MAT116 Project 2 Chapters 8 & 9

Size: px
Start display at page:

Download "MAT116 Project 2 Chapters 8 & 9"

Transcription

1 MAT116 Project 2 Chapters 8 & : 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 term of the loan? What interest rate were they paying? What kind of workout options are available? What role does inflation play in the value of the loan over time? 2 The Project In this short project, we will extend the results of Project 1 by looking at these factors and see if doing so changes our original decision in any way. We will look at two common workout options in more detail and build amortization tables to adjust the true full value of out loan. This will allow us to recomputed our expected values. These tables will be built using common ideas and formulas from the finance world such as annuities, future value, present value, etc. 3 1

2 8-2: Workout Options There are at least five types of loan workouts 4 Debt Forgiveness This means all or part of the debt is simply erased. For example, in 1999, the International Monetary Fund (IMF) and World Bank cancelled much of the debt owed by Mozambique. (http://news.bbc.co.uk/1/hi/special_report/1999/0 6/99/debt/ stm) 5 Graduated Payment In this situation, the payments are lowered in the beginning, then gradually increased as the borrower builds his or her earning ability. For example, college graduates usually have student loans to repay, but in their early twenties they typically do not have much earning power. So they could make small payments for a few years. As they get older, their salaries increase, so they are able to make larger payments. 6 2

3 Forbearance This means the borrower pays nothing for a period of time (for example, one year). During this time, the interest continues to accumulate. Then, when the time period is over, the borrower resumes payments. An example of this would be a small business owner during a recession. He or she could apply for a forbearance during the economic hard times. This is an option we will explore. 7 Deferment This is similar to forbearance, except interest does not accumulate during the deferment period. Typically, the interest is paid by a third party, such as a government program. For example, college graduates who join the Peace Corp qualify for a deferment, and the U.S. government will pay the interest of their student loans until their duty is finished. 8 Extended Repayment The length of time to pay back the loan is extended. For example, if the original loan is to be paid back in 5 years, then under this plan, the borrower would have 10 years to pay back the loan. This reduces the amount of each payment. This is the other option we will explore in this project. 9 3

4 9-1: Intro to Time Value of Money Problems Whenever we take out loans, make investments, or deal with money over time, common questions arise. How much will this be worth in 10 years? How will inflation eat into my retirement savings over the next 30 years? How much do I need to save each month to have $1,000,000 at age 60? How much do I need to pay each month on a 30 year mortgage of $400,000 if the interest rate is 5.25% These are called Time Value of Money (TVM) problems. 10 Common Notations P = Present Value: the current value of an investment or loan or sum of money. F = Future Value: The value of an investment, loan or sum of money in the future. r = Annual Percentage Rate m = Number of periods per year. : If you make monthly loan payments, them m = Common Notations (Cont d) i = Interest Rate per Period: : The annual interest rate is 6% and there are 12 payments per year. Then i =.06/12 =.005, or 0.5% per month. i = r/m t = Time, measured in years n = Total number of periods. : You have a 10 year loan that is paid monthly. Then you have n = 10 12=120 total periods. n = m t R = The amount of a Rent. This is the regular payment made on a loan or into an investment. 12 4

5 Suppose you deposit $1000 into an account that compounds interest quarterly. The annual rate of interest is 2.3% and you are going to keep it in the account for 4.5 years. At the end of this time, the account will be worth $ P = $1000 F = $ r =.023 m = 4 i =.023/4 = t = 4.5 n = 18 R = $ : Percent Increase/Decrease If a quantity increases by some percent, we can create a multiplier that helps us convert a beginning value to an ending value. To create the appropriate multiplier: Percent Increase: 1 + i Percent Decrease: 1 - i Why? 14 This year, the SCCC student population is 11,350. The administration estimates that will increase by 2% next year. How many students can we expect next year? The multiplier = = 1.02 New student population = 11,350(1.02) =

6 The current balance of my retirement account is $244,350. If the value of the account drops by 5.2% over the next year, what will be the new value? Multiplier = =.948 New Value = $244,350(.948) = $231, : Compound Interest When we invest money, interest may be applied to the account on a regular basis. For example, if we invest in account that pays interest monthly, we say the interest compounds monthly. Anything in the account at the time of compounding gets interest added to it. In the monthly case, we have 12 compoundings per year, with each compounding representing 1/12 of the total annual interest rate. 17 We invest $1000 in an account that compounds monthly. The annual interest rate is 3.6%. If we keep it in the account for 5 years, adding or removing nothing, how much will be in the account at the end of 5 years? First, we need to note that the interest per period is i =.036/12 =.003. Let s begin by building a table 18 6

7 Period # Previous Balance $0 $1000 $1000(1.003) $1000(1.003)(1.003) New Balance $1000 $1000(1.003) $1000(1.003)(1.003) $1000(1.003) (1.003) (1.003) 19 Period # New Balance $1000 $1000(1.003) $1000(1.003)(1.003) $1000(1.003) (1.003) (1.003) =$1000(1.003) 0 =$1000(1.003) 1 =$1000(1.003) 2 =$1000(1.003) 3 20 Period # New Balance 3 $1000(1.003) (1.003) (1.003) =$1000(1.003) 3 4 =$1000(1.003) 4 5 =$1000(1.003) 5 After 5 years, or 20 periods, we have 20 =$1000(1.003) 20 After n total periods, we have n =$1000(1.003) n 21 7

8 Compound Interest Formula If P dollars earn an annual interest rate of i per period for n periods, with no additional principal added or removed, then the future value (F) is given by: F = P(1+i) n 22 A bank account is opened with $4,000 in the account. It earns 6% annual interest. If it earns interest quarterly (four times per year), then what is in the account after 10 years? 23 Technology Question How could you use Excel to try to answer the previous question? Excel File Link 24 8

9 Suppose you invest $500 today at an annual rate of 1.5%, compounded daily. How long before the balance doubles? : Rule of 72 Given some investment that grows at an annual interest rate, r, (not expressed as a decimal), then the amount of time in years it takes for the investment to double is approximately: 72 r 26 How long will it take for an investment to double if it earns 5% annual interest? Note that estimating the doubling time does not require that we know how much is originally invested! 27 9

10 If you want your investment to double in 30 months, what annual interest rate do you need to secure? : Yield Because each compounding acts on the original balance and any interest that has been previously earned, at the end of the investment, the net interest earned will not be the same as the annual interest rate. The true interest rate earned is called the Yield. 29 Invest $100 for 1 years, compounded monthly at an annual rate of 12%. F = 100(1+.01) 12 = $ This represents a yield of 12.68%, which is higher than the original 12% stated above. The yield is often called the Annual Percentage Yield (APY ). Always ask what this is when you take out a loan time, compounding and bank fees can substantially increase your rate of interest and therefore your total payments due! 30 10

11 Yield Formula The formula for yield in the t = 1 year case is: y = n ( 1+ i) 1 31 Where Did Get That? new old P(1 + i) y = = old P n P[ (1 + i) 1] = P n = (1+ i) 1 n P 32 Who Cares? The APR, or yield, is helpful since it simplifies calculations. If we know the APR, then it does not matter how many times we compound per year because the APR will give us actual percentage increase at the end of a year

12 If $375 is invested with an APR of 5.22% for 8 years and 3 months, what is the Future value of the investment? : Annuities When additional payments or deposits (rents) are made at regular intervals into an investment, then we call these annuities: Ordinary Annuity: Payment is due at the END of each period. Annuity Due: Payment is due at the START of each period. This will complicate our Future Value calculations. 35 We invest in an account that compounds annually. The annual interest rate is 3%. If we add $800 at the end of each year (an ordinary annuity), how much will be in the account at the end of 5 years total? Let s look at a picture of what is going on here

13 Start Year 1 Year 2 Year 3 Year 4 Year 5 Investment Period $800 $800 $800 $800 $800 Each of these $800 investments earns interest for a different period of time. Hence, the value of each of these deposits is different at the end of the 5-year period. 37 Start The End End Year 1 End Year 2 End Year 3 End Year 4 End Year 5 Investment Period $800 $800 $800 $800 $800 This one is worth $800(1+.03) 4 at the End $ This one is worth $800(1+.03) 3 at the End $ This one is worth $800(1+.03) 2 at the End $ This one is worth $800(1+.03) 1 at the End $824 This one is worth $800 at the End $ We can add all of these up: $800(1.03) 4 + $800(1.03) 3 + $800(1.03) 2 + $800(1.03) 1 + $800 =$ $ $ $824 + $800 =$

14 A General Formula Now imagine if the monthly payments were deposited and monthly interest credited. We would then have 5 12 = 60 different deposits to find the values for so we can add them up. To avoid this inefficiency, we instead use the following formula, which is equivalent to going through that process. 40 A General Formula If R dollars are paid at the end of each period, with an interest rate of I per period, then the Future Value of the Annuity is: F = R ( 1+ i) i n 1 41 Check We invest in an account that compounds annually. The annual interest rate is 3%. If we add $800 at the end of each year (an ordinary annuity), how much will be in the account at the end of 5 years total? 5 (1 +.03) 1 F = = = 800 ( ) =

15 What is the future value if you invest $95 per month for 7 years at an annual rate of 3.75%? R = 95 i =.0375/12 n = 12 7 = 84 Note that I try to keep as many decimal places as possible until the end F = 95 ( /12) = = ( ) = / The Excel Table How can we build an Excel Table to keep track of all of this? Excel File Link 44 FV for Annuities Due When payments or deposits are made at the beginning of a period (rather than at the end as in the previous examples), an adjustment is needed. We can view each payment as if it were made at the end of the preceding period. This would require one more payment (n+1 total) than usual and would require that we subtract the last payment so we don t overpay

16 FV for Annuities Due F = R ( 1+ i) n+ 1 i 1 R 46 If $300 payments are made at the beginning of the month for 18 years (a college savings fund), what is the Future Value if the annual interest rate is 5.5% : Future Value (FV) on Excel The FV command will do these computations for us automatically. Command Format: =FV(rate, nper, pmt, [pv], [type]) This is i, the rate per period This is n, the total # of periods This is R, the amount of rent This is P, the Present Value Blank for ordinary annuity, 1 for annuity due 48 16

17 Excel s If $300 payments are made at the beginning of the month for 18 years (a college savings fund), what is the Future Value if the annual interest rate is 5.5% What is the future value if you invest $95 per month (paid at the end of the month) for 7 years at an annual rate of 3.75%? 49 Project Questions What does this have to do with Project 1? : PV of Annuities Suppose you have an ordinary 20-year annuity that you pay $500 into at the end of each quarter. The annual interest rate is 7%. What is the lump-sum of money which should be deposited at the start of the annuity that would produce the exact same amount of money at the end of the period, without any additional payments? This is know as the Present Value of the Annuity 51 17

18 Present Value of an Ordinary Annuity P = ( 1 i) 1 + R i n 52 Origin of the Formula Can you figure out how this comes from the formula for the Future Value of an Ordinary Annuity? 53 Suppose you set up an ordinary annuity account which is to last 10 years and earn 4% annual interest rate. If your rent payment is $150 per month, how much do you need to deposit as a lump sum up front to achieve the same end result without any regular payments? 54 18

19 Suppose you win the lottery and have a choice. $4,500,000 in one lump sum. (Present Value = $4.5M) $650,000 per year for 9 years (an annuity). Assume you can earn 5% annual interest on these payments if you deposit them into an account. Which one is better? Compute the Present Value of the second option and compare it to the first option : Present Value (PV) on Excel The PV command will do these computations for us automatically. Command Format: =PV(rate, nper, pmt, [fv], [type]) This is i, the rate per period This is n, the total # of periods This is R, the amount of rent This is the FV you want after the last pmt Optional Blank for ordinary annuity, 1 for annuity due 56 Excel Suppose you set up an ordinary annuity account which is to last 10 years and earn 4% annual interest rate. If your rent payment is $150 per month, how much do you need to deposit as a lump sum up front to achieve the same end result without any regular payments? 57 19

20 Loan Payments We use the formula for the Present Value of an Ordinary Annuity to determine the periodic amount of Rent (the payment) on a loan. Since P is the present value of the loan, then this makes since. 58 Loan Payment Formula P ( 1 i) 1 + i P 1 + i ( 1 i) ( 1 i) + P= R i ( 1 i) 1 + i = R n 1 + i = R n n 1 Start here with the original formula ( 1 i) n n Divide both sides by [ ] to get R alone Here is the formula for the loan payment. 59 Loan Payment Formula Using basic algebra, we can rewrite this as: R Pi = n 1 ( 1+ i) Pi = 1 1 ( 1+ i) n 60 20

21 Suppose you want to buy a home and take out a 30-year mortgage for $240,000. The annual interest rate is 5.75%. a) What is the monthly payment? b) What total amount of money do you pay over the life of the loan (assuming all regular payments are made)? c) How much of you total payments is interest? 61 (a) We use the formula to get $ per month. R Pi = n 1 ( 1+ i) Pi = 1 1 ( 1+ i) n 62 (b) and (c) The total amount of money we pay is: 360 $ = $ Hence, the amount of interest paid is: $504, $240,000 = $264,

22 9-10: Loan Payments (PMT) on Excel The PMT command will do these computations for us automatically. Command Format: =PV(rate, nper, pv, [fv], [type]) This is i, the rate per period This is n, the total # of periods This is the present value of the loan This is the FV you want after the last pmt Default=0 Blank for ordinary annuity, 1 for annuity due 64 Amortization Tables Am amortization table is a chart that shows the balance of the loan/account after each payment. Sometimes, it includes information on how much of each payment is interest and how much goes to pay off the loan balance. We will not need this extra information for our project, however. 65 Suppose you buy a house and take out a 15- year mortgage. The loan amount is $180,000 and the annual interest rate is 6%. Let s build an amortization table. Excel File Link 66 22

23 9-12: Adjusting for Inflation As we discussed earlier, inflation can seriously devalue a loan or asset over time. For example, if the average inflation rate is 3.5%, how much will $50 be worth in 5 years (in terms of today s dollars)? In other words, in five years how much can I buy with a $50 bill compared to what I can buy today? 67 F = P(1+i) n 50 = P(1+.035) 5 50 = P( ) 50/( ) = P $42.10 = P Important: Notice that we substituted $50 for F since that is what we know we will have in the future. We solve for P since we want to know what the Future $50 is worth in Present dollars. 68 Terms Nominal Dollars are those that have not been adjusted for inflation. Real Dollars = Present Dollars are those that have been adjusted for inflation and therefore reflect the spending power of some future amount of money in terms of today s dollars

24 A business takes out a loan of $1,000,000 over a 10-year period at 5% annual interest, compounded quarterly. They pay quarterly payments of both interest and principal. a) What is the total amount of payments made to the bank? b) What is the value of these payments to the bank, after adjusting each payment for inflation? 70 (a) and (b) Let s build an amortization table. Excel File Link 71 Focus on the Project Recall our Loan Data Name: John Sanders f = full loan value = $4,000,000 d = default value = $250,000 r = foreclosure amount = $2,100,000 From Project 1: E(Z) = $2,040,000 We re only going to consider the case where Y = 7 and not ranges of years as we did in Project

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 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

More information

DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS

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

More information

Chapter F: Finance. Section F.1-F.4

Chapter F: Finance. Section F.1-F.4 Chapter F: Finance Section F.1-F.4 F.1 Simple Interest Suppose a sum of money P, called the principal or present value, is invested for t years at an annual simple interest rate of r, where r is given

More information

2 The Mathematics. of Finance. Copyright Cengage Learning. All rights reserved.

2 The Mathematics. of Finance. Copyright Cengage Learning. All rights reserved. 2 The Mathematics of Finance Copyright Cengage Learning. All rights reserved. 2.3 Annuities, Loans, and Bonds Copyright Cengage Learning. All rights reserved. Annuities, Loans, and Bonds A typical defined-contribution

More information

Calculating Loan Payments

Calculating Loan Payments IN THIS CHAPTER Calculating Loan Payments...............1 Calculating Principal Payments...........4 Working with Future Value...............7 Using the Present Value Function..........9 Calculating Interest

More information

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%?

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

More information

Chapter 22: Borrowings Models

Chapter 22: Borrowings Models October 21, 2013 Last Time The Consumer Price Index Real Growth The Consumer Price index The official measure of inflation is the Consumer Price Index (CPI) which is the determined by the Bureau of Labor

More information

Chapter 4: Time Value of Money

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)

More information

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

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

More information

5. Time value of money

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

More information

Sample problems from Chapter 10.1

Sample problems from Chapter 10.1 Sample problems from Chapter 10.1 This is the annuities sinking funds formula. This formula is used in most cases for annuities. The payments for this formula are made at the end of a period. Your book

More information

Compound Interest Formula

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

More information

Compounding Quarterly, Monthly, and Daily

Compounding Quarterly, Monthly, and Daily 126 Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly,

More information

FIN 3000. Chapter 6. Annuities. Liuren Wu

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

More information

Module 5: Interest concepts of future and present value

Module 5: Interest concepts of future and present value Page 1 of 23 Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present and future values, as well as ordinary annuities

More information

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?

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

More information

first complete "prior knowlegde" -- to refresh knowledge of Simple and Compound Interest.

first complete prior knowlegde -- to refresh knowledge of Simple and Compound Interest. ORDINARY SIMPLE ANNUITIES first complete "prior knowlegde" -- to refresh knowledge of Simple and Compound Interest. LESSON OBJECTIVES: students will learn how to determine the Accumulated Value of Regular

More information

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

More information

TIME VALUE OF MONEY (TVM)

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

More information

Ordinary Annuities Chapter 10

Ordinary Annuities Chapter 10 Ordinary Annuities Chapter 10 Learning Objectives After completing this chapter, you will be able to: > Define and distinguish between ordinary simple annuities and ordinary general annuities. > Calculate

More information

Value of Money Concept$

Value of Money Concept$ Value of Money Concept$ Time, not timing is the key to investing 2 Introduction Time Value of Money Application of TVM in financial planning : - determine capital needs for retirement plan - determine

More information

Problem Set: Annuities and Perpetuities (Solutions Below)

Problem Set: Annuities and Perpetuities (Solutions Below) Problem Set: Annuities and Perpetuities (Solutions Below) 1. If you plan to save $300 annually for 10 years and the discount rate is 15%, what is the future value? 2. If you want to buy a boat in 6 years

More information

Discounted Cash Flow Valuation

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

More information

CHAPTER 1. Compound Interest

CHAPTER 1. Compound Interest CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.

More information

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 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

More information

1. Annuity a sequence of payments, each made at equally spaced time intervals.

1. Annuity a sequence of payments, each made at equally spaced time intervals. Ordinary Annuities (Young: 6.2) In this Lecture: 1. More Terminology 2. Future Value of an Ordinary Annuity 3. The Ordinary Annuity Formula (Optional) 4. Present Value of an Ordinary Annuity More Terminology

More information

Chapter 3 Mathematics of Finance

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

More information

Chapter 6. Time Value of Money Concepts. Simple Interest 6-1. Interest amount = P i n. Assume you invest $1,000 at 6% simple interest for 3 years.

Chapter 6. Time Value of Money Concepts. Simple Interest 6-1. Interest amount = P i n. Assume you invest $1,000 at 6% simple interest for 3 years. 6-1 Chapter 6 Time Value of Money Concepts 6-2 Time Value of Money Interest is the rent paid for the use of money over time. That s right! A dollar today is more valuable than a dollar to be received in

More information

Chapter The Time Value of Money

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

More information

Mathematics. Rosella Castellano. Rome, University of Tor Vergata

Mathematics. Rosella Castellano. Rome, University of Tor Vergata and Loans Mathematics Rome, University of Tor Vergata and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings

More information

1. % of workers age 55 and up have saved less than $50,000 for retirement (not including the value of a primary residence).

1. % of workers age 55 and up have saved less than $50,000 for retirement (not including the value of a primary residence). Toward Quantitative Literacy: Interesting Problems in Finance 2008 AMATYC Conference, Washington, D.C., Saturday, November 22, 2008 http://www.delta.edu/jaham Fill in the blanks. 1. % of workers age 55

More information

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 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

More information

Section 8.1. I. Percent per hundred

Section 8.1. I. Percent per hundred 1 Section 8.1 I. Percent per hundred a. Fractions to Percents: 1. Write the fraction as an improper fraction 2. Divide the numerator by the denominator 3. Multiply by 100 (Move the decimal two times Right)

More information

Chapter 2 Applying Time Value Concepts

Chapter 2 Applying Time Value Concepts Chapter 2 Applying Time Value Concepts Chapter Overview Albert Einstein, the renowned physicist whose theories of relativity formed the theoretical base for the utilization of atomic energy, called the

More information

Chapter 4. The Time Value of Money

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

More information

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;

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.

More information

Annuities and Sinking Funds

Annuities and Sinking Funds Annuities and Sinking Funds Sinking Fund A sinking fund is an account earning compound interest into which you make periodic deposits. Suppose that the account has an annual interest rate of compounded

More information

The time value of money: Part II

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

More information

Understand the relationship between financial plans and statements.

Understand the relationship between financial plans and statements. #2 Budget Development Your Financial Statements and Plans Learning Goals Understand the relationship between financial plans and statements. Prepare a personal balance sheet. Generate a personal income

More information

Statistical Models for Forecasting and Planning

Statistical Models for Forecasting and Planning Part 5 Statistical Models for Forecasting and Planning Chapter 16 Financial Calculations: Interest, Annuities and NPV chapter 16 Financial Calculations: Interest, Annuities and NPV Outcomes Financial information

More information

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

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

More information

Calculations for Time Value of Money

Calculations for Time Value of Money KEATMX01_p001-008.qxd 11/4/05 4:47 PM Page 1 Calculations for Time Value of Money In this appendix, a brief explanation of the computation of the time value of money is given for readers not familiar with

More information

Introduction to Real Estate Investment Appraisal

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

More information

Solutions to Problems: Chapter 5

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

More information

Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations

Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Real Estate Division Sauder School of Business University of British Columbia Introduction to the Hewlett-Packard

More information

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

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

More information

BEST INTEREST RATE. To convert a nominal rate to an effective rate, press

BEST INTEREST RATE. To convert a nominal rate to an effective rate, press FINANCIAL COMPUTATIONS George A. Jahn Chairman, Dept. of Mathematics Palm Beach Community College Palm Beach Gardens Location http://www.pbcc.edu/faculty/jahng/ The TI-83 Plus and TI-84 Plus have a wonderful

More information

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

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

More information

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 1. The simple interest per year is: $5,000.08 = $400 So after 10 years you will have: $400 10 = $4,000 in interest. The total balance will be

More information

TIME VALUE OF MONEY. Return of vs. Return on Investment: We EXPECT to get more than we invest!

TIME VALUE OF MONEY. Return of vs. Return on Investment: We EXPECT to get more than we invest! TIME VALUE OF MONEY Return of vs. Return on Investment: We EXPECT to get more than we invest! Invest $1,000 it becomes $1,050 $1,000 return of $50 return on Factors to consider when assessing Return on

More information

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

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

More information

Module 5: Interest concepts of future and present value

Module 5: Interest concepts of future and present value file:///f /Courses/2010-11/CGA/FA2/06course/m05intro.htm Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present

More information

TIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY

TIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY TIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction In this assignment we will discuss how to calculate the Present Value

More information

1 Interest rates, and risk-free investments

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)

More information

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

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

More information

The Institute of Chartered Accountants of India

The Institute of Chartered Accountants of India CHAPTER 4 SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY APPLICATIONS SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY- APPLICATIONS LEARNING OBJECTIVES After studying this chapter students will be able

More information

Bond valuation. Present value of a bond = present value of interest payments + present value of maturity value

Bond valuation. Present value of a bond = present value of interest payments + present value of maturity value Bond valuation A reading prepared by Pamela Peterson Drake O U T L I N E 1. Valuation of long-term debt securities 2. Issues 3. Summary 1. Valuation of long-term debt securities Debt securities are obligations

More information

Activity 3.1 Annuities & Installment Payments

Activity 3.1 Annuities & Installment Payments Activity 3.1 Annuities & Installment Payments A Tale of Twins Amy and Amanda are identical twins at least in their external appearance. They have very different investment plans to provide for their retirement.

More information

Texas Instruments BAII Plus Tutorial for Use with Fundamentals 11/e and Concise 5/e

Texas Instruments BAII Plus Tutorial for Use with Fundamentals 11/e and Concise 5/e Texas Instruments BAII Plus Tutorial for Use with Fundamentals 11/e and Concise 5/e This tutorial was developed for use with Brigham and Houston s Fundamentals of Financial Management, 11/e and Concise,

More information

TVM Applications Chapter

TVM Applications Chapter Chapter 6 Time of Money UPS, Walgreens, Costco, American Air, Dreamworks Intel (note 10 page 28) TVM Applications Accounting issue Chapter Notes receivable (long-term receivables) 7 Long-term assets 10

More information

This is Time Value of Money: Multiple Flows, chapter 7 from the book Finance for Managers (index.html) (v. 0.1).

This is Time Value of Money: Multiple Flows, chapter 7 from the book Finance for Managers (index.html) (v. 0.1). This is Time Value of Money: Multiple Flows, chapter 7 from the book Finance for Managers (index.html) (v. 0.1). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

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

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

More information

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

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

More information

A = P (1 + r / n) n t

A = P (1 + r / n) n t Finance Formulas for College Algebra (LCU - Fall 2013) ---------------------------------------------------------------------------------------------------------------------------------- Formula 1: Amount

More information

6: Financial Calculations

6: Financial Calculations : Financial Calculations The Time Value of Money Growth of Money I Growth of Money II The FV Function Amortisation of a Loan Annuity Calculation Comparing Investments Worked examples Other Financial Functions

More information

The Time Value of Money

The Time Value of Money CHAPTER 7 The Time Value of Money After studying this chapter, you should be able to: 1. Explain the concept of the time value of money. 2. Calculate the present value and future value of a stream of cash

More information

Exercise 1 for Time Value of Money

Exercise 1 for Time Value of Money Exercise 1 for Time Value of Money MULTIPLE CHOICE 1. Which of the following statements is CORRECT? a. A time line is not meaningful unless all cash flows occur annually. b. Time lines are useful for visualizing

More information

Math 120 Basic finance percent problems from prior courses (amount = % X base)

Math 120 Basic finance percent problems from prior courses (amount = % X base) Math 120 Basic finance percent problems from prior courses (amount = % X base) 1) Given a sales tax rate of 8%, a) find the tax on an item priced at $250, b) find the total amount due (which includes both

More information

PowerPoint. to accompany. Chapter 5. Interest Rates

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

More information

CHAPTER 5. Interest Rates. Chapter Synopsis

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)

More information

10. Time Value of Money 2: Inflation, Real Returns, Annuities, and Amortized Loans

10. Time Value of Money 2: Inflation, Real Returns, Annuities, and Amortized Loans 10. Time Value of Money 2: Inflation, Real Returns, Annuities, and Amortized Loans Introduction This chapter continues the discussion on the time value of money. In this chapter, you will learn how inflation

More information

Determinants of Valuation

Determinants of Valuation 2 Determinants of Valuation Part Two 4 Time Value of Money 5 Fixed-Income Securities: Characteristics and Valuation 6 Common Shares: Characteristics and Valuation 7 Analysis of Risk and Return The primary

More information

Section 5.1 - Compound Interest

Section 5.1 - Compound Interest Section 5.1 - Compound Interest Simple Interest Formulas If I denotes the interest on a principal P (in dollars) at an interest rate of r (as a decimal) per year for t years, then we have: Interest: Accumulated

More information

How to calculate present values

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

More information

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

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.

More information

Geometric Series and Annuities

Geometric Series and Annuities Geometric Series and Annuities Our goal here is to calculate annuities. For example, how much money do you need to have saved for retirement so that you can withdraw a fixed amount of money each year for

More information

Chapter 21: Savings Models

Chapter 21: Savings Models October 16, 2013 Last time Arithmetic Growth Simple Interest Geometric Growth Compound Interest A limit to Compounding Problems Question: I put $1,000 dollars in a savings account with 2% nominal interest

More information

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

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.

More information

Solutions to Time value of money practice problems

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,

More information

Main TVM functions of a BAII Plus Financial Calculator

Main TVM functions of a BAII Plus Financial Calculator Main TVM functions of a BAII Plus Financial Calculator The BAII Plus calculator can be used to perform calculations for problems involving compound interest and different types of annuities. (Note: there

More information

Dick Schwanke Finite Math 111 Harford Community College Fall 2013

Dick Schwanke Finite Math 111 Harford Community College Fall 2013 Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of

More information

Using the Finance Menu of the TI-83/84/Plus calculators KEY

Using the Finance Menu of the TI-83/84/Plus calculators KEY Using the Finance Menu of the TI-83/84/Plus calculators KEY To get to the FINANCE menu On the TI-83 press 2 nd x -1 On the TI-83, TI-83 Plus, TI-84, or TI-84 Plus press APPS and then select 1:FINANCE The

More information

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 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

More information

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION

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

More information

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

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

More information

The Time Value of Money

The Time Value of Money C H A P T E R6 The Time Value of Money When plumbers or carpenters tackle a job, they begin by opening their toolboxes, which hold a variety of specialized tools to help them perform their jobs. The financial

More information

FINANCIAL CALCULATIONS

FINANCIAL CALCULATIONS FINANCIAL CALCULATIONS 1 Main function is to calculate payments, determine interest rates and to solve for the present or future value of a loan or an annuity 5 common keys on financial calculators: N

More information

Present Value Concepts

Present Value Concepts Present Value Concepts Present value concepts are widely used by accountants in the preparation of financial statements. In fact, under International Financial Reporting Standards (IFRS), these concepts

More information

Finance Unit 8. Success Criteria. 1 U n i t 8 11U Date: Name: Tentative TEST date

Finance Unit 8. Success Criteria. 1 U n i t 8 11U Date: Name: Tentative TEST date 1 U n i t 8 11U Date: Name: Finance Unit 8 Tentative TEST date Big idea/learning Goals In this unit you will study the applications of linear and exponential relations within financing. You will understand

More information

Chapter 4: Managing Your Money Lecture notes Math 1030 Section C

Chapter 4: Managing Your Money Lecture notes Math 1030 Section C Section C.1: The Savings Plan Formula The savings plan formula Suppose you want to save money for some reason. You could deposit a lump sum of money today and let it grow through the power of compounding

More information

The values in the TVM Solver are quantities involved in compound interest and annuities.

The values in the TVM Solver are quantities involved in compound interest and annuities. Texas Instruments Graphing Calculators have a built in app that may be used to compute quantities involved in compound interest, annuities, and amortization. For the examples below, we ll utilize the screens

More information

Discounted Cash Flow Valuation

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?

More information

The Time Value of Money

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

More information

E INV 1 AM 11 Name: INTEREST. There are two types of Interest : and. The formula is. I is. P is. r is. t is

E INV 1 AM 11 Name: INTEREST. There are two types of Interest : and. The formula is. I is. P is. r is. t is E INV 1 AM 11 Name: INTEREST There are two types of Interest : and. SIMPLE INTEREST The formula is I is P is r is t is NOTE: For 8% use r =, for 12% use r =, for 2.5% use r = NOTE: For 6 months use t =

More information

Present Value and Annuities. Chapter 3 Cont d

Present Value and Annuities. Chapter 3 Cont d Present Value and Annuities Chapter 3 Cont d Present Value Helps us answer the question: What s the value in today s dollars of a sum of money to be received in the future? It lets us strip away the effects

More information

UNIT AUTHOR: Elizabeth Hume, Colonial Heights High School, Colonial Heights City Schools

UNIT AUTHOR: Elizabeth Hume, Colonial Heights High School, Colonial Heights City Schools Money & Finance I. UNIT OVERVIEW & PURPOSE: The purpose of this unit is for students to learn how savings accounts, annuities, loans, and credit cards work. All students need a basic understanding of how

More information

Present Value (PV) Tutorial

Present Value (PV) Tutorial EYK 15-1 Present Value (PV) Tutorial The concepts of present value are described and applied in Chapter 15. This supplement provides added explanations, illustrations, calculations, present value tables,

More information

EXAM 2 OVERVIEW. Binay Adhikari

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

More information

Chapter 4. Time Value of Money. Copyright 2009 Pearson Prentice Hall. All rights reserved.

Chapter 4. Time Value of Money. Copyright 2009 Pearson Prentice Hall. All rights reserved. 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

More information