Question

what is the future value of 1000 deposited for 5 years, if the interest rate is 10% per annuam compounded, daily, annually, monthly, contuously, weekly, weekly.

Answer 1

There are 5 different situations, where the interest rate has been compounded daily, annually, monthly, continuously and weekly.

First we need to find the effective interest rate in case of daily, monthly and weekly compounding.

Effective annual interest rate when compounding is done daily.

Effective interest rate = (1 + i/m) ^m -1

Where,

Nominal interest rate (i) = 10% per year

Number of compounding in a year (m) = 365

Let's put all the values in the formula

Effective interest rate = (1 + 0.1/365) ^365 - 1

                                              = (1 + 0.0002739726) ^365 - 1

                                              = (1.0002739726) ^365 - 1

                                              = 1.10516 - 1

                                              = 0.1052

So annual effective interest rate is 10.52% per year

Effective annual interest rate when compounding is done Monthly.

Nominal interest rate (i) = 10% per year

Number of compounding in a year (m) = 12

Let's put all the values in the formula

Effective interest rate = (1 + 0.1/12) ^12 - 1

                                              = (1 + 0.0083333333) ^12 - 1

                                              = (1.0083333333) ^12 - 1

                                              = 1.10471 - 1

                                              = 0.1047

So annual effective interest rate is 10.47% per year

Effective annual interest rate when compounding is done weekly.

Nominal interest rate (i) = 10% per year

Number of compounding in a year (m) = 52

Let's put all the values in the formula

Effective interest rate = (1 + 0.1/52) ^52 - 1

                                              = (1 + 0.0019230769) ^52 - 1

                                              = (1.0019230769) ^52 - 1

                                              = 1.10506 - 1

                                              = 0.1051

So annual effective interest rate is 10.51% per year

Now we will calculate the FV of the amount one by one

1.

Daily Compounding

Future value is calculated by compounding the Present cash flow

The formula is,

FV = Present value *(1 + r)^n

Where,

               Present value = $1000

               Time (n) = 5

               Interest rate [r] = 10.52%

FV = 1000*(1 + 0.1052)^5

      = 1000*(1.1052)^5

      = 1000*(1.6489382078)

      = 1648.94

So FV value of 1000 received after 5 years is 1648.94

2.

Annually Compounding

FV = Present value *(1 + r)^n

Where,

               Present value = $1000

               Time (n) = 5

               Interest rate [r] = 10%

FV = 1000*(1 + 0.1)^5

      = 1000*(1.1)^5

      = 1000*(1.61051)

      = 1610.51

So FV value of 1000 received after 5 years is 1610.51

3.

Monthly

               Present value = $1000

               Time (n) = 5

               Interest rate [r] = 10.47%

FV = 1000*(1 + 0.1047)^5

      = 1000*(1.1047)^5

      = 1000*(1.6452116268)

      = 1645.21

So FV value of 1000 received after 5 years is 1645.21

4.

Continuously compounded interest means that the principal is constantly earning interest and the interest keeps earning on the interest earned.

Formula is                                                                                                                                                          

A = P* e^(rt)                                                                                                                                                     

Where                                                                                                                                                 

Amount = 1000                                                                                                                                                

Interest rate = 0.1                                                                                                                                                           

Term = 5                                                                                                                                                             

e = a mathmatical constant having value of 2.7183                                                                                                                                                           

Let’s put all the values in the formula                                                                                                                                    

= 1000* 2.7183^(0.1 * 5)                                                                                                                                              

= 1000* 2.7183^(0.5)                                                                                                                                                     

= 1000* 1.64872678148928                                                                                                                                          

= 1648.73

5.

Weekly

FV = Present value *(1 + r)^n

Where,

               Present value = $1000

               Time (n) = 5

               Interest rate [r] = 10.51%

FV = 1000*(1 + 0.1051)^5

      = 1000*(1.1051)^5

      = 1000*(1.6481923519)

      = 1648.19

So FV value of 1000 received after 5 years is 1648.19

Hope that helps.

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