Sample Problems and Solutions for Loan Receivables

Introduction

Loan receivables are a crucial component of financial management for both lenders and borrowers. Understanding how to handle loan receivables involves grasping concepts such as amortization, interest calculations, and the management of default risks. This article will delve into several sample problems related to loan receivables and provide detailed solutions to each. These problems will illustrate the practical application of various financial principles and help enhance your understanding of managing loan receivables effectively.

Problem 1: Calculating Monthly Loan Payments

Scenario:
Suppose you have taken a loan of $50,000 with an annual interest rate of 6%, to be repaid over 5 years. You need to calculate the monthly payment amount.

Solution:
To determine the monthly payment, we use the formula for an amortizing loan:

$M=\frac{P\cdot r\cdot (1+r{)}^n}{(1+r{)}^n-1}$M=(1+r)n−1P⋅r⋅(1+r)n​

Where:

• $M$M = monthly payment
• $P$P = loan principal ($50,000)
• $r$r = monthly interest rate (annual rate / 12) = 6% / 12 = 0.005
• $n$n = total number of payments (years * 12) = 5 * 12 = 60

Plugging in the values:

$M=\frac{50,000\cdot 0.005\cdot (1+0.005{)}^{60}}{(1+0.005{)}^{60}-1}$M=(1+0.005)60−150,000⋅0.005⋅(1+0.005)60​

$M=\frac{50,000\cdot 0.005\cdot 1.34885}{1.34885-1}$M=1.34885−150,000⋅0.005⋅1.34885​

$M=\frac{50,000\cdot 0.00674225}{0.34885}$M=0.3488550,000⋅0.00674225​

$M=\frac{337.1125}{0.34885}\approx 967.55$M=0.34885337.1125​≈967.55

Thus, the monthly payment is approximately $967.55.

Problem 2: Determining the Total Interest Paid Over the Loan Term

Scenario:
Using the same loan details as in Problem 1, determine the total interest paid over the life of the loan.

Solution:
First, calculate the total amount paid over the term of the loan:

$\text{Total Payments}=M\cdot n$Total Payments=M⋅n

$\text{Total Payments}=967.55\cdot 60=58,053$Total Payments=967.55⋅60=58,053

The total interest paid is:

$\text{Total Interest}=\text{Total Payments}-P$Total Interest=Total Payments−P

$\text{Total Interest}=58,053-50,000=8,053$Total Interest=58,053−50,000=8,053

Thus, the total interest paid over the term of the loan is $8,053.

Problem 3: Calculating the Remaining Balance After 2 Years

Scenario:
For the same loan, calculate the remaining balance after 2 years of payments.

Solution:
To find the remaining balance, use the formula for the loan balance after a certain number of payments:

$B=P\cdot \frac{(1+r{)}^n-(1+r{)}^p}{(1+r{)}^n-1}$B=P⋅(1+r)n−1(1+r)n−(1+r)p​

Where:

• $B$B = remaining balance
• $p$p = number of payments made (2 years * 12) = 24

Plugging in the values:

$B=50,000\cdot \frac{(1+0.005{)}^{60}-(1+0.005{)}^{24}}{(1+0.005{)}^{60}-1}$B=50,000⋅(1+0.005)60−1(1+0.005)60−(1+0.005)24​

$B=50,000\cdot \frac{1.34885-1.12749}{0.34885}$B=50,000⋅0.348851.34885−1.12749​

$B=50,000\cdot \frac{0.22136}{0.34885}$B=50,000⋅0.348850.22136​

$B=50,000\cdot 0.6346=31,730$B=50,000⋅0.6346=31,730

Thus, the remaining balance after 2 years is approximately $31,730.

Problem 4: Assessing the Impact of Early Repayment

Scenario:
You decide to make an extra payment of $5,000 towards the principal after 3 years. How will this affect your loan balance and the total interest paid?

Solution:
First, calculate the new loan balance after 3 years:

$p=36\text{ (payments made)}$p=36 (payments made)

$B=50,000\cdot \frac{(1+0.005{)}^{60}-(1+0.005{)}^{36}}{(1+0.005{)}^{60}-1}$B=50,000⋅(1+0.005)60−1(1+0.005)60−(1+0.005)36​

$B=50,000\cdot \frac{1.34885-1.18376}{0.34885}$B=50,000⋅0.348851.34885−1.18376​

$B=50,000\cdot \frac{0.16509}{0.34885}$B=50,000⋅0.348850.16509​

$B=50,000\cdot 0.4738=23,690$B=50,000⋅0.4738=23,690

With an additional payment of $5,000:

New principal = $23,690 - $5,000 = $18,690

Recalculate the remaining payments based on the new principal. Assuming the same interest rate and remaining term of 24 months:

$M_{new}=\frac{18,690\cdot 0.005\cdot (1+0.005{)}^{24}}{(1+0.005{)}^{24}-1}$Mnew​=(1+0.005)24−118,690⋅0.005⋅(1+0.005)24​

$M_{new}=\frac{18,690\cdot 0.005\cdot 1.12749}{1.12749-1}$Mnew​=1.12749−118,690⋅0.005⋅1.12749​

$M_{new}=\frac{104.25}{0.12749}\approx 817.31$Mnew​=0.12749104.25​≈817.31

Thus, with the additional payment, your new monthly payment will be approximately $817.31. The total interest paid will decrease as a result of this early repayment.

Conclusion

Managing loan receivables requires a solid understanding of various financial calculations and their implications. By working through these sample problems, you can gain valuable insights into how to handle different scenarios involving loan receivables. Mastering these concepts will not only help in effective financial planning but also in making informed decisions regarding loan management.