x.i5rong>4-2. (Problem 4-6) Your daughter is currently 8 years old. You anticipa
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x.i5rong>4-2. (Problem 4-6) Your daughter is currently 8 years old. You anticipate that she will be going to college in ten years. You would like to have $100,000 in a savings account to fund her education at that time. If the account promises to pay a fixed interest rate of 3% per year, how much money do you need to put into the account today to ensure that you will have $100,000 in ten years? The introduction to Chapter 4 draws on the discussion of NPV in Chapter 3. The text puts a great deal of emphasis on drawing a timeline when approaching every problem so that students may better visualize the problems they are solving. Another distinguishing feature of this text is that this chapter puts virtually everything in the context of present value. Hence all future value problems are illustrated as they relate to present value. The chapter covers present and future values of a single sum, an uneven stream of cash flows, and an annuity; present value of a perpetuity; present value of a growing perpetuity; and present and future value of a growing annuity. 4.1?Timeline Drawing a timeline as the first step in solving a problem will often clarify any confusion about amount and/or timing of cash flows, making problems much clearer. The timeline is a very important part of calculating the time value of money. 4.2?The Three Rules of Time Travel There are three rules of time travel. They are: 1. It is only possible to compare or combine values at the same point in time. 2. To move a cash flow forward in time, you must compound it. 3. To move a cash flow backward in time, you must discount it. To compound cash flows, multiply the amount by (1 + r)n, where r is the periodic interest rate and n is the number of compounding periods. To discount cash flows, divide the amount by (1 + r)n, where r and n are as defined previously. The rules of time travel allow us to compare and combine cash flows that occur at different points in time. In order to compare or combine them, they must be either discounted or compounded. 4.3?The Power of Compounding: An Application This section shows the effect of investing a single sum over a long time period. Figure 4.1 provides a nice illustration of that concept. 4.4?Valuing a Stream of Cash Flows This section shows how to calculate the present value or future value of an uneven cash flow stream. Note that FV can be solved by calculating the present value (discounting each cash flow), then compounding that total n periods into the future. This information makes FV of an uneven cash flow stream solvable in a financial calculator. 4.5?The Net Present Value of a Stream of Cash Flows The logic developed in Chapter 3 is re-examined here in the context of the time value of money. 4.6?Perpetuities, Annuities, and Other Special Cases Several alternatives to the formulas developed earlier are addressed at this point in the chapter. In every case, the formula is derived using algebra: 1. Present value of a perpetuity. 2. If we have several cash flows of the same size occurring at regular intervals, starting either now or sometime in the future, this is present value or future value of an annuity. Annuities due are treated as ordinary annuities with one extra period of interest. 3. The authors introduce the concept of valuing an infinite stream of cash flows that grow at a constant rate each period, using the formula for present value of a growing perpetuity. This development is particularly useful in later chapters. 4. Several cash flows occurring at regular intervals which grow at a constant rate each period. Present value or future value of a growing annuity. 4.7?Solving Problems with a Spreadsheet Program Time value of money problems lend themselves well to spreadsheet solutions. The following functions are mentioned in the chapter: NPV, FV, PV, PMT, IRR, and NPER. Example 4.11 uses the annuity spreadsheet available on the book website to solve for future value. 4.8?Solving for Variables Other Than Present Value or Future Value This section explains how to solve for (1) cash flows, (2) internal rate of return, or (3) number of periods. Again, the derivation uses strictly algebra. Lecture Chapter 4, Time Value of Money In chapter 3 we looked at a one year perspective of determining the present value and future value of a specified amount of money. The goal was to be able to accurately compare alternatives by determining the monetary value at some point in time, that is, a single value. Chapter 4 extends this to more than one period which is obviously more typical of business decisions. This discussion?s intent is to summarize the content of this chapter to provide an overview to guide you through the textbook chapter. Multiple period analysis looks at the present value or a future value of investments over several time periods. This chapter typically considers periods to be years, but analyses using month, weeks, days or other are identical with only adjusts in interest rates needed.. For instance, an simple interest rate of 6% annually would be .5% monthly. But we will stay with years at the start and then provide an example of monthly periods. Two broad categories of investments cover most situations: perpetuities and annuities. Perpetuity One category of making investments is to make a single investment at a given rate of interest. If the interest earned each period on this investment is taken out, it is called a ?Perpetuity?. The invested amount does not change and the interest income taken out each year is constant as well. Perpetuities are used to either guarantee a fixed income forever for oneself or for someone else. The same one payment situation but with only a portion or none of the interest, or just part of the interest, removed each period, is called a ?Growing Perpetuity?. A growing perpetuity provides a growing fixed inco?o x.?i5r for someone else based on an upfront investment. Equations for computing perpetuity values follow: Equation Perpetuity PV =Amount initially Invested r = Interest Rate C = Interest value removed each period C = PV x r PV = C/r r = C/PV Growing perpetuity PV =Amount initially Invested r = Interest Rate g = Rate of withdrawal each year C = Interest value removed at end of first period C= PV x (r-g) PV = C / (r-g) r = (C- PV x g)/PV g = (PV x R ? C) / PV If Joe purchases a 6% perpetuity for $500,000, he will receive $500,000 x .06 = $30,000 annually, forever. This can be calculated by the equation C = PV x r from above table. If Joe wanted to receive $50,000 annually forever, the amount he would have to invest in a 6% perpetuity is $50,000/.06 or $833,333.33 using the equation PV = C/r. But maybe Joe wants a perpetuity that grows a little each year to keep up with inflation. Suppose he feels inflation will average 2.5% annually, how much can he receive at the end of year 1 if he purchases a 6% perpetuity for $1,000,000. C= PV x (r-g) C = $1,000,000 x (.06-.025) = 1,000,000 x .035 = $35,000 How much will he receive at the end of year 2? C2 = $35,000 x 1.035 = $36,225 Year 3? C3 = $36,225 * 1.035 = $37,492.88 Annuity If a specific amount is invested each and every period, and the interest is taken out, it is called an annuity. The value at any time is simply the amount C invested times the number of periods. Of more interest and more commonly used is a growing annuity where, identical to a growing perpetuity, the interest is not removed each period. Not only does the invested principal grow each period by the newly invested amount, but interest is earned on interest from previous periods (called compounding) plus the interest on the periodic investments. Real world applications can be from two perspectives, although the calculations are identical. For instance, people invest (save) for the education of their children and similar major expenses with a growing annuity. Examples in the book show how these can grow to be substantial amounts. From another perspective, annuities are used for loans for houses, cars and the like where the lending institution gets the funds, not you. The computations for annuities are more complex than perpetuities but can be done in a spreadsheet fairly easily. For growing annuities, five basic Excel functions are used to determine each variable with each requiring the other four to determine the value.Explanation / Answer
$74,409.39
Number of Periods (Nper) 10 Years Interest rate (Rate) 3% per year Present Value (PV) ? Future Value (FV) $100,000 Using Excel Sheet: Rate : 3% Nper : 10 years FV : -$100,000 Present Value (PV)$74,409.39
(OR) Present Value FV/(1+r)t Present Value $100,000 / (1.03)10 Present Value $74,409.39Related Questions
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