Compound Interest. Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate:

Save this PDF as:

Size: px
Start display at page:

Download "Compound Interest. Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate:"

Transcription

1 Compound Interest Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate: Table 1 Development of Nominal Payments and the Terminal Value, S. Nominal Interest S Year = (1.1) Year = 500(1.1)(1.1) Year = 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. ( r) n S = P 1 + 1

2 Example: A principal of is invested at 12% interest compounded annually. After how many years will it have exceeded ? ( ) 10P = P 1+ r ( ) 250,000 = 25, = 1.12 ln10 = n ln1.12 ln10 ln1.12 n n = n n 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: Table 2 Quarterly Progression of Interest Earned and End-of-Quarter Value, S. Quarter Interest Earned S

3 Generally: S = P 1+ r m nm where m is the amount of compounding per period n. Example: 10 invested at 12% interest for two years. What is the future value if compounded a) annually? b) semi-annually? c) quarterly? d) monthly? e) weekly? As the interval of compounding shrinks, i.e. it becomes more frequent, the interest earned grows. However, the increases become smaller as we increase the frequency. As compounding increases to continuous compounding our formula converges to: Example: S = rt Pe A principal of is invested at one of the following banks: a) at 4.75% interest, compounded annually 3

4 b) at 4.7% interest, compounded semi-annually c) at 4.65% interest, compounded quarterly d) at 4.6% interest, compounded continuously Which is the best bank to lodge the money? => a) 10,000(1.0475) = 10,475 b) 10,000( /2) 2 10, c) 10,000( /4) 4 10, d) 10,000e 0.046t 10, Example: Determine the annual percentage rate, APR, of interest of a deposit account which has a (simple) nominal rate of 8% compounded monthly. 1* Example: A firm decides to increase output at a constant rate from its current level of to over the next 5 years. Calculate the annual rate of growth required to achieve this growth. 4

5 ( r) ( r ) = = r = r 3.7% 5 5

6 Geometric Series Until now we have considered what happens to a lump-sum investment over a specific time-horizon. However, many investments occur over a prolonged period, such as life insurance, and debt is usually repaid periodically, such as with a mortgage. In order to deal with this feature of investments, we introduce geometric series. Consider the following sequence of numbers: 2, 8, 32, 128, This is called a geometric progression with a geometric ratio of four. Have we encountered anything like a geometric progression in our previous discussion? See Table 1 Example 500(1.1), 500(1.1)(1.1),, 500(1.1) 25 is a geometric progression with a geometric ratio of (1.1). How can we make use of this? A geometric series is the sum of the elements of a geometric progression: 6

7 Let a be the first term in the progression and let g be the geometric ratio, and let n be the number of terms in the series. Then, a g g n 1 1 Example: A person saves 100 in a bank account at the beginning of each month. The bank offers 12% compounded monthly. a) Determine the amount saved after 12 months. Series of payments, each of which has a different value at the end of the investment horizon: The final value of the 1 st instalment: 100(1.01) 12 The final value of the 2 nd instalment: 100(1.01) 11 The final value of the final instalment: 100(1.01) So: 100(1.01) 1, 100(1.01) 2,, 100(1.01) 12 Could simply add up each element in the series, a? g? n? 7

8 12 (1.01) 1 100(1.01) = (1.01) 1 b) After how many months does the amount saved first exceed 2000? Series? 100(1.01) 1, 100(1.01) 2,, 100(1.01) n So? n (1.01) (1.01) = 2000 (1.01) 1 n ( ) 2000 n ( ) = = n 1.01 = n ln( 1.01) = n = ln(1.198) 8

9 Example: The monthly repayments needed to repay a 100,000 loan, which is repaid over 25 years when the interest rate is 8% compounded annually? Two things happen: Each month: repayments, so the loan shrinks. Each year: interest applies, so the loan grows. To see a series, write down what happens each year: End of year Amount outstanding 1 100,000(1.08) 12x 2 [100,000(1.08) 12x] x 3 100,000(1.08) 3 12x(1.08) 2 12x(1.08) 12x After 25 years: 100,000(1.08) 25 12x [ ) a = 1 g = 1.08 n = 25 9

10 = So, after 25 years: x = 0 x = = One thing we can take note of is the development of the loan: Table 3 Development of Amount Outstanding End of Year Outstanding 1 98, , , The reduction in the size of the loan increases. Why? 10

11 Proof of Geometric Series Formula Begin with: S n = a + ag + ag ag n-1 Which states that a certain amount is equal to the sum of a geometric progression containing n terms, with geometric ratio g and starting value a. Then proceed by expressing gs n : gs n = ag + ag 2 + ag ag n-1 + ag n. Now, if we subtract S n from gs n we have: gs n S n = ag n a all other terms cancel Thus, by multiplying out the common terms (S n on the LHS & a on the RHS) and dividing across both sides by g 1, we obtain: S n g 1 = a g 1 n. Q.e.d. 11

12 Investment Appraisal Recall, if we want to know how much a specific amount of money will be given specific interest rates, we can use: ( r) n S = P 1 + S = or rt Pe So, if we knew S rather than P, i.e. we are interested in the present value of S, we could write: P = S( 1+ r) n P or rt = Se. Example: Find the present value of 1000 in four years time if the discount rate is 10% compounded (a) semi-annually (b) continuously a) P = 1000(1.05) 8 = b) 0.1*4 P = 1000e 12

13 Net Present Value Assume the interest rate regime implied by scenario a) applies, and you are offered an alternative investment opportunity, where you invest 600 and receive 1000 after four years. Which project would you opt for? To see this calculate the net present value (NPV) of the alternative: In order to receive 1000 in four years time one would have to invest in a bank today. However, the alternative scenario offers the same nominal return if only 1000 is invested: NPV: = Generally speaking, a project can be considered worthwhile if the NPV of it is positive, i.e. if I gain more from investing in it rather than in an equivalent project. Alternatively, we could calculate the Internal Rate of Return (IRR). This is that rate of growth that, when applied to the initial outlay, yields the final sum. If this rate of growth exceeds that of alternative investments the project is worthwhile (the NPV is positive). Example: 13

14 A project requiring an initial outlay of is guaranteed to produce a return of in three years time. Use a) NPV b) IRR methods to decide whether this investment is worthwhile if the prevailing market rate is 5%, compounded annually. What if the rate were 12%? a) The present value of in three years time is given by: ( ) 3 = P = Thus, the NPV of this project is: => positive NPV. b) For the IRR of the project, = = 15000(1 + r) => = 1+ r 3 => 0.1 = r This is larger than the current interest rate of 5%. 3 14

15 Example: Suppose it is possible to invest in only one of two 4 year projects, A or B. A requires an outlay of 1000 and yields 1200, while B requires an outlay of and yields If the market rate is 3% compounded annually, which of these projects is to be preferred? NPV A = 1200(1.03) = NPV B = 35000(1.03) = Both are viable, but B has higher value: Consider investing in A and placing the remaining on deposit: (1.03) 4 = Clearly project B would yield more. What of IRR recommendations? A ( + ) = r 1.2 = (1 + r) 4 15

16 1.2 1/4 = 1 + r B = r A = 30000(1+r) 4 r B =

17 Annuity An annuity is a financial instrument that (usually) provides annual nominal payments for a certain period. A perpetuity does this for eternity. Find the present value of an annuity which yields an income of 10,000 for ten years, with an interest rate of 7% compounded annually. What about a perpetuity? What we face is a stream of income, and we would like to know how much one should pay in order to obtain that income stream. Naturally, this arrangement should have the same value as the sum of the present values of ten individual contracts, each offering one payment in one of the ten years, i.e. we can break the series up into its constituent components, evaluate these, and add up their values, which we gain from the present value formula. P = S 1 ( + r) t Thus, the vale of the first payment is given by: P = 10000(1.07) -1 = That is, if we invested today, we would obtain 10,000 a year from now. 17

18 If we write this out for all ten years, we obtain a geometric progression, with starting value, a = 10000(1.07) -1, and a geometric ratio, g = (1.07) -1 < 0. Thus, (1.07) = What of a perpetuity? N.B.: As we are discounting, the power the geometric ratio, g, is raised to is negative. This implies that for g < 1 => 1 > g -n > 0 for all n > 0. Also, the larger is n, the closer to zero g -n becomes. In the case of a perpetuity, n = and g - = 0. S 0 1 = a = g 1 a = g 1 1 So, for a perpetuity with annual payments of 10,000 in an a g investment climate that has a discount rate of 7%: ( ) = 1 142, r 18

19 Similarly, we could calculate the net present value of a business proposition that provided the same income stream for an investment of 60,000: 70,235 60,000 = 10, What if payments are irregular? Comparing two projects, A & B, with the following irregular income streams: Year Revenue A Revenue B Total Which project should be chosen? What if the discount rate is 11%? Year (A) Discounted Revenue (B) 19

20 Total PDV With an initial outlay of 20,000, what are the respective NPVs? NPV A = = NPV B = = Example What is the IRR of a project that requires an initial outlay of 20,000 and produces a return of 8,000 at the end of year 1 and 15,000 at the end of year ( + r ) (1 + ) = r (1 + r ) = 8000(1 + r)

21 r + r = r Since, r r 0.15 = 0 We can solve this quadratic for r. 2 For: x + px + q = x 1,2 = p 2 ± p 4 2 q Thus, 2.56 x 1,2 = 0.8 ± x 1,2 = 0.8 ± => x = 8.9% 21

3. Time value of money. We will review some tools for discounting cash flows.

1 3. Time value of money We will review some tools for discounting cash flows. Simple interest 2 With simple interest, the amount earned each period is always the same: i = rp o where i = interest earned

Ch 3 Understanding money management

Ch 3 Understanding money management 1. nominal & effective interest rates 2. equivalence calculations using effective interest rates 3. debt management If payments occur more frequently than annual, how

Compound Interest Formula

Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt \$100 At

5.1 Simple and Compound Interest

5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

Time Value of Money 1

Time Value of Money 1 This topic introduces you to the analysis of trade-offs over time. Financial decisions involve costs and benefits that are spread over time. Financial decision makers in households

\$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be \$3500 after 9 months?

Simple Interest, Compound Interest, and Effective Yield Simple Interest The formula that gives the amount of simple interest (also known as add-on interest) owed on a Principal P (also known as present

ICASL - Business School Programme Quantitative Techniques for Business (Module 3) Financial Mathematics TUTORIAL 2A This chapter deals with problems related to investing money or capital in a business

APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS

CHAPTER 8 Current Monetary Balances 395 APPENDIX Interest Concepts of Future and Present Value TIME VALUE OF MONEY In general business terms, interest is defined as the cost of using money over time. Economists

Time Value of Money. 2014 Level I Quantitative Methods. IFT Notes for the CFA exam

Time Value of Money 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction...2 2. Interest Rates: Interpretation...2 3. The Future Value of a Single Cash Flow...4 4. The

Finding the Payment \$20,000 = C[1 1 / 1.0066667 48 ] /.0066667 C = \$488.26

Quick Quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? You want to receive \$5,000 per month in retirement.

5. Time value of money

1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned

FNCE 301, Financial Management H Guy Williams, 2006

Review In the first class we looked at the value today of future payments (introduction), how to value projects and investments. Present Value = Future Payment * 1 Discount Factor. The discount factor

With compound interest you earn an additional \$128.89 (\$1628.89 - \$1500).

Compound Interest Interest is the amount you receive for lending money (making an investment) or the fee you pay for borrowing money. Compound interest is interest that is calculated using both the principle

Future Value of an Annuity Sinking Fund. MATH 1003 Calculus and Linear Algebra (Lecture 3)

MATH 1003 Calculus and Linear Algebra (Lecture 3) Future Value of an Annuity Definition An annuity is a sequence of equal periodic payments. We call it an ordinary annuity if the payments are made at the

Stock and Bond Valuation: Annuities and Perpetuities

Stock and Bond Valuation: Annuities and Perpetuities Lecture 3, slides 3.1 Brais Alvarez Pereira LdM, BUS 332 F: Principles of Finance, Spring 2016 February 23, 2016 Important Shortcut Formulas Present

FinQuiz Notes 2 0 1 5

Reading 5 The Time Value of Money Money has a time value because a unit of money received today is worth more than a unit of money to be received tomorrow. Interest rates can be interpreted in three ways.

DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS

Chapter 5 DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS The basic PV and FV techniques can be extended to handle any number of cash flows. PV with multiple cash flows: Suppose you need \$500 one

Chapter 4. The Time Value of Money

Chapter 4 The Time Value of Money 1 Learning Outcomes Chapter 4 Identify various types of cash flow patterns Compute the future value and the present value of different cash flow streams Compute the return

PRESENT VALUE ANALYSIS. Time value of money equal dollar amounts have different values at different points in time.

PRESENT VALUE ANALYSIS Time value of money equal dollar amounts have different values at different points in time. Present value analysis tool to convert CFs at different points in time to comparable values

Discounted Cash Flow Valuation

Discounted Cash Flow Valuation Chapter 5 Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM FM SAMPLE QUESTIONS

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Copyright 2005 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions

Study Questions for Actuarial Exam 2/FM By: Aaron Hardiek June 2010

P a g e 1 Study Questions for Actuarial Exam 2/FM By: Aaron Hardiek June 2010 P a g e 2 Background The purpose of my senior project is to prepare myself, as well as other students who may read my senior

CHAPTER 8 INTEREST RATES AND BOND VALUATION

CHAPTER 8 INTEREST RATES AND BOND VALUATION Solutions to Questions and Problems 1. The price of a pure discount (zero coupon) bond is the present value of the par value. Remember, even though there are

APPENDIX 3 TIME VALUE OF MONEY. Time Lines and Notation. The Intuitive Basis for Present Value

1 2 TIME VALUE OF MONEY APPENDIX 3 The simplest tools in finance are often the most powerful. Present value is a concept that is intuitively appealing, simple to compute, and has a wide range of applications.

Bank: The bank's deposit pays 8 % per year with annual compounding. Bond: The price of the bond is \$75. You will receive \$100 five years later.

ü 4.4 lternative Discounted Cash Flow Decision Rules ü Three Decision Rules (1) Net Present Value (2) Future Value (3) Internal Rate of Return, IRR ü (3) Internal Rate of Return, IRR Internal Rate of Return

Continuous Compounding and Discounting

Continuous Compounding and Discounting Philip A. Viton October 5, 2011 Continuous October 5, 2011 1 / 19 Introduction Most real-world project analysis is carried out as we ve been doing it, with the present

2.1 The Present Value of an Annuity

2.1 The Present Value of an Annuity One example of a fixed annuity is an agreement to pay someone a fixed amount x for N periods (commonly months or years), e.g. a fixed pension It is assumed that the

380.760: Corporate Finance. Financial Decision Making

380.760: Corporate Finance Lecture 2: Time Value of Money and Net Present Value Gordon Bodnar, 2009 Professor Gordon Bodnar 2009 Financial Decision Making Finance decision making is about evaluating costs

Review Solutions FV = 4000*(1+.08/4) 5 = \$4416.32

Review Solutions 1. Planning to use the money to finish your last year in school, you deposit \$4,000 into a savings account with a quoted annual interest rate (APR) of 8% and quarterly compounding. Fifteen

How to calculate present values

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

THE TIME VALUE OF MONEY

QUANTITATIVE METHODS THE TIME VALUE OF MONEY Reading 5 http://proschool.imsindia.com/ 1 Learning Objective Statements (LOS) a. Interest Rates as Required rate of return, Discount Rate and Opportunity Cost

LO.a: Interpret interest rates as required rates of return, discount rates, or opportunity costs.

LO.a: Interpret interest rates as required rates of return, discount rates, or opportunity costs. 1. The minimum rate of return that an investor must receive in order to invest in a project is most likely

The Time Value of Money Part 2B Present Value of Annuities

Management 3 Quantitative Methods The Time Value of Money Part 2B Present Value of Annuities Revised 2/18/15 New Scenario We can trade a single sum of money today, a (PV) in return for a series of periodic

1. If you wish to accumulate \$140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%?

Chapter 2 - Sample Problems 1. If you wish to accumulate \$140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%? 2. What will \$247,000 grow to be in

Chapter 02 How to Calculate Present Values

Chapter 02 How to Calculate Present Values Multiple Choice Questions 1. The present value of \$100 expected in two years from today at a discount rate of 6% is: A. \$116.64 B. \$108.00 C. \$100.00 D. \$89.00

1 Interest rates, and risk-free investments

Interest rates, and risk-free investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 (\$) in an account that offers a fixed (never to change over time)

Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material. i = 0.75 1 for six months.

Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material 1. a) Let P be the recommended retail price of the toy. Then the retailer may purchase the toy at

Introduction to Real Estate Investment Appraisal

Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has

Discounted Cash Flow Valuation

6 Formulas Discounted Cash Flow Valuation McGraw-Hill/Irwin Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline Future and Present Values of Multiple Cash Flows Valuing

NPV calculation Academic Resource Center 1 NPV calculation PV calculation a. Constant Annuity b. Growth Annuity c. Constant Perpetuity d. Growth Perpetuity NPV calculation a. Cash flow happens at year

Topics Covered. Compounding and Discounting Single Sums. Ch. 4 - The Time Value of Money. The Time Value of Money

Ch. 4 - The Time Value of Money Topics Covered Future Values Present Values Multiple Cash Flows Perpetuities and Annuities Effective Annual Interest Rate For now, we will omit the section 4.5 on inflation

Chapter 4 Nominal and Effective Interest Rates

Chapter 4 Nominal and Effective Interest Rates Chapter 4 Nominal and Effective Interest Rates INEN 303 Sergiy Butenko Industrial & Systems Engineering Texas A&M University Nominal and Effective Interest

Chapter 6. Discounted Cash Flow Valuation. Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Answer 6.1

Chapter 6 Key Concepts and Skills Be able to compute: the future value of multiple cash flows the present value of multiple cash flows the future and present value of annuities Discounted Cash Flow Valuation

2. How would (a) a decrease in the interest rate or (b) an increase in the holding period of a deposit affect its future value? Why?

CHAPTER 3 CONCEPT REVIEW QUESTIONS 1. Will a deposit made into an account paying compound interest (assuming compounding occurs once per year) yield a higher future value after one period than an equal-sized

HOW TO CALCULATE PRESENT VALUES

Chapter 2 HOW TO CALCULATE PRESENT VALUES Brealey, Myers, and Allen Principles of Corporate Finance 11th Edition McGraw-Hill/Irwin Copyright 2014 by The McGraw-Hill Companies, Inc. All rights reserved.

Finding Rates and the Geometric Mean

Finding Rates and the Geometric Mean So far, most of the situations we ve covered have assumed a known interest rate. If you save a certain amount of money and it earns a fixed interest rate for a period

Time Value of Money. Background

Time Value of Money (Text reference: Chapter 4) Topics Background One period case - single cash flow Multi-period case - single cash flow Multi-period case - compounding periods Multi-period case - multiple

Basic Concept of Time Value of Money

Basic Concept of Time Value of Money CHAPTER 1 1.1 INTRODUCTION Money has time value. A rupee today is more valuable than a year hence. It is on this concept the time value of money is based. The recognition

Time-Value-of-Money and Amortization Worksheets

2 Time-Value-of-Money and Amortization Worksheets The Time-Value-of-Money and Amortization worksheets are useful in applications where the cash flows are equal, evenly spaced, and either all inflows or

Chapter 04 - More General Annuities

Chapter 04 - More General Annuities 4-1 Section 4.3 - Annuities Payable Less Frequently Than Interest Conversion Payment 0 1 0 1.. k.. 2k... n Time k = interest conversion periods before each payment n

Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Chapter Outline. Multiple Cash Flows Example 2 Continued

6 Calculators Discounted Cash Flow Valuation Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute

Ch. 11.2: Installment Buying When people take out a loan to make a big purchase, they don t often pay it back all at once in one lump-sum. Instead, they usually pay it back back gradually over time, in

TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION

TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION Professor Peter Harris Mathematics by Sharon Petrushka Introduction This problem will focus on calculating mortgage payments. Knowledge of Time Value

4 Annuities and Loans

4 Annuities and Loans 4.1 Introduction In previous section, we discussed different methods for crediting interest, and we claimed that compound interest is the correct way to credit interest. This section

Real Estate. Refinancing

Introduction This Solutions Handbook has been designed to supplement the HP-2C Owner's Handbook by providing a variety of applications in the financial area. Programs and/or step-by-step keystroke procedures

Chapter 8. 48 Financial Planning Handbook PDP

Chapter 8 48 Financial Planning Handbook PDP The Financial Planner's Toolkit As a financial planner, you will be doing a lot of mathematical calculations for your clients. Doing these calculations for

Calculator and QuickCalc USA

Investit Software Inc. www.investitsoftware.com. Calculator and QuickCalc USA TABLE OF CONTENTS Steps in Using the Calculator Time Value on Money Calculator Is used for compound interest calculations involving

CHAPTER 5. Interest Rates. Chapter Synopsis

CHAPTER 5 Interest Rates Chapter Synopsis 5.1 Interest Rate Quotes and Adjustments Interest rates can compound more than once per year, such as monthly or semiannually. An annual percentage rate (APR)

Capital Budgeting OVERVIEW

WSG12 7/7/03 4:25 PM Page 191 12 Capital Budgeting OVERVIEW This chapter concentrates on the long-term, strategic considerations and focuses primarily on the firm s investment opportunities. The discussions

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

CHAPTER 12 RISK, COST OF CAPITAL, AND CAPITAL BUDGETING

CHAPTER 12 RISK, COST OF CAPITAL, AND CAPITAL BUDGETING Answers to Concepts Review and Critical Thinking Questions 1. No. The cost of capital depends on the risk of the project, not the source of the money.

Chapter 4 Time Value of Money ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 4 Time Value of Money ANSWERS TO END-OF-CHAPTER QUESTIONS 4-1 a. PV (present value) is the value today of a future payment, or stream of payments, discounted at the appropriate rate of interest.

Chapter 03 - Basic Annuities

3-1 Chapter 03 - Basic Annuities Section 7.0 - Sum of a Geometric Sequence The form for the sum of a geometric sequence is: Sum(n) a + ar + ar 2 + ar 3 + + ar n 1 Here a = (the first term) n = (the number

CHAPTER 2. Time Value of Money 2-1

CHAPTER 2 Time Value of Money 2-1 Time Value of Money (TVM) Time Lines Future value & Present value Rates of return Annuities & Perpetuities Uneven cash Flow Streams Amortization 2-2 Time lines 0 1 2 3

Chapter Review Problems

Chapter Review Problems State all stock and bond prices in dollars and cents. Unit 14.1 Stocks 1. When a corporation earns a profit, the board of directors is obligated by law to immediately distribute

Prepared by: Dalia A. Marafi Version 2.0

Kuwait University College of Business Administration Department of Finance and Financial Institutions Using )Casio FC-200V( for Fundamentals of Financial Management (220) Prepared by: Dalia A. Marafi Version

15.407 Sample Mid-Term Examination Fall 2008. Some Useful Formulas

15.407 Sample Mid-Term Examination Fall 2008 Please check to be certain that your copy of this examination contains 18 pages (including this one). Write your name and MIT ID number on every page. You are

Chapter Two. THE TIME VALUE OF MONEY Conventions & Definitions

Chapter Two THE TIME VALUE OF MONEY Conventions & Definitions Introduction Now, we are going to learn one of the most important topics in finance, that is, the time value of money. Note that almost every

CARMEN VENTER COPYRIGHT www.futurefinance.co.za 0828807192 1

Carmen Venter CFP WORKSHOPS FINANCIAL CALCULATIONS presented by Geoff Brittain Q 5.3.1 Calculate the capital required at retirement to meet Makhensa s retirement goals. (5) 5.3.2 Calculate the capital

Excel Financial Functions

Excel Financial Functions PV() Effect() Nominal() FV() PMT() Payment Amortization Table Payment Array Table NPer() Rate() NPV() IRR() MIRR() Yield() Price() Accrint() Future Value How much will your money

MAT116 Project 2 Chapters 8 & 9

MAT116 Project 2 Chapters 8 & 9 1 8-1: The Project In Project 1 we made a loan workout decision based only on data from three banks that had merged into one. We did not consider issues like: What was the

Chapter 13 The Basics of Capital Budgeting Evaluating Cash Flows

Chapter 13 The Basics of Capital Budgeting Evaluating Cash Flows ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS 13-1 a. The capital budget outlines the planned expenditures on fixed assets. Capital budgeting

Time Value of Money (TVM)

BUSI Financial Management Time Value of Money 1 Time Value of Money (TVM) Present value and future value how much is \$1 now worth in the future? how much is \$1 in the future worth now? Business planning

1. Which of the following statements is an implication of the semi-strong form of the. Prices slowly adjust over time to incorporate past information.

COURSE 2 MAY 2001 1. Which of the following statements is an implication of the semi-strong form of the Efficient Market Hypothesis? (A) (B) (C) (D) (E) Market price reflects all information. Prices slowly

Check off these skills when you feel that you have mastered them.

Chapter Objectives Check off these skills when you feel that you have mastered them. Know the basic loan terms principal and interest. Be able to solve the simple interest formula to find the amount of

Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

Bond Price Arithmetic

1 Bond Price Arithmetic The purpose of this chapter is: To review the basics of the time value of money. This involves reviewing discounting guaranteed future cash flows at annual, semiannual and continuously

Risk-Free Assets. Case 2. 2.1 Time Value of Money

2 Risk-Free Assets Case 2 Consider a do-it-yourself pension fund based on regular savings invested in a bank account attracting interest at 5% per annum. When you retire after 40 years, you want to receive

1.1 Introduction. Chapter 1: Feasibility Studies: An Overview

Chapter 1: Introduction 1.1 Introduction Every long term decision the firm makes is a capital budgeting decision whenever it changes the company s cash flows. Consider launching a new product. This involves

Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis

Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................

FIN 5413: Chapter 03 - Mortgage Loan Foundations: The Time Value of Money Page 1

FIN 5413: Chapter 03 - Mortgage Loan Foundations: The Time Value of Money Page 1 Solutions to Problems - Chapter 3 Mortgage Loan Foundations: The Time Value of Money Problem 3-1 a) Future Value = FV(n,i,PV,PMT)

Chapter 6. Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams

Chapter 6 Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams 1. Distinguish between an ordinary annuity and an annuity due, and calculate present

The Basics of Interest Theory

Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

Interest Rate and Credit Risk Derivatives

Interest Rate and Credit Risk Derivatives Interest Rate and Credit Risk Derivatives Peter Ritchken Kenneth Walter Haber Professor of Finance Weatherhead School of Management Case Western Reserve University

Solutions to Time value of money practice problems

Solutions to Time value of money practice problems Prepared by Pamela Peterson Drake 1. What is the balance in an account at the end of 10 years if \$2,500 is deposited today and the account earns 4% interest,

Lecture Notes on the Mathematics of Finance

Lecture Notes on the Mathematics of Finance Jerry Alan Veeh February 20, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects

Fin 3312 Sample Exam 1 Questions

Fin 3312 Sample Exam 1 Questions Here are some representative type questions. This review is intended to give you an idea of the types of questions that may appear on the exam, and how the questions might

Time Value of Money CAP P2 P3. Appendix. Learning Objectives. Conceptual. Procedural

Appendix B Time Value of Learning Objectives CAP Conceptual C1 Describe the earning of interest and the concepts of present and future values. (p. B-1) Procedural P1 P2 P3 P4 Apply present value concepts

Time Value of Money. 15.511 Corporate Accounting Summer 2004. Professor S. P. Kothari Sloan School of Management Massachusetts Institute of Technology

Time Value of Money 15.511 Corporate Accounting Summer 2004 Professor S. P. Kothari Sloan School of Management Massachusetts Institute of Technology July 2, 2004 1 LIABILITIES: Current Liabilities Obligations

Chapter 3 Mathematics of Finance

Chapter 3 Mathematics of Finance Section 3 Future Value of an Annuity; Sinking Funds Learning Objectives for Section 3.3 Future Value of an Annuity; Sinking Funds The student will be able to compute the

CHAPTER 4 DISCOUNTED CASH FLOW VALUATION

CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. Assuming positive cash flows and interest rates, the future value increases and the present value

rate nper pmt pv Interest Number of Payment Present Future Rate Periods Amount Value Value 12.00% 1 0 \$100.00 \$112.00

In Excel language, if the initial cash flow is an inflow (positive), then the future value must be an outflow (negative). Therefore you must add a negative sign before the FV (and PV) function. The inputs

The Time Value of Money C H A P T E R N I N E

The Time Value of Money C H A P T E R N I N E Figure 9-1 Relationship of present value and future value PPT 9-1 \$1,000 present value \$ 10% interest \$1,464.10 future value 0 1 2 3 4 Number of periods Figure

How to Calculate Present Values

How to Calculate Present Values Michael Frantz, 2010-09-22 Present Value What is the Present Value The Present Value is the value today of tomorrow s cash flows. It is based on the fact that a Euro tomorrow

FI 302, Business Finance Exam 2, Fall 2000 versions 1 & 8 KEYKEYKEYKEYKEYKEYKEYKEYKEYKEYKEYKEYKEY

FI 302, Business Finance Exam 2, Fall 2000 versions 1 & 8 KEYKEYKEYKEYKEYKEYKEYKEYKEYKEYKEYKEYKEY 1. (3 points) BS16 What is a 401k plan Most U.S. households single largest lifetime source of savings is

Mortgage-Backed Sector of the Bond Market

1 Mortgage-Backed Sector of the Bond Market LEARNING OUTCOMES 1. Mortgage Loan: a. cash flow characteristics of a fixed-rate, b. level payment, and c. fully amortized mortgage loan; 2. Mortgage Passthrough