Math 1140 Lecture 26 Perpetuities Ana Nora Evans

Math 1140 Lecture 26 Perpetuities Ana Nora Evans 403 Kerchof Ana. NEvans@virginia. edu http: //people. virginia. edu/~ans 5 k Financial Mathematics

I think my grade on the quiz will be A) B) C) D) E) A B C D F Math 1140 - Financial Mathematics 2

14 13 12 10 8 8 8 6 4 5 3 2 0 A Math 1140 - Financial Mathematics B C D F 3

Grading Scale A 18 -20 B 16 -17 C 14 -15 D 12 -13 No curved grading! F 0 -11 Math 1140 - Financial Mathematics 4

8 students guessed their grade correctly. 3 students got a better grade than expected. 10 students were off by one grading unit. 11 students thought they will get C, but got F. Math 1140 - Financial Mathematics 5

If you got a D or an F, to improve your grade on exam 2 you must start studying NOW! Read class slides and textbook. Try to solve the examples without looking at the book. If you are unable to solve them by yourself come to the office hours or make an appointment for other times. Read the exam topics posted on the web. Solve practice problems posted on the web. Math 1140 - Financial Mathematics 6

Plan Today, I will present perpetuities. Wednesday, we start chapter 6. Friday and next Monday, we will have review sessions. Next Wednesday, I will present section 6. 2 in class. Math 1140 - Financial Mathematics 7

Exam 2 Chapter 3, Compound Interest, sections 3. 1, 3. 2, 3. 4, 3. 5, 3. 6, 3. 7 Chapter 4, Ordinary Annuities, sections 4. 1, 4. 2, 4. 3, 4. 4, 4. 5, 4. 6 Chapter 5, Other Annuities Certain, sections 5. 1, 5. 2, 5. 3, 5. 4, 5. 5, 5. 6 Lectures 11(partly) trough 26. Math 1140 - Financial Mathematics 8

Discounting of a note Equivalent interest rates Math 1140 - Financial Mathematics 9

Math 1140 - Financial Mathematics 10

Perpetuity A perpetuity is an annuity with infinitely many payments. Math 1140 - Financial Mathematics 11

How is this possible? The periodic payment is the interest earned during the previous rent period. Math 1140 - Financial Mathematics 12

Present Value The present value is located one period before the first payment. We are given: R, the rent payment i, the interest rate per rent period We want to calculate: P, the present value of the perpetuity Math 1140 - Financial Mathematics 13

Present Value R is the interest earned in one conversion period at rate i per conversion period on the present value of the annuity P. R = i. P Math 1140 - Financial Mathematics 14

A perpetuity makes annual payments of $20, 000, the first payment one year from now. If the nominal rate is 8. 5% convertible quarterly, what is the present value of the perpetuity? Math 1140 - Financial Mathematics 15

A perpetuity makes annual payments of $20, 000, the first payment one year from now. If the nominal rate is 8. 5% convertible quarterly, what is the present value of the perpetuity? Math 1140 - Financial Mathematics A) $20, 000/0. 085 B) $20, 000 x 0. 085 C) None of the above 16

A perpetuity makes annual payments of $20, 000, the first payment one year from now. If the nominal rate is 8. 5% convertible quarterly, what is the present value of the perpetuity? First we need to calculate the effective interest rate of 8. 5%(4). 1 + i = (1+0. 085/4)4 i = 0. 0877 Second, we apply the formula for present value of an annuity The correct answer is C. Math 1140 - Financial Mathematics 17

A perpetuity makes annual payments of $20, 000, the first payment thirteen years from now. If the nominal rate is 8. 5% convertible quarterly, what is the present value of the perpetuity? Math 1140 - Financial Mathematics 18

A perpetuity makes annual payments of $20, 000, the first payment thirteen years from now. If the nominal rate is 8. 5% convertible quarterly, what is the present value of the perpetuity? A) $20, 000/0. 085 B) $20, 000/ 0. 0877 C) ($20, 000/0. 085 )(1. 085)-13 D) ($20, 000/0. 085 )(1. 085)-12 E) ($20, 000/ 0. 0877)(1. 0877)-13 F) ($20, 000/ 0. 0877)(1. 0877)-12 (1+0. 085/4)4 – 1 = 0. 0877 Math 1140 - Financial Mathematics 19

A perpetuity makes annual The correct answer is: payments of $20, 000, the ($20, 000/ 0. 0877)(1. 0877)-12 first payment thirteen years from now. If the nominal rate is 8. 5% convertible quarterly, what is the present value of the perpetuity? (1+0. 085/4)4 – 1 = 0. 0877 Math 1140 - Financial Mathematics 20

Math 1140 - Financial Mathematics 21

Future Value of a Perpetuity Math 1140 - Financial Mathematics 22

Perpetuity Due The present value is located at the first payment. We are given: R, the rent payment i, the interest rate per rent period We want to calculate: P, the present value of the perpetuity Math 1140 - Financial Mathematics 23

Imagine you deposit P dollars in your account, paying monthly interest. Immediately after make the deposit you withdraw R dollars. How many dollars will earn you interest for the first month? P-R How much interest do you earn at a rate i per month? (P-R)i Math 1140 - Financial Mathematics 24

Present Value of a Perpetuity Due The present value of the annuity is P. No interest is earned before the first payment, R. The interest for the second payment is (P-R)i. The rent, R, is the interest earned during the previous conversion period. We have (P-R)i = R, and we solve for P. Math 1140 - Financial Mathematics 25

Present Value of a Perpetuity Due (P-R)i = R Math 1140 - Financial Mathematics 26

Math 1140 - Financial Mathematics 27

A University receives a gift of $278, 900 gift. It invests the money at 9%(1) and wants to provide yearly scholarships with the first beginning right away. How much is the scholarship? Math 1140 - Financial Mathematics What are we given? The present value P = $278, 900. The interest rate per rent period, i = 9%. What do we need to calculate? The rent, R. 28

A University receives a gift of $278, 900 gift. It invests the money at 9%(1) and wants to provide yearly scholarships with the first beginning right away. How much is the scholarship? Which type of perpetuity is this? Perpetuity due. Solve for R. Math 1140 - Financial Mathematics 29

Charge Wednesday Read section 5. 7 Friday Homework 9 due Exam 2 Wed, 2 Nov 2011, 7 -9 pm Make-up Thu, 3 Nov 2011, 7 -9 am Sign-up for make-up before Wed, 26 Oct 2011, 5 pm Math 1140 - Financial Mathematics 30