Question

You are given two loans, with each loan to be repaid by a single payment in the future. Each payment includes both principal and interest.
The first loan is repaid by a 3000 payment at the end of four years. The interest is accrued at an annual nominal rate of discount equal to 5% compounded semiannually.
The second loan is repaid by a 4000 payment at the end of five years. The interest is accrued at an annual nominal rate of interest equal to 5% compounded quarterly.
These two loans are to be consolidated. The consolidated loan is to be repaid by two equal installments of X, with interest accruing at an annual effective rate of 5%. The first payment is due immediately (i.e. at time t = 0 years), and the second payment is due at the end of the first year (i.e. at time t = 1 year).

Calculate X. Give your answer rounded to the nearest whole number.

(please make sure answer is right as all answers on chegg are incorrect)

Answer 1

Answer)

Calculation of amount of each payment (i.e. Value of X)

Amount of consolidated loan = Amount of Loan 1 + Amount of Loan 2

                                                       = 2,462.25 + 3120.04

                                                       = 5,582.29

The above consolidated loan is to be repaid in two annual installments of ‘X’ each (interest is charged 5% compounded annually). Therefore, the present value of these two payments will be equal to the amount of consolidated loan.

5,582.29 = Present value of X payable immediately + Present value of X Payable one year from now at 5% annually

5,582.29 = X + (X) x (present value of $ 1 at 5% at the end of 1 year)

5,582.29 = X + (X) x (0.95238)

5,582.29 = 1.95238 X

X = 2,859.22 or 2,859 (approximately)

Therefore the value if X is 2,859.

Working Note:

Calculation of value of Loan -1 (for which 3,000 is to be paid at the end of four years at 5% compounded semiannually)

The amount of loan can be calculated by calculating the present value of the repayment.

Amount of Loan = 3,000 X Present value of $ 1 at 2.5% for 8 periods

                             = 3,000 X 0.82075

                             = 2,462.25

Note: Since interest is compounded semiannually, the effective rate of interest will be half (i.e. 5%/2 = 2.5%) of the annual rate of interest and the number of periods will be doubled (i.e. 4 years X 2 = 8 semiannual periods).

Calculation of value of Loan -2 (for which 4,000 is to be paid at the end of five years at 5% compounded quarterly)

The amount of loan can be calculated by calculating the present value of the repayment.

Amount of Loan = 4,000 X Present value of $ 1 at 1.25% for 20 periods

                             = 4,000 X 0.78001

                             = 3,120.04

Note: Since interest is compounded quarterly, the effective rate of interest will be divided by 4 (i.e. 5%/2 = 1.25%) of the annual rate of interest and the number of periods will be four times (i.e. 5 years X 4 = 20 quarterly periods).