The Impact of the Extra-Costs on the Global Cost of Credit

In this note, we analyze the impact of the extra-costs payment schedule on the Effective Annual interest Rate (EAR), one of the most popular global cost measures of consumer credit loan payments. First, we prove that the EAR can be expressed by the financing credit interest rate with an extra-costs interest rate addendum, and we investigate the drivers of this latter. We show that the extra-costs interest rate decreases if extra-costs payments are postponed. Consequently, the EAR is minimum if extra-costs are charged in a lump sum at the expiry date of the contract and maximum if they are charged in a lump sum at the contract beginning time. To explain how the schedule of payments impacts on the EAR, we develop a sensitivity analysis through illustrative applications. We also highlight that EAR depends on the timing of extra-costs payments. In particular, we show that EAR decreases with the increase in the Modified Duration of the cash flow of extra-costs. The results of the paper are useful to provide decision-makers a better awareness about how to spread the extra-costs payments during the contract lifetime and, therefore, to define the structure of consumer credit loan payments to supervise the global cost of the financing.


Introduction
In financial practice, consumer payment credits contracts require payment of extra-costs spread over the contract lifetime. At the beginning of the fixed-term contract, the parties agree on the extra-costs amount and their temporal payment schedule over the contract lifetime. The agreed terms' effect on the global cost of credit is measured by the Effective Annual interest Rate (EAR) (see, for example, Disney & Gathergood, 2013). To make an evidence of the EAR drivers, we achieve an approximation that is equal to the financing credit interest rate plus an extra-costs interest rate. Our analysis shows the dependence of this latter addendum on extra-costs size and schedule of payments. This two-dimensional information can be summarized by the Modified Duration of the extra-costs payments stream. We show that the EAR decreases if the extra-costs payments accrued by the interests computed at the financing credit rate are postponed over time. As a consequence, the EAR is maximum if discounted extra-costs are charged in a lump sum at the contract beginning time and minimum if accrued extra-costs are charged in a lump sum at the contract expiry date. Formulae become more straightforward if the contract involves periodic payments of a constant amount. Some simulations assess the robustness of our results.
The remainder of the paper is organized as follows. In Section 2, the notation is set up. In Section 3, we achieve an EAR approximation. Section 4 presents our results through simulations and discusses the outcomes of some sensitivity tests. Section 5 concludes the note. The proofs are reported in the Appendix.

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F be the initial debt financing at time 0; • r be the annual compound financing contractual interest rate; • n be the number of contract's payments; for the ease of explanation, we assume that the payments are paid annually and so n coincides with the contract expiry date; • 0 s f ≥ be the contractual payment due at the time s, with 1,..., s n = , to pay back the initial debt F within the contract expiry date n. Each s f is the sum of the capital share and the interest share, and the payments are The discounted cash flow of the total cash flows is given by By definition, the Effective Annual Interest Rate (EAR) of the contract complies the condition (Note 1) In general, no closed formula exists for EAR, to overcome this limitation, we achieve an analytical approximated measure for EAR in the following Section.

An "Extra-Costs Interest Rate Addendum."
In this section, we develop an approximated measure of EAR given by the financing credit interest rate r and an extra-costs interest rate addendum.
THEOREM 1. An analytical approximated measure for the EAR Let the initial debt F be financed at the credit interest rate r. The financing contract includes the payments s f satisfying (1) and the cash flows of extra-costs s c at times 0,1,..., s n = , with 0 c F ≤ . Then the EAR admits the following approximated measure (Note 2) where ε is called the "extra-costs interest rate addendum," and is the Modified Duration of the contract payback of the initial debt F at the financing interest rate r; in Equation (3) becomes more straightforward if the contract involves n periodic global cash flows of equal amounts of payments. Let the credit reimbursement be made via amortization with level payments, i.e., the repayment is structured in these equal payments The Modified Duration of a series of cash flows with equal amounts of payments is independent on amounts' size (see Dierkes & Ortmann, 2015). Therefore, the Modified Duration D of the global cash flow is the same as the Modified Durations F D of the contract level payments and Extra costs D of the extra-costs payments Using Equation (4), Equation (3) where 0 F c − is the net financing cash flow at the initial time.

The Minimum and Maximum for
proxy EAR Let us assume that the parties agree on customizing the extra-costs payment timing, under the condition that if a payment is anticipated, then the due amount is discounted at the financing interest rate r. If the payment is deferred, then the due amount is accrued at the financing interest rate r. More formally, if the payment s c is paid: • m periods in advance concerning the scheduled time s, then the due amount to be paid is the discounted if extra-costs are paid in a lump sum at the final time n and at the initial time 0, respectively.

Numerical Illustrative Examples
In this section, we develop a sensitivity analysis of the impact of the extra-costs payment timing on the EAR. Without loss in generality, we consider credit contracts with level payments of equal amount s f f = , at all times 1,..., s n = . So, the Modified Duration F D of the credit cash flow F is given by Equation (4) where F D depends only on the financing interest rate r and the contract lifetime n, but not on the amount of the initial debt F. Example 1.
Due to Theorem 1, the EAR decreases as Extra-costs D lengthens.

Conclusions
In this note, we analytically approximate the consumer credit EAR with the financing contractual interest rate plus an extra-costs interest rate. Assuming that the due dated payments can be anticipated or postponed by discounting or accumulating, respectively, at the financing contractual interest rate, we analytically prove that EAR decreases if the extra-costs payments are postponed. Therefore, the maximum EAR is achieved if the extra-costs are lump sum paid at the contract starting time and likewise the minimum EAR if the extra-costs are lump sum paid at the contract expiry date. Another relevant result of this paper is to highlight how EAR depends on the timing of extra-costs payments. In fact, we prove that the extra-costs interest rate addendum in the approximation of EAR not only depends on the Modified Duration of contractual payments linked to the repayment of the initial debt but also it depends on the Modified Duration of extra-costs payments distributed over time.
In particular, taking into account that Modified Duration measures the sensitivity of any series of cash flows to changes in the yield rate used to value series, we show that EAR decreases with the increase in the Modified Duration of the cash flow of extra-costs.
To illustrate how our results can be applied to support the decision-makers in defining the structure of consumer credit loan payments to supervise the global cost of the financing, we develop a sensitivity analysis through numerical simulations in accordance to current observed rates consistent with the amount of the initial debt and a contract of 24 months lifetime. We consider consumer credit loans with level payments of equal amount and extra-costs paid in a lump sum at times ranging between the initial date and the expiry date of the contract. The numerical results show how the EAR decreases as well as the estimate of the exact EAR is increasingly precise as modified duration of extra-cost payments lengthens.  Norstrøm (1972) condition, the EAR is the unique solution of the equation NPV Global cost (x)=0. By construction NPV Global cost (x) in (2) is a strictly increasing function in x Note 2. In the following, the term proxy is also used as shorthand by approximated measure.

Appendix A Proof of Theorem 1
Given the annual compound financing interest rate r, let us approximate the function ( )  The derivative of (2) at x r = can be reformulated as

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