 0.21 |

k | 6.10*** | 5.11 |

GFC | − 0.07*** | 0.13 |

Currency in circulation | 0.00 | 0.00** |

Velocity of circulation | 7.01*** | 30.46*** |

Velocity × t | − 0.31 | − 1.23*** |

MAPE | 117 | 4.9 |

RMSE | 4055 | 129 |

## Concluding Remarks and Implications for Policy

^{13}The parameter estimates for both the Weibull as well as the GG functions for certain denominations (such as INR100 and INR500) are uninformative and more data is needed to make meaningful estimations of banknote life for these denominations. It is also likely that these estimations are sensitive to assumptions on hoarding, where high-value banknotes are known to be associated with the shadow economy (Drehmann et al. 2002; Kumar 2016). The use of survival models in currency management applications is sensitive to the ability to optimize the aggregate fit of individual survival functions to the underlying issuance and destruction data. Other studies in this domain typically use monthly issuances data to model life of a banknote, and here it is important to note that our results are constrained by the annualized Indian data available in the public domain. The purpose of this exercise is to illustrate the utility of the survival analysis methods in estimating the life of banknotes for India, while acknowledging limitations and outlining avenues for future work. For the survival analysis, we set initial disposal values for 2002 to be zero as this is our initial period where banknotes are issued. This implies that we explicitly assume a “burn-in” period for issuance and disposal of banknotes based on data availability and not necessarily informed by theoretical or policy-based assumptions. Another major limitation of our analysis is the inability to distinguish between INR 10 and INR 20 using the Weibull function. Location refers to a parameter of the GG function that measures the fit of the survival function to the data. Since it is not statistically significant in any of the models, we cannot comment on its interpretation, and it cannot be compared across models either (Aves 2019). We do not have additional covariates to include in this model as lack of convergence due to low frequency data (and therefore small sample size) is a major barrier to incorporating additional insights.