Home Simple & Compound Interest Present Value Ordinary Annuities Present Value Annuities General Annuities & Equivalent Rates Mortgages Review&Test

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.

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