Question

A ten year loan of $10,000 at 8% annual effective can be repaid using any of the four following methods: (i) Amortization method, with annual payments at the end of each year. (ii) Repay the principal at the end of ten years while paying the 8% annual effective interest on the loan at the end of each year. In addition, make level annual deposits at the end of each year into a sinking fund earning 6% annual effective so that the sinking fund accumulates to $10,000 at the end of the 10th year. (iii) Same as ii, except that the sinking fund earns 8% annual effective. (iv) Same as ii, except that the sinking fund earns 12% annual effective. Rank the annual payment amounts of each method.

Answer 1

ANSWER

(I) Amortization method, with annual payments at the end of each year.

Annual payments at the end of each year = Loan Amount/((1-(1+r)^-n)/r)

Annual payments at the end of each year = 10000/((1-(1+8%)^-10)/8%)

Annual payments at the end of each year = $ 1490.29

(II) Repay the principal at the end of ten years while paying the 8% annual effective interest on the loan at the end of each year. In addition, make equal annual deposits at the end of each year into a sinking fund earning 6% annual effective so that the sinking fund accumulates to 10,000 at the end of the 10th year.

Annual Interest Payment = Loan Amount*Interest Rate

Annual Interest Payment = 10000*8%

Annual Interest Payment = 800

Equal annual deposits at the end of each year into a sinking fund = Loan Amount/(((1+r)^n-1)/r)

Equal annual deposits at the end of each year into a sinking fund = 10000/(((1+6%)^10-1)/6%)

Equal annual deposits at the end of each year into a sinking fund = $ 758.68

Annual payments at the end of each year =Annual Interest Payment + Equal annual deposits at the end of each year into a sinking fund

Annual payments at the end of each year = 800 + 758.68

Annual payments at the end of each year = $ 1558.68

(III) Same as (II), except the sinking fund earns 8% annual effective.

Annual Interest Payment = Loan Amount*Interest Rate

Annual Interest Payment = 10000*8%

Annual Interest Payment = 800

Equal annual deposits at the end of each year into a sinking fund = Loan Amount/(((1+r)^n-1)/r)

Equal annual deposits at the end of each year into a sinking fund = 10000/(((1+8%)^10-1)/8%)

Equal annual deposits at the end of each year into a sinking fund = $ 690.29

Annual payments at the end of each year =Annual Interest Payment + Equal annual deposits at the end of each year into a sinking fund

Annual payments at the end of each year = 800 + 690.29

Annual payments at the end of each year = $ 1490.29

(IV) Same as (II), except the sinking fund earns 12% annual effective.

Annual Interest Payment = Loan Amount*Interest Rate

Annual Interest Payment = 10000*8%

Annual Interest Payment = 800

Equal annual deposits at the end of each year into a sinking fund = Loan Amount/(((1+r)^n-1)/r)

Equal annual deposits at the end of each year into a sinking fund = 10000/(((1+12%)^10-1)/12%)

Equal annual deposits at the end of each year into a sinking fund = $ 569.84

Annual payments at the end of each year =Annual Interest Payment + Equal annual deposits at the end of each year into a sinking fund

Annual payments at the end of each year = 800 + 569.84

Annual payments at the end of each year = $ 1369.84

Ranking:

IV ---------Rank 1

I and III-----------------Rank 2

II ----------------Rank 3

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