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UNIT 9 : 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

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