The Size of the Monthly Repayment $k$ That Amortizes a Loan Of

question 107

Short Answer

The size of the monthly repayment $k$ that amortizes a loan of $A$ dollars in $N$ years at an interest rate of $r$ per year, compounded monthly, on the unpaid balance is given by
$$k = \frac { A r } { 12 \left[ 1 - \left( 1 + \frac { r } { 12 } \right) ^ { - 12 N } \right] }$$
The value of $r$ can be found by performing the iteration
$$r _ { n + 1 } = r _ { n } - \frac { A r _ { n } + 12 k \left[ \left( 1 + \frac { r _ { n } } { 12 } \right) ^ { - 12 N } - 1 \right] } { A - 12 N k \left( 1 + \frac { r _ { n } } { 12 } \right) ^ { - 12 N - 1 } } .$$ A family secured a loan of $\$ 360,000$ from a bank to finance the purchase of a house. They have agreed to repay the loan in equal monthly installments of $\$ 2476$ over 25 years. Find the interest rate on this loan. Round the rate to one decimal place.

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Definitions:

Economic Profits

Profits exceeding the opportunity costs of all resources employed, including both explicit and implicit costs, signifying above-normal returns.

Long Run

An economic term referring to a period in which all factors of production and costs are variable, allowing for full adjustment to changes in the market.

Price-taking

A market condition in which individual buyers or sellers have no influence over the market price of a product, commonly seen in perfectly competitive markets.

Perfectly Competitive Industry

A market structure characterized by many sellers offering identical products, where no single seller can influence the market price.