The monthly payment, m dollars, for a 30 -year fixed rate...

The monthly payment, $m$ dollars, for a 30 -year fixed rate mortgage is a function of the total amount borrowed, $P$ dollars, and the annual interest rate, $r \% .$ In other words, $m=f(\boldsymbol{P}, \boldsymbol{r})$ (a) Interpret the following statement in the context of monthly payment: $f(300,000,5)=1610.46$ (b) If $P$ is held constant, is $m$ an increasing or a decreasing function of $r ?$ Why? (c) If $r$ is held constant, is $m$ an increasing or a decreasing function of $P ?$ Why?

The monthly payment, $m$ dollars, for a 30 -year fixed rate mortgage is a function of the total amount borrowed, $P$ dollars, and the annual interest rate, $r \% .$ In other words, $m=f(\boldsymbol{P}, \boldsymbol{r})$
(a) Interpret the following statement in the context of monthly payment: $f(300,000,5)=1610.46$
(b) If $P$ is held constant, is $m$ an increasing or a decreasing function of $r ?$ Why?
(c) If $r$ is held constant, is $m$ an increasing or a decreasing function of $P ?$ Why?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Function Evaluation in Financial Context

This concept involves interpreting a function that represents a financial quantity, where specific input values (such as principal and interest rate) yield an output (the monthly payment). It requires understanding that the numerical result is generated by substituting the given parameters into a formula that encapsulates the relationship between the borrowed amount, the interest rate, and the payment over the term of the mortgage.

Dependence on Interest Rate (Monotonicity)

This concept addresses how the monthly payment changes as the interest rate varies while the amount borrowed remains constant. Typically, an increase in the interest rate will result in higher monthly payments because the cost of borrowing money becomes more expensive. This is an example of analyzing the monotonic behavior of a function with respect to one of its variables.

Dependence on Principal (Monotonicity)

This concept is concerned with how the monthly payment changes in response to variations in the total amount borrowed, while the interest rate remains fixed. The relationship is generally directly proportional, so increasing the principal leads to a higher monthly payment. This analysis helps in understanding the sensitivity of the payment to changes in the loan amount.

Transcript

Please give Ace some feedback