Risk transfer versus cost reduction on two-sided microfinance platforms

Abstract

Microfinance can be an important tool for fighting global poverty by increasing access to loans and possibly lowering interest rates through microlending. However, the dominant mechanism used by online microfinance platforms, in which intermediaries administer loans, has profound implications for borrowers. Using an analytical model of microlending with intermediaries who disburse and service loans, we demonstrate that profit-maximizing intermediaries have an incentive to increase interest rates because much of the default risk is transferred to lenders. Borrower and lender interest rate elasticities can serve as disciplining mechanisms to mitigate this interest rate increase. Using data from Kiva.org, we find that interest rates do not affect lender decisions, which removes one of these disciplining mechanisms. Interest rates are high, around 38% on Kiva. In contrast, on an alternative microfinance platform that does not use intermediaries, Zidisha, interest rates are only around 10%, highlighting the dramatic impact of intermediaries on interest rates. We propose an alternative loan payback mechanism that still allows microfinance platforms to use intermediaries, while removing the incentive to increase interest rates due to the transfer of risk to lenders.

Appendices

Appendix A: Proof of Claim 1

Proof

The probability of borrowing Kiva loan up to the amount of K is given by

$$ y(r)= 1-{\Phi}_{\epsilon}\left( \beta_{1}\log\left( \frac{\overline{K}_{0}}{\overline{K}_{0}+K}\right)+\beta_{2}\log\left( \frac{L^{*}+(K_{0}^{*}-\overline{K}_{0})\frac{p_{K}}{p_{L}}}{L^{*}+(K_{0}^{*}-\overline{K}_{0})\frac{p_{K}}{p_{L}}-K\frac{p(r)}{p_{L}}}\right)\right). $$

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Taking the derivative of y(r) with respect to $\overline {K}_{0}$, we have

$$\begin{array}{@{}rcl@{}} \frac{\partial y(r)}{\partial\overline{K}_{0}} \!& = & \left( \frac{-\phi_{\epsilon}\left( \beta_{1}\log\left( \frac{\overline{K}_{0}}{\overline{K}_{0}+K}\right)+\beta_{2}\log\left( \frac{L^{*}+(K_{0}^{*}-\overline{K}_{0})\frac{p_{K}}{p_{L}}}{L^{*}+(K_{0}^{*}-\overline{K}_{0})\frac{p_{K}}{p_{L}}-K\frac{p(r)}{p_{L}}}\right)\right)}{1-{\Phi}_{\epsilon}\left( \beta_{1}\log\left( \frac{\overline{K}_{0}}{\overline{K}_{0}+K}\right)+\beta_{2}\log\left( \frac{L^{*}+(K_{0}^{*}-\overline{K}_{0})\frac{p_{K}}{p_{L}}}{L^{*}+(K_{0}^{*}-\overline{K}_{0})\frac{p_{K}}{p_{L}}-K\frac{p(r)}{p_{L}}}\right)\right)}\right)\cdot\\ & & \left( \!\beta_{1}\frac{K}{\overline{K}_{0}(\overline{K}_{0}\,+\,K)}\,+\,\beta_{2}\frac{K\frac{p_{K}}{p_{L}}\frac{p(r)}{p_{L}}}{\left[L^{*}\,+\,(K_{0}^{*}\,-\,\overline{K}_{0})\frac{p_{K}}{p_{L}}\right]\left[L^{*}\,+\,(K_{0}^{*}\,-\,\overline{K}_{0})\frac{p_{K}}{p_{L}}\,-\,K\frac{p(r)}{p_{L}}\right]}\right). \end{array} $$

The first term ($\frac {-\phi _{\epsilon }(.)}{1-{\Phi }_{\epsilon }(.)}$) is negative, and the second and third terms in the brackets are non-negative. Hence, the derivative is negative. □

Appendix B: Proofs of Proposition and Corollary

Proof of Proposition 1.

Total expected profits were given in Eq. 8 as:

$$ \pi(r)=y(r)\left[F(r)p(r)-\widetilde{K}+\left( \widetilde{K}-F(r)\frac{K}{T}-C\right)l(r)\right], $$

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in which profits for a commercially financed loan are:

$$\begin{array}{@{}rcl@{}} \pi^{C}(r) & = & F(r)p(r)-\widetilde{K}, \end{array} $$

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and expected profits for a loan offered through microfinance are:

$$\begin{array}{@{}rcl@{}} \pi^{M}(r) & = & \pi^{C}(r)+\left( \widetilde{K}-F(r)\frac{K}{T}-C\right)l(r). \end{array} $$

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For readability, we omit the function arguments in the proof. Taking the derivative of total expected profits yields:

$$\begin{array}{@{}rcl@{}} \frac{\partial\pi}{\partial r} & = & \eta_{y}\frac{y(r)}{r}\pi^{M}+y(r)[F^{\prime}(r)p(r)+F(r)\frac{\partial p(r)}{\partial r}+\widetilde{K}\frac{\partial l(r)}{\partial r}-F(r)\frac{K}{T}\frac{\partial l(r)}{\partial r}\\ &&-C\frac{\partial l(r)}{\partial r}-F^{\prime}(r)\frac{K}{T}l(r)]\\ & = & \frac{y(r)}{r}\left[\eta_{y}\left( \left( F(r)p(r)-\widetilde{K}\right)+\left( \widetilde{K}-F(r)\left( \frac{K}{T}\right)-C\right)l(r)\right)\right.\\ & & \left.+\left( rF^{\prime}(r)p(r)+rF(r)\frac{\partial p(r)}{\partial r}+\left( \widetilde{K}-F(r)\left( \frac{K}{T}\right)-C\right)\eta_{l}l(r)\right.\right.\\ &&\left.\left.-rF^{\prime}(r)\frac{K}{T}l(r)\right)\right]. \end{array} $$

Setting this partial derivative to zero, we can define:

$$\begin{array}{@{}rcl@{}} {\Theta}_{1}(r) & \equiv & p(r)+ \end{array} $$

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$$\begin{array}{@{}rcl@{}} & & \frac{1}{\eta_{y}F(r)+rF^{\prime}(r)}\left[-\eta_{y}\widetilde{K}+rF(r)\frac{\partial p(r)}{\partial r} \right. \end{array} $$

$$\begin{array}{@{}rcl@{}} &&+ \left. l(r)\left( \eta_{y}\left( \widetilde{K}-F(r)\left( \frac{K}{T}\right)-C\right)+\eta_{l}\left( \widetilde{K}-F(r)\left( \frac{K}{T}\right)-C\right) \right.\right. \end{array} $$

$$\begin{array}{@{}rcl@{}} &&\left.\left.-rF^{\prime}(r)\frac{K}{T}\right)\right], \end{array} $$

such that Θ₁(r_M) = 0 at the optimal interest rate r_M.

Without microfinance, we can set l(r) and η_l to 0. This yields:

$$\begin{array}{@{}rcl@{}} {\Theta}_{2}(r) & \equiv & p(r)+\frac{1}{\eta_{y}F(r)+rF^{\prime}(r)}\left[-\eta_{y}\widetilde{K}+rF(r)\frac{\partial p(r)}{\partial r}\right], \end{array} $$

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which is equal to zero at the optimal interest rate r_C without microfinance.

We can calculate the difference in these two functions as:

$$\begin{array}{@{}rcl@{}} {\Theta}_{2}-{\Theta}_{1} & = & -\frac{l(r)}{\eta_{y}F(r)+rF^{\prime}(r)}\left[\eta_{y}\left( \widetilde{K}-F(r)\left( \frac{K}{T}\right)-C\right)\right.\\ &&\left.+\eta_{l}\left( \widetilde{K}-F(r)\left( \frac{K}{T}\right)-C\right)-rF'(r)\frac{K}{T}\right]. \end{array} $$

Here, we make one final assumption:

Assumption 5

Define Θ ≡Θ₂ −Θ₁ . Θ is increasing and satisfies the single-crossing condition if ∃r|Θ(r) = 0.

This assumption is benign because the expression is negative at r = 0. We know the first term, $-\frac {l(r)}{\eta F+rF^{\prime }}$, is negative, because from Assumption 3, we have that F^′(r) ≤ 0, η_y is negative from Assumption 1, η_l is negative from Assumption 2, and F(r) is positive. Next, we know $\widetilde {K}-F\left (\frac {K}{T}\right )-C$ is positive, otherwise the FP would not post the loan and so the first and second terms in the square brackets are negative. The third term is the only non-negative term, because F^′(r) is negative. At the optimal interest rate without microfinance, r^C, we have that Θ₂(r^C) = 0. Therefore Θ₂ (r^C) −Θ₁ (r^C) < 0, at the optimal interest rate without microfinance if and only if the third (risk-transfer) term is smaller in magnitude than the combined first two terms. By the single crossing property, this means that for all r < r^C,Θ₂ (r^C) −Θ₁ (r^C) < 0, and so for the optimal rate with microfinance, r^M, we have that r^M < r^C. Conversely, if the third term is larger in magnitude than the other two terms, we have that Θ₂ (r^C) −Θ₁ (r^C) > 0,and therefore r^M > r^C.

□

Proof of Corollary 1.

From Eq. 9, and by the implicit function theorem, we have that:

$$\frac{dr_{M}}{dz}=-\frac{\frac{\partial}{\partial z}\frac{\partial\pi(r_{M})}{\partial r}}{\frac{\partial}{\partial r}\frac{\partial\pi(r_{M})}{\partial r}}=-\frac{\frac{\partial\eta_{y}}{\partial z}\pi^{M}}{r\frac{\partial^{2}\pi^{M}}{\partial r^{2}}+\frac{\partial\eta_{y}}{\partial r}\pi^{M}}>0 $$

The numerator is positive by definition, and the first term in the denominator is negative by Assumption 4. The second term in the denominator is negative above the inflection point in the choice probability expression since $\frac {\partial -\phi (r)}{\partial r}>0$ for y > .5 (see Eq. 3). Thus the entire expression is positive and the optimal interest rate increases with less elastic demand. □

Appendix C: Alternative Mechanism

Profits for a microfinance-funded loan under our alternative payment mechanism A are

$$\begin{array}{@{}rcl@{}} \pi^{Funded|A}(r) & = & F(r)\left( p(r)\right)-N\frac{K}{T}-C, \end{array} $$

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where the FP receives monthly payments p(r) from the borrower but pays back the lenders with the monthly contribution to principal, L/T, regardless of whether the borrower defaults. N is the total discounted number of payments (equal to T if there is no discounting by FPs). Note that T ≥ N ≥ F. The total expected profits for a loan posted under this alternative are:

$$\begin{array}{@{}rcl@{}} \pi^{A}(r) & = & \left( F(r)p(r)-\widetilde{K}\right)\left( 1-l(r)\right)+\left( F(r)\left( p(r)\right)-N\frac{K}{T}-C\right)l(r) \\ & = & \left( F(r)p(r)-\widetilde{K}\right)+\left( \widetilde{K}-N\frac{K}{T}-C\right)l(r) \\ & = & \pi^{C}(r)+\left( \widetilde{K}-N\frac{K}{T}-C\right)l(r). \end{array} $$

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The expected profit for a loan the FP offers to a borrower is:

$$ \pi(r)=y(r)\,\pi^{A}(r). $$

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Taking the derivative of profits with respect to r, we have:

$$\begin{array}{@{}rcl@{}} \frac{\partial\pi(r)}{\partial r} & = & \eta_{y}\frac{y(r)}{r}\pi^{A}+y(r)\left[F^{\prime}(r)p(r)+F(r)\frac{\partial p(r)}{\partial r}+\widetilde{K}\frac{\partial l(r)}{\partial r}\right.\\ &&\left.-N\frac{K}{T}\frac{\partial l(r)}{\partial r}-C\frac{\partial l(r)}{\partial r}\right]\\ & = & \frac{y(r)}{r}\left[\eta_{y}\left( \left( F(r)p(r)-\widetilde{K}\right)+\left( \widetilde{K}-N\left( \frac{K}{T}\right)-C\right)l(r)\right)\right.\\ & & \left.+\left( rF^{\prime}(r)p(r)+rF(r)\frac{\partial p(r)}{\partial r}+\left( \widetilde{K}-N\left( \frac{K}{T}\right)-C\right)\eta_{l}l(r)\right)\right]. \end{array} $$

Similar to before, we can define:

$$\begin{array}{@{}rcl@{}} {\Theta}_{3}(r) & \equiv & p(r)+ \end{array} $$

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$$\begin{array}{@{}rcl@{}} & & \frac{1}{\eta_{y}F(r)+rF^{\prime}(r)}\left[-\eta_{y}\widetilde{K}+rF(r)\frac{\partial p(r)}{\partial r}+ \right. \end{array} $$

$$\begin{array}{@{}rcl@{}} & & \left. l(r)\left( \eta_{y}\left( \widetilde{K}-N\left( \frac{K}{T}\right)-C\right)+\eta_{l}\left( \widetilde{K}-N\left( \frac{K}{T}\right)-C\right)\right)\right]. \end{array} $$

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Assumption 6

Θ ≡Θ₃ −Θ₂ is increasing and satisfies the single-crossing condition.

This assumption yields a unique solution r^A to Eq. 21 over the support of the parameters. This assumption allows us to compare the situations with and without microfinance with the alternative mechanism.

We can again calculate the differences in the Θ functions:

$$\begin{array}{@{}rcl@{}} {\Theta}_{3}-{\Theta}_{2} & = & \frac{l(r)}{\eta_{y}F(r)\,+\,rF^{\prime}(r)}\left[\!\eta_{y}\left( \!\widetilde{K}\,-\,N\left( \frac{K}{T}\right)\,-\,C\right)\!+\eta_{l}\left( \widetilde{K}-N\left( \frac{K}{T}\right)\!-C\right)\right]. \end{array} $$

These terms are the same as with the current microfinance payment mechanism with the exception that N appears on the right-hand side inside the braces instead of F, and the term that includes F^′, which was the risk-transfer term, is absent. Because both terms inside the brackets are negative in value, the interest rate must be lower due to Kiva, from the single-crossing assumption. The intuition is that by having the FPs pay back lenders irrespective of default, the upward force on rates from risk transfer has been removed.

Appendix D: Simulations

1.1 Current Mechanism

We use numerical simulations to demonstrate the dependence of the interest rate on the demand and demand elasticity with and without the presence of microfinance, in order to demonstrate microfinance’s effect on the optimal interest rate. For these simulations, we set the loan duration to be one year and loan amount of 0.61, which is the average loan size relative to the country’s GDP per capita in our data. For the first set of simulations, we assume that additional loan disbursement costs through microfinance are low, 10% of the loan amount, and that they do not increase with interest rates (if costs increased with interest rates, this would further push rates down). We assume that the cost of borrowing for the FP is 10% of the value of the loan (which we later vary). We set demand to be 0.5 for a range of demand elasticities between -0.5 and -1.0. The qualitative relationships are robust to changes in the parameter values, which are used for illustrative purposes.

We assume the probability of making each payment is a time-invariant function of r, δ(r), so that delinquent payments (which occur with probability 1 − δ(r)) are not made; the borrower continues to make future payments with probability δ(r):

$$\begin{array}{@{}rcl@{}} F(r)\!\equiv\!\sum\limits_{m = 1}^{T}\left( \rho\delta(r)\right)^{m} & \,=\,\frac{1-\left( \rho\delta(r)\right){}^{T + 1}}{1-\rho\delta(r)}\,-\,1\,=\,\frac{-\left( \rho\delta(r)\right){}^{T + 1}+\rho\delta(r)}{1-\rho\delta(r)} & \,=\,\frac{\rho\delta(r)\left( 1\,-\,\left( \rho\delta(r)\right){}^{T}\right)}{1\,-\,\rho\delta(r)}.\\ \end{array} $$

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These assumption are not needed, but they help with the interpretation of the simulations. Because δ(r) can be easily interpreted (as “1−monthly delinquency probability”), we will vary δ(r) in the simulations in order to vary F(r).

The FPs’ discount rate and the monthly delinquency probability have the same effect because a higher value of either will lower the net present value of the loan. For the simulations, we set the FPs’ discount rate to be ρ = 0.99 and alter the delinquency rate function. We specify this function as:

$$ \delta(r)= 1-\kappa_{1}\exp((r-.5)\kappa_{2}). $$

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This allows us to modify the level of delinquency rate as well as the curvature of the delinquency rate function by changing κ₁ and κ₂. To show the effect of the two κ parameters on the shape of the delinquency rate function, we plot the delinquency rate function for different values of the κ parameters in Fig. 4. The first figure shows the function for a lower initial delinquency rate at r = 0 with κ₁ = 0.01 for κ₂ = 5 and 10, and the second for a higher initial delinquency rate with κ₁ = 0.02 for the same values of κ₂. A larger κ₂ results in a greater curvature of the delinquency function (shown in Fig. 4 with the dashed line).

Fig. 4

Delinquency Rate Parameterization

Next, we compare the optimal APRs as a function of both demand and demand elasticity. The first derivative of the delinquency rate function (23) with respect to interest rates is − κ₁κ₂ exp((r − .5)κ₂), which measures the impact of interest rates on the level of delinquency risk. In Fig. 5, we show the optimal interest rates with and without microfinance as the magnitude of the first derivative becomes larger, which is achieved by increasing the value of κ₁ or κ₂. In Fig. 5, the delinquency rate is set at κ₁ = 0.01 in the left two graphs, with the bottom graph having the larger κ₂, i.e., delinquency increases more rapidly with interest rate. With a larger effect of rate on delinquency, there is a larger risk transfer effect. Hence the graphs of the optimal APRs with and without microfinance cross at a lower rate. In the right two figures, we use the same value for κ₂, but double κ₁ - the effect is to increase the disparity between the rates with and without microfinance - the rate at which the two curves cross is the same as with the lower κ₁ . Thus the greater the magnitude of the first derivative of delinquency with respect to interest rate, the greater the differences in the optimal interest rates with and without microfinance. The second derivative of delinquency with respect to interest rate determines the point at which demand is inelastic enough such that interest rates become higher with microfinance.

Fig. 5

Comparison of APRs between Microfinance Loans and Non-Microfinance Loans, High Default Rate

1.2 Proposed Alternative Mechanism

In this subsection, we again use simulations to calculate optimal interest rates, but under our proposed payback mechanism. In Fig. 6, we demonstrate how the optimal interest rate varies as a function of demand elasticity under our proposed alternative mechanism. In contrast to the current mechanism, as demand becomes more inelastic, the optimal APR under the proposed mechanism approaches the level without microfinance, but never exceeds it. The discrepancy in interest rates is driven by the reduction of lending costs, while the risk transfer effect is completely removed. As stated by Proposition 2, the optimal interest rate is always below that without microfinance.

Fig. 6

Comparison of APRs between Microfinance Loans and Non-Microfinance Loans, Alternative Mechanism

In Fig. 7, we compare the FP profits and optimal interest rates as a function of the additional costs of administering a loan through microfinance, at the observed demand elasticity in the data. We plot these variables both when using and when not using microfinance, and we use a thick, gray line to designate the profits and optimal interest rate for the option that maximizes profits. As the administrative costs increase, the profits when using microfinance go down for the FPs and the optimal interest rates go up. Essentially, any additional administrative cost of working with the microfinance platform reduces the downward pressure on rates that results from the lower costs of dispersing loans (due to the refinancing), and so the upward force due to risk transfer dominates. If costs become too high, the FP choses not to post the loan because its profits become higher when not using the microfinance site (at which point the optimal APR drops back down to what it would be without microfinance). From a consumer welfare perspective, it is the intermediate level of costs which cause concern under the current mechanism, since the FPs still choose to use the microfinance platform but the interest rates are higher when doing so. It is thus essential that microfinance sites keep administrative costs to a minimum for the FPs, otherwise rates may become higher than in the absence of microfinance.

Fig. 7

Comparison of FP Profit and APRs between Microfinance Loans and Non-Microfinance Loans as Administrative Costs Increase

At the observed demand elasticity in the data, and using our parameterization of the demand curve, the interest rate with microfinance exceeds that without microfinance (under the current mechanism) as soon as extra costs of administering microfinance loans exceeds 1.5% of the loan amount. In sharp contrast, under the alternative repayment mechanism, if costs with microfinance become high enough, the FP may decide not to use microfinance, but in all cases the optimal interest rate remains below the rate without microfinance.²³ In cases with minimal additional costs, rates would be considerably lower with microfinance than without microfinance, under our proposed mechanism.

Appendix E: Comparison of Kiva Loans before and after Zidisha

Table 6 Summary Statistics of Kiva Loan Characteristics before and after Zidisha