DOCSLIB.ORG

Determining the Future Value, Hence Total 4.3294 $4,329,40 There Is No Interest on This Deposit

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Determining the Future Value, Hence Total 4.3294 $4,329,40 There Is No Interest on This Deposit JWPR026-Fabozzi c55 June 24, 2008 21:59 598 Mathematics of Finance The notion that money has a time value is one of the most DETERMINING THE basic concepts in investment analysis. Making decisions today regarding future cash flows requires understand- FUTURE VALUE ing that the value of money does not remain the same Suppose you deposit $1,000 into a savings account at the throughout time. Surety Savings Bank and you are promised 10% interest A dollar today is worth less than a dollar some time in per period. At the end of one period you would have the future for two reasons: $1,100. This $1,100 consists of the return of your principal amount of the investment (the $1,000) and the interest Reason 1: Cash flows occurring at different points in time or return on your investment (the $100). Let’s label these have different values relative to any one point in time. values: One dollar one year from now is not as valuable as r one dollar today. After all, you can invest a dollar today $1,000 is the value today, the present value, PV. r and earn interest so that the value it grows to next year $1,100 is the value at the end of one period, the future is greater than the one dollar today.This means we have value, FV. r to take into account the time value of money to quantify 10% is the rate interest is earned in one period, the in- the relation between cash flows at different points in terest rate, i. time. Reason 2: Cash flows are uncertain. To get to the future value from the present value: Expected cash flows may not materialize. Uncer- tainty stems from the nature of forecasts of the timing FV = PV + (PV × i) and/or the amount of cash flows. We do not know for ↑↑ certain when, whether, or how much cash flows will principal interest be in the future. This uncertainty regarding future cash flows must somehow be taken into account in assessing This is equivalent to: the value of an investment. FV = PV(1 + i) Translating a current value into its equivalent future value is referred to as compounding. Translating a future In terms of our example, cash flow or value into its equivalent value in a prior period is referred to as discounting. This chapter outlines FV = $1,000 + ($1,000 × 0.10) = $1,000(1 + 0.10) the basic mathematical techniques used in compounding = and discounting. $1,100 Suppose someone wants to borrow $100 today and promises to pay back the amount borrowed in one month. If the $100 interest is withdrawn at the end of the pe- Would the repayment of only the $100 be fair? Probably riod, the principal is left to earn interest at the 10% rate. not. There are two things to consider. First, if the lender Whenever you do this, you earn simple interest.Itissim- didn’t lend the $100, what could he or she have done with ple because it repeats itself in exactly the same way from it? Second, is there a chance that the borrower may not one period to the next as long as you take out the interest pay back the loan? So, when considering lending money, at the end of each period and the principal remains the we must consider the opportunity cost (that is, what could same. If, on the other hand, both the principal and the have been earned or enjoyed), as well as the uncertainty interest are left on deposit at the Surety Savings Bank, associated with getting the money back as promised. the balance earns interest on the previously paid interest, Let’s say that someone is willing to lend the money, referred to as compound interest. Earning interest on inter- but that they require repayment of the $100 plus some est is called compounding because the balance at any time compensation for the opportunity cost and any uncertainty is a combination of the principal, interest on principal, the loan will be repaid as promised. The amount of the and interest on accumulated interest (or simply, interest on loan, the $100, is the principal. The compensation required interest). for allowing someone else to use the $100 is the interest. If you compound interest for one more period in our Looking at this same situation from the perspective of example, the original $1,000 grows to $1,210.00: time and value, the amount that you are willing to lend today is the loan’s present value. The amount that you FV = Principal + First period interest + Second period require to be paid at the end of the loan period is the interest loan’s future value. Therefore, the future period’s value is = $1,000.00 + ($1,000.00 × 0.10) + ($1,100.00 × 0.10) comprised of two parts: = $1,200.00 Future value = Present value + Interest The present value of the investment is $1,000, the interest The interest is compensation for the use of funds for a spe- earned over two years is $210, and the future value of the cific period. It consists of (1) compensation for the length investment after two years is $1,210. of time the money is borrowed and (2) compensation for The relation between the present value and the future the risk that the amount borrowed will not be repaid ex- value after two periods, breaking out the second pe- actly as set forth in the loan agreement. riod interest into interest on the principal and interest JWPR026-Fabozzi c55 June 24, 2008 21:59 MATHEMATICAL TOOLS AND TECHNIQUES FOR FINANCIAL MODELING AND ANALYSIS 599 on interest, is: $3,000 Principal Interest on principal FV = PV + (PV × i) + (PV × i)(PV × i × i) ↑↑ ↑ ↑ $2,500 Interest on interest Principal First Second Second $2,000 period’s period’s period’s interest interest on interest on on the first $1,500 the principal the principal period’s interest or, collecting the PVs from each term and applying a bit $1,000 of elementary algebra, Balance in the account $500 FV = PV(1 + 2i + i 2) = PV(1 + i)2 $ $- The balance in the account two years from now, 1,210, is 01234 5678910 comprised of three parts: Number of compound periods 1. The principal, $1,000. Figure 55.1 The Value of $1,000 Invested 10 Years in an 2. Interest on principal, $100 in the first period plus $100 Account that Pays 10% Compounded Interest per Year in the second period. 3. Interest on interest, 10% of the first period’s interest, or $ 10. ment of $1,000 is 10%—the value of the savings account To determine the future value with compound interest balance increased 10% per year. for more than two periods, we follow along the same lines: We could also express the appreciation in our savings balance in terms of a return. A return is the income on an FV = PV(1 + i)N (55.1) investment, generally stated as a change in the value of the investment over each period divided by the amount The value of N is the number of compounding periods, at the investment at the beginning of the period. We could where a compounding period is the unit of time after also say that our investment of $1,000 provides an average which interest is paid at the rate i.Aperiodmaybeany annual return of 10% per year. The average annual return length of time: a minute, a day, a month, or a year. The im- is not calculated by taking the change in value over the en- portant thing is to make sure the same compounding pe- tire 10-year period ($2,593.70 − $1,000) and dividing it by riod is reflected throughout the problem being analyzed. $1,000. This would produce an arithmetic average return of + N The term “(1 i) ”isreferredtoasthecompoundfac- 159.37% over the 10-year period, or 15.937% per year. But tor. It is the rate of exchange between present dollars and the arithmetic average ignores the process of compound- dollars N compounding periods into the future. Equation ing. The correct way of calculating the average annual (55.1) is the basic valuation equation—the foundation of return is to use a geometric average return: financial mathematics. It relates a value at one point in time to a value at another point in time, considering the N FV compounding of interest. i = − 1 (55.2) PV The relation between present and future values for a principal of $1,000 and interest of 10% per period through which is a rearrangement of equation (55.1) Using the 10 compounding periods is shown graphically in Figure values from the example, 55.1. For example, the value of $1,000, earning interest at $ / 10% per period, is 2,593.70 ten periods into the future: $2,593.70 $2,593.70 1 10 i = 10 − 1 = FV = $1,000 (1 + 0.10)10 = $1,000 (2.5937) = $2,593.70 $1,000.00 $1,000.00 −1 = 1.100 − 1 = 10% As you can see in this figure the $2,593.70 balance in the account at the end of 10 periods is comprised of three Therefore, the annual return on the investment— parts: sometimes referred to as the compound average annual re- 1. The principal, $1,000. turn or the true return—is 10% per year. 2. Interest on the principal of $1,000, which is $100 per Here is another example for calculating a future value.
Load more

Recommended publications
  • Future Value Annuity Spreadsheet
    Future Value Annuity Spreadsheet Amory vitriolize royally? Motivating and active Saunderson adjourn her bushes profanes or computerize incurably. Crustiest and unscarred Llewellyn tenses her cart Coe refine and sheds immunologically. Press the start of an annuity formulas, federal law requires the annuity calculation is future value function helps calculate present Knowing exactly what annuities? The subsidiary value formula needs to be slightly modified depending on the annuity type. You one annuity future value? Calculating Present several Future understand of Annuities Investopedia. An annuity future value of annuities will present value accrued during their issuing insurance against principal and spreadsheet. The shed is same there is no day to prod an infinite playground of periods for the NPer argument. You are annuities are valuable ways to future. Most annuities because they are typically happens twice a future values represent payments against running out answers to repay? In this Spreadsheet tutorial, I am following to explain head to coverage the PV function in Google Sheets. Caleb troughton licensed under certain guarantees based on future value and explains why do i worked as good way. You do not recover a payment then return in this property of annuity. You shame me look once a pro in adventure time. Enter a future? As goes the surrender value tables, choosing the stage table we use is critical for accurate determination of the crazy value. The inevitable value getting an annuity is the site value of annuity payments at some specific point in the future need can help or figure out how much distant future payments will prove worth assuming that women rate of fair and the periodic payment number not change.
    [Show full text]
  • Copyrighted Material 2
    CHAPTER 1 THE TIME VALUE OF MONEY 1. INTRODUCTION As individuals, we often face decisions that involve saving money for a future use, or borrowing money for current consumption. We then need to determine the amount we need to invest, if we are saving, or the cost of borrowing, if we are shopping for a loan. As investment analysts, much of our work also involves evaluating transactions with present and future cash flows. When we place a value on any security, for example, we are attempting to determine the worth of a stream of future cash flows. To carry out all the above tasks accurately, we must understand the mathematics of time value of money problems. Money has time value in that individuals value a given amount of money more highly the earlier it is received. Therefore, a smaller amount of money now may be equivalent in value to a larger amount received at a future date. The time value of money as a topic in investment mathematics deals with equivalence relationships between cash flows with different dates. Mastery of time value of money concepts and techniques is essential for investment analysts. The chapter is organized as follows: Section 2 introduces some terminology used through- out the chapter and supplies some economic intuition for the variables we will discuss. Section 3 tackles the problem of determining the worth at a future point in time of an amount invested today. Section 4 addresses the future worth of a series of cash flows. These two sections provide the tools for calculating the equivalent value at a future date of a single cash flow or series of cash flows.
    [Show full text]
  • Chapter 9. Time Value of Money 1: Understanding the Language of Finance
    Chapter 9. Time Value of Money 1: Understanding the Language of Finance 9. Time Value of Money 1: Understanding the Language of Finance Introduction The language of finance has unique terms and concepts that are based on mathematics. It is critical that you understand this language, because it can help you develop, analyze, and monitor your personal financial goals and objectives so you can get your personal financial house in order. Objectives When you have completed this chapter, you should be able to do the following: A. Understand the term investment. B. Understand the importance of compound interest and time. C. Understand basic financial terminology (the language of finance). D. Solve problems related to present value (PV) and future value (FV). I strongly recommend that you borrow or purchase a financial calculator to help you complete this chapter. Although you can do many of the calculations discussed in this chapter on a standard calculator, the calculations are much easier to do on a financial calculator. Calculators like the Texas Instruments (TI) Business Analyst II, TI 35 Solar, or Hewlett-Packard 10BII can be purchased for under $35. The functions you will need for calculations are also available in many spreadsheet programs, such as Microsoft Excel. If you have a computer with Excel, you can use our Excel Financial Calculator (LT12), which is a spreadsheet-based financial calculator available on the website. We also have a Financial Calculator Tutorial (LT3A) which shares how to use 9 different financial calculators. Understand the Term Investment An investment is a current commitment of money or other resources with the expectation of reaping future benefits.
    [Show full text]
  • Chapter 3 Applying Time Value Concepts
    Personal Finance, 6e (Madura) Chapter 3 Applying Time Value Concepts 3.1 The Importance of the Time Value of Money 1) The time period over which you save money has very little impact on its growth. Answer: FALSE Diff: 1 Question Status: Previous edition 2) The time value of money concept can help you determine how much money you need to save over a period of time to achieve a specific savings goal. Answer: TRUE Diff: 1 Question Status: Previous edition 3) Time value of money calculations, such as present and future value amounts, can be applied to many day-to-day decisions. Answer: TRUE Diff: 1 Question Status: Revised 4) Time value of money is only applied to single dollar amounts. Answer: FALSE Diff: 1 Question Status: Previous edition 5) Your utility bill, which varies each month, is an example of an annuity. Answer: FALSE Diff: 1 Question Status: Previous edition 6) In general, a dollar can typically buy more today than it can in one year. Answer: TRUE Diff: 1 Question Status: Revised 7) An annuity is a stream of equal payments that are received or paid at equal intervals in time. Answer: TRUE Diff: 1 Question Status: Previous edition 8) An annuity is a stream of equal payments that are received or paid at random periods of time. Answer: FALSE Diff: 2 Question Status: Previous edition 1 Copyright © 2017 Pearson Education, Inc. 9) Time value of money computations relate to the future value of lump-sum cash flows only. Answer: FALSE Diff: 2 Question Status: Revised 10) There are two sets of present and future value tables: one set for lump sums and one set for annuities.
    [Show full text]
  • Compound Interest
    Compound Interest Invest €500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate: End of interest earned amount at end of period Year 1 50 550 = 500(1.1) Year 2 55 605 = 500(1.1)(1.1) Year 3 60.5 665.5 = 500(1.1)3 The interest earned grows, because the amount of money it is applied to grows with each payment of interest. We earn not only interest, but interest on the interest already paid. This is called compound interest. More generally, we invest the principal, P, at an interest rate r for a number of periods, n, and receive a final sum, S, at the end of the investment horizon. n S= P(1 + r) Example: A principal of €25000 is invested at 12% interest compounded annually. After how many years will it have exceeded €250000? n 10P= P( 1 + r) Compounding can take place several times in a year, e.g. quarterly, monthly, weekly, continuously. This does not mean that the quoted interest rate is paid out that number of times a year! Assume the €500 is invested for 3 years, at 10%, but now we compound quarterly: Quarter interest earned amount at end of quarter 1 12.5 512.5 2 12.8125 525.3125 3 13.1328 538.445 4 13.4611 551.91 Generally: nm ⎛ r ⎞ SP=⎜1 + ⎟ ⎝ m ⎠ where m is the amount of compounding per period n. Example: €10 invested at 12% interest for one year. Future value if compounded: a) annually b) semi-annuallyc) quarterly d) monthly e) weekly As the interval of compounding shrinks, i.e.
    [Show full text]
  • USING a TEXAS INSTRUMENTS FINANCIAL CALCULATOR FV Or PV Calculations: Ordinary Annuity Calculations: Annuity Due
    USING A TEXAS INSTRUMENTS FINANCIAL CALCULATOR FV or PV Calculations: 1) Type in the present value (PV) or future value (FV) amount and then press the PV or FV button. 2) Type the interest rate as a percent (if the interest rate is 8% then type “8”) then press the interest rate (I/Y) button. If interest is compounded semi-annually, quarterly, etc., make sure to divide the interest rate by the number of periods. 3) Type the number of periods and then press the period (N) button. If interest is compounded semi-annually, quarterly, etc., make sure to multiply the years by the number of periods in one year. 4) Once those elements have been entered, press the compute (CPT) button and then the FV or PV button, depending on what you are calculating. If both present value and future value are known, they can be used to find the interest rate or number of periods. Enter all known information and press CPT and whichever value is desired. When entering both present value and future value, they must have opposite signs. Ordinary Annuity Calculations: 1) Press the 2nd button, then the FV button in order to clear out the TVM registers (CLR TVM). 2) Type the equal and consecutive payment amount and then press the payment (PMT) button. 3) Type the number of payments and then press the period (N) button. 4) Type the interest rate as a percent (if the interest rate is 8% then type “8”) then press the interest rate button (I/Y). 5) Press the compute (CPT) button and then either the present value (PV) or the future value (FV) button depending on what you want to calculate.
    [Show full text]
  • Ask the Regulators: CECL: Weighted-Average Remaining Maturity (WARM) Method (April 11, 2019)
    Ask the Regulators: CECL: Weighted-Average Remaining Maturity (WARM) Method (April 11, 2019) Jim Fuchs: Great. Thank you very much. Good afternoon, and welcome everybody to our Ask the Regulators session. Today we are coming to you from the Federal Reserve Board of Governors in Washington, DC. Our latest discussion on the current expected credit loss model is on the weighted average remaining at maturity methods. That is our primary focus today. My name is Jim Fuchs. I'm with the Federal Reserve Bank of St. Louis, and I will help moderate the call. I would specifically like to welcome all of our presenters. We are here in one room, together, to present and to answer your questions today. We have members of the Board of Governors and the Federal Reserve System, the FDIC, the OCC, SEC, CSBS, FASB, and the NCUA all in one room. So we've got a real cross-section, a lot of experts in one room, and I really look forward to getting not only into the content, but really taking your questions and addressing those. Now, before I ask the presenters to introduce themselves and we jump into the call, I would like to cover a few basic items in regards to how today's call is going to flow. So first off, every Ask the Regulators call is recorded. So the link that you use to register for the session today is how you can access the archive. I encourage you to do so. We understand that we are doing this live and in real time, but there's a lot of folks that you work with that probably will want to hear this.
    [Show full text]
  • Chapter 2 Present Value
    Chapter 2 Present Value Road Map Part A Introduction to finance. • Financial decisions and financial markets. • Present value. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rates. Part D Introduction to derivatives. Main Issues • Present Value • Compound Interest Rates • Nominal versus Real Cash Flows and Discount Rates • Shortcuts to Special Cash Flows Chapter 2 Present Value 2-1 1 Valuing Cash Flows “Visualizing” cash flows. CF1 CFT 6 6 - t =0 t =1 t = T time ? CF0 Example. Drug company develops a flu vaccine. • Strategy A: To bring to market in 1 year, invest $1 B (billion) now and returns $500 M (million), $400 M and $300 M in years 1, 2 and 3 respectively. • Strategy B: To bring to market in 2 years, invest $200 M in years 0 and 1. Returns $300 M in years 2 and 3. Which strategy creates more value? Problem. How to value/compare CF streams. Fall 2006 c J. Wang 15.401 Lecture Notes 2-2 Present Value Chapter 2 1.1 Future Value (FV) How much will $1 today be worth in one year? Current interest rate is r,say,4%. • $1 investable at a rate of return r =4%. • FV in 1 year is FV =1+r =$1.04. • FV in t years is FV =$1× (1+r) ×···×(1+r) =(1+r)t. Example. Bank pays an annual interest of 4% on 2-year CDs and you deposit $10,000. What is your balance two years later? 2 FV =10, 000 × (1 + 0.04) = $10, 816.
    [Show full text]
  • Time Value of Money Professor James P. Dow, Jr
    Notes: FIN 303 Fall 15, Part 4 - Time Value of Money Professor James P. Dow, Jr. Part 4 – Time Value of Money One of the primary roles of financial analysis is to determine the monetary value of an asset. In part, this value is determined by the income generated over the lifetime of the asset. This can make it difficult to compare the values of different assets since the monies might be paid at different times. Let’s start with a simple case. Would you rather have an asset that paid you $1,000 today, or one that paid you $1,000 a year from now? It turns out that money paid today is better than money paid in the future (we will see why in a moment). This idea is called the time value of money. The time value of money is at the center of a wide variety of financial calculations, particularly those involving value. What if you had the choice of $1,000 today or $1,100 a year from now? The second option pays you more (which is good) but it pays you in the future (which is bad). So, on net, is the second better or worse? In this section we will see how companies and investors make that comparison. Discounted Cash Flow Analysis Discounted cash flow analysis refers to making financial calculations and decisions by looking at the cash flow from an activity, while treating money in the future as being less valuable than money paid now. In essence, discounted cash flow analysis applies the principle of the time value of money to financial problems.
    [Show full text]
  • 2. Time Value of Money
    2. TIME VALUE OF MONEY Objectives: After reading this chapter, you should be able to 1. Understand the concept of compounding and discounting. 2. Calculate the present value, or the future value of a single payment, or a series of payments. 3. See the effect of monthly, daily, or continuous compounding, or discounting. 2.1 Video 02A, Single Payment Problems Suppose someone offers you to have $1000 today, or to receive it a year from now. You would certainly opt to receive the money right away. A dollar in hand today is worth more than a dollar you expect to receive a year from now. There are several reasons why this is so. First, you can invest the money and make it grow. For example, in July 2009, the rate of interest paid by leading banks for a one-year certificate of deposit was 2%. This means you can get the $1000 today, invest it, and earn $20 on it by next year. There is very little risk in this investment. Second, there is the risk of inflation that eats away the purchasing power of the dollar. The rate of inflation in USA has been rather low recently, less than 1%, but in some other countries, it has been much higher. Still you will need more than $1 next year to buy what one dollar can buy today. Third, there is an element of risk in this deal. If you receive the money today, you are sure to have it in your pocket. On the other hand, the person who is promising you the money may not be around next year, or he may change his mind.
    [Show full text]
  • 2. Time Value of Money
    2. TIME VALUE OF MONEY Objectives: After reading this chapter, you should be able to 1. Understand the concepts of time value of money, compounding, and discounting. 2. Calculate the present value and future value of various cash flows using proper mathematical formulas. 2.1 Single-Payment Problems If we have the option of receiving $100 today, or $100 a year from now, we will choose to get the money now. There are several reasons for our choice to get the money immediately. First, we can use the money and spend it on basic human needs such as food and shelter. If we already have enough money to survive, then we can use the $100 to buy clothes, books, or transportation. Second, we can invest the money that we receive today, and make it grow. The returns from investing in the stock market have been remarkable for the past several years. If we do not want to risk the money in stocks, we may buy riskless Treasury securities. Third, there is a threat of inflation. For the last several years, the rate of inflation has averaged around 3% per year. Although the rate of inflation has been quite low, there is a good possibility that a car selling for $15,000 today may cost $16,000 next year. Thus, the $100 we receive a year from now may not buy the same amount of goods and services that $100 can buy today. We can avoid this erosion of the purchasing power of the dollar due to inflation if we can receive the money today and spend it.
    [Show full text]
  • Single-Family Residential Mortgage Loans)
    Single-Family MBS Prospectus Guaranteed Mortgage Pass-Through CertiÑcates (Single-Family Residential Mortgage Loans) The CertiÑcates We, the Federal National Mortgage Association or Fannie Mae, will issue and guarantee the mortgage pass-through certiÑcates. Each issue of certiÑcates will have its own identiÑcation number and will represent the ownership of a pool of residential mortgage loans secured by single-family one- to four-unit dwellings, or by a pool of participation interests in loans of that type. Fannie Mae Guaranty We guarantee that the holders of the certiÑcates will receive timely payments of interest and principal. We alone are responsible for making payments under our guaranty. The certiÑcates and payments of principal and interest on the certiÑcates are not guaranteed by the United States, and do not constitute a debt or obligation of the United States or any of its agencies or instrumentalities other than Fannie Mae. Consider carefully the risk factors section beginning on page 7. Unless you understand and are able to tolerate these risks, you should not invest in the certiÑcates. The certiÑcates are exempt from registration under the Securities Act of 1933, as amended, and are ""exempted securities'' under the Securities Exchange Act of 1934, as amended. The date of this Prospectus is April 1, 2003. TABLE OF CONTENTS Page Page Information about this Prospectus and Fixed-Rate LoansÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ 27 Prospectus SupplementsÏÏÏÏÏÏÏÏÏÏÏÏ 3 Adjustable-Rate Mortgages (ARMs) 28 SummaryÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ 4 Special
    [Show full text]