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 UNIT 11  :  MATHEMATICS OF INVESTMENT

 LESSON 4: PRESENT VALUE ANNUITY

 

Definition 1: A sequence of payments made at regular intervals is called an annuity. 

Definition 2: When we calculate the present values of the sequence of payments made at regular intervals this is called the Present Value of the annuity. 

 

A Present Value annuity has the following properties.

Text Box: ·	The first payment is always at the end of the first interest period of the annuity.  This will be numbered 1 on your time line.
·	The accumulated sum of the present values of each payment at the end of the annuity’s term is called the present value of the annuity. This will be at the number 0 on your time line.
·	 
·	This accumulated sum forms a geometric series (see below).
·	A time line is very helpful in illustrating an annuity.
·	You can use either the geometric series formula OR the present value of an annuity formula (see below) to find the present value of an annuity.
 

 

 

 

 

 

 

 

 

 

 

 

 

Text Box: P = principal [amount borrowed                                
       or invested]
n = number of interest periods
i  = interest rate per interest period as  
      as a decimal
A = accumulated amount (due or payable)
Text Box: Formulas:
 

 


                                                                                                                       

 

 

 

 

 

 

Text Box: a = the regular payment of the annuity 
n = number of payments or terms
Text Box: Geometric Series Formula:
 

 

 

 

 

 

 

 

 

Text Box: R = the regular payment of the annuity
n  = the number of payments or terms 
i   = interest rate per interest period
PV = the Present Value of the annuity at the time of the last payment
Text Box: Present Value of an Annuity Formula:
 

 

 

 

 

 

 

 

 


Example 1: Finding the Present Value of an annuity.

Heywood recently won $5 000 000 in the lottery.  He plans to purchase an annuity that will pay him $50 000 every year for 25 years and spend the rest.  How much of his winnings would he need to pay today for that annuity  if  interest is 7.6%/a, compounded annually?  Include the series and a time line diagram in the solution.

Solution:

We calculate the present values of the 25 future payments of $50 000 each.  Notice the arrows go to the left for a present value annuity.

 

Interest Period   0          1         2          3                                                                                                                                         23   24    25     

                                                                                                                                                                           

Payment( 1000’s)           50       50                                                                                                                                                   50    50    50

                                                           

50000(1.076)-1                                                                                                                                                                                                     

50000(1.076)-2                                                                                                                                                                                                                                                                                                                                                     

       .

       .

50000(1.076)-23     

                                                                                                                                                                                   

50000(1.076)-24                                                                                                                                                                            

 

                                                                                                                                                                             

50000(1.076)-25

 

This forms the following geometric series:  Note – write the last term first.

50 000(1.076)-25 + 50000(1.076)-24 +  . . . + 50000(1.076)-2 + 50000(1.076)-1

 

 

 

Hence Michael needs $552 492.55 of his winnings to purchase the annuity. He would have $5 000 000 - $ 552 492.55 = $4 447 507.45 left to spend.

 

Alternate Solution:

 

Notice this present value amount is much less than the total amount paid out over 25 years which would be 25 x $50 000=$1 250 000.

 

 

Example 2: Finding the Payment for a Present Value annuity.

Richard Rummy recently retired from Rofasco Inc.  He received a retirement gratuity from profit sharing of $300 000.  He wishes to purchase an annuity that will pay him a fixed income every month for 25 years.  If interest is 8.4%/a, compounded monthly, determine his monthly income

 

Solution:

Let the monthly payment be $R, with the first payment at the end of the first month.

 

Interest Period   0          1         2          3                                                                                                                                         298   299   300     

                                                                                                                                                                           

Payment( 1000’s)           R         R        R                                                                                                                                          R      R      R

                                                           

        R(1.007)-1                                                                                                                                                                                                     

        R(1.007)-2                                                                                                                                                                                                                                                                                                                                                     

       .

       .

       R(1.007)-298     

                                                                                                                                                                                   

       R(1.007)-299                                                                                                                                                                           

 

                                                                                                                                                                             

      R(1.007)-300

 

 

This forms the following geometric series:

            R(1.007)-300 + R(1.007)-299 + . . . + R(1.007)-2 + R(1.007)-1    

 

Hence  Richard will receive a monthly income of $2395.50 for 25 years from his gratuity by purchasing an annuity.

 

Example 3: Borrowing money – finding the monthly payment.

Tanya plans to borrow $30 000 to use as a down payment on the purchase of her dream home.   She will repay the loan with monthly payments over a 10 year period.  How much will the monthly payment be if  interest is 4.2%/a, compounded monthly?  Include a time line diagram in the solution.

Solution:

As in the previous question, the money in question is borrowed now – at point 0 on the time line.  Hence this is a PV annuity question

 

Interest Period   0          1         2          3                                                                                                                                         118   119  120     

                                                                                                                                                                           

Payment( 1000’s)           R         R        R                                                                                                                                          R      R      R

                                                           

        R(1.0035)-1                                                                                                                                                                                                   

        R(1.0035)-2                                                                                                                                                                                                                                                                                                                                                   

       .

       .

       R(1.0035)-118     

                                                                                                                                                                                   

       R(1.0035)-119                                                                                                                                                                          

 

                                                                                                                                                                             

      R(1.0035)-120

 

 

This forms the following geometric series:

            R(1.0035)-120 + R(1.0035)-119 + . . . + R(1.0035)-2 + R(1.0035)-1        

 

 

 

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