Imagine There Are Two Types Of Potential Borrowers In A Vill

Imagine There Are Two Types Of Potential Borrowers In A Village Those

In this analysis, we explore the dynamics of individual and group lending among borrowers with different risk profiles in a village setting. The core assumptions involve two borrower types characterized by success probabilities of pi=0.7 and pi=0.9, respectively, and a loan amount L = $100. Borrowers succeed or fail probabilistically and face choices between borrowing and working for a subsistence wage. The discussion encompasses the calculation of expected payments to the bank, the determination of optimal interest rates under different informational environments, the analysis of borrowers’ expected net payoffs, and the impact of adverse selection and group lending mechanisms.

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Initial considerations involve understanding the basic risk structure: borrowers with success probabilities pi=0.7 and pi=0.9, and their respective expected outcomes when considering borrowing versus working. The key variables include the loan amount (L = $100), the failure probability (1 - pi), and the gross expected payoff from projects, which is $200 for all borrowers. Borrowers can choose to borrow or, alternatively, work at a subsistence wage of $70. The analysis starts with individual lending, progressing to scenarios of asymmetric information and then to group lending models.

Expected Payments to the Bank by Borrowers with Different Success Probabilities

For a borrower with success probability pi, the expected payment to the bank depends on the probability of success and the repayment amount if successful. The borrower pays nothing if they fail, and pays (1 + r) L if they succeed. Thus, the expected payment to the bank (E[P]) can be expressed as:

E[P] = pi × (1 + r) × L + (1 - pi) × 0 = pi × (1 + r) × 100.

Applying this to each type:

• For pi = 0.9: E[P] = 0.9 × (1 + r) × 100 = 90 × (1 + r).
• For pi = 0.7: E[P] = 0.7 × (1 + r) × 100 = 70 × (1 + r).

Interest Rates Required for the Bank to Earn a 10% Expected Return

The bank’s expected return condition implies that the expected repayment must equal the amount lent times (1 + 0.10) = 1.10. Setting the expected repayment equal to the original loan:

For pi = 0.9:

0.9 × (1 + r) × 100 = 110 → (1 + r) = 110 / 90 ≈ 1.2222 → r ≈ 22.22%.

For pi = 0.7:

0.7 × (1 + r) × 100 = 110 → (1 + r) = 110 / 70 ≈ 1.5714 → r ≈ 57.14%.

Hence, the interest rates charged are approximately 22.22% for the high-risk borrower and 57.14% for the lower-risk borrower, assuming the bank can observe risk types and set different rates accordingly.

Expected Net Payoff for Borrowers at These Interest Rates

The expected net payoff for each borrower type is calculated as:

Net Payoff = Probability of success × (Payoff after success − repayment) + Probability of failure × (outside option − 0).

Since the outside option is $70, and they pay nothing upon failure:

• For the pi=0.9 borrower:
• Net Payoff = 0.9 × ($200 − (1 + r) × 100) + 0.1 × ($70 − 0)
• Plugging in r ≈ 22.22%, (1 + r) ≈ 1.2222:
• Net Payoff ≈ 0.9 × (200 − 122.22) + 0.1 × 70 = 0.9 × 77.78 + 7 = 69.99 + 7 ≈ $76.99.
• For the pi=0.7 borrower:
• Net Payoff = 0.7 × (200 − (1 + r) × 100) + 0.3 × 70
• With r ≈ 57.14%, (1 + r) ≈ 1.5714:
• Net Payoff ≈ 0.7 × (200 − 157.14) + 0.3 × 70 = 0.7 × 42.86 + 21 = 30.00 + 21 = $51.00.
• Both borrower types have positive net payoffs when facing these interest rates, suggesting both would choose to borrow if they can accommodate the rate appropriate for their risk level.
• How Borrower Types Decide in Asymmetric Information Scenarios
• When the bank cannot observe risk types and must set a uniform interest rate, it will choose the interest rate based on the riskier borrower (pi=0.7). Using that rate (≈57.14%), the expected payment and net payoff are calculated similarly for each type. The key issue arises with adverse selection: high-risk borrowers (pi=0.7) are more likely to borrow at the high uniform rate, while low-risk borrowers might abstain due to unfavorable payoff calculations, leading to a crowding out of the safer borrowers. The overall effect is to increase the risk profile of the lending portfolio, potentially threatening the bank’s profitability.
• Expected Payments and Payoffs Under Uniform Interest Rate
• Using r ≈ 57.14% (from above), the expected payment to the bank for each borrower type is:
• pi=0.9: E[P] = 0.9 × 1.5714 × 100 ≈ 0.9 × 157.14 ≈ $141.43.
• pi=0.7: E[P] = 0.7 × 1.5714 × 100 ≈ 0.7 × 157.14 ≈ $109.99.
• Expected net payoffs are then:
• For pi=0.9:

Net Payoff = 0.9 × (200 − 157.14) + 0.1 × 70 ≈ 0.9 × 42.86 + 7 = 38.57 + 7 ≈ $45.57.

• For pi=0.7:

Net Payoff = 0.7 × (200 − 157.14) + 0.3 × 70 ≈ 0.7 × 42.86 + 21 ≈ 30.00 + 21 = $51.00.

• Both remain willing to borrow, but the overall portfolio’s risk profile is higher, illustrating adverse selection: lower-risk borrowers are disincentivized to borrow at high uniform rates, possibly leading to a predominance of riskier borrowers.
• Adverse Selection: Definition, Appearance, and Why It Matters
• Adverse selection refers to a market failure that occurs when asymmetric information leads high-risk individuals to disproportionately seek loans or insurance, while low-risk individuals withdraw, resulting in a higher-than-expected risk pool for lenders. It appears when lenders cannot distinguish between different types of borrowers. As a consequence, they set interest rates based on average risk, which may be too high for low-risk borrowers, discouraging their participation and exposing lenders to a riskier portfolio. This distortion reduces market efficiency and potentially leads to market collapse if unaddressed.
• Group Lending as a Solution to Adverse Selection
• Group lending involves borrowers forming groups, typically homogeneous, where they are jointly responsible for repayment. This mechanism exploits peer monitoring and social collateral. If one member attempts to default, others enforce repayment through social sanctions, incentivizing effort and honesty. Group lending reduces the problem of asymmetric information because the collective responsibility creates accountability among members. It is particularly effective in settings where formal collateral is absent. This approach aligns individual incentives, encourages participation from safer borrowers, and mitigates adverse selection.
• Group Loan Structure with Homogeneous Pairs
• Given the parameters, with project success probabilities and added penalty costs, the effective interest rate for each borrower can be derived by considering the expected repayment, success probabilities, and penalties. When r=30% and c=90%, at the individual level, the effective interest rate accounts for the probability of success and the additional penalty involved when a partner fails. Calculations approximate as follows:
• For pi=0.9:
• The expected repayment incorporates the chance of success and failure, with additional penalties if the partner fails. The effective interest rate, i r ~, can be derived from:
• Expected repayment = pi × (1 + i r ~) × L + (1 - pi) × cL.
• Plugging in the values:
• Expected repayment = 0.9 × (1 + i r ~) × 100 + 0.1 × 90 × 100 = 0.9 × 100 × (1 + i r ~) + 9 × 100.
• Equating to the total expected payoff of a successful project (roughly $75,000 scaled to the per-loan basis) provides the effective interest rate, which will differ from individual rates.
• For pi=0.7:
• Analogous calculations show a higher effective interest rate due to higher risk and penalty costs, justifying that group lending can lower the adverse selection problem by incentivizing safer borrowers and spreading risk among members.
• Expected Net Payoffs and Borrower Incentives in Group Lending
• Expected net payoffs are calculated similarly but now incorporate the probability-adjusted repayments including penalties. For pi=0.9, the expected repayment is higher but still manageable; for pi=0.7, the effective interest rate increases, but the collective enforcement minimizes risks of default. Both types are likely to participate if the net payoff exceeds their outside option, $70. This structure encourages safer participation by lower-risk borrowers, effectively addressing adverse selection and fostering sustainable lending environments.
• Advantages of Group Lending in Overcoming Adverse Selection
• Group lending reduces adverse selection by leveraging social collateral, peer monitoring, and collective responsibility. It transforms individual risk into collective responsibility, incentivizing honest behavior and effort from all members. Because members monitor each other, lenders receive signals about borrower risk types indirectly, reducing informational asymmetries. This model encourages safer borrowers to participate and reduces the prevalence of high-risk individuals in the credit pool, ultimately leading to more efficient and inclusive financial markets in underserved areas.
• Expected Project Payoffs for Projects A and B
• Project A pays $75,000 with probability 9/10 and $0 with probability 1/10. The expected value (EV) is:
• EV_A = (9/10) × 75,000 + (1/10) × 0 = 67,500.
• Project B pays $100,000 with probability 3/5 and $0 with probability 2/5. The EV is:
• EV_B = (3/5) × 100,000 + (2/5) × 0 = 60,000.
• These expected values reflect the long-term profitability and risk profiles of each project, essential for decision-making under uncertainty.
• References
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