^{ 1 }EDC Business School

^{1}

^{ * }Date of publication: 16/06/2021

Financial institutions^{2}

Unmatured deposits are an important source of funds and profits for banks. The amount of these deposits is fairly stable. It is accounted for in the M1 aggregate by Central Banks. These amounts are usually reinvested by banks on treasury markets or directly in customer loans (with a margin over treasury market rates). The profit generated by these deposits at the end of the accounting year is the product of the cumulative amount of deposits multiplied by the difference between the cash market rates less the usually lower interest rate paid to the client. Thus, profits can represent a large part of banks' profits. Indeed, deposit remuneration schemes can take different forms. The remuneration policy influences the behavioral models of clients, and therefore the valuation of deposits (see e.g. ^{3}

This paper focuses on the financial hedge of the net interest margin, defined as the income generated by the investment of demand deposit amounts on interbank markets while paying a deposit rate to customers. Our contribution is to analyze the interest rate margin of a quadratic hedging strategy (see

This paper is organized as follows. In section 2, we provide the main statistical properties of demand deposits both in the Euro and US zones, while recalling some characteristics of Euribor and Libor rates during the last decade. Section 3 illustrates the hedging issue of the margin interest rate by examining the quadratic hedging strategy in the static case for bank deposits. Finally, section 4 concludes.

In July 2002, a draft regulation was implemented by the European Commission to require listed institutions to adopt IFRS accounting standards that advocate a valuation method based on the fair value principle. Within this framework, the IAS 39 devoted to hedge accounting is based on accounting methods (macro-hedge versus micro-hedge). For this purpose, the proposal to revise the hedge accounting component has been sent to the International Accounting Standards Board (IASB) by the European Banking Federation (EBF). They support an accounting treatment for the macro-hedging practices of European banks. We consider the macro-hedge that is performed by asset-liability managers of European banks which allows reducing capital volatility. The macro-hedge is announced as the most flexible approach and the least expensive. Studies also show that it is the best overall understanding of the risk and the modeling of risk. For IAS 39, which is a micro-hedge meeting international accounting standards, it is suggested to use derivative products on interest rates. The demand deposits are proxied M1 in the period from September 1997 to July 2019. They measure the amount of money circulating in economy, usually presented as end-of-month values expressed in domestic currency. In France for example, the structure of household deposits is strongly influenced by regulated savings products (such as Livret A, PEL). On average, in nominal terms deposit amounts located in France are at a higher level than in other European countries.

Descriptive statistics and graphic representations in the Euro zone

In this section, we aim at reducing the interest rate margin's variance, for some given maturity T. In what follows, we define a payoff corresponding to a hedging strategy of the interest rate margin for the quarter [T,T+ ]. Yet, most banking practices tend to design hedging strategies on interest rate securities to alleviate the volatility of the net interest income at historical cost. The asset liability manager must hedge both interest rate and demand deposit to ensure a positive interest rate margin. For this purpose, we introduce a quadratic hedging criterion allowing us to get an explicit hedging strategy that we further analyze. We note that, this kind of hedging strategy is based on mean-variance approach.

3.1Static quadratic hedging for interest rate margins

We assume that the demand deposit amount follows:

where (

) is a standard Brownian motion. The trend

and the volatility

are assumed to be constant. Additionally, we assume that the forward market rate at date T for the time period of the interest rate margin -- a quarter -- follows a Market Rate Model , as defined in

Where

denotes the forward market rate and

is a standard Brownian motion under some historical probability measure P. For model simplicity,

and

are assumed to be constant and we denote the related interest rate risk premium by

. We will thereafter be able to account for higher average returns when investing in long term bonds than in short-term assets. The framework can readily be extended with extra -- but reasonable -- computation when both

and

depend upon the forward market rate. In what follows, we consider the following optimization problem:

Where

is the set of linear payoffs with respect to the market rate, namely:

where

is a constant. Such hedging strategy is based on Forward Rate Agreements (FRAs) contracted at initial date. The problem deals with risk minimization with the interest rate margin as the only objective, such that there is no minimal return constraint on the final income.

Proposition The solution of problem is given by:

Proof. We have:

The previous term is a polynomial function of order 2 with respect to

. Its minimum is reached at:

Corollary The expectation of the hedging cost is equal to:

The expectation and the standard deviation of the hedging error are respectively given by:

with:

Corollary Under assumptions of the processes

and

, we get:

We deduce in particular that, the optimal static hedging strategy

has the following form with respect to the forward rate

:

with:

Consequently, if the correlation

is negative, the optimal static quadratic hedging is decreasing with respect to the forward market rate if and only if the intercept

is positive. If the correlation

is positive, it is the converse.

3.2 Numerical illustrations

_{L} and σ_{L} for the value of the intercept for the Euro zone, we take α=0.011 otherwise we get a negative expectation of the IRM.

- Euro zone case

Bank deposits management has become an important issue for banking institutions. In this paper, we deal with the mitigation of the risk contained in interest rate margins of demand deposits for the case of Euro and US zones. This paper computes the optimal strategy in order to hedge their interest rate risk. For this purpose, we have modelled the deposit rates as a linear function of the market rate, both in the case of the Euro Zone and the US during the period lying from January 2003 to April 2019. Assuming that the demand deposits carry some source of risk called business risk orthogonal to market risk, we have provided an explicit formula for the mean-variance hedging of the interest rate margins in the static case. Using a quadratic hedging strategy in a static framework instead of using options greek letters is not very expensive for the risk manager. We then provided numerical analysis of sensitivities of the hedging strategy, showing how the opposite signs of correlation and intercept of the two zones impact the hedge. Banks must set a remuneration that must be positive (rate + spread >0) while facing competition (see e.g. Banque de France, 2005). However, for the Euro zone, as a result of the cut in rates applied by the ECB, deposit-taking banks had to cope with a lower rate of remuneration on deposits, recently leading them to suppress the remuneration of accounts. One possible further extension would be to assume that deposit rate can be influenced by other macroeconomic aggregates (for example, inflation rate or the growth rate of GDP).

webstat.banque-france.fr

fred.stlouisfed.org

See www.euribor-rates.eu

4See Federal Reserve Bank of Saint Louis, https://fred.stlouisfed.org/series/M2OWN