Find i​ (the rate per​ period) and n​ (the number of​ periods) for the following loan at the given annual rate. Annual payments of ​$4,700 are made

Question

Find i​ (the rate per​ period) and n​ (the number of​ periods) for the following loan at the given annual rate.

Annual payments of ​$4,700 are made for 10 years to repay a loan at 4.85​% compounded annually.

i= ______ (type an integer or a decimal_

n = __________

Answer 1

Given that the payments are made once a year and that the interest is compounded annually too, then we know that each period is 1 year and the:

1. Rate per period (i) is just the annual rate given in decimal form:

i = 4.85% = 4.85 / 100 = 0.0485

1. Number of periods (n) equals the number of years (since 1 payment per year):

n = 10

So, rate per period (i) is 0.0485 and number of periods (n) is 10.

Now, if you want to further find the original loan amount (present value), since the loan is being repaid through equal annual payments over time, we can apply the present value of an ordinary annuity formula here which is:

PV = PMT x ((1-(1+i)^-n)/i) where,

PV - original loan amount

PMT - regular payment

i - interest rate per period

n - number of periods

Lets apply it by plugging in the values for this problem:

PMT = 4700

i = 0.0485

n = 10

PV = 4700 x ((1 - (1 + 0.0485)^-10)) / 0.0485)

= 4700 x 7.79

PV = 36,613

So, the original amount is approximately $36,613.

Answer 2

i = r = 4.85% = 0.0485; n = 1.

Then value of loan PV = PMT[1 - (1 + i)^-nt]/i = 4700[1 - (1 + 0.0485)^-10]/0.0485 = $36557.98