Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance

Chapter Four Ordinary Annuities Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 2

1 Future Value of an Ordinary Annuity An annuity is simply a sequence of payments (usually equal), dispersed or received, at equal intervals of time. These intervals of time, or payment intervals, are tied to a compound interest rate, and generally the payment and interest intervals match. For example, a home mortgage is paid monthly using a rate that is compounded monthly. The periodic payment for an annuity often goes by the term periodic rent. Using the word rent indicates that the rent we pay for a house or apartment is also an annuity. Like our previous loans, the life of an annuity is called the term. The term of an annuity runs from the beginning of the first rent period to the end of the last rent period. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 3

4 Annuities can be classified in several ways: 1- The first classification deals with the term of the annuity. Annuities certain begin and end at a set point in time. A mortgage is the best example of an annuity certain. Contingent annuities have a beginning or ending date that depends on some event, life insurance is typically a contingent annuity, because the end of the payments depends on an event. A perpetuity is an annuity with a specific starting time but an infinite number of payments. Perpetuities provide periodic income from a sum of money without using any of the principal.

2- The second classification deals with the placement of the periodic rent. An ordinary annuity (annuity immediate) places the payments at the end of each rent period, and an annuity due places the payments at the beginning of each rent period. From a different perspective these annuities are the same sequence of payments but are just valued at different points in time. The annuities to be studied in this chapter will all be ordinary annuities certain. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 5

3. The third classification depends on the alignment of the compound interest conversion periods and payment intervals. With a simple annuity the interest is compounded at the same frequency as the payments are made. With a general annuity the payments and conversion periods do not align. The interest conversions may occur more or less often than the payments. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 6

Besides the periodic rent, every annuity also has a present value or a future value, or both. The present value is located at the beginning of the term for loans or mortgages. This location makes sense because the borrower receives this "bundle" of money at the start of the loan and the payments occur in the months that follow. The future value is located at the end of an annuity's term. This value is important for savings or retirement accounts. Here the saver makes periodic payments that accumulate and earn interest until some future date where the "bundle" is located. The future value of an annuity also goes by the term amount (or accumulated value) just as it did for the promissory notes, investments, and savings accounts in the previous chapters. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 7

Summary of Basic Definitions with Notation i interest rate/period nominal rate divided by periods per year, i = i(m)/m n number of payments number of payments in the term of the annuity R payment dollar value of the periodic payment or rent S n future value sum of all the payments valued at the date of the last payment Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 8

The notation s  n|i is read s angle n at i and is called the amount of 1 per period. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 9

Example 1.1 page 218 A man has $75 per month deposited in his company's credit union. His credit union pays 7.5%(12) on employees' deposits. What will his account be worth in 5 years? We are looking for the future value of 60 payments (n = 5 × 12). The monthly rate i = 0.625%. Using the TVM registers on the EL 733A financial calculator, we will let n = 60, i = 0.625, PMT = $75, and compute FV (future value). Before you start, do a 2 nd CE (clear) or put PV (present value) = 0. S n = R s  n|i = 75 s  60|.625%  Sn = $ Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 10

Example 1.2 page 218 A wise father begins an education fund for his infant daughter by depositing $125 on March 15, If he continues to make quarterly payments of $125 and the fund pays 6%(4), what amount is available when the payment is made on September 15, 2004? Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 11 Use denominate numbers to find the number of payments ( ) minus ( ) gives 18 years and 6 months. The number of quarters is 4 × 18+2=74. Because there is a payment on each of the dates, the Fence Post Principle will come into play, and n = = 75. We will use the future value formula S n since we are looking for the amount at the end.

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 12

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 13 Example 1.3 page 219 A man starts an IRA at age 40 by making a $2000 contribution into a mutual fund. If he continues to deposit $2000 per year until his last one at age 65, how much will be in his fund at that time? Assume the stock market yields 11.5%( 1) on the long run. The starting and ending ages of the man behave just like yearly dates. The number of years is = 25, but the Fence Post Principle requires us to add 1, so n = 26 payments. Again the future value of this annuity is needed.

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 14 S n = $277, available at age 65 (includes the $2000 deposit that year)

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 15 Exercise 23 page 220 On Sarah's 10th birthday, her mother deposited $100 into a savings account earning 7%. Her mother continued such $100 deposits, making the last one on Sarah's 23rd birthday. At that time Mom withdrew it all for a wedding gift for Sarah. How large was the gift?

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 16 S n = R s  n|i  S 48 = 100 s  14|7%(1) = 100[{(1.07) 14 -1}/.07] = $

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 17 2 Present Value of an Ordinary Annuity The present value (or discounted value) of an ordinary annuity (annuity immediate) is a sum of money at the beginning of the term that is equivalent to the sequence of payments that follow. There are two ways to find a formula for the present value of an ordinary annuity. First, in a fashion similar to the development of the future value formula, we can find the sum of the present values of the individual periodic payments of the annuity. The second approach is to find the present value of the amount of the annuity as expressed by the future value formula. We will use this method, and, as before, we first summarize the meaning of the symbols used.

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 18 Summary of Basic Definitions with Notation i interest rate/period nominal rate divided by periods per year, i = i(m) ÷ m n number of payments number of payments in the term of the annuity R payment dollar value of the periodic payment or rent A n present value sum of the present values of all the payments

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 19

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 20 Example 2.1 page 222 Find the investment needed on December 1, 2004, to produce a $1200 per month income starting on January 1, 2005, for 15 years. Assume the average rate of return to be 9.5%(12). Make a time line diagram with the dates and money. The number of payments n= 12 × 15= 180. The Fence Post Principle does not come into play because there is not a payment at both the beginning and ending dates.

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 21 A n = 1200a  180|9.5%/12 % = $114,917.80

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 22 Example 2.2 page 223 CarMax advertises a vehicle for $2000 down and $400 per month for 2 years financed at 10.5%(12). What is the cash price of this vehicle? Make a time line diagram with the dates and money. The $2000 down payment is a cash value that we will add after we have figured the present value of the payment annuity. The 2- year term with monthly interest will give n=2 ×12 = 24, while i = 10.5% ÷ 12 =.875%.

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 23 A n =400a  24| 0.875% = $ Cash price = $ $2000 = $10,625.14

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 24 Example 2.3 page 223 A 30-year mortgage on a home has a monthly payment of $ If this mortgage is sold at the end of the 9th year to a buyer desiring a yield of 14%(12), what did the purchase cost the buyer? In this problem we need to find the present value of the remaining payments. The problem is similar to discounting a promissory note, except the future value is a sequence of payments instead of a maturity value at some one date. There are = 21 years remaining, so n=21 × 12=252. The interest rate of the original mortgage is not important, because the buyer wants to find the present value at a rate i = 14 ÷ 12 = 1.16%. An = Ra  n|i = a  252|14%/12 = $47, purchase price (provided it is accepted )

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 25 Exercise 11 page 224 Is it better to pay $10,000 in cash or to pay $ per month for a year at 8%(12)?

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 26 A n = R a  n|i A 12 = a  12|8%(12) = [{1 – (1 +.08) -12 } /.08] =$ < It is better to pay $1700 per month for a year and save $ now.

Thank you 27