Question

Question 1:

. If we invest $500 in a bank where it will earn 8 percent interest compounded annually, how much will it be worth at the end of 7 years?

. what is the present value of $500 to be recieve10 years from today if the discount rate is 6 percent?

. How many years will take $500 to grow to $1,039.5 if invested at 5% compounded annually?

. If we invest $5000 in a bank where it will earn 10 percent compounded annually, how much will it be worth at the end of 10 years?

. what is the present value of $800 to be recieve10 years from today if the discount rate is 10 percent?

. what is the interest rate that will make 500 to grow to $1,948.00 if invested for 12 years?

Answer 1

1)

i)

For compound interest accumulated amount A is:

A = P x (1+r/m) ^mxt

P = Principal amount = $ 500

r = Interest rate = 0.08

m = Annual compounding frequency = 1

t = Number of years invested = 7

A = $ 500 x (1+0.08)⁷

   = $ 500 x (1.08)⁷

   = $ 500 x 1.71382426877952

= $ 856.91213438976 or $ 856.91

The accumulated amount will be $ 856.91 at the end of 7 years.

ii)

Present value, PV can be computed as:

PV = FV/(1+r) ⁿ

FV = Future value = $ 500

r = Interest rate = 0.06

n = Number of periods = 10

PV = $ 500/ (1+0.06)¹⁰

     = $ 500/ (1.06)¹⁰

= $ 500/ 1.79084769654285

= $ 279.197388457559 or $ 279.20

Present value is $ 279.20

iii)

A = P x (1+r/m) ^mxt

A = Accumulated amount = $ 1,039.50

P = Principal amount = $ 500

r = Interest rate = 0.05

m = Annual compounding frequency = 1

t = Number of years invested

$ 1,039.50 = $ 500 x (1+0.05) ^t

(1+0.05) ^t = $ 1,039.50 /$ 500

(1.05) ^t = 2.079

Taking log of both sides and solving for t, we get:

t x Log (1.05) = log 2.079

t x 0.02118929907= 0.31785448933

t = 0.31785448933/0.02118929907

= 15.00070806 or 15 years.

It will take 15 years for $ 500 to grow to $ 1,039.50

iv)

A = P x (1+r/m) ^mxt

P = $ 5,000; r = 0.1; m =1; t = 10

A = $ 5,000 x (1+0.1)¹⁰

   = $ 5,000 x (1.1)¹⁰

   = $ 5,000 x 2.5937424601

   = $ 12,968.7123005 or $ 12,968.71

Accumulated amount will be $ 12,968.71 at the end of 10 years.

v)

PV = FV/(1+r) ⁿ

FV = $ 800; r = 0.1; n = 10

PV = $ 800/ (1+0.1)¹⁰

     = $ 800/ (1.1)¹⁰

= $ 800/ 2.5937424601

= $ 308.434631543625 or $ 308.43

Present value is $ 308.43

vi)

A = P x (1+r/m) ^mxt

A = $ 1,948; P = $ 500; m =1; t = 12

$ 1,948 = $ 500 x (1+ r)¹²

(1+ r)¹² = $ 1,948/ $ 500

(1+ r)¹² = 3.896

1+ r = (3.896)^1/12 = (3.896) ^0.083333333

      = 1.12000057512913

r = 1.12000057512913 – 1

   = 0.12000057512913 or 12 %

Required interest rate is 12 %.