Modelling of non-maturing deposits for interest rate risk management

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for Interest Rate Risk Management

Rany Shaheen

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: Rany Shaheen

Student nr: 11388013

Email: RanyNow@gmail.com

Date: September 22, 2017

Supervisor: Prof. Dr. Sami Umut Can Second reader: Prof. Dr. Roger Laeven

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Abstract

Non-maturing deposits are a foundational component of banking prac-tice yet are quite challenging to model with respect to their behaviour in a given time horizon. Two issues are of concern: the liquidity risk of such deposits as well as their interest rate risk. The paper tackles the later issue, despite interrelatedness with the first. Previous research establishes a useful theoretical framework that integrates three com-ponents: market rates, client (deposit) rates, and volumes of deposits. However, the empirical side required for realistic estimation seems to be lacking, particularly, there is no one unifying approach, and vari-ous methods exist which differ in their way of application and results. The paper introduces the relevant business and theoretical contexts, and provides an empirical model based on well founded statistical techniques from Time Series modelling using State Space Exponen-tial Smoothing with a comparison to a standard ARIMA model.

Keywords Banking, Risk Management, Quantitative Risk, ALM, Non-Maturing Deposits, NMDs, Interest Rate Risk, Banking Book Risk , Linear Stochastic Process Applications, Stochas-tic processes, Time Series, ARIMA, State Space Models, Exponential Smoothing, Financial Econometrics.

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Introduction

With history stretching back to thousands of years ago in ancient Mesopotamia, forms of banking activity preceded even the invention of coinage (Davies, 2002). Cuneiform tablets from the region dating to those ancient times tell us of certain ”Grandsons of Egibi” (c. 700 BC) who were perhaps the earliest private banking families to exist on a known and well documented record by an operating name. While not the oldest in this regard, The House of Egibis collected deposits and gave loans. Fast forward to the 21st century, banking have advanced in line with tremendous developments of human civilisation as the centres of economic activity continued to sprout and shift around the world. Yet; despite the remarkable evolution in the practice of banking since those ancient times, especially regarding information technology, the most unchanged function of a bank as a financial institution remains: to accept deposits and provide credit.

Banks act as financial intermediaries by channelling funds between lenders and bor-rowers (Mishkin, 2010). In particular, banks formulate various deposit products to fit client needs while using such deposits as a relatively cheap source of funding enabling banks to acquire profit generating assets such as mortgages, or business loans ( Apos-tolik et al., 2009). The deposits therefore are a standard item on the liability side of the balance sheet of any banking institution with two main types of such deposits be-ing accounted fore: savbe-ings deposits and transactional deposits. The later is commonly referred to as current accounts in the UK and checkable/checking accounts in the US. In either type, the client is free to withdraw their funds at anytime, and almost in any amount under usual working conditions. Hence, deposits have zero contractual maturity which earned them the name of non-maturing deposits (NMDs) in the sense that such deposits lack a contractual maturity. That comes in contrast to other common financial assets/liabilities which have a known and fixed maturity. Note that there is a deposit type known as ”Certificates of Deposits” (CDs) and quite similarly ”Fixed Term De-posit Accounts” where both come with a fixed maturity that is specified contractually thus acting as a bond and hence fall out of the scope of the paper.

However; despite their nature and name, NMDs are widely observed to have a posi-tive residual maturity which is the result of funds remaining undrawn for some varying lengths of time as clients choose to delay their consumption or to fit varying future

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2 Rany Shaheen — NMD Modelling for IRR Management

uidity needs, or to simply safeguard their unused funds at the bank’s vault. Moreover, note that savings accounts and transactional accounts differ in that the later accumulate no (or very low) interest as they are typically used to carry out personal or business transactions for day-to-day liquidity needs while the former reward the client a periodic interest accumulated using a variable client rate offered at the discretion of the bank. Such variable savings rate is used by the bank to attract new deposits or to incentivise clients into keeping their existing funds undrawn from the bank and thus increasing their effective maturity. Extending effective maturity of funds on the liability side is of paramount benefit to the bank due to the naturally arising maturity mismatch between assets and liabilities in the banking model. Essentially, increased effective maturity re-duces the maturity mismatch (gap) and increase the potential for profitability of the bank since most if not all assets such as mortgages, retail and wholesale loans are of medium to long maturity.

Nevertheless, such simple setup of a client deposit with the embedded optionality of withdrawals being conducted at any time and for almost any amount up to the balance of the relevant account is inherently a risky liability for the involved bank. Specifically, the bank does not know what amount shall the client withdraw and at what (future) time. Thus, future cash flows are inherently stochastic in nature. Moreover, variable client rates offered on savings accounts are usually determined through an internal bank decision process which is largely based on the prevailing market rates controlled by the central banking authority in a given monetary zone in which deposits are issued, as well as on the types of assets held by the bank as assets vary in return, maturity, and risk. Another decisive factor for savings rate is competition by other banks which could put the funds managed by the bank at risk of being transferred by clients elsewhere if such transfers earns them higher interest. Therefore, a clear problem arises for the bank: the future cash flows of NMDs are unknown and future conditions affecting savings rates are unknown. The paper shall consider the management of interest rate risk (IRR) associated with NMDs.

So how does IRR of NMDs arise? And how could a bank manage it? As the preced-ing paragraph implies, IRR of NMDs arises primarily from two sources: 1.) Yield Curve Risk: temporal variations in interest rates through parallel and nonparallel shifts of the yield curve, 2.) Embedded Optionality of NMDs: whereby clients are free to withdraw or deposit at will since contractual maturity is zero (Balthazar, 2006). The first source relates to determination of the (discounted) present value of future deposit balances and the present value of their cash flows, as well as the determination of rates offered by banks to clients on their savings and hence the value of interest payments as applicable to client accounts currently and in the future. The second source relates to both the timing and the amounts of cash flows as deposit accounts continue to evolve in fluctua-tions stemming from subsequent inflows and outflows in uncertain ways which may put pressure on bank’s assets. Therefore, IRR directly affects the earnings a bank may ex-perience in relation to NMDs. Furthermore, increasing measures of banking regulation

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specifically name interest rate risk on NMDs as a risk category to be properly treated and to take adequate measures to control such risk (BIS, 2016). Therefore, the paper shall tackle the issue of modelling NMDs in terms of projecting future cash flows as well as measuring their interest rate risk exposure using the existing common measure of duration.

The paper shall be organised as follows: First, a brief review is given of relevant literature on the topic as well as a brief summary of the relevant regulatory requirements pertaining to the problem. Second, a theoretical framework is proposed to model NMDs and capture their link to interest rate. Third, an empirical model is constructed for which a data set will be used to apply and test the model for which the results shall be discussed in the following section. Finally, the paper terminates with a conclusion followed by a section on remaining issues and challenges which merit further research.

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Chapter 2

Literature Review

The topic of the paper is a standard banking problem, ubiquitously discussed in busi-ness publications and many textbooks on banking and banking risk management but to a lesser extent in academic journals. A thorough background on banking risks is given by for example van Greuning and Bratanovic (2009) from the angle of corporate gov-ernance of banks and their risk management, and by Apostolik et al. (2009) for more practical risk management methods and techniques. A more insightful source on the Basel regulation of banks and the taxonomy of bank risks and modelling is given by

Balthazar (2006). More quantitative texts also exist, but they tend to be of more gen-eral nature and lack the specificity of interest rate risk in the banking book, or do not treat the specific nature of non-maturing deposits in such context. Nonetheless, an out-standing coverage of non-maturing accounts is given byStraßer (2014) with a its main focus on dynamic replication methodology. Dynamic replication although well founded in the modern financial theory, remains a computationally ad-hoc procedure for finding market instruments with a known price that memic or replicate the cash flows of the asset or liability that one wishes to price, in this case it is the non-maturing deposit portfolio.

Several academic articles exist which treat the problem but a noteworthy theoret-ical framework is given by de Jong and Wielhouwer (2003) despite a lack of empirical completeness where the calibrated parameters according to the authors may not be suitable for meaningful real world inference. Indeed, the theoretical framework offered is not complex as shall be seen in this paper, but issues arise in empirical models for estimation which we shall attempt in the paper. An article by Baˇsiˇc (2015) focuses on modelling deposit volumes under observed high outflow rates and using multi-factor analysis as well as Time Series techniques while Strnad (2009) treat non-maturing de-posits from the aspect of fair price valuation using replicating portfolio technique and not specific theoretical or empirical models are provided. A multi-factor flexible affine model is proposed in Dewachter et al. (2006), a working paper published by National Bank of Belgium, part of the Euro System. The theoretical framework is somewhat complex but they use it to study aggregate data of Belgian banks which may not be suitable for an individual bank, plus the factors used may or may not be relevant in

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other countries.

An interesting article by Nystr¨om (2008) covers the issue of the paper in a com-prehensive theoretical framework, tackling the three components of market rates, client rates, and deposit volumes. However, the empirical estimates focus on deposit migra-tion under narrowly defined scenarios which reveal a linear decay of deposits under one-sided scenarios of up or down market rates which might not be realistic. Nonethe-less, the theoretical framework is impressive and uses stochastic discount factor pricing model under risk-neutral valuation techniques. Another interesting model using multi-stage stochastic programming techniques from the field of Operations Research is given byFrauendorfer and Sch¨urle(2003) which focuses on management of NMDs within the liquidity risk context and not interest rate risk. Kalkbrener and Willing (2004) study both contexts of liquidity risk and interest rate risk of NMDs after developing an ar-bitrage free theoretical pricing model which was based on an earlier work by Jarrow and van Deventer(1998) but when estimating the NMDs and the interest rate risk they do rely on replicating portfolio rather than constructing an empirical model which is different from what we do in this paper. Finally, the only relevant paper we found to apply State Space Exponential Smoothing techniques was by Marˇcek (2003) although the aim of their article was not within the context of interest rate risk management.

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Chapter 3

Theoretical Framework

3.1 Market Value of NMDs as a Liability

As a starting point to modelling the problem, consider the liability of the bank associ-ated with NMDs. The bank is liable for the accumulassoci-ated balance of the (yet undrawn) deposits of clients as well as the interest amounts (if any) to be earned on the balance under the bank’s offered savings rate henceforth called the client rate. In the following, an analytic expression shall be derived along similar lines to de Jong and Wielhouwer

(2003). Let t denote time and i(t) denote the client rate offered by the bank. Further-more, let B(t) be the balance in nominal value accumulated until time t, and the bank’s liability as a function of time to be L(t). Therefore, if we assume the client rate i(t) to be compounded per unit of time period then L(t) in nominal value at time t is simply given by:

L(t) = B(t) + i(t) · B(t) = B(t)(1 + i(t)) . (3.1)

Thus, L(t) would be the zero-maturity value that the bank must pay to the client shall they demand their entire account deposits at time t. However, as introduced earlier, this usually does not happen since clients may withdraw part or none of their deposits for sometime which extends the effective maturity beyond the contractual zero. Therefore, the NMDs will have a residual maturity and an economic value that may very well differ from the nominal value of NMDs. Such economic value is also known as ”Market Value” (MV) and it is stemming from the time value of money implicit in the banking model whereby the funds on NMDs are invested to generate bank’s income of which a part is used to compensate clients through interest payments for entrusting their deposits with the bank. Thus, the bank is interested in the market value of liabilities which follows by properly discounting the future cash flows of NMDs using the prevailing market rates.

Suppose we wish to compute the market value of a bank’s NMD liability L(0,T )

for the period running from now, that is t = 0, till time horizon t = T while assuming continuous time framework. To that end, let r(t) denote the market short rate applicable at (continuous) time t hence the corresponding discount factor for the time value of money in continuous time is given by e−t·r(t). Moreover, note that the client may conduct

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withdrawals of their balance during the time period [0, T ] and such withdrawn amounts from the balance do not earn interest payments and once withdrawn are no longer a liability to the bank which will be liable only for the remaining part of balance as well as for interest payments on that remaining part. Furthermore, and without loss of generality, let those withdrawals represent the change in balance level with resect to time hence the derivative dB(t)_dt . This is not restrictive since if such derivative at time t were negative it would mean the client added a fresh deposit to their balance rather than withdrawing from it, and the sum of all such infinitesimal changes over time would constitute the accumulated deposits. Hence, the market value of bank liability regarding NMDs in such setup is given by:

M V [L_{(0,T )}] = T Z 0 e−t·r(t) B(t) · i(t) − dB(t) dt dt . (3.2)

That is the market value of bank liability with respect to NMDs is the accumulation of discounted values of interest earning deposits at every time point less the change in deposit levels. Nevertheless, given the stochastic nature of market rates, client rates, and balances of NMDs; the expectation at time zero must be taken for expression (3.2) and hence: EM V [L_{(0,T )}] = E   T Z 0 e−t·r(t) B(t) · i(t) − dB(t) dt dt   . (3.3)

Note that the expression in (3.3) is the same as the expression given by de Jong and Wielhouwer (2003) with minor notational differences. In addition, the integral may be computed further using the fundamental theorem of calculus and integration by parts as follows: EM V [L(0,T )] = E   T Z 0 e−t·r(t) B(t) · i(t) − dB(t) dt dt   = E   T Z 0 e−t·r(t)· B(t) · i(t) dt − T Z 0 e−t·r(t) dB(t) dt dt   = E   T Z 0 e−t·r(t)B(t) · i(t) dt − h e−t·r(t)B(t) iT 0 − T Z 0 r(t) · e−t·r(t)· B(t)dt   = E   T Z 0 e−t·r(t)· B(t) · [ i(t) − r(t) ] dt −he−t·r(t)B(t)iT 0   = E   T Z 0 e−t·r(t)· B(t) · [i(t) − r(t)] dt  − E h e−T ·r(T )· B(T )i+ B(0) ,

where B(0) being the initial balance is known at time zero and hence comes out of the expectation. Furthermore, we may assume that B(0) is exactly the amount invested in

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the asset portfolio as funded by such NMDs. Therefore, the expected net present value (NPV), call it E[V ], of such position for the bank would be:

E[V ] = B(0) − EM V [L(0,T )] = E h e−T ·r(T )· B(T )i+ E   T Z 0 e−t·r(t)· B(t) · [r(t) − i(t)] dt   . (3.4)

Expression (3.4) can be intuitively interpreted for NPV from the perspective of the bank for its position in holding NMDs form now until horizon T whereby such NPV is nothing but the expected discounted value of the level of NMDs at horizon T plus the accumulation of discounted bank income stemming from spread of market rates above the client rates at every point in time within the horizon. Moreover, note that if we let T in (3.4) get sufficiently large, i.e. T → ∞, then the (continuous) discount factor e−T ·r(T ) will tend to zero. Moreover, it is reasonable to assume that the balance B(T ) would itself get very small or even zero as clients would naturally consume their deposits within an extended time horizon. Therefore for sufficiently long horizon the first term goes to zero and the NPV of NMDs for the bank is given by:

E[V ] = E   ∞ Z 0 e−t·r(t)· B(t) · [r(t) − i(t)] dt   (3.5)

The resulting analytic expression in (3.5) has a simple intuitive interpretation: the net present value of the bank’s position with respect to NMDs being invested in some asset portfolio is the expectation of the discounted value of all future accumulated spread income resulting from investing the available balance at market rate r(t) while paying client rate i(t) which is in practice almost always below market rates. Obviously, for simplicity of exposition attention is restricted here to bank’s profit from the liability side with regards to a benchmark of market rates. In practice, the funds of NMDs are invested in some asset portfolio with the intention to generate higher return than the market rate, which is indeed typically higher but if losses occur on assets then returns can well be lower. From (3.5), it immediately follows that if the bank offers a client rate equaling the market rate then its net profit on liability side is zero, and negative (that is a loss) if the client rate is higher than the market rate.

3.2 Component View of Market Value of NMD Liability

For (3.4) and (3.5) to be useable, three components must be determined analytically with their probability distributions: the market short rate r(t), the client rate offered by the bank i(t), and the balance level of accumulated deposits B(t). If all such components were known to some reasonable extent, it is then straightforward not only to compute the market value of NMDs but also to obtain risk factor analysis measures as is done in quantitative risk practice see for example McNeil et al. (2015) for extensive coverage. In particular; and for the purpose of the paper, the determination of interest rate risk

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(IRR) of NMDs is a simple exercise in taking the derivative of the expressions in (3.4) and (3.5) with respect to market rates which gives the sensitivity of market value of NMDs with respect to a change in market rates. Furthermore, viewing each component as a risk factor, one can also determine the sensitivity of the bank’s market value to the change in balance level, as well as the change in client rates by taking derivatives as well. Higher order measures such as convexity become also possible, and such measures can greatly aid in optimising the profitability of the bank associated with NMDs. Therefore, for net market value we have the following interesting measures:

Sensitivity to Market Rate := ∂E[V ]

∂r(t) , (3.6)

Sensitivity to Client Rate := ∂E[V ]

∂i(t) , (3.7)

Sensitivity to Balance Level := ∂E[V ]

∂B(t) , (3.8)

However, the issue remains on how to determine each of those factor components as well as obtaining an empirically viable form for estimation purposes. In the following subsections, each component shall be briefly discussed while as stated in the introduc-tion, the paper shall only focus on the determination of the development of the stochastic balance component.

3.2.1 Market Rate Component

Market rates may be modelled using one of several well documented approaches. Lit-erature abounds on the issue of interest rate and term structure modelling, for an up to date overview of such models see for examplePaseka et al. (2012) and the extensive coverage or theory and practice in the excellent textbook byBrigo and Mercurio(2007). For brevity and relevance, two common models which are straightforward to implement empirically will be mentioned in this section. The classic Vasicek model (Vasicek,1977) is a one factor short rate model and is based on assuming a single source of market risk modelled by a Wiener Process Wt. Vasicek model thus has the following form of

stochastic differential equation:

drt= (θ − α rt) dt + σ dWt ,

Alternatively, the Hull-White model (Hull and White,1990) might be used which is an extension of Vasicek’s model by allowing one or more of the parameters to vary with time not just the short rate itself, hence a general form is given by:

drt= (θt− αtrt) dt + σt dWt ,

However, for the purpose of the paper, market rates shall be treated as exogenously given rather than being empirically estimated and in the sense that a yield curve is known at the present time up to at least the last liquid point (LLP) at which long term securities are traded which will be taken as the horizon T , or up to some pre-determined

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time horizon cap in line with the bank’s asset portfolio for which benchmark market returns are known to a reasonable extent. For the purpose of the paper, standard shocks to market rates shall be applied and considered as a gauge for the IRR exposure as is done in practice by banks and as advised in the Basel II and III regulatory framework as mentioned earlier (BIS, 2016). More on such application shall be presented in the empirical chapter of the paper but here it suffices to say that symmetric up and down shocks of multiples of one basis point (bps) shall be applied to the currently available yield curve. Therefore, the resultant change in the market value will be observed and analysed. Furthermore, various forms of these shocks can also be applied in asymmetric ways, or to different horizons on the yield curve. In fact, Basel regulation specifies six standard interest rate scenarios for use by banks. Such methodology relies on the concept of Duration or one of its variants which shall be explained next.

Duration Measures

In order to circumvent lack of an analytic form linking the price of an asset to the market interest rate, an empirical measure of interest rate sensitivity is necessary. Furthermore, an empirical measure of the effective maturity of an NMD account balance is necessary as well to estimate a realistic maturity of NMD funds as they remain undrawn beyond their zero contractual maturity. Interestingly, the two concepts are quite related. Although Duration was first introduced by Frederick Macaulay (Macaulay, 1938;Hull, 2012) in what later became known as the Macaulay Duration as a measure of the average time until maturity of some asset, a bond typically, the concept was later extended to what is known as Modified Duration measuring the effect of a change in (applicable) interest rates on the value of an asset. In continuous time, the two concepts are equal numerically, but differ slightly in discrete time while remaining two different concepts measuring two different things. To make things concrete, definitions of the various Duration concepts relevant to the paper shall be defined below. Let P be the price of some asset and r be the applicable periodic interest rate (not necessarily continuous time rate), then we have the following concepts (Gajek et al.,2005):

• Duration: relative rate of change in price with respect to market rate, assuming P to have an analytic form as a function of r,

D(P ) := −d ln P dr = − 1 P dP dr (3.9)

• Monetary Duration: rate of change of price with respect to rate r; similar to (3.6),

DMonetary(P ) := −

dP

dr (3.10)

• Macaulay Duration: original duration concept by Macaulay (1938) which he de-fined as a weighted average time to maturity for a stream of future cashflows Ct,

∀t ∈ T := set of (future) time points, with Ct independent of r,

DMac(P ) := X t∈T t · Ct (1 + r)t · 1 P , (3.11)

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where time is weighted by the present value of cashflow Ct as a proportion of the

total value of the asset P as evident from the expression.

• Modified Duration: same as (3.9) while assuming a known stream of future cash-flows Ct , ∀t ∈ T := set of (future) time points, with Ct independent of r, thus

price of asset is given by:

P :=X

t∈T

Ct

(1 + r)t

Hence the Modified Duration for such asset is given by:

D(P ) = −1 P dP dr = 1 P X t∈T t · Ct (1 + r)t+1 = 1 (1 + r) X t∈T t · Ct (1 + r)t · 1 P (3.12)

Thus, Modified Duration is indeed a modified version of the Macaulay Duration in expression (3.11) by a factor of 1/(1 + r) :

D(P ) = 1

(1 + r)· DMac(P ) (3.13)

The above definitions assume knowledge of a deterministic interest rate, as well as exact values for future cash flows which is sufficient for applications like coupon bond pricing. However, for many applications in reality either rates are unknown or stochastic in nature and cashflows can also be stochastic not deterministic or at least not known in any analytic form. This is precisely the case of NMD balance cashflows: market rates as well as client rates are stochastic at best, and the exact cashflows are stochastic depending on client behaviour in withdrawals or deposits. Therefore, to use any of the measures above first an estimation of market rates as well as client rates is necessary, as well as estimation of the projected cashflows of NMD balances. Due to greater estimation uncertainty arising from both sides, namely that of rates as well as projected cashflows, the following definition of Effective Duration will prove very useful for the empirical part of the paper (Gajek et al.,2005):

• Effective Duration: A Taylor series based approximation of (3.9) for lack of analytic form for asset price P ,

DE(P ) ≈

P (r − ∆r) − P (r + ∆r)

2 · ∆r · P (r) , (3.14)

which is the change in price P (r) under a symmetric rate shock of ∆r down and up relative to the initial price under r and total shock size being (2 · ∆r).

3.2.2 Client Rate Component

As introduced earlier, banks offer a (variable) interest rate to clients on part of NMD accounts/products particularly for the common savings category. Transactional accounts

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typically earn no or very low interest, or even incur fees to allow clients to use the various services coming with such accounts particularly in the extremely low interest rate environment nowadays. Nonetheless, client interest rate aims to attract new deposits on existing accounts or to attract new clients as a way to raise more funding which allows the bank to acquire more profit generating assets. Although the bank decides internally on the level of client rates and their adjustment, the bank is nonetheless bound by several factors affecting such decision and client rates henceforth exhibit certain characteristics that are markedly different from market rates (Straßer, 2014; Rosen,

2007). Thus, even though the market rate remains a basis benchmark that underlies the determination of client rates of every bank, the profile and properties of the asset portfolio of the bank will be the also decisive factors particularly in terms of returns of the assets as well as their maturities which translate into bank’s income and timing of such income respectively. Therefore, for a theoretical abstraction of the client rate component, consider the following model:

i(t) = f ( rA(t, m), r(t, m), x ) , (3.15)

where the client rate i(·) at time t is related through some functional f (·) to the following:

• rA(t, m) := returns per asset category A in the asset portfolio of the bank as

determined at time t, per maturity m,

• r(t, m) := market rates corresponding to each maturity m, and

• x := any other factors specific to the bank that are thought to influence the determination of client rates.

Such general theoretical model in (3.8) is obviously hardly useable if the functional form of f (·) is not specified further. Nevertheless, it provides a reasonable starting point for an observational or data-driven empirical model. However, in the reminder of the paper, no empirical model for the client rate shall be developed as it falls outside the scope of the paper. For the interested reader, there are various theoretical and empirical approaches presented in the academic literature see for example Paraschiv (2011) and

Wal˚as (2013) as well as Straßer (2014) and Rosen (2007). Due to data availability and specificity per bank, we can expect each bank to have its own specifications for determining client rates empirically as fits the relevant factors unique to the bank. For the purpose of the empirical investigation of the paper, the bank’s data set used for modelling NMD balances shall rely on the same bank’s existing client rate model as input, which is proprietary and hence not presented herein.

3.2.3 NMD Balance Component

The balance of NMDs, the empirical focus of the paper, develops along client’s behaviour of depositing or withdrawing funds across time and to a lesser extent with the interest payments from the bank. Projection of the evolution of the balance as introduced earlier

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is a vital task for a bank’s risk management team since the funds are invested in assets with varying maturities while the NMDs have zero contractual maturity and clients may elect to withdraw them at anytime. Therefore, in order to manage the funds in terms of client demand for liquidity and more specifically to the purpose of the paper to assess exposure to IRR, a model must be devised to determine the expected cash flow pertaining to NMD accounts. For a generic theoretical model of NMD balance consider:

B(t) = g( B(t), r(t), i(t), x ) , (3.16)

where the balance level B(·) at time t is related through some function g(·) to the following:

• B(t) := the balance itself through its history and/or its dynamics: dB(t) ,

• r(t) := current market rates,

• i(t) := client rate, and

• x := other factors that may affect the client depository behaviour.

Regardless of the specific form the model may take, which can be investigated empiri-cally, the inclusion of other factors within the variable x gives greater flexibly to a risk manager to consider what factors are behind the development of NMD balances. In particular, factors affecting the evolution of NMDs may be categorised as follows:

• Systemic: relating to the wider economic system in which the bank is a market participant rather than a market maker. Examples: the market rates (already included separately), GDP/Stock Market Index, and Inflation,

• Idiosyncratic: factors relating to the bank as an institution. For example: client rate policy (already taken separately), mission/business model, reputation, current financial health,

• Client Indigenous: factors relating to the individual client and hence purely be-havioural and unexplained by the above factors: a pattern of deposit consumption, or client characteristics that influence such pattern of consumption.

Systemic factor analysis is very useful in the context of banks operating across various countries, especially when multiple currency zones are involved while the bank-specific factors termed idiosyncratic prove useful when comparing across banks within the same economic system. Finally, client-indigenous behavioural factors are useful to distinguish among clients if they are large enough for instance, or to give a client profile for a certain group of customers deemed to share common characteristics which can be quite subjective from one risk manager to another. For the sake of analysis of the paper, only the client-side behaviour in relation to NMD balances shall be treated.

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3.3 Brief Background on Time Series Analysis

A time series is a sequence of data points ordered by time of occurrence. Thus, an observed value of some variable is paired with a time point or period at which it was observed. Time series analysis techniques have grown into a standard tool in the study of stochastic dynamics arising in many fields and applications from physical sciences to economics and finance (Box et al.,2016). A particular motivation is to build models that are able to forecast some quantity of interest with reasonable statistical power (Tsay,

2010). Literature has grown tremendously on the topic as has the scope of applications widened, but Hamilton (1994) remains a classic and comprehensive reference with ex-tensive theoretical coverage, same goes for Box et al.(2016) who give a comprehensive coverage updated with state of the art academic developments into the topic, while

Enders(2015) andTsay(2010) are rather applied and somewhat less technical yet com-prehensive enough. This section shall summarise the general structure of SARIMAX and ETS models as relevant to our application for NMD balance data analysis. SARI-MAX stands for a Seasonal, Auto-Regressive, Integrated, Moving-Average process with possibly some exogenous regressors denoted as X in the acronym, while ETS stands for Error, Trend, and Seasonality and otherwise known as Exponential Smoothing tech-niques. The former is well known in Time Series Econometrics and has well established theory and applications while the later started out as a rule of thump method to mod-elling physical phenomena for example in geophysics, but later gained much academic support culminating in advanced techniques termed Innovations State Space (see Hyn-dman et al.(2008) for excellent introductory into the topic). Since so many natural and economic phenomena exhibit one or more of the components that are captured in a SARIMAX or ETS model; namely a stochastic component (called error or innovation), trend, and seasonality, such general Time Series Analysis framework has found appli-cation in virtually every field where statistical analysis across time could be applied. A time series model aims to identify and isolate the said components inherent to the nature of the quantity being studied, particularly the predictable (trend and/or season) from the stochastic (error or innovation), and the endogenous effects from the exogenous ones, so that a reasonable forecast may be obtained for the variable(s) under study.

3.3.1 SARIMAX Time Series Models

A fundamental assumption behind SARIMAX is stationarity of data being analysed. A time series must be stationary in order to have a valid SARIMA representation. If it is not stationary, some transformation procedure must be applied depending on the context and nature of the data to make it stationary. But what does it mean that a time series is stationary? Several standard theoretical definitions are given in the literature, and each component of the model can be given its own stationarity conditions, but in general stationarity of a time series means that while a time series variable being studied may evolve with time but the attributes of the said time series do not change with time

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(in the sense of its distribution, or moments). In particular, one definition of stationarity that is suitable for almost all practical purposes is known as Weak Stationarity or Covariance Stationarity. A time series {Yt} is said to be Weakly or Covariance stationary

if ∀ t, t − s, j:

• E[Yt] = µ

• Var[Yt] = σ2

• Cov[Yt, Yt−s] = Cov[Yt− j, Yt−j−s] = γs ,

where µ, σ2and γsare all finite constants which essentially says that the mean, variance,

and the per-interval covariance are all finite constants not dependant on time. Note that if one more more of those conditions were not met, a suitable transformation may be necessary to make the series stationary. Essentially, any SARIMAX model must obey stationarity to be a valid model, regardless of its particular structure. Regarding the structure, dropping the X from SARIMAX which signifies potential presence of exogenous regressors into the time series model, the general SARIMA model is typically described in terms of numeric arguments as: SARIMA(p, d, q)(P, D, Q)m where:

p := auto-regressive (AR) order , d := differencing order (I) q := moving-average (MA) order, P := seasonal AR order D := seasonal differencing order, Q := seasonal MA order m := data frequency per season.

The mentioned literature covers each and every component extensively, something which goes beyond the scope the paper. Nevertheless, a stochastic variable Yt following the

theoretical SARMAX(p, q)(P, Q)m model (notice we removed the integrated process

from the model for simplicity) can be described generically as:

Yt= α (constant term)

+

p

X

k=1

βk· Yt−k (Auto-Regressive component: AR(p))

+

q

X

l=1

θl· εt−l (Moving Average component: MA(q))

+

P

X

K=1

ψk· Yt−m−k (Seasonal AR component: SAR(P ))

+

Q

X

l=1

ωl· εt−m−l (Seasonal MA component: SMA(Q))

+

C

X

c=1

γc· Xc,t (X component: C-exogenous regressors)

+ εt . (Innovations component)

Innovations are assumed to be uncorrelated, with zero-mean and σ2 (constant) vari-ance, something known as White Noise in the econometric literature. Furthermore, the

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integrated process component has to do with the order of differencing applied to the stochastic variable under study. Continuing with Yt, it is said to be an integrated process

with order (d) which is simply denoted as I(d), if the d-th difference was applied to the variable. Thus, the model with I(1) implies that ∆Yt is the quantity being modelled.

Same goes for Seasonal Integrated process SI(D) but with the differencing applied with respect to the values at frequency m prior to Yt , thus an SI(1) would be a model for

∆mYt:= Yt− Yt−m

Obviously, SARIMAX category of models are necessarily linear and in fact they came out as an extension of classic multivariate linear regression modelling. SARIMAX models can have many parameters depending on which components exist and what order each component may have. In particular, number of parameters is equal to: p + q + P + Q + C , where C := number of regressors (or columns) of X. Moreover, SARIMAX models give equal weight to each single observation in the series, and thus are very sensitive to data quality, and particularly to length of series since the most distant observations in the series are of equal importance in estimation as the most recent ones. Estimation problems can arise especially for seasonality effects because the higher the value of m the much more observations are needed to make a reasonable and significant estimate. Long memory in SARIMAX models can be problematic for forecasting as well as the illusive nature of the distribution of innovations.

3.3.2 State Space Exponential Smoothing (ETS) Time Series Models

Exponential Smoothing is an alternative time series modelling technique to SARIMA type models and did not develop simply from classic linear regression. Exponential Smoothing encompasses an entire class of models which specifically developed with forecasting in mind and build on the exponential smoothing technique in its wide sense of giving more weight to the recent observations than the more distant past observations in a time increasing manner (Hyndman et al., 2008). Several authors came with their own models which later turned to be quite related and fall under the same class of models. Nevertheless, the estimation techniques of such models based on the State Space approach for innovations gave exponential smoothing much more power in describing and forecasting various phenomena in wide applications.

Therefore, two aspects for our brief introduction to such methodology must be men-tioned. First, Exponential Smoothing essentially aims to decompose a given time series in terms of three components: Error, Trend, and Seasonal; hence the acronym ETS which we shall use in this paper interchangeably with the term Exponential Smoothing, in a similar manner to ARIMA family of models. Thus, the basic idea in such modelling framework is that any given stochastic variable Yt is assumed to have the following

general ETS form, known as the measurement equation:

Yt= f (E, T, S) , (3.17)

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with respect to its arguments E, T , and S corresponding to the components of error, trend, and seasonality. In a similar fashion to the order numbers in ARIMA models, the functional form can be described using a three letter combination corresponding to how each component appears in the functional from. Each component therefore is assigned a particular label as follows (Hyndman and Athanasopoulos,2013):

N ⇔ None (the component is absent; does not apply to the Error) A ⇔ Additive (the component is added)

M ⇔ Multiplicative (the component is multiplied).

Additionally, the trend component has two special letter labels as follows:

Ad⇔ Trend is additive but with a dampening parameter

Md⇔ Trend is multiplicative with a dampening parameter

Such attributes of the components combine together to produce a total of 30 ETS possible models under such framework. Each component further has its own state equa-tion. To illustrate, and to shed some light on the meaning of state space, let us consider the simplest model given by ETS(A,N,N) which signifies an additive error with no trend component, nor a seasonal one. The ETS(A,N,N) for the stochastic variable Ytis given

by the following measure equation:

Yt= lt−1+ εt ;

where lt−1is taken to be the level from which stochastic variable evolves in the next time

period with the (additive) interaction of the new period’s error component (innovation). Thus, we speak of the state equations to describe the state of error feeding into the level if any, into the trend and seasonal components, and here there is only one equation namely that of the level given by:

lt= lt−1+ α εt ;

Note that the system of the two equations above is taken as to fully describe the ETS(A,N,N) model in the state space form. The state equations can be thought of as a transition equations. Such formulation is based on the original idea of a Kalman Filter (Kalman, 1960; Kalman and Bucy,1961). To put it simply, we may observe the values of Yt but are unaware of the ”actual” level lt that Yt takes, thus state of art

estimation technique followed assumes such two-layered approach applying a Kalman filter to get a better estimate for parameters and to untangle the dynamics of the time evolution of the stochastic variable Yt better than in standard techniques used for

in-stance in SARIMA models. Such ideas of Kalman filter found wide application in signal processing in engineering and physics but can also be useful in economics.

To illustrate further, suppose that Yt has an ETS(A,A,A) model, that is of an all

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mea-18 Rany Shaheen — NMD Modelling for IRR Management

surement and state equations:

Yt= lt−1+ bt−1+ st−m+ εt , measurement

lt= lt−1+ bt−1+ α εt , level state

bt= bt−1+ β εt , trend state

st= st−m+ γ εt ; seasonal state

where m is the frequency per season. Note that the trend feeds into the level, and given that the model is all additive, all three states are added to give the measurement equa-tion. In the empirical part, ETS(M,N,A) and ETS(A,N,A) shall be used and introduced respectively.

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Empirical Model for NMDs

Following the theoretic framework of the previous chapter, empirical models shall be developed in this chapter for cashflow forecast of of NMD balances based on techniques from Time Series Analysis. Furthermore, an alternative empirical model inspired by human mortality modelling from Actuarial Science literature will be given. Due to the fact that the construction of models was partly driven by observing the nature of data at hand, the first section shall briefly describe the data used throughout. Further computations of the resulting effective maturity and effective duration shall be left for the results chapter.

4.1 Data

The data used in the investigation was supplied by some European bank operating in branches across multiple countries. Due to a promise of confidentiality, the identity of the said bank as well as some other information shall remain anonymous. In particular, the data shall be in various forms of aggregation of (time) series of NMD balance amounts, deposit outflows, and client rates for one branch of the bank. The data corresponds to a span of roughly 25 years of bank activity, taken on a monthly basis. As for the current yield curve and other market rates data and the resultant discounts factors, all was also supplied thanks to the bank but originating from a professional commercial data service centre which supplies current market data.

4.2 Towards an Empirical Model

The objective of modelling NMD accounts is project their cashflows in terms their amounts and timing. While cashflows of NMD accounts can be classified in terms of direction as either inflows (new deposits) or outflows (withdrawals), from risk manage-ment perspective a bank is only interested in projecting the outflows. Why? In short, because outflows correspond to the liability of the bank, and (futrue) inflows are not a current liability risk per se. In more detail, it is helpful to view an NMD account as a coupon bond : a security which pays (periodic) coupon payments until maturity at which

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the principal amount is fully paid. For an NMD account, and in the spirit of expres-sion (3.1), the bank is liable at end of (discrete) period t for the (accumulated) balance level Bt as recorded at end of period t; assumed to be the shortest interest-awarding

period, right after the periodic interest has been credited to the account (hence Bt is

inclusive of the intra-period interest). Thus, Btconstitutes the principal of a NMD

ac-count thought of as bond. On the other hand, the interest payment by the bank on the remaining balance is equivalent to the coupon payment liability of the bank. However, given the optionality embedded into the NMD accouns, if the client chooses to with-draw any (partial) funds, such withwith-drawal will alter the principal amount! Essentially, this is where the complexity of NMD modelling stems from. Extending to the following time period (t + 1), let us for simplicity assume that the client shall not conduct any withdrawals nor any deposits. Then, from the end of period t (which in discrete time setting coincides with the beginning of period (t + 1) ) to the end of period (t + 1) the bank is liable to crediting the intra-period interest payment applicable during (t + 1) hence denoted as [it+1· Bt] where the new balance at end of period (t + 1) shall be:

Bt+1= Bt+ it+1· Bt (assuming no client activity)

∴ Bt+1= Bt(1 + it+1) . (4.1)

Obviously, if we let the client deposit new funds (inflow) during period (t + 1) it shall not affect the liability value of credited interest since banks pay it only on the starting balance if it remains till end of period. Moreover, the new deposit inflow will only enlarge the starting balance of the following period. However, if within the period there was an outflow in the form of a withdrawal larger than the intra-month new inflow, the bank is liable to pay out the amount withdrawn above the newly deposited as well as crediting the earned interest. Therefore, for the sake of examining the current liability, focusing on the net outflow within the period let us assume that there is a net positive outflow constituting the only change to the balance thus equal to −∆Bt+1−where (T + 1 − ) is

the last moment at which the client could make a transaction, and immediately before the end of the period at (t + 1) whereby the exact end of the period marks the point of crediting earned interest on the remaining balance of the client from last period. Then, from end of period t to the end of period (t + 1) we have:

∆Bt+1−= Bt+1−− Bt ⇒ Bt+1−= Bt+ ∆Bt+1− ;

assuming a positive net withdrawal implies that: Bt+1−< Bt, and thus: ∆Bt+1−< 0

, thus the balance at the proper end of period shall be Bt+1− plus earned interest:

∴ Bt+1= (Bt+ ∆Bt+1−) + it+1· (Bt+ ∆Bt+1−)

Bt+1= (Bt+ ∆Bt+1−) + it+1· (Bt+ ∆Bt+1−)

∴ Bt+1= Bt (1 + it+1) + ∆Bt+1− (1 + it+1) . (4.2)

Thus, by comparison of (4.2) to (4.1) we see that in the case of a positive net withdrawal (that is ∆Bt+1 is negative) the balance and hence bank liability at end of period is

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reduced by an amount equal to that positive withdrawal and the would be interest payment on that withdrawn amount. Nevertheless, the withdrawal is an amount paid from the bank to the client, while the would-be-earned interest payment is withheld by the bank since the account does not earn interest on what is withdrawn. Therefore, in modelling NMD balance evolution we disregard the incoming new inflows of deposits and we take into account the outflows only. Essentially, from the perspective of the current period, the assumption is that the current balance level shall be fixed up to any future withdrawals and thus shall be the only source of funding for all future cash outflows and hence a run-off scenario of NMD accounts is generated. To model the positive outflow only, which perhaps can be done in many ways, several approaches are possible, such as:

• Model ∆B_t on the positive side only (asymmetric model) (introduces non linear-ity),

• Model ∆Bt taken as is: positive and negative sides, then disregard the negative

side when performing liability calculations (the approach taken by the paper).

Once a projection of cashflows is made, various interesting quantities shall be computed and analysed, notably the market value of NMDs as assets M V [B], Macaulay Duration for approximation of residual maturity of NMDs using expression (3.11), and Effective Duration for an approximation of market rate sensitivity using expression (3.14). Nev-ertheless, the next section shall consider another practical issue which arises regarding the suitability of ∆Bt as the quantity to be modelled and more specifically what form

of aggregation of NMD account data (or lack of thereof) must such quantity take, as well as categorisation issues.

4.3 Categorisation and Aggregation of NMD Data for the

Empirical Model

As outlined in the introduction, there are two main types of NMD accounts: trans-actional and savings. For practical purposes, there could arise other subtypes of such accounts depending on the nature of such accounts in terms of its offered products and services. For the bank under our case study two types of NMD accounts exist: sav-ings accounts and current accounts which are less ambiguously termed transactional. Each of the two categories is further divided into private (non-business) vs. business (non-personal). Furthermore, the bank operates in multiple countries, hence each coun-try is taken separately for obvious reasons of differing economic environments which agrees with the theoretic framework in subsection (3.2.3). The data used for the paper corresponds to only one country and for the private savings NMD accounts only. In fol-lowing subsections, two different approaches of looking at NMD balance consumption are considered but to get a first impression of the data, consider Figure 4.1 showing the evolution of aggregate private savings portfolio for country A in terms of the balance

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levels at each end of month shown in blue corresponding to the left axis, while the in-dividual (positive) outflows per account were aggregated and superimposed in orange and corresponding to the right axis.

Figure 4.1: Private Savings NMD Portfolio Across Time

4.3.1 Outflow Rate Approach to Modelling NMD Balance Run-off

As evident from Figure 4.1, we see that the plotted portfolio has grown exponentially in the time span shown with some off-trend fluctuations. The aggregate of individual account outflows seems to have more or less grown around a similar exponential trend with much wilder fluctuations. Due to the similar growth in the aggregated outflows and balance levels, it is quite suggestive that the outflow follows the level of balance thus the possibility to model NMDs in terms of an outflow rate. More concretely, let:

B_tk:= balance of individual account k at end of period t,

Bt:= aggregate of all individual account balances at end of period t,

∴ Bt=

X

∀k

B_tk

Next, define the outflow level per period as the amount withdrawn which would reduce the balance level from period to period. Note that a client may deposit and withdraw within the same period, thus we take the net effect by only observing the end of period balance level compared to the balance level at end of previous period. Therefore:

W_t+1k := cash outflow of individual account k within time period (t + 1) ;

∴ Wt+1k =    0 if B_t+1k ≥ Bk t ; B_tk− Bk t+1 if Bt+1k < Btk ; = max{0, Bk_t − B_t+1k } = − min{0, B_t+1k − B_tk} ∴ Wt+1k = − min{0, ∆Bt+1k } ; (4.3)

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thus, a cash outflow in a period is expressed in terms of the change in balance ∆B_tkthat is funded entirely by the balance of the immediately preceding period. Furthermore, note that if the change in an account’s balance is positive it indicates a cash inflow which is disregarded as not being sourced at the bank, but at the client side, while the negative sign in front of the min function ensures positivity of the outflow for ease of interoperation. Hence, the aggregate outflow expression is given as sum of all such individual outflows:

Wt+1:= aggregate of all individual cash outflows at end of period (t + 1) ,

Wt+1:=

X

∀k

W_t+1k . (4.4)

Finally, the aggregate outflow rate is defined as the aggregate outflow in a period relative to the aggregate balance of the preceding period:

Qt+1:=

Wt+1

Bt

. (4.5)

Modelling of Qt therefore shall yield a forecast of the proportion of balance that the

bank can expect will be demanded by clients as an outflow in a future period, and the remaining portion of balance amount can therefore be directly deduced as: (1 − Qt+1).

Moreover, to gauge the effect of the monthly market rate (taken to be the one month EURIBOR) and the effect of the client rate, define the incentive rate as the difference between the two said rates in such away that the spread of the market rate above the client rate is thought to be the driving factor behind increased cash outflows of the next month. Hence, the following model is proposed:

Qt= α + p X k=1 βk· Qt−k+ γ · (rt−1− it−1) + q X l=1 θl· εt−l+ εt (4.6)

Figure 4.2: Private Savings Outflow Rate Across Time

Thus, QT is assumed to follow a constant, an ARIMA(p,q) process, and an effect

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Figure 4.3: ACF and PACF of NMD Private Savings Portfolio

established for Qt shown in Figure 4.2. Although the outflow rate seems fairly stable

in mean, variance seems to to clustering which suggesting a perhaps non-stationary behaviour. Data is obviously not stationary, evident from the seemingly exponential growth in aggregate balance levels.

However, the Auto-Correlation Function (ACF) in Figure 4.3 as well as Partial ACF in Figure 4.4 reveal the insignificance of the presence of any AR or MA process at the 90% level. Thus, despite some time irregularity in the variance of the data series, the mean seems to be highly stable and indeed the best model of such series based on statistical information criteria of AIC and BIC. Thus, the following equation describes the form of the resulting model for Qt:

Qt= 3.63% + 21.86%(rt−1− it−1) (4.7)

Unfortunately, the residuals of the model as shown in Figure 4.4 do not resemble Gaus-sian Distribution. More formally, using the Shapiro test for a null of GausGaus-sian distribu-tion rejects the null with p-value of less than 1% that the residuals come from Gaussian distribution. The outliers could be one reason there is skewness in the histogram of residuals shown in Figure 4.5 , and consultation with the bank regarding those outliers reveal an internal fund transfer decision by the bank was behind all those outliers, thus effectively those outliers in outflow did not originate with client behaviour. Furthermore, although the effect of the spread of client rate below that of the market rate seems to be significant statistically, it is however not so significant economically speaking. In particular, the mean value of such spread is about 0.27% that is less than 30bps, which

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if multiplied by the γ parameter of spread in the model would imply a mean value of 0.00064% of the effect of the spread on the outflow rate. To put things in persoective, such value would make a difference of only e635.64 per eMillion, or e635, 636.60 per eBillion.

Figure 4.4: Residuals of Model in Expression (4.7) and Their Histogram

Exponential Smoothing Model of Outflow Rate

Using an Exponential Smoothing Time Series model (ETS), observe that Qt seems to

have no trend but an error structure and some seasonality which the ARIMA model did not capture as significant. Nevertheless, intuition into the model structure agrees with using statistical information criterion to pick the ”best” model within the ETS framework yields an ETS(M,N,A) model, that is a model with multiplicative error, no trend, and an additive seasonality component:

Qt= (lt−1+ st−m) (1 + εt) , (4.8)

where its estimated level and seasonality components are as follows:

lt= lt−1+ 0.0668 (lt−1+ st−12) εt, (4.9)

st= st−12+ 0.0033 (lt−1+ st−12) εt, (4.10)

And initial states were estimated as:

l0 = 0.0397

s(0, ... 11)= (−0.0023, 0.0026, −0.0071, −0.0076, 0.0244, −0.0065,

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Figure 4.5: ETS Model Forecast for Qt

As the case with ETS estimation in general, the estimates for the initial states is highly dependent on the data sample chosen, so it is important to note that the listed values above correspond to using the entire data sample shown in Figure 4.2 . The obtained model yields a forecast which is shown in Figure 4.8 together with the 95% interval lower and upper estimates.

4.3.2 SARIMAX Model Based on Aggregate Balance Levels

To empirically model the balance levels, the balance data as given has an obvious exponential trend, as in Figure 4.1 reproduced in figure 4.6(a.), and therefore must be transformed properly. Thus, taking the natural log of balance, ln(Bt), transforms the

data into a linear form with a trend as evident by Figure 4.6(b.). Subsequently, a time series model will be based on the change in logarithm of balance (hence an I(1) process is assumed) given by ∆ ln Btand take reduction in balance (that is the negative values of

∆Bt) as a run-off criterion. Thus, proceeding with the following empirical model which

includes the parameter η to capture the effect of spread of client rate below the market rate (alternatively the incentive for the client to withdraw based on such spread):

∆ ln Bt= a + p X k=1 bk· ∆ ln Bt−k+ η · (rt−1− it−1) + q X l=1 cl· εt−l+ εt (4.11)

Using information criteria (AIC and BIC) as well as parameter significance tests to pick the most significant model, we get:

∆ ln Bt= 0.0147 + 0.2867 · ∆ ln Bt−1 . (4.12)

Thus, the change in logarithm of balance seems to follow a random walk with a drift (AR(1) with intercept ), and the logarithm of original balance is integrated with order 1 since we took one difference, thus the logarithm of the balance follows according to this model an ARIMA(1,1,0) with no seasonality. Furthermore, η seems to be zero or not

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Figure 4.6: Monthly Data series of (a.) NMD Aggregate Balance Level, (b.) Logarithm Transform of NMD Balance, and (c.) Change in logarithm of NMD Balance

significantly different from zero which is somewhat surprising since it means that there is no statistical evidence that outflow behaviour is significantly influenced by the spread of the client rate below the market rates. Important to note that the residuals of such model did not quite follow the normal distribution, and a quasi-maximum likelihood therefore was assumed for the estimate of the model. Furthermore, the series of residuals shows some long memery structure which could not be captured in any ARIMA-sense model. Nevertheless, the resulting model of the aggregate balance taken as the quantity to be modelled does not lead to a useful result in projecting the run-off scenario. More specifically, and due to the evident growth in the vast majority of periods in the plot 4.6(a.), the model (perhaps correctly) predicts an increasing growth trajectory which is not a problem if we wish to see the growth of the portfolio as a whole, but since we are interested in the risk management perspective of the run-off of funds of the existing NMD accounts, the result is not useful. The model parameters in expression 4.12 reveal why this is the case, since both parameters are positive which implies that the change in ln Btwill always be positive, unless some highly sophisticated error structure is imposed

on the unobserved innovations of the model. Figure 4.7 shows an in-sample prediction based on expression 4.17 where the 1-step forecast was made for all periods until the red dotted vertical line, followed by 10 year ahead forecast where we let the forecast be made from the previous periods’ prediction rather than use the actual observations

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for updating the prediction. As evidently clear from the plot in Figure 4.7, the 10 year forecast diverges greatly from that of the observed values. Moreover, the divergence is in the upward exponential growth trend, which if the risk manager would rely on, it would lead to incorrect (or even disastrous) risk management decisions. Therefore, we can safely conclude that aggregate balance modelling is not an adequate approach for the purpose risk management of NMDs, and shall be ignored for the reporting of final results of the paper.

4.3.3 Vintage-Based Approach to Modelling of NMD Balance Run-off

In terms of the discussion of the previous subsections, recall that a bank’s risk manager is interested in projecting the run-off scenario of existing funds. In other words, given a balance level Bt at end of period t, it shall be assumed that no new funds are added,

and that all future outflows are the run-off of Btas clients consume their existing funds

until reaching perhaps a zero balance, which we can also consider as a decay of balance. However, the growth in aggregate balance of a certain NMD portfolio, as that shown in Figure 4.1 and in Figure 4.7, hardly gives any clue to a decay behaviour we seek as the portfolio seems to be ever growing. Furthermore, the model in expression 4.12 which was developed in the discussion of the previous section reflects this reality of a portfolio that does not seem to decay.

A closer look at the nature of such growth reveals that it can be attributed to one of two sources: 1.) growth of funds on existing NMD accounts, and 2.) growth of number of accounts as new accounts are opened at each period (with their own new deposits of funds). Figure Figure 4.8 reveals very well such insight into the growth structure for the private savings portfolio, shown with a coloured-breakdown of the different segments of accounts whereby each segment is an aggregate of balances which correspond to a group of accounts that were opened in one particular year. Obviously, as older accounts may grow or decay in funds, new accounts are opened at every subsequent period and they have their own growth behaviour.

Therefore, modelling of ∆Bt may not quite represent a run off scenario given the two

dimensional growth in the portfolio, which distorts the actual decay of existing accounts, particularly older ones. Thus, consider an alternative approach whereby the individual account balances are aggregating by their period of origin. The period might be chosen arbitrarily, but suitably. In line with the presentation of Figure 4.8, taking the period to be the year in which an account is opened, we shall call it a vintage year and hence term the balance corresponding to those accounts in a given year as a vintage balance, or simply a vintage. Such idea allows for separating the two dimensions of growth and com-position of the portfolio by tracking the time development of each cohort of accounts, where the cohort itself is based on its age which gives a better basis to understanding the time-behaviour of deposits and isolating the growth factor stemming from the con-tinued increase in number of accounts from the evolution of funds on older accounts.

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Figure 4.8: NMD Balance Breakdown by Year of Origin of Underlying Accounts

Symbolically, a balance vintage shall be defined as follows:

Bk_x,t:= balance at end of period t of NMD account k which originated in year x ; Bx,t:= balance at end of period t of all client accounts originating at year x;

∴ Bx,t=

X

∀k

B_x,tk , (4.13)

with ∆Bx,tbeing the change in a vintage balance and thus the quantity to be modelled.

Further justification to using the vintage method of aggregation is based on the common economic dynamics of accounts within a vintage. Such economic dynamics are at play in determining the client behaviour in regards to the availability of funds for consumption or depositing. Clients opening an account within a given year are exposed to the same economic, particularly financial, conditions such as GDP growth, business cycle dynamics, etc, which may lead a cohort to exhibiting similar behavioural traits in deposits and withdrawals. Furthermore, a new client in a later year will exhibit a different behavioural pattern than that of a client who has been already a customer for several years. In what follows, two models shall be estimated for the NMD balance development based on vintage aggregation. Each vintage corresponding to year x will have its own parameter sets. First an ARIMA model shall be estimated, then followed by an ETS one.

ARIMA Time Series Model

Assuming a general ARIMA(p, d, q) model structure, each NMD balance vintage was fitted as a time series following standard ARIMA methodology, and using information criterion (AIC and BIC). As it turns out, the vintages were quite different as not only did each vintage indeed had a unique parameter set, but each vintage had a different ARIMA order. Auto-Regressive (AR) process had an order p ranging from 0 to 5, while

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Moving Average processes (MA) had an order q ranging from 0 to 1. Furthermore, differencing order, that is Integrated process, had an order d from 1 to 3. Moreover, the seasonality orders also showed variation, although to a lesser extent and ranged from 1 to 3 for SAR and SMA, but remained zero for seasonal difference. Nevertheless, some numerical issues arose when estimating some of the vintages but when forcing a lower order estimate it was fine. Overall, the resulting ARIMA models are too heterogenous to make a useful general model result with varying significance levels of the estimated parameters. However, in the results chapter an effective maturity and duration of the entire portfolio shall be reported in order to compare against the other methods, while the graphic results of the empirical models per vintage shall be shown to the interested reader in a collective plot in the appendix.

ETS Time Series Model

Following with the vintage approach to modelling evolution of NMD balance, a state space exponential smoothing model was considered. Starting with a general ETS as-sumption per vintage, the resulting models showed much more homogenous results per vintage than the ARIMA approach. The most striking feature was that ETS model for ∆Bx,t with selection based on statistical information criterion as well as statistical

significance, returned a homogenous model of ETS(A, N, A) for all vintages except the last one corresponding to year 2015 which could be due to data series of that vintage being very short (12 months in particular). Therefore, we may conclude that there is an overwhelming statistical evidence that a differenced vintage data series follows an ETS(A, N, A) model, which is that of an additive error, no-trend (due to differencing), and additive seasonality. The following general equations describe such structure:

∆Bx,t = lx,t−1+ sx,t−m+ εt, (4.14)

that is the change in the balance of vintage x is the previous period’s level of change, plus a(n) (additive) seasonality component as well as an (additive) error component; where m = 12 since the data series is of monthly data and the level and seasonality has the following theoretical forms:

lx,t= lx,t−1+ αx εt , (4.15)

sx,t= sx,t−m+ γx εt. (4.16)

Once again, we observe that the state space exponential smoothing techniques were much superior in comparison to ARIMA in empirically modelling the NMD vintage data. Parameters are much fewer while being able to capture complex structures of seasonality and errors (innovations), as well as correctly capturing the lack of trend for the differenced data. The graphical forecast plots shall be given in the appendix.

Modelling of non-maturing deposits for interest rate risk management

for Interest Rate Risk Management

Rany Shaheen

Contents

Introduction

Chapter 2

Literature Review

Chapter 3

Theoretical Framework

3.1

Market Value of NMDs as a Liability

3.2

Component View of Market Value of NMD Liability

3.3

Brief Background on Time Series Analysis

Empirical Model for NMDs

4.1

Data

4.2

Towards an Empirical Model

4.3

Categorisation and Aggregation of NMD Data for the

Empirical Model