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.

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

1.1 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) + S–P(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.

1.2 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.

1.3 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).