# Credit card balances and repayment under competing minimum payment regimes

## Abstract

Although much research has been done on financial literacy and the use of credit cards, ours is the first to focus on the mathematics of competing credit card minimum payment regimes. The lack of prior research is surprising because the Office of the Comptroller of the Currency does not specify exactly what form a credit card minimum payment should take, and yet we show that competing credit card minimum payment regimes have significantly different impacts on both the rate of accumulation and level of credit card debt and also on the time taken to pay that debt off. We give closed-form solutions for the balance owing under competing repayment regimes and two spending scenarios. In addition, we discuss the policy implications for credit card and banking regulators and the personal finance implications for individual consumers using credit cards.

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1. Note that in some countries, for example, Germany, the entire credit card balance is due each month, with no minimum (Crack and Roberts forthcoming). This corresponds to the proportional minimum payment case p = 1.

2. Jiang and Dunn (2013, p. 396) focus on the same residual balance but label it “credit card debt.” Note that some credit card companies refer to the difference between the credit limit and the amount owing as the “balance.” In that case, the balance is the unused credit remaining on the card.

3. Jiang and Dunn (2013, p. 404) present a formula and accompanying discussion that taken together are equivalent to Eq. (1) but with $$q_{D} = 0$$.

4. Jiang and Dunn (2013, p. 404) point out that after recent bankruptcy reform (GPO 2005a), many lenders increased $$p_{D}$$ in Eq. (1) from around 2 % to around 4 %—but this change is not a requirement of the act. They also find empirically that a one-percentage-point increase in the required minimum credit card payoff increases the average payoff rate by 1.9 %. They attribute this, at least partially, to a behavioral finance “anchoring” effect.

5. From the series expansion in the proof of Eq. (5), we get $$\frac{{\partial^{2} B_{\text{proportion}} (t)}}{\partial p\partial i} = - S\left( {1 - p} \right)\sum\limits_{j = 1}^{t - 1} {\left[ {j\left( {j + 1} \right)\lambda^{j - 1} } \right]}$$, where $$\lambda = \left( {1 + i} \right)\left( {1 - p} \right)$$. So clearly $$\frac{{\partial^{2} B_{\text{proportion}} (t)}}{\partial p\partial i} < 0\quad {\text{for}}\;0 < p < 1$$ with equality only when p = 1.

6. It should be noted that credit card APRs have been reported to be sticky in their sensitivity to the general level of interest rates (e.g., Stavins 1996; Ayadi 1997; Chang et al. 2010). This stickiness reduces problems with a proportional repayment regime when interest rates are rising, but compounds problems when interest rates are falling.

7. One difference between the convergent/divergent behavior of the 10-year spending residual balance and the perpetual-spend residual balance is that in the case $$p = p^{*}$$, $$B_{\text{proportion}} (t)$$ is equal to a steady finite limit (equal to $$B_{\text{proportion}} (120)$$) in the former case, but diverges to infinity in the latter case. See the summary in Table 1.

8. The APR level corresponding to $$i^{*}$$ is also the level of APR at which the proportional and minimax surfaces diverge in Fig. 4. Using our base-case numbers, this $$i^{*}$$ corresponds to an APR of about 12.24 %.

9. Given a change in monthly interest rate of $$\varDelta i$$, a Taylor series expansion of $$B_{\text{proportion}}$$ yields the simple and very accurate (i.e., no truncation is used) result that $$B\left( {i + \varDelta i} \right) \approx B + {{B^{2} \left( {\varDelta i} \right)} \mathord{\left/ {\vphantom {{B^{2} \left( {\varDelta i} \right)} {\left[ {S - B\left( {\varDelta i} \right)} \right]}}} \right. \kern-0pt} {\left[ {S - B\left( {\varDelta i} \right)} \right]}}$$. In the minimax case, we can use this expression for $$B_{\text{calibrate}}$$ rather than for $$B_{\text{minimax}}$$ directly to get a good approximation. Note that we used the derived result $$\frac{{\partial^{n} B}}{{\partial i^{n} }} = \frac{{n!B^{n + 1} }}{{S^{n} }}$$ in the Taylor series expansion.

10. Of course, if we recalibrate the proportion p as a function of the APR (i.e., let the calibrated proportion $$p_{\text{calibrate}}$$ change as the APR changes using Eq. (8)), then the sensitivities of these three quantities to the APR are nearly identical.

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Correspondence to Timothy Falcon Crack.

## Appendix: Proofs of Eq. (5), Eq. (6), and Eq. (7)

### Proof of Eq. (5)

Assume the proportional repayment in Eq. (3). Assume a monthly spend of S. Then, by definition we know that the residual balance owing evolves as B(t) = T(t) − P(t) = B(t − 1)(1 + i) + SP(t). Substitute in for P(t) to find B(t) = B(t − 1)(1 + i)(1 − p) + S(1 − p). Recursive substitution and imposition of B(0) = 0 yields $$B(t) = \frac{S}{(1 + i)}\sum\nolimits_{j = 1}^{t} {\left[ {\left( {1 + i} \right) \, \left( {1 - p} \right)} \right]^{j} } .$$ Define λ = (1 + i)(1 − p) and rewrite the summation as a closed-form expression and the first part of Eq. (5), for 1 ≤ t ≤ 120 follows immediately for λ ≠ 1. With no new balances, the balance grows as $$B(t) = B(120) \cdot \left[ {\left( {1 + i} \right) \, (1 - p)} \right] \, ^{t - 120} = B(120) \cdot \lambda^{t - 120}$$ from then on. In the case λ = 1, the summation above immediately yields the results in Eq. (5).

For completeness, note that it also follows immediately that the total owing at the end of each month just prior to the monthly payment is given by $$T_{\text{proportion}} (t) = S \cdot \frac{{\left( {1 - \lambda^{ \, t + 1} } \right)}}{1 - \lambda }$$, for 1 ≤ t ≤ 120.

### Proof of Eq. (6)

There are two ways to arrive at Eq. (6). We may begin with $$B_{\text{proportion}} (t) = \frac{S}{1 + i} \cdot \frac{{\lambda \left( {1 - \lambda^{t} } \right)}}{1 - \lambda }$$ for 1 ≤ t ≤ 120 from Eq. (5), and simply allow t to go to infinity. As long as $$\lambda = \left( {1 + i} \right) \, \left( {1 - p} \right) < 1$$, then $$B_{\text{proportion}} (t) \to \frac{S}{1 + i} \cdot \frac{\lambda }{1 - \lambda } = \frac{{S\left( {1 - p} \right)}}{{1 - \left( {1 + i} \right) \, \left( {1 - p} \right)}} = B_{\text{proportion}}$$, as required.

Alternatively, we may use an economic argument to assert that if the residual balance reaches an asymptote, then it must be that the payment made at time t equals (and cancels out) the growth in the amount owing over month t. That is, p[B(t − 1) · (1 + i) + S] = B(t − 1) · i + S. We may remove the dependence on t because it is an asymptotic result: p[B · (1 + i) + S] = B · i + S. Solving for B yields Eq. (6) immediately.

### Proof of Eq. (7)

Assuming a perpetual monthly spend that exceeds the dollar amount D in Eq. (1), the asymptotic balance for the minimax minimum payment regime depends only upon the last two terms inside the maximum calculation in Eq. (1): p D  · T(t) and q D  · T(t) + B(t − 1) · i, respectively. One of these dominates, depending upon the level of the APR. It is easy to show that for 0 ≤ i ≤ $$i^{*}$$, where $$i^{*} = \frac{{p_{D} - q_{D} }}{{1 - p_{D} }}$$, the term p D  · T(t) dominates and the asymptotic residual balance is worked out as in the proof in Eq. (6) above. In the case i > $$i^{*}$$, then the term q D  · T(t) + B(t − 1) · i dominates, and a similar argument yields the second term in Eq. (7).

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