Question $4$ Amortization

What level periodic payment, $x,$ is necessary to amortize and pay interest on a loan of $$1\mathrm{.}00$

over $N$ periods?

You need to find out what an amortized loan is$!$ The idea is the payments you make are such that when

you get to period $N$ you've paid back the principal amount borrowed $($which is the loan's present value$)$ and

you've paid interest on the loan $($in fact, part of $x$ goes toward principal and part is interest on the loan's

unpaid balance in each period $-$ the principal paid back increases and the amount allocated to interest

decreases as the payment periods unfold over time$).$ This structure is commonly found in home mortgage

loans and car loans.

This is a problem to find. Namely, find the unknown payment per period, $x.$

$x$ is the unknown

What is the problem's data? That is$,$ what information is given to you in the problem's statement?

The present value, here given as $$1.$

The interest rate, $r>0.$

The number of periods, $N.$

The triple $(1,r,N)$ specify the problems parameters. That is$,$ the variables $r$ and $N$ take on a fixed value

in computing a numerical answer. We'll solve for $x$ in terms of these parameters, as well as the given present

value, $$1.$ In fact, it might be better to think of the problem as having $3$ parameters, $(PV,r,N)$ where

$PV=1$ in this instance. Solve the problem for the one dollar present value and use your answer to write the

formula for an arbitrarily given $PV>0($e$.$g$.$ $$1000).$ Note that $N$ is a positive integer. The interest rate is

a percentage per unit of the time period $($for example, think of the period as measured in weeks, months or

years$).$

Part $1.$ What is the condition that connects the data and the unknowns?

Part $2.$ Use your condition to solve for the value of $x.$

Part $3.$ Part two assumed a present value of $$1.$ What is the value of the unknown $x$ when

the present value is $$1000.$

Briefly explain your answer.