Question

A year ago Alisha loaned $12,000 to Chang at the rate of 4% compounded quarterly. Chang was required to repay this amount in three equal yearly installments, with payments starting from the end of the first year. How much interest did Aisha earn from this transactions?

Answer 1

First we need to find effective annual interest rate.

The effective annual interest rate is the interest rate that is actually earned or paid on an investment, loan or other financial product due to the result of compounding over a given time period. It is also called the effective interest rate, the effective rate or the annual equivalent rate

So if nominal interest rate (i), number of compounding in a year is (m), effective interest will be

Effective interest rate = (1 + i/m) ^m -1

Where,

Nominal interest rate (i) = 4% per year

Number of compounding in a year (m) = 4

Let's put all the values in the formula

Effective interest rate = (1 + 0.04/4) ^4 - 1

                                              = (1 + 0.01) ^4 - 1

                                              = (1.01) ^4 - 1

                                              = 1.0406 - 1

                                              = 0.0406

So annual effective interest rate is 4.06% per year

Now we will calculate yearly loan payment amount

If the loan amount is P, rate on interest (monthly is r, and loan term is n the EMI will be

EMI = P*r[(1 +r)^n]/ [(1+ r)^n- 1]

Where,

              Loan amount (P) = $12000

                Time (n) = 3 Period

               Interest rate [r] = 4.06% /Period

Let's put all the values in the formula to calculate EMI

EMI = 12000*0.0406[(1 +0.0406)^3]/ [(1+ 0.0406)^3- 1]

        = 487.2[(1.0406)^3]/ [(1.0406)^3- 1]

        = 487.2[1.1268120034]/ [1.1268120034- 1]

      = 487.2[1.1268120034]/ [0.1268120034]

        = 487.2[8.88568884008342]

        = 4329.11

So Annual payment will be $4329.11