Question

6. For each problem, fill in the formula, label your answer, and round it to two decimal places. a) Find the amount in a savings plan after 3 years with an APR of 3% and monthly deposits of $100. b) Calculate the monthly payment for a student loan of $25,000 at a fixed APR of 6% if it is to be paid off in 10 years. c) Your goal is to create a college fund for your child. Suppose you find a fund that offers an APR of 5%. How much should you deposit monthly to accumulate $80,000 in 18 years? d) $13,0000 is invested for 5 years in an account with an APR of 4%. How much will be in the account if interested is compounded quarterly? How much if it is compounded continuously? Quarterly: Continuously:

Answer 1

6. a). The future value (F) of an annuity is given by F = (P/r)[(1+r)ⁿ-1], where P is the periodic payment , r is the rate per period and n is the number of periods.

Here, P = $100, r = 3/1200 = 0.0025 and n = 3*12 = 36.

Therefore, F = (100/0.0025)[(1.0025)³⁶-1] = 40000*0.0940514 = $ 3762.06 ( on rounding off to the nearest cent).

Thus, the amount in savings after 3 years will be $ 3762.06.

b). The formula used to calculate the fixed monthly payment (P) required to fully amortize a loan of $ L over a term of n months at a monthly interest rate of r is

P = L[r(1 + r)ⁿ]/[(1 + r)ⁿ - 1].

Here, L = $ 25000, r = 6/1200 = 0.005 and n = 12*10 = 120.

Then, P = 25000*0.005 *[(1.005)¹²⁰ ]/[ (1.005)¹²⁰ -1] = 125*1.819396734/0.819396734 = $ 277.55 ( on rounding off to the nearest cent).

Thus, the required fixed monthly payment is $ 277.55.

c). The future value (F) of an annuity is given by F = (P/r)[(1+r)ⁿ-1], where P is the periodic payment , r is the rate per period and n is the number of periods.

Here, F = $ 80000, r = 5/1200 = 1/240 and n = 12*18 = 216.

Then 80000 = 240P[(1+1/240)²¹⁶ -1] = 240P * 1.455008425 so that P = 80000/240*1.455008425 = $ 229.09 ( on rounding off to the nearest cent).

Thus, the amount required to be deposited monthly is $ 229.09.

d). i. The formula for calculating the maturity value (F) of an investment, when the interest is compounded, is F =                 P[(1 + r/t)^nt ] where P is the principal, r is the annual interest rate in decimals, t is the number of times the interest is compounded in an year and n is number of years.

Here, P = $ 130000, r = 0.04, t = 4 and n = 5. Hence, F = 130000*(1+0.04/4)^5*4= 130000*[(1+0.01)²⁰ = 130000* 1.22019004 = $ 158624.71( on rounding off to the nearest cent).

Thus, if the interest is compounded quarterly, then there will be $ 158624.71 in the account after 5 years.

ii. The formula for calculating the maturity value (F) of an investment, when the interest is compounded continuously, is F = Pe^rn, where P is the principal, r is the annual interest rate in decimals and n is number of years.

Here, P = $ 130000, r = 0.04 and n = 5. Hence, F = 130000*e^0.04*5 = 130000* e^0.20 = 130000* 1.221402758 = 158782.36            ( on rounding off to the nearest cent).

Thus, if the interest is compounded continuously, then there will be $ 158782.36 in the account after 5 years.