Question

(a) Compute the future value of $500 after 5 years if it earns an annual interest rate of 6% compounded quarterly.

(b)Determine the monthly compounding interest rate that would cause $400 to grow to $644.89 in four years.

(c)A 20-year-old student saves $10 a day for her retirement. Every day she places $10 in a drawer. At the end of each year, she invests the accumulated savings ($3,650) into a brokerage account with an expected annual return of 8%. The last investment she will make is at the age of 64.

i. If she keeps saving in this manner, how much will she accumulate at the age of 64?

ii. If a 40-year-old investor also begins saving in the same manner, how much would he have at the age of 65?

iii. How much would the 40-year-old investor have to save each year to accumulate the same amount at the age of 65 as the 20-year-old student?

(d) Determine the value of the following cash flows at the end of year 4 with an annual interest rate of 8% compounded quarterly.

Answer 1

(a) Formula for Future value (FV) = PV(1+r/n)^nt

where

PV is present value

r is annual interest rate (in decimal)

n is number of times the interest is compounded per unit t

t is the time money is invested or borrowed for

So, according to question, PV = $500, r = 0.06, n = 4, t = 5

Hence,

FV = 500(1+0.06/4)^4*5

FV = 500(1+0.015)²⁰

FV = 500(1.015)²⁰

FV = 673.42

Future value is $ 673.42.

(b) According to question,

FV = $644.89, PV = $400, n = 12, t = 4 and we have to find the value of r i.e, monthly interest rate

So, using formula of future value

FV = PV(1+r/n)^nt

644.89 = 400(1+r/12)^4*12

1.612 = (1+r/12)⁴⁸

By trial method, r = 1.612.

(c) i. as she saves $3650 in brokerage account with interest 8%.

So, using formula of future value of annuity(FV) = P ([(1+r)ⁿ - 1)/r]

where P = periodic payment

r = rate per period

n = number of periods

According to question,

P = $3650

r = 0.08

n = 44 (64-20 = 44)

Hence, total amount she accumulates at the age of 64 is 3650[((1+0.08)⁴⁴-1)/ 0.08] = 1302866.21

Total amount she accumulates at the age of 64 is $1,302,866.21.

(c) ii.

s she saves $3650 in brokerage account with interest 8%.

So, using formula of future value of annuity(FV) = P ([(1+r)ⁿ - 1)/r]

where P = periodic payment

r = rate per period

n = number of periods

According to question,

P = $3650

r = 0.08

n = 25 (65-40 = 25)

Hence, the total amount at the age of 45 is 3650[((1+0.08)²⁵-1)/0.08] = 266836.68

The total amount at the age of 45 is $266,836.68.