# THE VALUE OF MONEY PROBLEM #3: ANNUITY. Professor Peter Harris Mathematics by Dr. Sharon Petrushka. Introduction

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1 THE VALUE OF MONEY PROBLEM #3: ANNUITY Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction Earlier, we explained how to calculate the future value of a single sum placed on deposit with interest compounding over a period of time. This assignment will take future value applications a step further, to deal with a series of deposits. An annuity is a series of payments which are equal and made for a given number of periods. As an example, if you were going to deposit \$2,000 into the bank every year for ten years, this would create a 10 year annuity. These payments can be made at the end of the period,( known as ordinary annuity), or at the beginning of every period, (known as an annuity due). If we invest \$100 a year for 8 years at the end of each year, what will the value of our investment be at the end of 8 years? To sove this problem, we need the Future Value of an Annuity which is the focus of this section.

2 Mathematics: Mathematically, the Future Value of an ordinary annuity is calculated as follows: N Annual Payment!"#\$%' (\$) FVA = [3] where FVA = Future Value of the annuity = Interest rate N = Number of time Periods Example 1: If you invest \$1,000 at the end of every year and receive a return of 5%, how much money will you have at the end of four years? FVA = N Annual Payment!"#\$%' (\$) \$***! "#\$%+*,' (\$) +*, FVA = = At the end of four years we will have \$4, What do you think will happen if we increase the value of? We have seen in the previous two lessons, that if increasing causes an increase in the numerator of the expression, then the value being computed will increase in value, while if increasing causes an increase in the denominator of the expression, then the value being computed will decrease. Here, however appears in both the numerator and the denominator of formula [3] so we can not be sure if the value being computed will increase or decrease. Let us investigate by increasing from 5% to 6%. Example 2 : Let Annual Payment = 1,000, and N = 4 as in Example 1, but let us increase 6%. Then \$***! "#\$%+* 6' 4(\$) FVA = = The FVA here is \$4, which is larger than the value in Example 1. +* 6 to We see then, that increasing the value of results in an increase in the value of FVA.

3 If the payments are made at the beginning of the year (Annuity Due) then the formula for the Future Value of the Annuity Due is found by multiplying the right side of formula [3] by \$%. Thus N Annual Payment! ( \$%'!"#\$%' (\$) FV Annuity Due = Example 3: If you invest \$1,000 at the beginning of every year at 5%, how much money will you have at the end of four years? FV Annuity Due = 1000! #\$%+*,'! "#\$%+*,' (\$ ] +*, -. -,/,+01

4 Using the TI-83: Returning to our previous example, how much money will we have at the end of 4 years, if we invest \$1,000 at 5% at the end of each year? How about if we invested at the beginning of each period? Press [2nd] [ FINANCE ] [ENTER] to display the TMV Solver. Enter the following: N = 4 I = 5 PV = 0.00 PMT = 1000 FV = 0.00 This is the value we are calculating for. P/Y = 1 C/Y = 1 PMT = END BEGIN As payments are made at the end of each period. Next, place the cursor on the variable you are looking to solve. In this case it is the FV (Future Value). Press [Alpha] [Solve]. The answer is computed and stored for the appropriate TVM variable. In this case, will be displayed on the FV amount with an indicated square on the left column, designating the solution variable as follows: FV = This represents the value after year four from the five annual payments of \$1,000 made at the end of each year. If the payments are made at the beginning of each period, then the calculation required to find the amount of money we will have at the end of four years at 5% interest will be the same as above except that we set PMT: END BEGIN As payments are made at the beginning of each period. Since the money is invested for a longer period, we obtain a greater future value after the fifth year. FV = \$

5 Business Application: A 20 year old college student would like to have \$100,000 in ten years. How much will she have to deposit at the end of each year to meet her financial goal? Assume a 10% annual rate of return. What if she invested at the beginning of each year? Solution: This is a Future Value Annuity problem and we will solve for FV. We enter the following on the TI - 83: N = 10 I = 10 PV = 0.00 PMT = 0.00 This is the value we are looking for. FV = P/Y = 1 C/Y = 1 PMT: END BEGIN Next Press [Alpha] [Solve] and we get œ PMT = She needs to invest \$6, per year at the end of each year for ten years and earn 10% interest per year in order to meet her goal of \$100,000 accumulation in ten years. If the deposits were made the beginning of each year, the variables entered above would also apply here, with the exception that we highlight BEGIN after PMT. PMT: END BEGIN As payments are made at the beginning of each period. After inputting the data, we get the solution: PMT = -5704

6 Additional Problems: 1. If you invest \$1,000 a year for twenty years and earn 11% per year, how much will you have at the end of twenty years? (Assume payments are made at the beginning of the year). 2. Same as number 1 except payments are made at then end of each year. 3. Call your insurance company and find out their Research Problem: guaranteed fixed rate of return on their annuities. Given this, how much money will you need to invest at the beginning of each year in order to have \$1,000,000 forty years from now?

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How to calculate present values Back to the future Chapter 3 Discounted Cash Flow Analysis (Time Value of Money) Discounted Cash Flow (DCF) analysis is the foundation of valuation in corporate finance