Question

Find the future value of the following annuities. The first payment in these annuities is made at the end of Year 1, so they are ordinary annuities. Round your answers to the nearest cent. (Notes: If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can "override" the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in many situations, to see how changes in input variables affect the output variable. Also, note that you can leave values in the TVM register, switch to Begin Mode, press FV, and find the FV of the annuity due.)

1. $800 per year for 10 years at 8%.
   $  
   
2. $400 per year for 5 years at 4%.
   $  
   
3. $800 per year for 5 years at 0%.
   $  

Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

4. $800 per year for 10 years at 8%.
   $  
   
5. $400 per year for 5 years at 4%.
   $  
   
6. $800 per year for 5 years at 0%.
   $  

Answer 1

Method I

Let us first solve the Question without the use of financial calculator:

Part A :Calculation of Future value of Annuity if the payments are done at the end of each year

Formula:

FV= PMT * [ ((1+i)ⁿ-1) / i }

Here, FV= future Value

PMT= payment made annually

   n= No. of years

i = interest rate

Solution

a.

Here PMT=800$ , i=8%(=0.08) , n=10 (years)

FV= 800*[ ( (1+0.08)¹⁰ - 1 ) / 0.08]

FV=800*14.4866 (rounded to 4 digits after 0. )

FV=11,589.25$

b.

Here PMT=800$ , i=4%(=0.04) , n=5 (years)

FV= 800*[ ( (1+0.04)⁵ - 1 ) / 0.04]

FV=800*5.4163 (rounded to 4 digits after 0. )

FV=4333.06$

c.

Here PMT=800$ , i=0%(=0.00) , n=5 (years)

It is to be noted that since i=0 the formula will not hold good ,

further, when interest rate is zero, FV of annuity =PMT*n

FV=800*5

FV=4000.00$

Part B :Calculation of Future value of Annuity if the payments are done at the beginning of each year

Formula:

FV= PMT * [ ((1+i)ⁿ-1) / i ] * (1+i)

Here, FV= future Value

PMT= payment made annually

   n= No. of years

i = interest rate

Solution

a.

Here PMT=800$ , i=8%(=0.08) , n=10 (years)

FV= 800*[ ( (1+0.08)¹⁰ - 1 ) / 0.08] * (1+0.08)

FV=800*14.4866*1.08

FV=800*15.6455 (rounded to 4 digits after 0. )

FV=12516.4$

b.

Here PMT=800$ , i=4%(=0.04) , n=5 (years)

FV= 800*[ ( (1+0.04)⁵ - 1 ) / 0.04] *(1.04)

FV=800*5.4163*1.04

FV=800*5.6330 (rounded to 4 digits after 0. )

FV=4506.40$

c.

Here PMT=800$ , i=0%(=0.00) , n=5 (years)

It is to be noted that since i=0 the formula will not hold good ,

further, when interest rate is zero, FV of annuity =PMT*n

FV=800*5

FV=4000.00$

Method II

Now let us solve the Question with the use of financial calculator:

Part A :Calculation of Future value of Annuity if the payments are done at the end of each year

Terms in financial calculator :

N= Number of payment periods , I/Y = Interest rate per period (here per year) ,PV= present value

PMT=Annuity amount per period ,FV= Future value.

Before we begin the operations, to clear memory , press CLR TVM in the calculator

Part A :Calculation of Future value of Annuity if the payments are done at the end of each year

Solution

a.

Input PMT= - 800 , I/Y=8 , N=10 , PV= 0

Then compute FV ,it shall give

FV=11,589.25$

b.

Input PMT= - 800 , I/Y=4 , N=5 , PV= 0

Then compute FV ,it shall give

FV=4333.06$

c.

Input PMT= - 800 , I/Y=0 , N=5 , PV= 0

Then compute FV ,it shall give

FV=4000.00$

Part B :Calculation of Future value of Annuity if the payments are done at the beginning of each year

Before we do this, we have to change the calculator to bgn mode ,which might differ based on the calculator used.

Display of the calculator should show BGN

Solution

a.

Input PMT= - 800 , I/Y=8 , N=10 , PV= 0

Then compute FV ,it shall give

FV=12516.4$

b.

Input PMT= - 800 , I/Y=4 , N=5 , PV= 0

Then compute FV ,it shall give

FV=4506.40$

c.

Input PMT= - 800 , I/Y=0 , N=5 , PV= 0

Then compute FV ,it shall give

FV=4000.00$