1. Lecture 23 Annuities Ana Nora Evans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia.edu/~ans5k Math 1140 Financial Mathematics

2. Math 1140 - Financial Mathematics Due Now Project description Team member evaluations Homework 6 2

3. Math 1140 - Financial Mathematics 1 + x + x 2 + … + x n = A)(x n -1 )/(x-1) B)(x n+1 -1 )/(x-1) C)(x n-1 -1 )/(x-1) D)(x n +1 )/(x+1) E)(x n+1 +1 )/(x+1) The correct answer is B. 3

4. Math 1140 - Financial Mathematics An annuity is a sequence of payments, usually equal, received at equal intervals of time. The equal intervals are called payment intervals or rent period. The periodic rent is the periodic payment. The term of an annuity is runs from the beginning of the first rent period to the end of the last rent period. A compound interest rate is used to move the money on the timeline. 4

5. Math 1140 - Financial Mathematics Present value (price) of an annuity is the sum of all payments moved to the beginning of the term. Future value (amount) of an annuity is the sum of all payments moved to the end of the term. 5 The monies are moved on the timeline using compound interest.

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7. Rent Period vs Compounding Period A simple annuity is an annuity for which the rent period and the compounding period are the same. A general annuity is an annuity for which the rent period and the compounding period are different. 7

8. Math 1140 - Financial Mathematics Have you seen a simple annuity before? A)Yes B)No 8

9. Math 1140 - Financial Mathematics Examples of Simple Annuities HW7 #1 Dilbert makes annual deposits of $5000 on the first day of each year for 20 years. If the effective rate of interest is 7%, how much is in the account immediately after the last deposit? HW7#2 On January 1, 2004 Ralph opens an IRA with a $2,000 deposit. He continues to deposit $2,000 at the beginning of each year until January 1, 2044, when he makes the final deposit. If the account earns an effective rate of interest of 9%, how much is in the account on the day of the last deposit? 9

10. Math 1140 - Financial Mathematics Have you seen a general annuity before? A)Yes B)No 10

11. Math 1140 - Financial Mathematics Example of General Annuity HW7#4 Luke makes deposits of $50,000 on January 1 of years 2000, 2005, 2010, 2015, and 2020 into an account paying 10% interest convertible quarterly. How much is the account worth on January 1, 2040? 11

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13. Math 1140 - Financial Mathematics On June 1 of the years 1994, 1998, 2002,..., 2022, you celebrate the World Cup by making a deposit of $2,500 into an account that pays 9.1% effective interest. How much is in the account on June 1, 2022? Another Example of a General Annuity 13

14. Math 1140 - Financial Mathematics Have: effective interest rate 9.1% Want: Interest rate per four years, x. (Section 3.4) 1.Calculate a term that is a multiple of both compounding periods 2.Calculate the maturity values of $1 using both interest rates with the term calculated in 1. 3.Set the two maturity values equal to each other and solve the equation for x. Step 1 – Calculate the interest rate per 4 years 14

15. Math 1140 - Financial Mathematics 1. Calculate a term that is a multiple of both compounding periods The compounding period for the effective interest rate of 9.1% is one year. The compounding period of the desired interest rate is four years. A common multiple of 1 and 4 is 4. The term is four years. 15

16. Math 1140 - Financial Mathematics 2. Calculate the maturity values of $1 using both interest rates with the term calculated in 1. S=P(1+i) n For the effective interest rate of 9.1%: S=$1(1+0.091) 4 For the interest rate x per four years: S=$1(1+x) 16

17. Math 1140 - Financial Mathematics 3. Set the two maturity values equal to each other and solve the equation for x. $1(1+0.091) 4 =$1(1+x) x = (1+0.091) 4 -1 x = 0.416768858961 17

18. Math 1140 - Financial Mathematics On June 1 of the years 1994, 1998, 2002,..., 2022, you celebrate the World Cup by making a deposit of $2,500 into an account that pays 9.1% effective interest. How much is in the account on June 1, 2022? After step 1: R = 2,500 n = 8 i = x = 0.416768858961 Apply the future value of an ordinary annuity formula: Step 2 Apply Future Value Formula 18

19. Math 1140 - Financial Mathematics 19 Questions?

20. Math 1140 - Financial Mathematics Classification based on the position of the payments An ordinary annuity (annuity immediate) is an annuity with the payments placed at the end of each rent period. An annuity due is an annuity with the payments placed at the beginning of each rent period. A deferred annuity is an annuity whose first payment is made two or more rent periods after the beginning of the term. A forborne annuity is an annuity whose last payment is made two or more rent periods before the end of the term. 20

21. Math 1140 - Financial Mathematics Ordinary Annuity (Annuity Immediate) An ordinary annuity is an annuity with payments placed at the end of each rent period. Examples: HW7#1, HW7#2, HW7#5, HW7#6. 21

22. Math 1140 - Financial Mathematics Annuity Due An annuity due is an annuity with payments placed at the beginning of each rent period. Where did you see this before? HW7 Bonus #2. We’ll talk more about it next class. 22

23. Math 1140 - Financial Mathematics Deferred Annuity A deferred annuity is an annuity whose first payment is made two or more rent periods after the beginning of the term. Where did you see this before? Examples: HW7#7 We’ll talk more about this on Friday. 23

24. Math 1140 - Financial Mathematics Forborne Annuity A forborne annuity is an annuity whose last payment is made two or more rent periods before the end of the term. Where did you see this before? Examples: HW7#3, HW7#4 We’ll talk more about this on Friday. 24

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26. Math 1140 - Financial Mathematics Wednesday Read sections 5.1, 5.2 Friday Homework 8 Due Quiz from compound interest and annuities. Charge 26