# Modeling nonmaturing deposits: a framework for interest and liquidity risk management

## Emil Avsar and Benjamin Ruimy

#### Need to know

• We present a generic framework for modeling nonmaturing deposits that can be used by banks for interest rate risk and balance sheet management.
• The framework allows the practitioner to calibrate both the rate dependence of the cashflows between different types of deposits as well as the vintage specific run-off of each type of deposits.
• A concrete model calibrated on UK industry corporate sight and time deposit balance series published by the Bank of England is presented together with forecasts obtained under different interest rate scenarios.

#### Abstract

We present a generic framework for modeling nonmaturing deposits that can be used by banks for interest and liquidity risk management, funds transfer pricing and dynamic balance sheet management. The framework is split up into the modeling of the fraction of funds held in different deposit categories, which captures cashflows between products, and the modeling of the sight and time deposit rates. Combining these different components, together with assumptions on market share for individual institutions as well as the growth of the money supply, the framework then predicts the institution-specific balance composition of deposits under different interest rate scenarios. We calibrate the model framework on UK industry corporate sight and time deposit balance series published by the Bank of England, and present forecasts obtained under different interest rate scenarios.

## 1 Introduction

The banking book of a bank consists of the advances and loans to and deposits from customers (corporate and retail) for which net interest income (NII) is a major source of profitability.

A major source of customer deposits is so-called nonmaturing deposits (NMDs), which are deposits without any contractual maturity. From its 2018 annual statement we find, for example, that for Barclays £287 billion was held in NMDs, which make up 75% of the total number of customer deposits, accounting for almost a third of the value of the balance sheet. For interest rate risk purposes we can categorize the NMDs into fixed-rate products and managed-rate deposits (MRDs). Fixed-rate products almost exclusively refer to non-interest-bearing current accounts (NIBCAs), which do not pay interest to customers (ie, they pay a fixed rate of 0%), while MRDs are usually interest-bearing savings accounts where the deposit rate is set by the relevant business and is not contractually fixed or floating based on some index. NIBCAs and MRDs are generally instant access accounts, which means that a customer can withdraw or top up an account anytime. However, some banks can offer MRDs that have a limited number of withdrawals over a fixed period of time. An alternative to NMDs for customers is time deposits (TDs), which refers to products where funds are locked up until the contractual maturity, paying either a guaranteed fixed rate or a floating rate.

Fixed-rate NMDs such as NIBCAs are subject to interest rate risk for banks as the product can earn a floating rate (eg, the base rate, if customer deposits are placed at the central bank) whereas the product rate (ie, what is paid to customers) is fixed (eg, at 0%). As a result the bank’s NII is exposed to falling rates. MRDs are also exposed to interest rate risk as the product rate does not necessarily move one-to-one with its related floating market rate. For example, if on average the business passes on 80% of changes in the base rate to the customer rate, then 20% of the portfolio is effectively a fixed-rate product. In addition, the uncertainty around the duration of balances implies that the bank must accurately assess the correct notional to hedge at any time, which exposes the bank to behavioral-assumption risk.

Interest rate risk arising from the banking book will be centrally managed by the treasury asset and liability management (ALM) desk given a certain risk appetite. To manage the interest rate risk posed by NMDs with unknown maturity, the ALM desk will usually implement a rolling set of structural hedges with the aim of stabilizing the NII.

To satisfactorily manage such hedges it is imperative for a bank to properly model the expected behavior of NMDs under different rate cycles. However, given the environment of low interest rates that has dominated in many countries in the past decade, many institutions will find themselves in a situation where customer data is readily available only in a declining or flat interest rate environment, significantly increasing model uncertainty and risk. To overcome this data limitation our modeling framework is built by combining both external, industry data (with a long available history covering a rising interest rate environment) and internal customer data.

In this paper we develop a methodology that allows banks to assess the interest rate sensitivity of their NMD products. The main contribution is the modeling of the fraction of deposits held in different NMDs, which allows us to estimate the potential cashflows between different products. This itself is a crucial task for managing interest rate risk sensitivity as the different types of deposit products have different rate characteristics. In addition, our methodology demonstrates how internal account-level data can be used in conjunction with industry-level data to also model specific runoff profiles that are important for liquidity risk management, capturing simultaneously the account age dependence and the overall interest rate sensitivity of the portfolio in a consistent manner.

The paper is organized as follows. In Section 2 we give an overview of the literature on NMDs and explain how our approach, which models proportion among the different deposit categories, differs and allows interest rate sensitivity to be captured. Then, in Section 3 we describe the UK industry data used for building the framework. We then go on to describe the model architecture at a high level in Section 4, after which we describe the different model components in more detail; Section 5.1 describes the model for the fraction of funds allocated into NIBCAs, while Section 5.2 describes the model for the fraction of funds allocated to sight deposits. Section 6 describes the models for the balance-weighted average savings and TD rates, and Section 7 illustrates how the stock runoff of a given NMD portfolio can be determined by a bank using its own internal data together with the industry-level models for the total balances in each deposit category. We present the model outputs in Section 8 under different interest rate scenarios as well as scenarios for the growth of the money supply. Our conclusions are presented in Section 9.

## 2 Overview of the literature

As explained in Bardenhewer (2012), models of NMDs are generally categorized in the literature as so-called option-adjusted spread (OAS) or replicating portfolio models (see, for example, Brunqvist (2018) for an overview of the latter). While these two approaches may have somewhat different starting points, one of the main tasks faced by both types of models will be to accurately estimate the actual cashflows of the nonmaturing product. While a generic formalism using the principles of arbitrage-free pricing can be formulated, as in Jarrow and Van Deventer (1998), in this paper we focus on an empirical model that captures the potentially nonoptimal decisions made by customers.

Generally, balances are modeled using standard ordinary least squares (OLS) regression calibrated on aggregate balance data (see, for example, Bardenhewer 2012; Brunqvist 2018; Janosi et al 1999; Goosse et al 1999; de Jong and Wielhouwer 1999; O’Brien 2000; Laurent 2004; Kalkbrener and Willing 2004; Paraschiv and Schürle 2010; Sheehan 2012). The models will usually specify a linear dependence on interest rates (in a given transformation of the balances, such as logarithmic differences). In order to obtain reasonable in-sample fits, the models usually require the introduction of an autoregressive term on the balance, which then dominates the regression.

Market interest rates are typically modeled using a mean-reverting diffusive-type model such as one- or two-factor Vasicek or Cox–Ingersoll–Ross-type models. The deposit rates are modeled as functions of the market interest rates by a linear-regression or error-correction-type model (see, for example, Paraschiv and Schürle 2010; Castagna and Miste 2016; Castagna and Scaravaggi 2017); see also Kordel (2017) for a broader overview on the modeling of deposit rates for NMDs.

There exists also a somewhat different category of models that we might refer to as nonlinear behavioral models (see, for example, Selvaggio 1996; Frachot 2001; Nystrom 2008; Castagna and Manenti 2013). The starting point in these models is that customers target a certain long-term balance to keep in an NMD account determined by a “strike” level for interest rates that is unique to each individual customer. A certain distribution of strike levels is then assumed to describe the portfolio at the aggregate level, resulting in a model that is nonlinear. Nystrom (2008) in particular constructs a detailed model in which customers have the choice of keeping their deposits in a non-interest-paying transactional account as well as a number of interest-paying savings accounts.

In this approach it is assumed that the customer keeps a certain fraction of their income in a transactional account at every period. With this setup, cash cannot be accumulated into the transactional account, and the outstanding balance on such accounts can at most grow in line with income. This framework therefore misses the possibility of cash being increasingly accumulated in NIBCAs, which as a matter of fact is precisely what has happened since the financial crisis, as we can observe outstanding balances on NIBCAs that have grown considerably faster than income.

Our approach differs from the literature by the fact that we do not directly model the volumes but rather model the fraction of balances kept in each account type. The only other paper where we have found a similar approach is Iwakuma and Hibiki (2015), where in formula (3) in that paper a linear model for the ratio of TDs to sight deposits is presented. The advantage of this approach is that, for one thing, it is easier to isolate the rate dependencies, as the decision to allocate a certain fraction of deposits into, say, a NIBCA versus a savings account is most likely driven by the rate offered on the latter. Moreover, to understand the volume growth observed in a given NMD product, it is not enough to look at the volumes kept in this product in isolation as there are two different parts determining the balance evolution; one is how total deposits in the economy (the broad money supply) has evolved, while the other determines the dynamics of customers’ preferences to keep deposits in that particular category over time. A major advantage of our framework is that it takes into account cashflows between the different deposit types.

## 3 Industry data

The Bank of England (BoE) publishes on its statistical database data on sectoral deposits and Divisia money.11 1 URL: https://http://www.bankofengland.co.uk/statistics. The data consist of monthly or quarterly observations of amounts outstanding or changes in sectoral components of M4 money supply. UK-resident banks or monetary financial institutions (MFIs) directly report the data to the BoE using Form BE. Banks with private-sector holdings above £1 billion report the data on a monthly basis. All data are reported in sterling. The BoE publishes only aggregate data, and it is not possible to obtain data on individual institutions.

We make use of the following time series from the BoE.

LPMBF95:

monthly changes in MFIs’ sterling non-interest-bearing sight deposits (including transit and suspense) from private nonfinancial corporations (in millions of pounds sterling) seasonally adjusted.

LPMB2F7:

monthly changes in UK resident banks’ (including the central bank but excluding mutuals) sterling interest-bearing sight deposits from private nonfinancial corporations (in millions of pounds sterling) seasonally adjusted.

LPMZ3TM:

monthly changes in MFIs’ sterling interest-bearing sight deposits from private nonfinancial corporations (in millions of pounds sterling) seasonally adjusted.

LPMVVHM:

monthly changes in MFIs’ sterling M4 liabilities to private nonfinancial corporations (in millions of pounds sterling) seasonally adjusted.

Here, the first series represents the monthly changes in the UK industry NIBCA balances, the second and third series the changes in the interest-bearing sight deposit (savings) balances and the fourth series the changes in the total deposit base of private nonfinancial corporations. For the interest-bearing accounts we use two different series, as the series LPMB2F7 spans October 1997 to November 2013, while the series LPMZ3TM spans January 2009 to the present. For the data prior to 2009 we therefore use the former series. In addition, in order to reconstruct the amounts outstanding for each month, we use the most recent values from the series LPMBF92, LPMZ3TL and LPMVVHL for NIBCAs, savings and M4, respectively. It should be noted that the difference between the M4 balances and the sum of NIBCA and savings balances can almost entirely be attributed to time deposits.

The reason we reconstruct the amounts outstanding from the series for monthly changes in this way is that the series for the former published by the BoE contain various adjustments over time (such as changes in reporting from individual institutions) and thus can contain jumps. These adjustments are already taken into account by the BoE when constructing the time series of changes, and we therefore end up with more reliable series for the amounts outstanding when reconstructing them from the former.

Figure 1 shows the historical UK corporate NIBCA, savings and TD balances, together with the balance-weighted average rates offered on savings and TDs.

We can see that NIBCA balances have historically had an increasing trend except for a dip around 2002.

Starting from 2003, NIBCA balances have kept growing, with two different regimes being discernible. Prior to the financial crisis the average savings rate offered was never below 200 basis points (bps); then, around 2008 it dropped from 400bps to below 50bps in less than a year. Since then, rates have remained very low, which has resulted in cash being accumulated into NIBCAs as there has been very little monetary incentive to move cash into interest-bearing products.

If we look at the savings balances, we find that they decreased by approximately 10% in the space of a year and a half around the time of the crisis. After that, the balances started to grow again, despite the low interest rate environment. Historically, the biggest decreases in balances are seen in TDs, with a drop of around 35% from December 2007 to January 2014. It is reasonable to assume that the main reason for this drop is that there has been almost no incentive to lock money away when rates are at such low levels. Moreover, corporations have generally been more cautious since the crisis and might have a higher preference toward liquid products such as sight deposits.

As mentioned above, we need to look separately into the dynamics of the money supply and the allocation into each deposit category to better understand the historical behavior. The growth of the money supply depends on factors determining the overall state of the economy as well as on monetary policy. Along with the time series for the deposit balances, the BoE also publishes the time series of the total sterling M4 liabilities of UK MFIs to private nonfinancial corporations. In this paper we will not be presenting a model for the evolution of M4 balances but we will rather take these as given and model the allocated fractions into each of the aforementioned deposit categories.

Figure 2 shows the time series of the ratios of the different deposit volumes to M4. Looking at the NIBCA-to-M4 ratio we see that prior to the crisis, even though the absolute balances held in NIBCAs were constantly growing, the percentage of funds allocated to NIBCAs was actually decreasing. After rates were cut aggressively in response to the financial crisis, we see that the NIBCA-to-M4 ratio began a period of strong and steady growth, tripling in 10 years. In the same period we see that it was mainly the allocation to TDs that decreased, while the ratio of savings to M4 became comparatively flat after an initial strong increase just after the crisis, coinciding with an equally strong decrease in the TD ratio.

## 4 Overview of framework architecture

An overview of our modeling framework is shown in Figure 3. The two main time series modeled are

• the NIBCA-to-M4 ratio (described in Section 5.1), with the average savings rate as the main driver; and

• the NIBCA-plus-savings-to-M4 ratio (described in Section 5.2), with the spread between the average TD and savings rates as the main driver.

The savings and TD ratios can then be immediately deduced from these two ratios. Combining these with a model for the total M4 liabilities then allows us to forecast the total balances for each deposit category.

## 5 The allocation into the different deposit types

### 5.1 NIBCA-to-M4 ratio

The time series for the fraction of funds deposited in NIBCAs is shown in Figure 4 together with the balance-weighted average rate paid on savings products. We will be modeling the monthly changes in this fraction.

When the average savings rate is low, there is little incentive for customers to move money away from NIBCAs into savings, while conversely when rates are high we expect little to no excess deposits to be held in NIBCAs. As can be seen from Figure 4, in the period prior to the financial crisis of 2008 the average savings rate was high, and the NIBCA fraction slowly but steadily declined toward 12%. After interest rates were then cut we observe a strong and persistent growth in the NIBCA fraction. If interest rates were now suddenly to rise, it is reasonable to expect that a lot of this excess cash held in NIBCAs would instead be reallocated to savings and TDs. As also argued in Nystrom (2008), we do not necessarily expect the customer sensitivity to interest rates to be linear. For very low rates customers would be rather insensitive to their movements, as, for example, it makes very little difference whether the savings rate is 25bps or 35bps; in both cases there is almost no incentive to move money away from NIBCAs anyway. Eventually, however, we expect customers to become sensitive to the level of rates, at which point they will start to reallocate excess deposits into interest-bearing products. Alternatively, if rates are already rather high we would again expect customers to have little sensitivity to further changes in the rates, as, for example, it would make very little difference to a customer whether rates are 8% or 9%; in both cases there is almost no incentive to keep excess cash in a NIBCA anyway.

We also expect that the changes in the NIBCA ratio depend on its current level, since if the latter is high then a lot of excess deposits are kept in NIBCAs, and a subsequent rise in interest rates would then potentially cause a big drop. On the other hand, if the current level is already small, then even if rates were to rise, making NIBCAs less attractive, the NIBCA ratio would not really decrease much as there are no excess deposits to move away to interest-bearing products, and it should be kept in mind that customers will always need to hold a certain minimum fraction of deposits in NIBCAs to cover their short-term liquidity needs.

In order to capture these effects we introduce a mean-reversion term into the model such that, when rates are high enough, the fraction of funds allocated to NIBCAs converges toward a long-term target value, representing the equilibrium level of deposits that customers need to hold in NIBCAs to cover their short-term liquidity needs.

To describe the expected nonlinear dependence on the average savings rate, we use a linear basis function. For simplicity we use a basis of linear functions, especially since the low number of data points spanning different interest rates makes it rather hard to properly fit more complicated basis functions. The basis functions we use are then given by

 $b_{k}(x)=x-(x-\xi_{k})_{+},$ (5.1)

where $\xi_{k}$ is a so-called knot point and $(z)_{+}=\max(z,0)$. Thus, we include a term

 $\sum_{k=1}^{K}\beta_{k}b_{k}(d^{\mathrm{s}}_{t-1}),$ (5.2)

Here, we assume that $\xi_{1}<\xi_{2}<\cdots<\xi_{K}$.

Note that, if $d^{\mathrm{s}}_{t-1}>\xi_{K}$, then

 $b_{k}(d^{\mathrm{s}}_{t-1})=\xi_{k}$ (5.3)

for all $k$, and thus this term in (5.2) equals

 $\sum_{k=1}^{K}\beta_{k}\xi_{k},$ (5.4)

which is independent of the rate $d^{\mathrm{s}}_{t}$. The change in the NIBCA ratio is therefore insensitive to the deposit rate when the latter is large. In addition, we also require that the rate dependence is monotonically decreasing as a function of $d^{\mathrm{s}}_{t}$, since the incentive to allocate funds to NIBCAs should not increase when the savings rate is increasing.

The other component of the model is the mean-reversion term, which has the form

 $\gamma(\theta-\lambda_{t-1}^{\mathrm{nibca}})\chi(d^{\mathrm{td}}_{t-1},\zeta_% {1},\zeta_{2}),$ (5.5)

where $\gamma$ and $\theta$ are the speed and level of mean reversion, respectively, $\lambda_{t}^{\mathrm{nibca}}$ is the NIBCA fraction of deposits to M4 at time $t$, $d_{t}^{\mathrm{td}}$ is the balance-weighted average corporate TD rate at time $t$ and $\chi$ is the following activation function that depends on two further knot points, $\zeta_{1}<\zeta_{2}$:

 $\chi(d_{t},\zeta_{1},\zeta_{2})=\begin{cases}0,&d_{t}\leq\zeta_{1},\\ \\ \displaystyle\frac{d_{t}-\zeta_{1}}{\zeta_{2}-\zeta_{1}},&\zeta_{1}\leq d_{t}% \leq\zeta_{2},\\ \\ 1,&\zeta_{2}\leq d_{t}.\end{cases}$

The mean-reversion term therefore starts being active from $\zeta_{1}$, and has a full effect above $\zeta_{2}$. In between, the effect is linear; this choice ensures that the entire rate dependency is still piecewise linear, which is consistent with the choice of linear basis functions.

In order to choose the knot points, we note that historically we have observed two different regimes; the first is a high interest rate environment in which the NIBCA ratio was steadily decreasing, and the second a low interest rate environment in which the ratio has grown with a strong trend. One of the challenges we face here is that rates historically dropped from 5% to 0.5% during the financial crisis, and we have actually never observed any rates between 50bps and 200bps. There exist then several combinations of knots in this range that give rise to models with the same in- and out-sample fit qualities, but with slightly different dependencies, thus leading to slightly different predictions. We are therefore in a situation where all of these models with different combinations of knots are plausible candidates.

Given this situation, we perform an averaging over these different candidate models.

The final model is then an ensemble average of the models that have the same structure as that presented above, for which we consider all combinations of knots $(\xi_{1},\xi_{2})$, $\xi_{1}<\xi_{2}$, and $(\zeta_{1},\zeta_{2})$, $\zeta_{1}<\zeta_{2}$, selected from the set

 $E=\{50,75,100,125,150,175,200,250\}\text{bps}.$

Denote the set of all pairs of combinations of the set $E$ by $P_{2}(E)$. We then select among these the 25 best models based on the root-mean-squared relative error calculated via five-fold cross validation. The final model is then obtained by averaging the predictions of these 25 models. We did not find that the inclusion of three or more knots led to any improvements in the modeling accuracy.

For the final model we rewrite the mean-reversion term (5.5) in an equivalent form to allow us to fit both the speed and the level using OLS:

 $\displaystyle\Delta\lambda_{t}^{\mathrm{nibca}}$ $\displaystyle=\frac{1}{|\mathcal{A}|}\sum_{(\xi^{\alpha}_{1},\xi^{\alpha}_{2})% \in P_{2}}\sum_{(\zeta^{\alpha}_{1},\zeta^{\alpha}_{2})\in P_{2}}\bm{1}(\alpha% \in\mathcal{A})(\beta_{0}^{\alpha}+\beta^{\alpha}_{1}(d^{\mathrm{s}}_{t-1}-(d^% {\mathrm{s}}_{t-1}-\xi^{\alpha}_{1})_{+})$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\beta^{\alpha}_{2}(d^{% \mathrm{s}}_{t-1}-(d^{\mathrm{s}}_{t-1}-\xi^{\alpha}_{2})_{+})$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+(\gamma_{1}^{\alpha}+% \gamma_{2}^{\alpha}\lambda_{t-1}^{\mathrm{nibca}})\chi(d^{\mathrm{td}}_{t-1},% \zeta_{1}^{\alpha},\zeta_{2}^{\alpha})),$ (5.6)

where $\Delta\lambda_{t}=\lambda_{t}-\lambda_{t-1}$, $\gamma_{1}=\gamma\theta$ and $\gamma_{2}=-\gamma$, $d^{\mathrm{s}}_{t}$ is the balance-weighted average savings rate at time $t$, $\mathcal{A}$ denotes the total set of models retained, $|\mathcal{A}|$ is the size of this set (fixed to 25 in this case) and $\bm{1}(\mathcal{B})$ is the indicator function with the value 1 if its argument $\mathcal{B}$ is true and 0 otherwise. All remaining coefficients are then determined by running a constrained OLS regression for each $\alpha$ with the constraints

 $\displaystyle\beta^{\alpha}_{1}+\beta^{\alpha}_{2}$ $\displaystyle\leq 0,$ (5.7) $\displaystyle\beta^{\alpha}_{2}$ $\displaystyle\leq 0,$ (5.8) $\displaystyle\gamma_{1}^{\alpha}$ $\displaystyle\geq 0,$ (5.9) $\displaystyle\gamma_{2}^{\alpha}$ $\displaystyle\leq 0.$ (5.10)

Constraint (5.7) ensures that the difference in the fraction is a decreasing function of the deposit rate when the latter is smaller than $\xi^{\alpha}_{1}$. Constraint (5.8) similarly ensures that the difference is a decreasing function of the rate when the latter is between $\xi^{\alpha}_{1}$ and $\xi^{\alpha}_{2}$. The abovementioned basis functions ensure that there is no dependence on the rate terms when rates are above $\xi^{\alpha}_{2}$. Finally, constraints (5.9) and (5.10) ensure that the mean-reversion speed and level are positive.

Figure 5 shows the rate dependency of the NIBCA ratio model for specific values of the knot points $(\xi_{1},\xi_{2})$ and $(\zeta_{1},\zeta_{2})$, namely, $(\xi_{1},\xi_{2})=(50,150)$bps and $(\zeta_{1},\zeta_{2})=(50,175)$bps for the lower surface and $(\xi_{1},\xi_{2})=(100,175)$bps and $(\zeta_{1},\zeta_{2})=(50,175)$bps for the surface on top, using a level of 15% for the ratio.

Figure 6 shows the rate dependency of the ensemble average model. We see that changes in the ratio are not particularly sensitive to the average savings rate when the latter is below 50bps, fairly sensitive when it is between 50bps and 250bps and not sensitive above 250bps. The top surface shows the dependency when the level of the ratio is 15% whereas the surface below shows the dependency when the ratio is 30%. We can observe that the larger the current ratio is, the more it will decrease when rates are high, as there are more excess deposits held in NIBCAs that will flow into interest-bearing products.

The in-sample fit of model (5.6) is shown in Figure 7. The figure shows the fit of the average model (orange solid line) together with the fits of the two most extreme individual models in the average: the upper extreme (green dashed line) and the lower extreme (red dashed line). As can be seen, all individual models have a very similar in-sample fit. The root-mean-squared error is 0.85%, which demonstrates a very good fit quality overall.

The five-fold cross validation of the model is shown in Figure 8. As can be seen, the predictive accuracy of the model is rather high, as on average the out-sample root-mean-squared error is only 0.75%. Note that here in each respective fold we take the starting value of the level as given, and then construct the levels from the predicted changes obtained from the model trained on the remaining four folds.

### 5.2 NIBCA-plus-savings-to-M4 ratio

The historical time series for the fraction of funds deposited to sight deposits, either NIBCAs or savings, is shown in Figure 9 (orange line).

As can be seen, this fraction has kept growing since 2003, the growth being interrupted only during a brief period leading up to the financial crisis. The growth prior to the crisis is mainly due to the growth of savings whereas the postcrisis growth is driven mainly by NIBCAs.

Note that this fraction is more or less equal to 1 minus the fraction allocated to TDs. As such, the main drivers are expected to be the average TD and savings rates. The spread between these rates measures the attractiveness of TDs relative to sight deposits. If this spread is large, then customers will be more willing to lock away their cash to earn a better interest. When the spread is tightened we expect the allocation to sight deposits to increase as customers would prefer to keep their money liquid. Figure 9 also shows the evolution of the spread between the TD and savings rates (green line). Historically, from 2003 to the crisis the spread tightened, leading to savings products being relatively more attractive than TDs, thus increasing the allocation into savings. When interest rates dropped with the onset of the crisis, the fraction of sight deposits kept growing, mainly due to the growth of NIBCAs. The model uses the simple changes in the spread, with different sensitivities when the changes are positive or negative. The effects of these variables are lagged. The negative changes in the spread have an impact both at time $t$ and with a two-month lag whereas the positive changes contribute with a two-month lag. Moreover, given that the ratio of sight deposits to M4 contains the ratio of NIBCA to M4, we use the predicted changes in the latter to explain changes in the former. The logit transformation is used to ensure that the ratio remains bounded between 0 and 1. The reason we have not used the logit transformation for the NIBCA-to-M4 ratio is due to the mean-reversion term in (5.6), as we want to link the changes in the fraction to its level at the same scale. This means that the NIBCA-to-M4 ratio can eventually grow beyond unity, if rates remain low. However, we have checked that this happens only after several decades, as the growth rate is rather low.

The final model for the sight-deposit-to-M4 ratio then reads

 $\displaystyle\Delta\operatorname{logit}(\lambda_{t}^{\mathrm{sight}})$ $\displaystyle=\beta_{0}\Delta\operatorname{logit}(\lambda_{t}^{\mathrm{nibca}}% )+\beta_{1}(s_{t_{k}}-s_{t_{k-1}})^{-}$ $\displaystyle\qquad+\beta_{2}(s_{t_{k-2}}-s_{t_{k-3}})^{-}+\beta_{3}(s_{t_{k-2% }}-s_{t_{k-3}})^{+},$ (5.11)

where $s_{t}$ is the spread between the average TD rate and the savings rate at time $t$ and the logit function is defined as

 $\operatorname{logit}(x)=\log\bigg{(}\frac{x}{1-x}\bigg{)}.$ (5.12)

Figures 9 and 10 show, respectively, the in- and out-sample performance of the model. For the latter figure, we note that as we are only testing the accuracy of (5.11) the changes in the NIBCA fraction are computed using the actual values rather than predicted values.

## 6 Deposit rates

As the models for the fractions depend on the average savings and TD rates, we now turn to the modeling of these two variables. We follow here a rather standard approach using error correction models (see also Castagna and Miste 2016; Castagna and Scaravaggi 2017; Kordel 2017).

Typically, deposit rates change in response to changes in the central bank rate, in our case the BoE base rate. Business tends to speak of the “pass-through” rate, which determines the fraction of change in the base rate that is passed on to the deposit rate. Thus, in the short term it makes sense to model the changes in deposit rates in terms of the changes in the base rate. Assuming, however, that the differences of the rates are stationary means that the rates themselves would be integrated of order 1, $I(1)$. However, if rates are individually $I(1)$ without any constraining relations they can deviate from each other arbitrarily, which is not observed in reality. Thus, it is usually assumed that there exists a co-integration relation between the rates themselves, and this naturally leads to an error-correction-type model where first we fit the equilibrium relation between the deposit rate and the base rate and then use the fitted equilibrium relation to modify the model for the differences so that the two rates never drift too far away from each other.

The model for the average savings rate is then given by

 $\displaystyle\Delta d^{\mathrm{s}}_{t}$ $\displaystyle=\beta_{1}\Delta{b}_{t}+\beta_{2}\varepsilon_{t},$ (6.1) $\displaystyle\varepsilon_{t}$ $\displaystyle=d^{\mathrm{s}}_{t-1}-d^{*}_{t-1},$ (6.2) $\displaystyle d^{*}_{t}$ $\displaystyle=\beta_{3}{b}_{t},$ (6.3)

where $d^{*}_{t}$ is the equilibrium deposit rate, which is estimated using the long-run relationship against the base rate; ${b}_{t}$ is the BoE base rate; and $\varepsilon_{t}$ is the deviation between $d^{\mathrm{s}}_{t}$ and the equilibrium rate $d^{*}_{t}$.

The in-sample fit of the model is shown in Figure 11. As can be seen the model has a reasonable fit against the historical time series. There are some divergences in 2002–3 that do not revert, which are then partially compensated by the large decline in 2009, and the error correction helps bring back the predictions to the actual data over the remainder of the time series. It is common also to have different coefficients for the positive and negative changes in the market rate, respectively. This is rather easy to fit but we have not found it to significantly improve the fit quality, and for simplicity we therefore assume that changes in the deposit rate react symmetrically to changes in the market rate, even though it is natural to assume that banks adjust their pricing somewhat asymmetrically as they would be more prone to quickly pass falls in the rates to customers rather than the opposite way around.

For the average TD rate we use a very similar model. The difference is that we also introduce a dependence on the six-month London Interbank Offered Rate (Libor 6M) since TD rates would generally depend on not only the overnight rate but also the term structure of rates. The model is then given by

 $\displaystyle\Delta d^{\mathrm{td}}_{t}$ $\displaystyle=\beta_{1}\Delta l^{6\mathrm{M}}_{t}+\beta_{2}\varepsilon_{t},$ (6.4) $\displaystyle\varepsilon_{t}$ $\displaystyle=d^{\mathrm{td}}_{t-1}-r^{*}_{t-1},$ (6.5) $\displaystyle r^{*}_{t}$ $\displaystyle=\beta_{3}l^{6\mathrm{M}}_{t},$ (6.6)

where $r^{*}_{t}$ is the equilibrium time deposit rate, which is estimated using the long-run relationship against the Libor 6M; $l^{6\mathrm{M}}_{t}$ is the GBP Libor 6M; $\varepsilon_{t}$ is the deviation between the TD rate, $d^{\mathrm{td}}_{t}$, and the equilibrium rate, $r^{*}_{t}$.

The in-sample fit of the model is shown in Figure 12. Just as for the average savings rate, the model fits well against the historical data. There is a slight overshoot in the model right after the crisis dip in 2009, but this gets corrected by the mean-reversion term in subsequent periods.

## 7 Stock runoff

The framework explained so far can be used to forecast the total balances in a given NMD portfolio; for example, in the simplest case, for a given institution’s balances, $\smash{V_{t}}$, we can assume a constant market share in relation to the respective industry balance, $\smash{V_{t}^{\mathrm{industry}}}$, to predict $\smash{\hat{V}_{t}=\rho V_{t}^{\mathrm{industry}}}$, for a constant $\rho$. Obviously, each specific institution will have its own tailored assumption on the market share based on considerations such as business strategy, expected management actions, etc.

The evolution of the total balances, $V_{t}$, is driven by both the evolution of existing stock and the origination (“churn”) of new accounts. To understand the bank’s liquidity profile and to support processes such as funds transfer pricing (FTP) it is important to understand how existing stock runs off in time, ie, it is important to understand the balance evolution of the present portfolio without the inclusion of new business. To describe the stock profile we take the snapshot of the portfolio at a given time, $t$, and we then consider the balance evolution only of those accounts that were present at time $t$.

The stock profile, however, cannot be deduced purely from the industry-level data, which is aggregated and includes contributions from both existing stock and new origination. Therefore, internal account-level data must also be used. Here, we illustrate how this can be achieved.

Let $C(t,t_{\mathrm{o}})$ denote the time-$t$ aggregate balance of all those accounts that originated in period $t_{\mathrm{o}}$. We call $C$ a cohort. We let $S(t,t^{\prime})$ denote the time-$t$ aggregate balance of all those accounts that originated prior to or at time $t^{\prime}$. We then have

 $\displaystyle S(t,t^{\prime})$ $\displaystyle=\sum_{a=t-t^{\prime}}^{\infty}C(t,t-a)$ $\displaystyle=\sum_{a=t-t^{\prime}}^{\infty}\lambda(t,t-a)V_{t}$ $\displaystyle=\varLambda(t,t^{\prime})V_{t},$ (7.1)

where $\lambda(t,t-a)$ is the time-$t$ fraction of total balance held by the cohort that originated at time $t-a$ and $\varLambda(t,t^{\prime})$ is the time-$t$ fraction of the total balance held by the stock defined as of $t^{\prime}$, ie, the stock that consists of all cohorts that originated on or before $t^{\prime}$. We note that $\varLambda(t,t)=1$ since $S(t,t)=V_{t}$ by definition.

We now make the assumption that the fraction $\lambda(t,t-a)$ depends only on the “time-on-book”, $a$, and not on $t$ itself. This means that the total fraction $\varLambda(t,t^{\prime})$ is a function of $t-t^{\prime}$ only. Thus, we have

 $S(t,t^{\prime})=\varLambda(t-t^{\prime})V_{t}.$ (7.2)

The fraction of balance held by the stock $S(t,t^{\prime})$ therefore only depends on the time elapsed since $t^{\prime}$, which we might refer to as the “age” of the stock. While the validity of this assumption must be tested on an individual bank’s proprietary account data, we have found this simple assumption to work rather well in our personal experience.

To fit the cohort fractions $\lambda$, and hence the total fraction $\varLambda$, we first calculate for each period the proportion of total portfolio balance held by each cohort present in the portfolio. For each cohort $C$, we then get a fraction profile $\lambda_{C}(a)$ as a function of time-on-book, $a=t-t_{C}$, where $t_{C}$ is the unique origination period of the cohort. We then fit an average profile, $f(a)$, by penalized nonparametric regression, minimizing

 $\sum_{C}\sum_{a=0}^{T-t_{C}}(\lambda_{C}(a)-f(a))^{2}+\eta\int\mathrm{d}s\,(f^% {\prime\prime}(s))^{2},$ (7.3)

where $T$ is the last observed period for the historical portfolio and $\eta$ is a smoothing parameter that penalizes curvature of the fitted function. For each fixed $\eta$ the solution is given by a natural cubic spline with knots at the unique values of the exogenous variable (Green and Silverman 1994) – in this case with knots at every observed historical time-on-book, $a$. Finally, the optimal value for $\eta$ can be chosen by standard $K$-fold cross validation.

Given the fitted function, $\hat{f}(a)$, we are then able to predict the stock runoff for a given portfolio as

 $\hat{S}(t,t^{\prime})=\bigg{(}\sum_{a=t-t^{\prime}}^{\infty}\hat{f}(a)\bigg{)}% \hat{V}_{t}.$ (7.4)

This completes the construction of the stock profile. Note that the resulting profile is both time and “age” dependent, with these dependencies factorized. The age dependence sits in the first factor in (7.4), which is fitted using the internal account-level data, while the time dependence sits in the second factor in (7.4), which is the predicted total portfolio balance that in turn was deduced from the industry-level model.

Given the predicted stock profile, we can calculate the weighted average life (WAL) of the portfolio, which is a measure of the average time it takes for the present portfolio balances to run off:

 $\displaystyle\mathrm{WAL}_{t}$ $\displaystyle=\sum_{s=1}^{\infty}s\frac{\hat{S}(s+t-1,t)-\hat{S}(s+t,t)}{\hat{% S}(t,t)}$ $\displaystyle=\sum_{s=0}^{\infty}\frac{\hat{S}(s+t,t)}{\hat{S}(t,t)}.$ (7.5)

In practice, when calculating the WAL a cut, $s_{\mathrm{c}}$, will usually be applied such that $\smash{\hat{S}(t+s,t)=0}$ for all $s\geq s_{\mathrm{c}}$, effectively reducing the computed WAL.

We note that $\smash{\hat{S}(s+t,t)}$ might not necessarily decrease with increasing $s$, at least for some periods, as it represents the expected balance evolution of a fixed set of accounts including both deposits and withdrawals. Even though the fitted fraction $\smash{\hat{\varLambda}(t-t^{\prime})}$ will tend to be monotonically decreasing, the growth of the total portfolio might compensate this decrease to give a stock profile that increases with time. This is actually exactly what has happened for NIBCAs in recent years. Note that the definition in the second line of (7.5) still makes sense in this case, even though the interpretation of the first line as analogous to the duration of a bond is problematic because of negative cashflows.

## 8 Results

Here, we present the model outputs obtained under different interest rate scenarios as well as different assumptions for the growth of M4.

### 8.1 Input scenarios

As we do not explicitly model the total M4 liabilities, we will here construct three different custom scenarios for this variable. These input scenarios are obtained by replicating three different periods of growth observed historically: precrisis from 2003 to 2007, crisis from 2007 to 2011 and postcrisis from 2011 to 2015. Specifically, we first calculate a three-month moving average of the historical M4 series, after which we compute the month-on-month percentage change. We then take these changes from the three respective time periods mentioned above to produce the scenarios for the forecast period. Figure 13 shows the three scenarios for M4 together with its history.

To generate scenarios for market interest rates we make use of the BoE yield curve data. Our baseline scenario for the base rate and Libor 6M is constructed using the overnight indexed swap and UK commercial bank liability forward curves, respectively. On top of the baseline scenario we also construct three additional rate scenarios. These are “parallel up”, “short rate up” and “steepener” as defined by Basel Committee on Banking Supervision (2016). The specified shocks are applied to the spot curves, after which the relevant implied forward rates are computed to build the corresponding scenarios for the base rate and Libor 6M, as show in Figure 14.

### 8.2 Forecast of rates

Figure 15 shows the outputs for the weighted average savings and TD rates under the aforementioned rate scenarios. When forecasting the model predictions for these rates we floor the savings rate to 0 and the TD rate to the savings rate. The predicted savings rate is therefore always nonnegative, and never above the predicted TD rate.

### 8.3 Forecast of fraction of deposits

Figure 16 shows the evolution of the ratio of NIBCA to M4 and savings to M4 under the rate scenarios. The ratio of NIBCA to M4 grows slowly under the baseline scenario, whereas it drops under scenarios with rising rates, reflecting the preference of customers to move the excess cash out of NIBCAs and into interest-paying accounts. The fraction of savings decreases in downward-rate scenarios, while it increases in upward-rate scenarios.

### 8.4 Forecast of volumes

The volumes of NIBCA and savings both depend on the rates but also on the growth of the money supply. In the following we present the results obtained for the aforementioned scenarios for M4 as well as interest rates.

#### 8.4.1 Volumes with M4 similar to precrisis environment

Figure 17 shows the forecast of deposit volumes under the precrisis environment, defined by a strong growth in money supply, which results in both NIBCA and savings balances growing noticeably under baseline and downward-rate scenarios. However, we can see that outflows from NIBCAs into savings are predicted in the rising-rate scenarios even though the money supply is growing.

#### 8.4.2 Volumes with M4 similar to crisis environment

Figure 18 shows the forecast with an M4 that behaves roughly as it did during the crisis. We can see that NIBCA volumes would drop under a high interest rate environment, and would keep growing under a low interest rate environment. Savings volumes flatten under the baseline scenario, similarly to what happened during the crisis. The growth under the upward-rate scenarios is small.

#### 8.4.3 Volumes with M4 similar to postcrisis environment

Figure 19 show the forecast when the M4 growth is similar to its postcrisis growth rate. The results are similar to the precrisis growth scenario, with somewhat less growth overall.

## 9 Conclusions

We have presented a framework for modeling NMDs that can be used by banks for interest and liquidity risk management purposes. The framework uses publicly available data from the BoE, but is not restricted to the United Kingdom, as it would also apply for any country where the central bank publishes balance and rate data on NMDs, if the three main categories of deposits – NIBCAs, interest-bearing sight deposits and TDs are separately recorded. In addition, it requires that the balance-weighted average savings and TD rates are published. Given the BoE series, we were able to construct models for the fraction of balances held in each deposit category, linking the dynamics of these fractions to the average savings and TD rates. The main equations are given by (5.6) for the NIBCA fraction and (5.11) for the NIBCA-plus-savings fraction. We have also presented out-sample backtests based on cross validation that demonstrate the predictive accuracy of these models.

Using the presented models for the industry balances, it is possible to obtain predictions for a bank’s own portfolio balances for the different products, eg, NIBCAs, given assumptions on market share. In the simplest case, for an established bank that is a major player in the market, a constant market share can be assumed. In our own experience we have found this to work rather well. On the other hand, for a bank that is actively seeking to expand its market share, a perhaps linearly increasing share can be assumed. Obviously, each bank must have its own tailored assumptions on its market share. We have also shown how a bank’s own internal data can be combined with the industry-level models to predict the runoff of existing stock. This gives a measure of the retention rate of deposits and allows banks to calculate metrics such as the WAL.

One limitation of our methodology is that we do not take into account level variables beyond the account age, and assume that the dependence on interest rates and age factorizes. As mentioned above we have found this simple assumption to work rather well in our own experience, but this may not always be the case, particularly if product changes have occurred over time that might break such a simple factorization. Moreover, we rely on predictions on the growth of the total M4 money supply, which is not modeled explicitly.

For future improvements, it would be desirable to integrate the modeling approach here with a framework for the optimization of NII. This would involve a setup where, given the model forecast balances for NIBCAs and MRDs alongside interest rate paths, hedging portfolios are constructed with the aim of optimizing the expected future value of a certain objective related to the NII. This could, for example, be the minimization of the expected downward deviations from a target margin, as in Frauendorfer and Schuerle (2012), or a similar setup.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

## Acknowledgements

We thank Anthony Owen for his support and for helpful comments on the manuscript.

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