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UNIT 9 : MATHEMATICS OF INVESTMENT

LESSON 4: PRESENT VALUE ANNUITY HOMEWORK QUESTIONS CONT’D

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: (Continued)

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?

Solution:

Interest Period 0 1 2 3 23 24 25

Payment 1500 1500 1500 1500 1500

1500(1.065)^-1

1500(1.065)^-2

.

.

1500(1.065)^-23

1500(1.065)^-24

1500(1.065)^-25

Hence $18 296.82 should be deposited now to provide for this scholarship.

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?

Solution:

Interest Period 0 1 2 3 8 9 10

Payment R R R R R R

R(1.054)^-1

R(1.054)^-2

.

.

R(1.054)^-8

R(1.054)^-9

R(1.054)^-10

This forms the following geometric series:

R(1.054)^-10 + R(1.054)^-9 + . . . + R(1.054)^-2 + R(1.054)^-1

Hence $13 203.22 will be available each year to minor hockey.

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?

Solution:

Interest Period 0 1 2 3 418 419 420

Payment R R R R R R

R(1.005)^-1

R(1.005)^-2

.

.

R(1.005)^-418

R(1.005)^-419

R(1.005)^-420

This forms the following geometric series:

R(1.007)^-420 + R(1.007)^-419 + . . . + R(1.007)^-2 + R(1.007)^-1

Hence Betty will receive a monthly income of $11 403.79 for 35 years from her lottery winnings.

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?

Solution:

Interest Period 0 1 2 3 58 59 60

Payment R R R R R R

R(1.008)^-1

R(1.008)^-2

.

.

R(1.008)^-58

R(1.008)^-59

R(1.008)^-60

This forms the following geometric series:

R(1.008)^-60 + R(1.008)^-59 + . . . + R(1.008)^-2 + R(1.008)^-1

Hence the monthly payments will be $418.91.

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.

Solution:

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

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.

Solution:

Interest Period 0 1 2 3 118 119 120

Payment R R R R R R

R(1.0085)^-1

R(1.0085)^-2

.

.

R(1.0085)^-118

R(1.0085)^-119

R(1.0085)^-120

This forms the following geometric series:

R(1.0085)^-120 + R(1.0085)^-119 + . . . + R(1.0085)^-2 + R(1.0085)^-1

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