Loan Repayment

Loan repayment is a typical application of sequences. Understanding the mathematics of repayment plans helps make wise financial decisions.

Two Repayment Methods

1. Equal Principal and Interest

The monthly payment is fixed and includes both principal and interest.

Features:

• The monthly payment is the same
• Interest is more in the early period, principal is more in the later period
• The total interest is higher

2. Equal Principal

The monthly principal payment is fixed, and interest decreases.

Features:

• The monthly payment decreases
• Greater payment pressure in the early period
• The total interest is lower

The Mathematical Model of Equal Principal and Interest

Let the total loan be $P$, the monthly interest rate be $r$, the number of repayment periods be $n$, and the monthly payment be $A$.

The outstanding principal $P_k$ after the $k$th month satisfies the recurrence relation:

$P_{k+1} = P_k(1+r) - A$

This is a recurrence relation of the form $a_{n+1} = qa_n + d$!

Solving gives:

The Mathematical Model of Equal Principal

Each month the principal repayment is $\frac{P}{n}$, and the interest in the $k$th month is the interest on the outstanding principal.

The payment in the $k$th month:

$A_k = \frac{P}{n} + \left[P - (k-1) \cdot \frac{P}{n}\right] \cdot r = \frac{P}{n}[1 + (n-k+1)r]$

This is an arithmetic sequence!

Practical Applications

Example 1: Mortgage Calculation

Borrow 1 million yuan at an annual rate of 4.9% for 30 years. Compare the two repayment methods.

Equal principal and interest:

Monthly rate $r = \frac{4.9\%}{12} \approx 0.00408$, number of periods $n = 360$

$A = \frac{1000000 \times 0.00408 \times 1.00408^{360}}{1.00408^{360} - 1} \approx 5307 \text{ yuan/month}$

Total repayment: $5307 \times 360 \approx 1,910,520$ yuan Total interest: about 910,000 yuan

Equal principal:

First month payment: $\frac{1000000}{360} + 1000000 \times 0.00408 \approx 6861$ yuan Last month payment: $\frac{1000000}{360} + \frac{1000000}{360} \times 0.00408 \approx 2789$ yuan

Total repayment: about 1,886,000 yuan Total interest: about 890,000 yuan

Early Repayment

If the loan is paid off in a lump sum in the $m$th month, how much must be paid?

Equal principal and interest: the outstanding principal is

$P_m = P \cdot \frac{(1+r)^n - (1+r)^m}{(1+r)^n - 1}$

Equal principal: the outstanding principal is

$P_m = P - m \cdot \frac{P}{n} = P\left(1 - \frac{m}{n}\right)$

Practice Problems

Exercise 1

Borrow 500,000 yuan at an annual rate of 5.4% for 20 years. With the equal-principal-and-interest method, what is the monthly payment?

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Summary

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