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 UNIT 11  :  MATHEMATICS OF INVESTMENT

 LESSON 4: PRESENT VALUE ANNUITY HOMEWORK QUESTIONS  PAGE 1

 

Quick Review

Present Value of an  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. 

When a lump sum of money is deposited or borrowed today in order to receive a series of payments in the future, this is a PV annuity

A Present Value annuity has the following properties.

Text Box: ·	When a lump sum of money is deposited or borrowed today in order to receive a series of payments in the future, this is a PV annuity
·	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:
 

 

 

 

 

 

 

 

 


Homework Questions: (Solutions below)

1.  Evaluate each of the following annuities using the geometric series formula.  Remember to write the series backwards – last term first.

     Include a complete time line diagram for # a).

a) 

Interest Period   0          1         2          3                                                                                                                                         23   24    25  years   

                                                                                                                                                                           

Payment                         500      500                                                                                                                                                500   500  500

Rate of interest is 6.65%/a, compounded annually

 

b) 

Interest Period   0                     1                       2                                                                                                                             23            24  years

                                                                                                                                                                           

Payment                         250     250       250      250                                                                                                                          250  250  250

Rate of interest is 6.6%/a, compounded  semi-annually

 

c) 

Interest Period   0                                              1                                              2              .  .  .                                         6                              7  years                     

                                                                                                                                                                            

Payment                       800       800       800      800      800      800      800     800                                                         800   800  800   800  800

Rate of interest is 8.4%/a, compounded quarterly

 

2.  Evaluate each of the following annuities using the PV annuity formula above.    Include a complete time line diagram for each.

a)  150(1.06)-12 + 150(1.06)-11 + . . . + 150(1.06)-2 + 150(1.06)-1

 

b)  300(1.045)-19 + 300(1.045)-18 + . . . + 300(1.045)-2 + 300(1.045)-1

 

 

3.  Find the present value of each of the following annuities.  The interest rate is 4.8%/a, compounded monthly.  The first payment will be at the end of the first month.

a)  $500 per month for 48 months.

 

b)  $750 per month for 12 ½ years.

 

c)  $250 per month for 20 ¼ years.

 

4.  Find the present value of each of the following annuities.  The interest rate is 5.4%/a, compounded quarterly.  The first payment will be at the end of the first 3 month period.

a) $1000 every 3 months for 10 years.

 

b)  $1500 every ¼ year for 15 ½ years.

 

c)  $800 every ¼ year for 20 years.

 

5.  The Witmer foundation wishes to establish an academic athletic scholarship to be awarded each year for 25 years.  The scholarship will be worth $1500 per year.  How much should be deposited now in a trust fund that pays 6.5%/a, compounded annually?

 

6.  Mr. I. M. Generus donated $100 000 to minor hockey in his home town.  It is to be paid out over a 10 year period starting one year from now.  How much will be paid out each year if interest is 5.4%/a, compounded annually?

 

7.  Betty won $2 000 000 in a recent lottery.  If she uses the funds to purchase an annuity over 35 years, what monthly payment will she receive if interest is 6%/a, compounded monthly?

 

8.  Mrs. Peres purchased a car for $19 900 including all taxes.  She wishes to finance the purchase over 5 years.  If interest is 9.6%/a, compounded monthly, what will her monthly payment be?

 

9.  Find the purchase price of an annuity that pays $4000 every 6 months for 15 years if interest is 6.6%/a, compounded semi-annually.

 

10.  Mr. Cameron purchased a new tractor for his farm for $80 000.  He paid $5000 down and financed the rest over 10 years at 10.2%/a, compounded monthly. Determine his monthly payment and the finance charge.

 

Solutions:

1.  Evaluate each of the following annuities using the geometric series formula.  Remember to write the series backwards – last term first.

     Include a complete time line diagram for # a).

a) 

Interest Period   0          1         2          3                                                                                                                                         23   24    25     

                                                                                                                                                                           

Payment                         500      500                                                                                                                                                500   500  500

Rate of interest is 6.65%/a, compounded annually

 

Solution:

We calculate the present values of the 25 future payments of $500 each.  Notice the arrows go to the left for a present value annuity.

 

Interest Period   0          1         2          3                                                                                                                                         23   24    25     

                                                                                                                                                                           

Payment                        500       500                                                                                                                                                 500  500  500

                                                           

500(1.0665)-1                                                                                                                                                                                                       

500(1.0665)-2                                                                                                                                                                                                                                                                                                                                                       

       .

       .

500(1.0665)-23      

                                                                                                                                                                                   

500(1.0665)-24                                                                                                                                                                              

 

                                                                                                                                                                             

500(1.0665)-25

 

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

500(1.0665)-25 + 500(1.0665)-24 +  . . . + 500(1.0665)-2 + 500(1.0665)-1

 

 

 

 

Alternate Solution:

 

b) 

Interest Period   0                     1                       2                                                                                                                             23            24  years

                                                                                                                                                                           

Payment                         250     250       250      250                                                                                                                          250  250  250

Rate of interest is 6.6%/a, compounded  semi-annually

 

Solution:

Interest Period   0                      1                      2                                                                                                                            23            24  years       

                                                                                                                                                                           

Payment( 1000’s)          250      250     250      250                                                                                                                         250    250  250

                                                           

   250(1.033)-1                                                                                                                                                                                                      

   250(1.033)-2                                                                                                                                                                                                                                                                                                                                                      

       .

       .

   250(1.033)-46     

                                                                                                                                                                                   

   250(1.033)-47                                                                                                                                                                             

 

                                                                                                                                                                             

   250(1.033)-48

 

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

250(1.0033)-48 + 250(1.0033)-47 +  . . . + 250(1.0033)-2 + 250(1.0033)-1

 

 

 

 

c) 

Interest Period   0                                              1                                              2              .  .  .                                         6                              7  years                     

                                                                                                                                                                            

Payment                       800       800       800      800      800      800      800     800                                                         800   800  800   800  800

Rate of interest is 8.4%/a, compounded quarterly

 

Solution:

 

Interest Period   0                                             1                                                                                                               6                              7  years

                                                                                                                                                                           

Payment( 1000’s)          800      800      800      800                                                                                                        800   800   800  800  800           

                                                           

800(1.021)-1                                                                                                                                                                                                        

800(1.021)-2                                                                                                                                                                                                                                                                                                                                                        

       .

       .

800(1.021)-26     

                                                                                                                                                                                   

800(1.021)-27                                                                                                                                                                               

 

                                                                                                                                                                             

800(1.021)-28

 

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

800(1.021)-28 + 800(1.021)-27 +  . . . + 800(1.021)-2 + 800(1.021)-1

 

 

 

 

 

 

 

2.  Evaluate each of the following annuities using the PV annuity formula above.    Include a complete time line diagram for each.

a)  150(1.06)-12 + 150(1.06)-11 + . . . + 150(1.06)-2 + 150(1.06)-1

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         10    11    12     

                                                                                                                                                                           

Payment                         150       150                                                                                                                                                 150  150  150

                                                           

    150(1.06)-1                                                                                                                                                                                                       

    150(1.06)-2                                                                                                                                                                                                                                                                                                                                                       

       .

       .

    150(1.06)-10     

                                                                                                                                                                                   

    150(1.06)-11                                                                                                                                                                              

 

                                                                                                                                                                             

    150(1.06)-12

 

 

 

b)  300(1.045)-19 + 300(1.045)-18 + . . . + 300(1.045)-2 + 300(1.045)-1

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         17    18     19     

                                                                                                                                                                           

Payment                        300      300                                                                                                                                                 300   300   300

                                                           

300(1.045)-1                                                                                                                                                                                                        

300(1.045)-2                                                                                                                                                                                                                                                                                                                                                        

       .

       .

300(1.045)-17     

                                                                                                                                                                                   

300(1.045)-18                                                                                                                                                                               

 

                                                                                                                                                                             

300(1.045)-19

 

 

 

 

 

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         27    28    29     

                                                                                                                                                                           

Payment                        200       200                                                                                                                                                 200   200  200

                                                           

200(1.07)-1                                                                                                                                                                                                          

200(1.07)-2                                                                                                                                                                                                                                                                                                                                                          

       .

       .

200(1.07)-27     

                                                                                                                                                                                   

200(1.07)-28                                                                                                                                                                                 

 

                                                                                                                                                                              

200(1.07)-29

 

 

 

 

 

 

3.  Find the present value of each of the following annuities.  The interest rate is 4.8%/a, compounded monthly.  The first payment will be at the end of the first month.

a)  $500 per month for 48 months.

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         46    47     48     

                                                                                                                                                                           

Payment                        500       500                                                                                                                                                 500   500  500

                                                           

500(1.004)-1                                                                                                                                                                                                        

500(1.004)-2                                                                                                                                                                                                                                                                                                                                                        

       .

       .

500(1.004)-46     

                                                                                                                                                                                   

500(1.004)-47                                                                                                                                                                               

 

                                                                                                                                                                             

500(1.004)-48

 

 

 

 

 

 

 

b)  $750 per month for 12 ½ years.

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         148  149  150     

                                                                                                                                                                           

Payment( 1000’s)           750      750                                                                                                                                                750   750  750

                                                           

750(1.004)-1                                                                                                                                                                                                        

750(1.004)-2                                                                                                                                                                                                                                                                                                                                                        

       .

       .

750(1.004)-148     

                                                                                                                                                                                   

750(1.004)-149                                                                                                                                                                              

 

                                                                                                                                                                             

750(1.004)-150

 

 

 

 

 

c)  $250 per month for 20 ¼ years.

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         241  242   243     

                                                                                                                                                                           

Payment                        250       250                                                                                                                                                 250  250   250

                                                           

250(1.004)-1                                                                                                                                                                                                        

250(1.004)-2                                                                                                                                                                                                                                                                                                                                                        

       .

       .

250(1.004)-241     

                                                                                                                                                                                   

250(1.004)-242                                                                                                                                                                              

 

                                                                                                                                                                             

250(1.004)-243

 

 

 

 

 

4.  Find the present value of each of the following annuities.  The interest rate is 5.4%/a, compounded quarterly.  The first payment will be at the end of the first 3 month period.

a) $1000 every 3 months for 10 years.

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         38     39    40     

                                                                                                                                                                           

Payment                        1000    1000                                                                                                                                             1000  1000 1000

                                                           

1000(1.0135)-1                                                                                                                                                                                                     

1000(1.0135)-2                                                                                                                                                                                                                                                                                                                                                     

       .

       .

1000(1.0135)-38     

                                                                                                                                                                                   

1000(1.0135)-39                                                                                                                                                                            

 

                                                                                                                                                                             

1000(1.0135)-40

 

 

 

 

 

b)  $1500 every ¼ year for 15 ½ years.

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         60    61     62     

                                                                                                                                                                           

Payment                      1500       1500                                                                                                                                             1500 1500  1500

                                                           

1500(1.0135)-1                                                                                                                                                                                                     

1500(1.0135)-2                                                                                                                                                                                                                                                                                                                                                     

       .

       .

1500(1.0135)-60      

                                                                                                                                                                                   

1500(1.0135)-61                                                                                                                                                                            

 

                                                                                                                                                                             

1500(1.0135)-62

 

 

 

 

c)  $800 every ¼ year for 20 years.

 

Solution:

 

Interest Period   0          1         2          3                                                                                                                                         78    79    80     

                                                                                                                                                                           

Payment                         800      800                                                                                                                                                 800   800  800

                                                           

800(1.0135)-1                                                                                                                                                                                                       

800(1.0135)-2                                                                                                                                                                                                                                                                                                                                                       

       .

       .

800(1.0135)-78     

                                                                                                                                                                                   

800(1.0135)-79                                                                                                                                                                              

 

                                                                                                                                                                             

800(1.0135)-80

 

 

 

                                                                                                                                         

 

                                               

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