Question

13. How long will it take $800 to double if it earns the following rates? Compounding occurs once a year. Round your answers to two decimal places.

1. 6%.

   year(s)

2. 9%.

   year(s)

3. 19%.

   year(s)

4. 100%.

   year(s)

14. Find the future values of these ordinary annuities. Compounding occurs once a year. Do not round intermediate calculations. Round your answers to the nearest cent.

1. $600 per year for 10 years at 14%.

   $  

2. $300 per year for 5 years at 7%.

   $  

3. $800 per year for 5 years at 0%.

   $  

4. Rework parts a, b, and c assuming they are annuities due.

   Future value of $600 per year for 10 years at 14%: $  

   Future value of $300 per year for 5 years at 7%: $  

   Future value of $800 per year for 5 years at 0%: $  

Answer 1

13)Principal Amount = $ 800

Future Value =2*$ 800= $ 1600

We know that Future Value = Present value ( 1+i)^n

Here I = Rate of interest

n = No.of years

a)When ROI is 6%

$ 1600= $ 800( 1+0.06)^n

$ 1600/ $ 800 = ( 1.06)^n

$ 2 = 1.06^n

Applying log on both sides

log 2= log 1.06^n   [ log m^n = n log m]

log $ 2 = n log 1.06

0.30103 = n 0.025306

0.30103/ 0.025306 = n

n = 11.90 Years

It takes around 11.9 years to double the amount.

b)When ROI is 9%

$ 1600= $ 800( 1+0.09)^n

$ 1600/ $ 800 = ( 1.09)^n

$ 2 = 1.09^n

Applying log on both sides

log 2= log 1.09^n [ log m^n = n log m]

log $ 2 = n log 1.09

0.30103 = n 0.037426

0.30103/ 0.037426 = n

n = 8.04

It takes around 8.04 years to double the amount.

c)When ROI is 19%

$ 1600= $ 800( 1+0.19)^n

$ 1600/ $ 800 = ( 1.19)^n

$ 2 = 1.19^n

Applying log on both sides

log 2= log 1.19^n

log $ 2 = n log 1.19

0.30103 = n 0.075547

0.30103/ 0.075547 = n

n = 3.98

It takes around 3.98 years to double the amount.

d)When ROI is 100%

$ 1600= $ 800( 1+1)^n

$ 1600/ $ 800 = ( 2)^n

$ 2 = 2^n

Applying log on both sides

log 2= log 2^n

log $ 2 = n log 2

0.30103 = n 0.30103

n= 1

It takes around 1 years to double the amount.

14) We know that Future Value of Ordinary Annuity = C [{ ( 1+i)^n -1}/i]

Here C = Cash flow per period

I = Interest rate per period

n= No.of payments

a: $ 600 per year, for 10 years at 14%

Future value = $ 600[ { ( 1+0.14)^10 -1 } /0.14]

= $ 600[ { ( 1.14)^10 - 1} /0.14]

= $ 600[ { 3.707221-1} /0.14]

= $ 600[ 2.707221/0.14]

= $ 600*19.3373

= $ 11602.38

Future value is $ 11602.38

b: $ 300 per year, for 5 years at 7%

Future value = $ 300[ { ( 1+0.07)^5 -1 } /0.07]

= $ 300[ { ( 1.07)^5 - 1} /0.07]

= $ 300[ { 1.402552-1} /0.07]

= $ 300[ 0.402552/0.07]

= $ 300*5.750739

= $ 1725.22

Future value is $ 1725.22

c: $ 800 per year, for 5 years at 0%

Since there is no interest accrued , then the future value is nothing but whatever the amount paid

Future Value= $ 800*5 = $ 4000

Future value is $ 4000

d) We know that future value of Annuuity due = Future value of Ordinary Annuity ( 1+i)

Case 1: $ 600 per year, for 10 years at 14%

future value of Annuuity due = Future value of Ordinary Annuity ( 1+i)

= $ 11602.38 ( 1.14)

= $ 13226.71

Hence future value is $ 13226.71

Case 2: $ 300 per year, for 5 years at 7%

future value of Annuuity due = Future value of Ordinary Annuity ( 1+i)

= $ 1725.22 ( 1.07)

= $ 1845.99

Hence future value is $ 1845.99

Case 3: $ 800 per year, for 5 years at 0%

Since there is no interest accrued , then the future value is nothing but whatever the amount paid

Future Value= $ 800*5 = $ 4000

Hence future value is $ 4000.

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