(Compound interest with? non-annual periods?) You just received a bonus of ?$3,000.

a.  Calculate the future value of ?$3,000?, given that it will be held in the bank for 5 years and earn an annual interest rate of 4 percent.

b.  Recalculate part ?(a?) using a compounding period that is? (1) semiannual and? (2) bimonthly.

c.  Recalculate parts ?(a?) and ?(b?) using an annual interest rate of 8 percent.

d.  Recalculate part ?(a?) using a time horizon of 10 years at an annual interest rate of 4 percent.

e.  What conclusions can you draw when you compare the answers in parts ?(c?) and ?(d?) with the answers in parts ?(a?) and ?(b?)?

 

a. FV = 3000 (1 + 0.04)⁵

= 3000 (1.04)⁵

= 3000 * 1.2167

= 3650.1

 Future value of ?$3,000? for 5 years @ an annual interest rate of 4% = $3,650.1

b.

(1) FV = 3000 (1 + 0.02)¹⁰

= 3000 (1.02)¹⁰

= 3000 * 1.2189

= 3656.7

 Future value of ?$3,000? for 5 years @ an annual interest rate of 4% compounded semi-annually = $3,656.7

(2) FV = 3000 (1 + 0.0067)³⁰

= 3000 (1.0067)³⁰

= 3000 * 1.2218

= 3665.4

Future value of ?$3,000? for 5 years @ an annual interest rate of 4% compounded bi-monthly = $3,665.4

c.

(1) FV = 3000 (1 + 0.08)⁵

= 3000 (1.08)⁵

= 3000 * 1.4693

= 4407.9

 Future value of ?$3,000? for 5 years @ an annual interest rate of 8% compounded semi-annually = $4,407.9

(2) FV = 3000 (1 + 0.04)¹⁰

= 3000 (1.04)¹⁰

= 3000 * 1.4802

= 4440.6

 Future value of ?$3,000? for 5 years @ an annual interest rate of 8% compounded semi-annually = $4,440.6

(3) FV = 3000 (1 + 0.0133)³⁰

= 3000 (1.0133)³⁰

= 3000 * 1.4864

= 4459.2

Future value of ?$3,000? for 5 years @ an annual interest rate of 8% compounded bi-monthly = $4,459.2

d. FV = 3000 (1 + 0.04)¹⁰

= 3000 (1.04)¹⁰

= 3000 * 1.4802

= 4440.6

 Future value of ?$3,000? for 10 years @ an annual interest rate of 4% = $4,440.6

e. The future value of ?$3,000? calculated in parts ?(c?) and ?(d?) for 5 years @ an annual interest rate of 8% and for 10 years @ an annual interest rate of 4% respectively is more than the future value of ?$3,000 calculated in parts ?(a?) and ?(b?)? for 5 years @ an annual interest rate of 4%. Therefore, an annual interest rate of 8% compounded bi-monthly for a time period of 5 years provide maximum benefit. Also, The future value of ?$3,000? for 5 years @ an annual interest rate of 8% and future value of ?$3,000? for 5 years @ an annual interest rate of 4% for 10 years yield equal future value i.e. $4,440.6

Step-by-step explanation

FV = PV (1 + I) ^t

Here, PV = the present value of the investment

FV = the future value of the investment after the number of periods, the deposit is invested

I = the interest earned on the investment

t = the number of time periods, the deposit remains invested

For calculation of interest rate in part (b) and (c) :-

While compounding semi-annually, the annual interest rate is divided by 2 and the number of time periods is multiplied by 2

& in case of bi-monthly compounding, the annual interest rate is divided by 6 and the number of time periods is multiplied by 6.