# 3. Time value of money

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 1 Simple interest 2 3. 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 r = interest rate (per period) p o = principal Therefore, at the end of n periods, we will have (principal plus interest) p t = p o + trp o = p o(1 + tr) Example simple interest 3 Compound interest 4 If we invest \$100 at 10% simple interest for 7 years, how much will we have? With compound interest, we earn interest not only on the principal but also on the interest earned in previous periods: = 170 p 1 = p o + rp o = p o(1 + r) p 2 = p 1(1 + r) = p o(1 + r) 2 p 3 = p 2(1 + r) = p o(1 + r) 3. p t = p t 1(1 + r) = p o(1 + r) t

2 Example compound interest 5 Simple vs compound interest 6 If we invest \$100 at 10% compound interest for 7 years, how much will we have? How much of the interest earned in the previous example was from the principal, and how much was earned on previous periods interest? The total will be = , of which = is interest on interest. Compounding over many periods 7 Present value and future value 8 The value of an investment at present is often referred to as the present value (PV). Its value in the future is often referred to as its future value (FV). Thus, one might also write the formula for compound interest as FV t = PV (1 + r) t Because of compounding, small differences in interest rate can make a large difference after many periods.

3 Discounting 9 Four variables 10 Computing the present value of a future cash flow is often referred to as discounting the cash flow. By rearranging the previous formula, we get PV = FVt (1 + r) t There are four variables in the equation FV t = PV (1 + r) t. Given values for any three, we can solve for the fourth. It is not hard to do this algebraically. But, it is easier to use our financial calculators. Calculators 11 Example compound interest 12 There are a couple of things to be careful about when using your calculators: Be sure that you are in end mode rather than begin mode. This means that payments are at the end rather than the beginning of each period. This is the standard convention unless noted otherwise. Be sure that the number of payments per period is set to 1. Suppose I invest \$100 initially. After 8 years, the investment is worth \$190. What interest rate did I earn (assume annual compounding)? n = 8 pmt = 0 pv = -100 fv = 190 rate = 8.35%

4 Example compound interest 13 Example compound interest 14 Suppose I invest \$100 initially. The investment earns 8% compounded annually. How long until the investment is worth \$200? Suppose I have an investment that will pay off \$1000 after 20 years. What is the present value of this investment if I discount at 11% per year? r = 8 pmt = 0 pv = -100 fv = 200 n = fv=1000 n=20 r=11% pmt=0 pv = Example compound interest 15 Rule of I invest \$300 initially. The investment earns 10% compounded annually. How much is it worth in 15 years? pv=-300 r=10 pmt=0 n=15 A good rule of thumb is that time required for an investment to double multiplied by the rate is about 72. Example: About how long is needed for an investment at a 6% annual rate (compounded annually) to double? fv=

5 Time periods 17 Example time periods 18 You have to be sure when doing these problems that the time units are consistent between I invest \$200 at an annual rate of 8% compounded weekly. How much do I have after 3 years? compounding period interest rate Note: Be careful about rounding! period of investment If not, it is usually best to convert everything to the same units as the compounding frequency. Note: For the purposes of this course, always assume a year is comprised of 12 months (each of equal length), 52 weeks, and 365 days. pv=-200 r=8/52 n=3*52 pmt=0 fv= Effective annual rate (EAR) 19 Annual percentage rate (APR) 20 There are many different ways to quote rates: Rate: annual, monthly, weekly,... Compounding: annual, monthly, weekly,... Truth-in-lending laws in the US require that lenders disclose the APR. The APR is just the nominal rate quoted on an annual basis. It says nothing about the compounding interval. To compare different rates, it is convenient to standardize them. The effective annual rate (EAR) is the equivalent annual rate based on annual compounding.

6 Examples EAR/APR 21 IMPORTANT 22 What is the APR for a loan with a daily rate of 0.03% compounded daily? What is the EAR? What is the EAR for an annual rate of 11% compounded weekly? Unless otherwise noted, interest rates (and other kinds or rates) will be quoted as nominal annualized rates (i.e., APR). Note: Be careful about rounding!! APR = = = 10.95% If I start with \$1, then the future value after 1 year will be FV = = so EAR = 11.6% The weekly rate must be 11/52, so if I start with \$1, after one year I will have FV = (1 +.11/52) 52 = so, EAR = 11.61%. Continuous compounding 23 Continuous compounding formula 24 You could think of compounding over shorter and shorter intervals. In the limit, as the compounding interval becomes infinitely short, we refer to this as continuous compounding: FV t = lim PV (1 + r/m)mt m = PV e rt As with the standard compounding problems, there are four variables. Given any three, we can solve for the fourth. The formula is obtained using L Hopitals rule: ( lim x [ ( = exp lim x x) x log )] x x [ log ( )] x = exp lim x 1 x 1/x 2 1+1/x = exp lim x 1/x 2 [ ] 1 = exp lim x x = e

7 Example continuous compounding 25 Example continuous compounding 26 Suppose we have \$100 invested at 12% annual interest. How much do we have at the end of two years if the interest is compounded continuously? Suppose I invest some sum of money with 8% interest compounded continuously. At the end of 5 years, I have \$200. How much did I invest? FV = 100 exp(0.24) = = P V exp(5.08) P V = 200/ exp(5.08) = Example continuous compounding 27 Present value with unequal cash flows 28 Suppose I invest \$100 with continuously compounded interest. At the end of three years, I have \$185. What is the interest rate? Suppose I am to receive cash flows of \$300 after one year, \$500 after two years, and \$700 per year for the next three years. What is the present value of these cash flows if I discount at 8% per year? 185 = 100 exp(3 r) 185/100 = exp(3 r) log(185/100) = 3 r r = 1 log(185/100) = 20.5% 3 Discount the cash flows individually and add them up: Or, use cash flow function on our calculators. cf = 0, 300, 500, 700, 700, 700 r=8 PV = npv = Note: You don t have to re-enter 700 three times. Just hit the cash flow button three times in a row.

8 Future value with unequal cash flows 29 Interest rate with unequal cash flows 30 Suppose I am to receive cash flows of \$300 after one year, \$500 after two years, and \$700 per year for the next three years. What is the future value of these cash flows at the end of the fifth year 8% annual interest? Suppose I am to receive cash flows of \$300 after one year, \$500 after two years, and \$700 per year for the next three years. If the present value of these cash flows is \$2400, what is the interest rate? (Use cash flow function and then compute IRR using calculator.) Compound the cash flows individually and add them up: FV = = Or, use cash flow function on our calculators (use NFV button if there is one, otherwise, use NPV and then multiply by ). cf = -2400, 300, 500, 700, 700, 700 irr = 5.91% Perpetuity 31 Deriving the perpetuity formula 32 A perpetuity is a stream of cash flows that continues forever. The perpetuity formula is based on the identity P V = C R Example: What is the present value of a perpetuity that pays \$100 per quarter (use a discount rate of 12%)? PV = 100/.03 = This is referred to as a geometric series. With a little algebra, we can get C 1 + r + C (1 + r) x + x 2 + = 1 1 x = C 1 + r ( 1 1 ) 1 + r + (1 + r) 2 + = C 1 + r 1 1 1/(1 + r) = C r

9 Annuities 33 Deriving the annuity formula 34 A common situation is where a fixed number of equal cash flows are paid out. This is referred to as an annuity. The present value of an annuity can be calculated using the following formula: (Or, use calculator). PV = C [ r 1 ] 1 (1 + r) n We can derive the formula using the perpetuity formula: An annuity with payments of size C at the end of years 1 through n is equal to: the present value of a perpetuity with payments of size C minus the present value of a perpetuity with payments of size C beginning in year n+1. I.e., P V = C R C R 1 (1 + r) n = C [ ] R 1 1 (1 + r) n Annuity with final lump sum payment 35 Annuities continued 36 A common problem involves There are five variables involved in a standard problem: an annuity with payments of size C at the end of years 1 through n present value plus a final lump sum payment, F V n. interest rate The present value of this stream of cash flows is: P V = C [ R 1 1 (1 + r) n ] + F Vn (1 + r) n number of periods periodic payments final lump sum payment Given any four, you should be able to solve for the fifth.

10 Example annuity 37 Example annuity 38 Consider an investment that pays \$100 at the end of each of the next 20 years. What is the present value of these cash flows if I discount at 9% (APR)? Suppose that I have a loan for \$100,000 at an annual rate of 9% that I wish to pay off with 5 equal annual payments. What is the required payment? PV = 100/ / / / = 100 (1/ / / ) [ ] 1 1/ = = [ ( 1 1 ) 5 ] , 000 = C 0.09 C = 100, 000/3.98 = 25, 709 or, or, pmt=100 r=9 fv=0 n=20 pv= r=9 n=5 fv=0 pv= pmt = 25,709 Example annuity 39 Example annuity 40 If I borrow \$1000 at 12% and make annual payments of \$150 for 10 years, what is the balance on the loan after making the payment at the end of year 10? I invest \$1000 in some project. The investment pays back \$120 per year for 10 years. I then sell the investment for \$500. What is the rate of return? pv=-1000 pmt=150 n=10 r=12 fv = pv=-1000 pmt=120 fv=500 n=10 r=8.65

11 Reminder 41 Example loan 42 Be careful that the time units match up for compounding interval time to maturity Suppose that I buy a house for \$300,000. If I put down \$50,000 and take out a 30-year loan at an annual rate of 9% (compounded monthly) for the remainder. Assume equal monthly payments, and that the loan is paid off after the last payment. What are the monthly payments? payment period. Of the first payment, how much goes toward the principal? What is the remaining balance on the loan after 10 years? What would the payments be for a 15 year loan? Solution 43 Calendar time vs calculator time 44 The monthly rate is r monthly =.09/12 = Interest: i = = 1875 Dave is to receive a perpetuity with annual cash flows of \$100 beginning five years from today. If the appropriate discount rate is 12%, what is the value of those cash flows at time t=5? At time t=4? Today? 30 year loan: ( ) r = C r C = r ( ) r = t=5: FV5 = /.12 = t=4: FV4 = 100/.12 = t=0: FV4/(1.12)^4 = Toward principal: = Balance after 10 years: pmt= , n=240, fv=0, r=.75, pv=?=223,575. I.e., after 10 years, you have only paid off 17,000. Alternate solution: pmt= , n=120, pv=250000, r=.75, solve for fv=?=223, year loan: ( ) r = C r C = r ( ) r =

12 45 Jill is thinking of buying an annuity that pays out 10 annual cash flows beginning 5 years from today. If Jill thinks the appropriate discount rate is 8% and is willing to pay \$500 today for this annuity, what must the amount of the cash flows be? A note on cash flow timing The timing of the cash flows is very important: 46 p4 = 500*1.08^4 = Now, Plug this into the calculator as PV= n=10 r=8 FV=0 and solve for PMT=?= The usual convention (unless stated otherwise) is that the cash flows occur at the end of the period. Also, we will generally assume that the compounding frequency is the same as that of the cash flows unless otherwise stated. Annuity due 47 Example annuity due 48 Recall that the usual convention is that cash flows occur at the end of each period. An annuity with cash flows at the beginning of each period is called an annuity due. Annuity due value = Ordinary annuity value (1 + r) (Or, you can switch your calculator to begin mode.) I borrowed \$1000 which I wish to pay off in five years making equal monthly payments. The annual interest rate (APR) is 12% and the payments are due at the beginning of each month. What are the payments? Two ways to do this: (1) switch to begin mode!! pv=-1000 n=60 r=1 fv=0 pmt = Or, (2) in end mode, pv = / 1.01 = n = 60 r = 1 fv = 0 pmt = 22.02

13 Growing perpetuity 49 Example growing perpetuity 50 A growing perpetuity is a series of cash flows which grow at rate g. If C 1 is the size of the first cash flow, then John is to receive a series of quarterly cash flows which are to grow at an annual rate of 6% and continue forever. If the appropriate discount rate is 10%, and the present value of the cash flows is \$100,000, what is the first cash flow? P V = C1 r g C 1 = P V (r g) = (.10.06)/4 = 1000 Growing annuity 51 Example growing annuity 52 A growing annuity is like a growing perpetuity, except only n cash flows are paid. The formula can be derived in a manner similar to the annuity formula, as the difference between a growing annuity starting immediately and one starting at time n: John is to receive a series of 10 annual cash flows beginning in one year. The first cash flow will be \$100, and the cash flows grow at 10%. If the appropriate discount rate is 15%, what is the present value of the cash flows? P V = C1 r g [ 1 ( ) n ] 1 + g 1 + r [ ( ) ] =

14 Some common types of loans 53 Exercise Loan comparison 54 Pure discount loan The borrower receives the money today and repays the principal plus accumulated interest in a single lump sum at some time in the future. Interest-only loan The borrower repays the interest each period and repays the entire principal in a lump sum at some point in the future. Amortized loans The borrower pays the interest each period plus some amount toward the principal. The loan is paid off when the entire principal has been paid down. The most common structure is for the borrower to make equal payments. Balloon loan The borrower makes a payment each period (usually of equal size) and pays off the balance at some point in the future. Suppose that I buy a house for \$300,000, putting \$50,000 down and borrowing \$250,000. I plan to take out a 30 year loan. The annual rate is 9% (compounded monthly). What are my monthly payments if I take out a pure discount loan? How much will I have paid for the house (principal plus accumulated interest)? What if I take out an interest only loan? How about an amortized loan (equal payments)? Which of these is the better deal? Solution 55 Discounting cash flows using a spreadsheet 56 Pure discount : r monthly = 0.09/12 = See posted examples. FV 360 = 250, = 3, 682, 644 So, 359 payments of zero followed by one of \$3,682,644. Total amount paid is that plus \$50,000 Interest only : 359 payments of = 1875 plus a final payment of 251,875 for a total of , , 000 = 975, 000. Amortized : ( ) r = C r C = r ( ) r = Total payments: = 774, 162

### 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

### 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

### Chapter. Discounted Cash Flow Valuation. CORPRATE FINANCE FUNDAMENTALS by Ross, Westerfield & Jordan CIG.

Chapter 6 Discounted Cash Flow Valuation CORPRATE FINANCE FUNDAMENTALS by Ross, Westerfield & Jordan CIG. Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute

### Chapter 28 Time Value of Money

Chapter 28 Time Value of Money Lump sum cash flows 1. For example, how much would I get if I deposit \$100 in a bank account for 5 years at an annual interest rate of 10%? Let s try using our calculator:

### Chapter 5 Time Value of Money

1. Future Value of a Lump Sum 2. Present Value of a Lump Sum 3. Future Value of Cash Flow Streams 4. Present Value of Cash Flow Streams 5. Perpetuities 6. Uneven Series of Cash Flows 7. Other Compounding

### Chapter 4 Discounted Cash Flow Valuation

University of Science and Technology Beijing Dongling School of Economics and management Chapter 4 Discounted Cash Flow Valuation Sep. 2012 Dr. Xiao Ming USTB 1 Key Concepts and Skills Be able to compute

### 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

### 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

### 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

### The Time Value of Money

The Time Value of Money Time Value Terminology 0 1 2 3 4 PV FV Future value (FV) is the amount an investment is worth after one or more periods. Present value (PV) is the current value of one or more future

### 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

### Chapter 5 Time Value of Money 2: Analyzing Annuity Cash Flows

1. Future Value of Multiple Cash Flows 2. Future Value of an Annuity 3. Present Value of an Annuity 4. Perpetuities 5. Other Compounding Periods 6. Effective Annual Rates (EAR) 7. Amortized Loans Chapter

### 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

### Integrated Case. 5-42 First National Bank Time Value of Money Analysis

Integrated Case 5-42 First National Bank Time Value of Money Analysis You have applied for a job with a local bank. As part of its evaluation process, you must take an examination on time value of money

### EXAM 2 OVERVIEW. Binay Adhikari

EXAM 2 OVERVIEW Binay Adhikari FEDERAL RESERVE & MARKET ACTIVITY (BS38) Definition 4.1 Discount Rate The discount rate is the periodic percentage return subtracted from the future cash flow for computing

### 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

### 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.

### 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

### 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.

### 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

### 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

### 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

### Compound Interest Chapter 8

8-2 Compound Interest Chapter 8 8-3 Learning Objectives After completing this chapter, you will be able to: > Calculate maturity value, future value, and present value in compound interest applications,

### 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.

### CHAPTER 9 Time Value Analysis

Copyright 2008 by the Foundation of the American College of Healthcare Executives 6/11/07 Version 9-1 CHAPTER 9 Time Value Analysis Future and present values Lump sums Annuities Uneven cash flow streams

### Chapter 6 Contents. Principles Used in Chapter 6 Principle 1: Money Has a Time Value.

Chapter 6 The Time Value of Money: Annuities and Other Topics Chapter 6 Contents Learning Objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate present and future values

### FIN Chapter 5. Time Value of Money. Liuren Wu

FIN 3000 Chapter 5 Time Value of Money Liuren Wu Overview 1. Using Time Lines 2. Compounding and Future Value 3. Discounting and Present Value 4. Making Interest Rates Comparable 2 Learning Objectives

### Learning Objectives. Upon completion of this unit, students should be able to:

UNIT IV STUDY GUIDE Time Value of Money Learning Objectives Reading Assignment Chapter 6: Time Value of Money Key Terms 1. Amortized loan 2. Annual percentage rate 3. Annuity 4. Compounding 5. Continuous

### 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

### The Time Value of Money

The Time Value of Money 1 Learning Objectives The time value of money and its importance to business. The future value and present value of a single amount. The future value and present value of an annuity.

### The time value of money: Part II

The time value of money: Part II A reading prepared by Pamela Peterson Drake O U T L I E 1. Introduction 2. Annuities 3. Determining the unknown interest rate 4. Determining the number of compounding periods

### 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

### Future Value or Accumulated Amount: F = P + I = P + P rt = P (1 + rt)

F.1 Simple Interest If P dollars (called the principal or present value) earns interest at a simple interest rate of r per year (as a decimal) for t years, then: Interest: I = P rt Examples: Future Value

### 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

### Sharp EL-733A Tutorial

To begin, look at the face of the calculator. Almost every key on the EL-733A has two functions: each key's primary function is noted on the key itself, while each key's secondary function is noted in

### Ch. Ch. 5 Discounted Cash Flows & Valuation In Chapter 5,

Ch. 5 Discounted Cash Flows & Valuation In Chapter 5, we found the PV & FV of single cash flows--either payments or receipts. In this chapter, we will do the same for multiple cash flows. 2 Multiple Cash

### TIME VALUE OF MONEY (TVM)

TIME VALUE OF MONEY (TVM) INTEREST Rate of Return When we know the Present Value (amount today), Future Value (amount to which the investment will grow), and Number of Periods, we can calculate the rate

### Chapter 4. Time lines show timing of cash flows. Time Value Topics. Future value Present value Rates of return Amortization

Time Value Topics Chapter 4 Time Value of Money Future value Present value Rates of return Amortization 1 2 Determinants of Intrinsic Value: The Present Value Equation Net operating Required investments

### 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

### 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

### 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

### Chapter 7 SOLUTIONS TO END-OF-CHAPTER PROBLEMS

Chapter 7 SOLUTIONS TO END-OF-CHAPTER PROBLEMS 7-1 0 1 2 3 4 5 10% PV 10,000 FV 5? FV 5 \$10,000(1.10) 5 \$10,000(FVIF 10%, 5 ) \$10,000(1.6105) \$16,105. Alternatively, with a financial calculator enter the

### 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

### FinQuiz Notes 2 0 1 4

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.

### Chapter 4: Time Value of Money

FIN 301 Homework Solution Ch4 Chapter 4: Time Value of Money 1. a. 10,000/(1.10) 10 = 3,855.43 b. 10,000/(1.10) 20 = 1,486.44 c. 10,000/(1.05) 10 = 6,139.13 d. 10,000/(1.05) 20 = 3,768.89 2. a. \$100 (1.10)

### Chapter 4. Time Value of Money

Chapter 4 Time Value of Money Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. 2. Understand the concept of future value

### 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

### Time Value of Money. If you deposit \$100 in an account that pays 6% annual interest, what amount will you expect to have in

Time Value of Money Future value Present value Rates of return 1 If you deposit \$100 in an account that pays 6% annual interest, what amount will you expect to have in the account at the end of the year.

### Grade 7 & 8 Math Circles. Finance

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 7 & 8 Math Circles October 22/23, 2013 Finance A key point in finance is the time value of money, a concept which states that a dollar

### Future Value. Basic TVM Concepts. Chapter 2 Time Value of Money. \$500 cash flow. On a time line for 3 years: \$100. FV 15%, 10 yr.

Chapter Time Value of Money Future Value Present Value Annuities Effective Annual Rate Uneven Cash Flows Growing Annuities Loan Amortization Summary and Conclusions Basic TVM Concepts Interest rate: abbreviated

### International Financial Strategies Time Value of Money

International Financial Strategies 1 Future Value and Compounding Future value = cash value of the investment at some point in the future Investing for single period: FV. Future Value PV. Present Value

### Topics. Chapter 5. Future Value. Future Value - Compounding. Time Value of Money. 0 r = 5% 1

Chapter 5 Time Value of Money Topics 1. Future Value of a Lump Sum 2. Present Value of a Lump Sum 3. Future Value of Cash Flow Streams 4. Present Value of Cash Flow Streams 5. Perpetuities 6. Uneven Series

### 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

### Financial Management Spring 2012

3-1 Financial Management Spring 2012 Week 4 How to Calculate Present Values III 4-1 3-2 Topics Covered More Shortcuts Growing Perpetuities and Annuities How Interest Is Paid and Quoted 4-2 Example 3-3

### 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

### PowerPoint. to accompany. Chapter 5. Interest Rates

PowerPoint to accompany Chapter 5 Interest Rates 5.1 Interest Rate Quotes and Adjustments To understand interest rates, it s important to think of interest rates as a price the price of using money. When

### Chapter 5 & 6 Financial Calculator and Examples

Chapter 5 & 6 Financial Calculator and Examples Konan Chan Financial Management, Spring 2016 Five Factors in TVM Present value: PV Future value: FV Discount rate: r Payment: PMT Number of periods: N Get

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

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.

### 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?

### FIN 3000. Chapter 6. Annuities. Liuren Wu

FIN 3000 Chapter 6 Annuities Liuren Wu Overview 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams Learning objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate

### 2. How many months will it take to pay off a \$9,000 loan with monthly payments of \$225? The APR is 18%.

Lesson 1: The Time Value of Money Study Questions 1. Your mother, who gave you life (and therefore everything), has encouraged you to borrow \$65,000 in student loans. The interest rate is a record-low

### 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

### Section 8.3 Notes- Compound Interest

Section 8.3 Notes- Compound The Difference between Simple and Compound : Simple is paid on your investment or principal and NOT on any interest added Compound paid on BOTH on the principal and on all interest

### Ehrhardt Chapter 8 Page 1

Chapter 2 Time Value of Money 1 Time Value Topics Future value Present value Rates of return Amortization 2 Time lines show timing of cash flows. 0 1 2 3 I% CF 0 CF 1 CF 2 CF 3 Tick marks at ends of periods,

### 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

### 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

### 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

### CALCULATOR TUTORIAL. Because most students that use Understanding Healthcare Financial Management will be conducting time

CALCULATOR TUTORIAL INTRODUCTION Because most students that use Understanding Healthcare Financial Management will be conducting time value analyses on spreadsheets, most of the text discussion focuses

### Time Value of Money Practice Questions Irfanullah.co

1. You are trying to estimate the required rate of return for a particular investment. Which of the following premiums are you least likely to consider? A. Inflation premium B. Maturity premium C. Nominal

### NOTES E. Borrower Receives: Loan Value LV MATURITY START DATE. Lender Fixed Fixed Fixed Receives: Payment FP Payment FP Payment FP

NOTES E DEBT INSTRUMENTS Debt instrument is defined as a particular type of security that requires the borrower to pay the lender certain fixed dollar amounts at regular intervals until a specified time

### The Time Value of Money

The following is a review of the Quantitative Methods: Basic Concepts principles designed to address the learning outcome statements set forth by CFA Institute. This topic is also covered in: The Time

### 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

### Chapter 4. The Time Value of Money

Chapter 4 The Time Value of Money 4-2 Topics Covered Future Values and Compound Interest Present Values Multiple Cash Flows Perpetuities and Annuities Inflation and Time Value Effective Annual Interest

### Chapter The Time Value of Money

Chapter The Time Value of Money PPT 9-2 Chapter 9 - Outline Time Value of Money Future Value and Present Value Annuities Time-Value-of-Money Formulas Adjusting for Non-Annual Compounding Compound Interest

### Hewlett-Packard 10BII Tutorial

This tutorial has been developed to be used in conjunction with Brigham and Houston s Fundamentals of Financial Management 11 th edition and Fundamentals of Financial Management: Concise Edition. In particular,

### 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)

### 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

### Solutions to Problems: Chapter 5

Solutions to Problems: Chapter 5 P5-1. Using a time line LG 1; Basic a, b, and c d. Financial managers rely more on present value than future value because they typically make decisions before the start

### CHAPTER 4. The Time Value of Money. Chapter Synopsis

CHAPTER 4 The Time Value of Money Chapter Synopsis Many financial problems require the valuation of cash flows occurring at different times. However, money received in the future is worth less than money

### 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

### Overview of Lecture 4

Overview of Lecture 4 Examples of Quoted vs. True Interest Rates Banks Auto Loan Forward rates, spot rates and bond prices How do things change when interest rates vary over different periods? Present

### PRACTICE EXAMINATION NO. 5 May 2005 Course FM Examination. 1. Which of the following expressions does NOT represent a definition for a n

PRACTICE EXAMINATION NO. 5 May 2005 Course FM Examination 1. Which of the following expressions does NOT represent a definition for a n? A. v n ( 1 + i)n 1 i B. 1 vn i C. v + v 2 + + v n 1 vn D. v 1 v

### Key Concepts and Skills

McGraw-Hill/Irwin Copyright 2014 by the McGraw-Hill Companies, Inc. All rights reserved. Key Concepts and Skills Be able to compute: The future value of an investment made today The present value of cash

### Exam 2 Study Guide. o o

1. LS7a An account was established 7 years ago with an initial deposit. Today the account is credited with annual interest of \$860. The interest rate is 7.7% compounded annually. No other deposits or withdrawals

### Compound Interest Calculations

Compound Interest Calculations The HP-12C Compound Interest calculations involves the,,, and keyboard keys. These keys correspond to the so called Financial Registers and represent the variables in the

### 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

### Chapter 1: Time Value of Money

1 Chapter 1: Time Value of Money Study Unit 1: Time Value of Money Concepts Basic Concepts Cash Flows A cash flow has 2 components: 1. The receipt or payment of money: This differs from the accounting

### Annuities, Sinking Funds, and Amortization Math Analysis and Discrete Math Sections 5.3 and 5.4

Annuities, Sinking Funds, and Amortization Math Analysis and Discrete Math Sections 5.3 and 5.4 I. Warm-Up Problem Previously, we have computed the future value of an investment when a fixed amount of

### Discounted Cash Flow Valuation

BUAD 100x Foundations of Finance Discounted Cash Flow Valuation September 28, 2009 Review Introduction to corporate finance What is corporate finance? What is a corporation? What decision do managers make?

### TIME VALUE OF MONEY. In following we will introduce one of the most important and powerful concepts you will learn in your study of finance;

In following we will introduce one of the most important and powerful concepts you will learn in your study of finance; the time value of money. It is generally acknowledged that money has a time value.

### CHAPTER 6 DISCOUNTED CASH FLOW VALUATION

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and

### 1.3.2015 г. D. Dimov. Year Cash flow 1 \$3,000 2 \$5,000 3 \$4,000 4 \$3,000 5 \$2,000

D. Dimov Most financial decisions involve costs and benefits that are spread out over time Time value of money allows comparison of cash flows from different periods Question: You have to choose one of

### 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 5 Discounted Cash Flow Valuation

Chapter Discounted Cash Flow Valuation Compounding Periods Other Than Annual Let s examine monthly compounding problems. Future Value Suppose you invest \$9,000 today and get an interest rate of 9 percent

### Chapter 4 Time Value of Money

Chapter 4 Time Value of Money Solutions to Problems P4-1. LG 1: Using a Time Line Basic (a), (b), and (c) Compounding Future Value \$25,000 \$3,000 \$6,000 \$6,000 \$10,000 \$8,000 \$7,000 > 0 1 2 3 4 5 6 End

### Chapter 3 Understanding Money Management. Nominal and Effective Interest Rates Equivalence Calculations Changing Interest Rates Debt Management

Chapter 3 Understanding Money Management Nominal and Effective Interest Rates Equivalence Calculations Changing Interest Rates Debt Management 1 Understanding Money Management Financial institutions often

Chapter 4 Time Value of Money Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. 2. Understand the concept of future value