Interest Rates: Loans, Credit Cards, and Annuties. Interest Rates: Loans, Credit Cards, and Annuties 1/43

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Interest Rates: Loans, Credit Cards, and Annuties. Interest Rates: Loans, Credit Cards, and Annuties 1/43"

Transcription

1 Interest Rates: Loans, Credit Cards, and Annuties Interest Rates: Loans, Credit Cards, and Annuties 1/43

2 Last Time Last time we discussed compound interest and saw that money can grow very large given enough time, or a high enough interest rate. We ll see how this is relevant for discussing loans. Home loans, which often run for 30 years, are over a long enough period of time that rates of interest are very significant. We ll see why. Interest Rates: Loans, Credit Cards, and Annuties 2/43

3 Loans If you borrow money, how is the monthly payment determined? What does it even mean to say you get a car loan at an annual interest rate of 6%? Interest Rates: Loans, Credit Cards, and Annuties 3/43

4 Say you get a $20,000 car loan at 6% per year for 5 years. From the loan company s point of view, here are two scenarios: 1 The company invests the $20,000 at 6% per year, compounded monthly. It would have $26,977 after 5 years. 2 The company gives you the loan, and then invests each payment you make at 6% per year, compounded monthly. To say you have a 6% loan for 5 years means the loan company would have the same amount of money in each of the two scenarios. Interest Rates: Loans, Credit Cards, and Annuties 4/43

5 Interest Rates: Loans, Credit Cards, and Annuties 5/43

6 If the company invests each payment P at an annual interest rate r, then the last payment does not generate interest, so is worth exactly P. To simplify writing we ll abbreviate r/12 by q. This is the monthly interest rate. The second to last payment generates 1 month interest, so is worth P (1 + q) at the end of the loan. The third to last payment generates 2 months interest, so is worth P (1 + q) 2 at the end of the loan. And so on. If you have n payments, then the first payment generates n 1 months interest, so is worth P (1 + q) n 1 at the end. If L is the loan amount, if the company invested the money instead of giving it to you, after n months it would have L (1 + q) n. Interest Rates: Loans, Credit Cards, and Annuties 6/43

7 So, the payment satisfies the equation L (1 + q) n = P + P (1 + q) + + P (1 + q) n 1 = P (1 + (1 + q) + + (1 + q) n 1) since both sides represent how much money the loan company would have at the end of your loan in the two different scenarios we mentioned above. Fortunately, expressions like the one on the right occur often, and people have found formulas to simplify them. Interest Rates: Loans, Credit Cards, and Annuties 7/43

8 For example, suppose we consider the expression Then s = n 1 s + 3 n = n n = ( n 1) Rearranging gives 3 n 1 = 2s, and so s = 3n 1 2 = 1 + 3s Interest Rates: Loans, Credit Cards, and Annuties 8/43

9 More generally, if a is any number (other than 1), then 1 + a + a a n 1 = an 1 a 1 Applying this to the loan situation, with a = 1 + q gives us ( 1 + (1 + q) + + (1 + q) n 1 ) = (1 + q)n 1 q Thus, our loan formula simplifies as L (1 + q) n = P (1 + (1 + q) + + (1 + q) n 1) ( (1 + q) n ) 1 = P q Interest Rates: Loans, Credit Cards, and Annuties 9/43

10 Solving for P gives ( (1 + q) L (1 + q) n n ) 1 = P q P = Lq (1 + q)n (1 + q) n 1 or, if we divide the top and bottom of the fraction by (1 + q) n, P = Lq 1 (1 + q) n Interest Rates: Loans, Credit Cards, and Annuties 10/43

11 Loan Formula To summarize, if we borrow L at an interest rate of r per year, and make n payments, then the monthly payment P is P = Lr ( 12 1 ( 1 + r ) ) n 12 Using the Interest Calculator spreadsheet or an online financial calculator is a good way to do these calculations. Last time we saw the Moneychimp.com calculator that can do these sort of calculations without having to use the formulas. Interest Rates: Loans, Credit Cards, and Annuties 11/43

12 Clicker Questions Q What is the monthly payment for a $20,000 car loan at an annual interest rate of 6% for 5 years? The loan formula is P = Lr ( 12 1 ( 1 + r ) ) n 12 but it is a lot easier to do in the MoneyChimp calculator. Interest Rates: Loans, Credit Cards, and Annuties 12/43

13 Answer A The monthly payment (not including insurance, taxes, license, etc.) would be $ Q What if the loan was at 8%? A The monthly payment would now be $ Q A 2014 Mercedes-Benz SLS AMG convertible starts at $208,000. If you borrowed $200,000 at 6% for 5 years to buy one, what would your monthly payment be? A It would be 10 times the first answer, or $3, Interest Rates: Loans, Credit Cards, and Annuties 13/43

14 If you bought the Mercedes, your total payments for the loan would be $3, = $231, This means you d pay about $32,000 in interest in order to borrow $200,000 for 5 years at 6%. Interest Rates: Loans, Credit Cards, and Annuties 14/43

15 Home Loans With home loans typically for 30 years, interest is much more an issue than for car loans. Getting a 15 year loan can save a huge amount of money! Interest Rates: Loans, Credit Cards, and Annuties 15/43

16 Clicker Question Let s suppose you borrow $150,000 to buy a house. Let s first compare doing this now to the early 1980s, when interest rates were much higher. How much more money in interest do you think you d pay with a 15% loan versus a 4% loan over the lifetime of the loan? A $4,000 more B $40,000 more C $400,000 more D $4,000,000 more E $40,000,000 more Interest Rates: Loans, Credit Cards, and Annuties 16/43

17 Answer C You d pay about $400,000 more in interest. Let s see why. We ll use the Interest Calculator spreadsheet to do this. The results do not include real estate taxes and insurance, which can be a few hundred dollars a month. Rate Monthly Payment Total Payments Interest Paid 4% $ $257,804 $107,804 6% $ $323,757 $173,757 10% $1,316,36 $473,889 $323,889 15% $1, $682,800 $532,800 Interest Rates: Loans, Credit Cards, and Annuties 17/43

18 Mortgage rates of 15% were common in in the 1980s. By 1990 rates were down around 10%. Currently the lowest rates are at or under 4%. If you get a 15 year home loan, generally you ll get a better interest rate. Let s compare a 30 year loan at 10% to a 15 year loan at 9%. We ll continue to consider a $150,000 loan. Again, we ll use the Interest Calculator spreadsheet for this. The results are Rate Monthly Payment Total Payments 10% for 30 years $1,316,36 $473,889 9% for 15 years $1, $273,852 Interest Rates: Loans, Credit Cards, and Annuties 18/43

19 Let s do the same but for a 30 year loan at 6% and a 15 year loan at 4.5%. The results are Rate Monthly Payment Total Payments 6% for 30 years $ $323, % for 15 years $1, $206,548 Interest Rates: Loans, Credit Cards, and Annuties 19/43

20 Rate Monthly Payment Total Payments 10% for 30 years $1,316,36 $473,889 9% for 15 years $1, $273,852 6% for 30 years $ $323, % for 15 years $1, $206,548 In the first example, going from a 30 year loan to a 15 year loan would save about $200,000 over the life of the loan. In the second example the savings isn t as much, but it is still about $115,000. While affording the higher monthly payment may not always be possible, if you can do it you ll save a lot of money in the long run. Interest Rates: Loans, Credit Cards, and Annuties 20/43

21 Refinancing While this isn t as relevant now since interest rates are so low, at times interest rates drop a few points after you buy a house. If you borrow $150,000 for 30 years at 7%, and you have the opportunity to refinance with a 15 year loan at 4%, is it a good idea? At least, what are the monthly payments? Rate Monthly Payment Total Payments 7% for 30 years $ $359,263 4% for 15 years $ $199,716 Refinancing would increase your monthly payment by about $115, but you d save about $160,000 over the life of the loan. Interest Rates: Loans, Credit Cards, and Annuties 21/43

22 The calculation above represents what would happen if you refinance at the very beginning of your loan. For a little more realistic example, suppose you had a 7% loan and paid on it for 5 years, at which time you could refinance for 4% for a 15 year loan. To do this calculation we d need to know how much you still owe. The Loan Schedule tab of the Interest Calculator spreadsheet does this kind of calculation. After 5 years you d still owe $141,000. Rate Monthly Payment Total Payments 7% for 25 more years $ $299,386 4% for 15 years $ $187,733 Interest Rates: Loans, Credit Cards, and Annuties 22/43

23 To get the amount we d pay with the 7% loan, we take the $359,263 we d pay over 30 years, and multiply by 25/30 to see how much we d pay in 25 years. Or, to come close to this amount, we could see how much we d pay by borrowing $141,000 for 25 years at 7%. Since there are up-front costs in refinancing, typically a few thousand dollars, whether this is a good idea depends on how long you ll keep the house and how far into your mortgage you are. If you plan to keep your house for at least 5 years, refinancing when you can get a lower interest rate is often a good deal. Interest Rates: Loans, Credit Cards, and Annuties 23/43

24 Credit Cards Credit cards work the same as loans.the main difference is the high interest rate most charge. Interest rates up to 20% per year have been common. If you pay off your credit card in full each month, then you don t get charged interest. Using a credit card this way amounts to treating it as a debit card from your checking account. What happens if you don t pay it off in full? More particularly, what if you pay the minimum payment each month? Interest Rates: Loans, Credit Cards, and Annuties 24/43

25 The minimum payment on a credit card bill is typically the larger of a fixed amount and a certain percentage of your balance. Suppose your minimum payment is the larger of $20 or 1.5% of your balance. Let s suppose the credit card company charges you 15% interest on unpaid balances. Let s also suppose you have a $10,000 credit limit, and you max out your credit card. Your next statement then shows a $10,000 balance. Interest Rates: Loans, Credit Cards, and Annuties 25/43

26 Clicker Question Q What is your minimum payment on a $10,000 balance, when the credit card company requires you to pay at least the larger of $20 or 1.5% of your balance? A Since 1.5% of $10,000 is your minimum payment is $150. $10, = $150 Interest Rates: Loans, Credit Cards, and Annuties 26/43

27 Let s now think what will happen if you make the minimum payment each month on your credit card. Your credit card company will charge you 15% = 1.25% interest on an unpaid balance each month. 12 You pay your minimum payment of $150 the first month. The next statement you ll have a starting balance of $9,850 since you paid $150 of your $10,000 balance. The company will then charge you 1.25% of that in interest. Interest Rates: Loans, Credit Cards, and Annuties 27/43

28 Clicker Questions Q If your balance is $9,850 and you pay 1.25% interest, how much interest do you owe that month? A You owe in interest charges that month. $9, = $123 Q What will be your next monthly payment, if you pay the minimum payment? A Your new balance is $9,850 + $123 = $9,973, and you will have to pay 1.5% of that for your minimum payment. So, your minimum payment is $9, = $ Interest Rates: Loans, Credit Cards, and Annuties 28/43

29 We can continue seeing what happens when we pay the minimum payment each month. The result for the first year will be the following table. Interest Rates: Loans, Credit Cards, and Annuties 29/43

30 After 12 months of payments you ll have paid around $1,774 with $1,336 of that in interest. You will still owe around $9,700. One problem with this, besides the large amount of interest you pay, is that you won t be able to use your credit card very much, because your balance is close to your credit limit. To continue, we could keep going month by month. However, with some algebra we can be more efficient. This will help to see what will happen after several years. Interest Rates: Loans, Credit Cards, and Annuties 30/43

31 Suppose the original balance in some month (after the first) is B. We then get charged B interest. The new balance is then B + B = B Our minimum payment is 1.5% of this, or B Our next month s balance is then the previous month s new balance minus the minimum payment. That is, the next month s original balance is new balance payment = = B B = B = B Interest Rates: Loans, Credit Cards, and Annuties 31/43

32 Using the same reasoning, the original balance two month s later is (B ) = B (0.9973) 2 In general, n months later, the original balance would be B (0.9973) n Interest Rates: Loans, Credit Cards, and Annuties 32/43

33 This helps to figure out interest and payments years at a time. The results are summarized in the spreadsheet Credit Card Calculations.xlsx. These calculations are the reasons credit card companies are happy for people to pay the minimum payment. Interest Rates: Loans, Credit Cards, and Annuties 33/43

34 Annuities An annuity is a repeating payment, typically of a fixed amount, over a period of time. An annuity is like a loan in reverse; rather than paying a loan company, a bank or investment company pays you the monthly payment. There are two typical calculations for annuities. 1 Paying into an annuity 2 Collecting from an annuity Interest Rates: Loans, Credit Cards, and Annuties 34/43

35 Paying into an annuity means investing or saving money in order to receive an annuity in the future. Collecting from annuity then happens once you have invested enough money to receive an annuity. The MoneyChimp website isn t so easy to use for annuities. We ll use a bankrate.com website for collecting from annuity calculations. The Interest Calculator spreadsheet is a convenient way to do all these calculations. You can use the compound interest calculator on MoneyChimp.com to do paying into an annuity calculation with some trial and error. Interest Rates: Loans, Credit Cards, and Annuties 35/43

36 Paying Into an Annuity Q Suppose you need to have $100,000 saved 20 years from now. If you can invest at 6% per year, how much do you need to put away each month? A We d have to invest $216 each month to end up with $100,000 in 20 years if we received 6% on our money. Q What if instead you want to end up with $500,000 in 20 years? How much do you have to invest? A You want to end up with 5 times more money, so you need to invest 5 times more each month, or $1,082. Interest Rates: Loans, Credit Cards, and Annuties 36/43

37 We saw that if you want to end up with $500,000 by saving for 20 years at a 6% annual return, you need to invest $1,082 each month. Q How much do you need to invest to end up with $500,000 if you save for 30 years? A You d need to invest $498. Saving for a longer time means you have to save far less money each month. Interest Rates: Loans, Credit Cards, and Annuties 37/43

38 Collecting From an Annuity Q Suppose you ve saved $250,000 that you put into an annuity paying 5% per year, and you wish to collect from it for 20 years. How much monthly income will you receive? A You ll receive $1,643. If you do this sort of calculation 30 years before you plan to retire, you need to realize the amount you ll receive will be in future dollars and will sound better than the same amount today. Q What if you want to collect for 30 years? A You will collect $1,336 each month for 30 years. Interest Rates: Loans, Credit Cards, and Annuties 38/43

39 Suppose you ve invested $250,000 into an annuity paying 6% and want to be paid from it indefinitely. How much will you receive? Collecting indefinitely is treated as collected forever. This is called a perpetual annuity. Surely you d get almost nothing if they let you collect forever. Let s look at some data, where we take a larger and larger number of years to collect from the annuity. Interest Rates: Loans, Credit Cards, and Annuties 39/43

40 If you invest $250,000 in an annuity paying 5% per year, this table shows your monthly income depending on how long you collect. Number of Years Monthly to Collect Income 30 $1, $1, $1, $1, $1, $1, $1, $1,042 In fact, you can collect a reasonable amount indefinitely. Interest Rates: Loans, Credit Cards, and Annuties 40/43

41 Collecting From a Perpetual Annuity If you have L invested in an annuity paying an annual rate of r, then you ll collect L r 12 each month from a perpetual annuity. For example, in the previous example, L = $250, 000 and r = So, the monthly return would be $250, = $1, This is an easier calculation and doesn t require you to estimate how long you ll collect. Interest Rates: Loans, Credit Cards, and Annuties 41/43

42 One way to see that collecting from a perpetual annuity is a reasonable thing in terms of getting a decent amount of money is to realize that what you are doing is simply withdrawing the interest earned each period. By withdrawing the interest the principal stays untouched. Doing this allows you to earn income each month but keep the principal unchanged. This is a reasonable retirement income method, as long as you have enough principal to survive on the interest and any pension/social security you have. Interest Rates: Loans, Credit Cards, and Annuties 42/43

43 Next Week and Assignment 9 Next time we will discuss issues of statistics. Assignment 9 is on the website. It is due next Thursday. Interest Rates: Loans, Credit Cards, and Annuties 43/43

Interest Rates: Credit Cards and Annuities. Interest Rates: Credit Cards and Annuities 1/24

Interest Rates: Credit Cards and Annuities. Interest Rates: Credit Cards and Annuities 1/24 Interest Rates: Credit Cards and Annuities Interest Rates: Credit Cards and Annuities 1/24 Last Time Last time we discussed loans and saw how big an effect interest rates were on a loan, especially a home

More information

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

Paying off a debt. Ethan D. Bolker Maura B. Mast. December 4, 2007

Paying off a debt. Ethan D. Bolker Maura B. Mast. December 4, 2007 Paying off a debt Ethan D. Bolker Maura B. Mast December 4, 2007 Plan Lecture notes Can you afford a mortgage? There s a $250,000 condominium you want to buy. You ve managed to scrape together $50,000

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

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

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

Example. L.N. Stout () Problems on annuities 1 / 14

Example. L.N. Stout () Problems on annuities 1 / 14 Example A credit card charges an annual rate of 14% compounded monthly. This month s bill is $6000. The minimum payment is $5. Suppose I keep paying $5 each month. How long will it take to pay off the

More information

Finance 197. Simple One-time Interest

Finance 197. Simple One-time Interest Finance 197 Finance We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for

More information

Time Value Conepts & Applications. Prof. Raad Jassim

Time Value Conepts & Applications. Prof. Raad Jassim Time Value Conepts & Applications Prof. Raad Jassim Chapter Outline Introduction to Valuation: The Time Value of Money 1 2 3 4 5 6 7 8 Future Value and Compounding Present Value and Discounting More on

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

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

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

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

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

The following is an article from a Marlboro, Massachusetts newspaper.

The following is an article from a Marlboro, Massachusetts newspaper. 319 CHAPTER 4 Personal Finance The following is an article from a Marlboro, Massachusetts newspaper. NEWSPAPER ARTICLE 4.1: LET S TEACH FINANCIAL LITERACY STEPHEN LEDUC WED JAN 16, 2008 Boston - Last week

More information

Chapter 5 Discounted Cash Flow Valuation

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

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

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

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

4 Annuities and Loans

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

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

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

CHAPTER 4. The Time Value of Money. Chapter Synopsis

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

More information

International Financial Strategies Time Value of Money

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

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

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

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 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

Real Estate Investment Newsletter November 2003

Real Estate Investment Newsletter November 2003 Maximizing Returns on Equity Why and How In this newsletter I will explain some financial management concepts that provide a framework for maximizing your wealth accumulation over time. Proper application

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

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

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 Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount

More information

Managing Home Equity to Build Wealth By Ray Meadows CPA, CFA, MBA

Managing Home Equity to Build Wealth By Ray Meadows CPA, CFA, MBA Managing Home Equity to Build Wealth By Ray Meadows CPA, CFA, MBA About the Author Ray Meadows is the president of Berkeley Investment Advisors, a real estate brokerage and investment advisory firm. He

More information

The Time Value of Money

The Time Value of Money The Time Value of Money Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original

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

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 2 Present Value

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

More information

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

Chapter 4: Managing Your Money Lecture notes Math 1030 Section D Section D.1: Loan Basics Definition of loan principal For any loan, the principal is the amount of money owed at any particular time. Interest is charged on the loan principal. To pay off a loan, you must

More information

Get out of debt faster

Get out of debt faster Get out of debt faster Unleash the money-saving power of balance transfer credit cards. The Beginner s Guide to Balance Transfers Table of Contents creditcard.com.au Introduction 1 Balance Transfer Basics

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

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

Future Value Sinking Fund Present Value Amortization. P V = P MT [1 (1 + i) n ] i

Future Value Sinking Fund Present Value Amortization. P V = P MT [1 (1 + i) n ] i Math 141-copyright Joe Kahlig, 15C Page 1 Section 5.2: Annuities Section 5.3: Amortization and Sinking Funds Definition: An annuity is an instrument that involves fixed payments be made/received at equal

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

Perpetuities and Annuities EC 1745. Borja Larrain

Perpetuities and Annuities EC 1745. Borja Larrain Perpetuities and Annuities EC 1745 Borja Larrain Today: 1. Perpetuities. 2. Annuities. 3. More examples. Readings: Chapter 3 Welch (DidyoureadChapters1and2?Don twait.) Assignment 1 due next week (09/29).

More information

housing information www.housing-information.org Reverse Mortgages A project of Consumer Action

housing information www.housing-information.org Reverse Mortgages A project of Consumer Action housing information www.housing-information.org Reverse Mortgages A project of Consumer Action One of the major benefits of buying a home is the opportunity to build equity, or ownership, in the property.

More information

Midterm 1 Practice Problems

Midterm 1 Practice Problems Midterm 1 Practice Problems 1. Calculate the present value of each cashflow using a discount rate of 7%. Which do you most prefer most? Show and explain all supporting calculations! Cashflow A: receive

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

Chapter 21: Savings Models

Chapter 21: Savings Models October 18, 2013 Last Time A Model for Saving Present Value and Inflation Problems Question 1: Suppose that you want to save up $2000 for a semester abroad two years from now. How much do you have to put

More information

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

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

More information

Time Value of Money. Background

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

More information

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.

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

More information

Numbers 101: Cost and Value Over Time

Numbers 101: Cost and Value Over Time The Anderson School at UCLA POL 2000-09 Numbers 101: Cost and Value Over Time Copyright 2000 by Richard P. Rumelt. We use the tool called discounting to compare money amounts received or paid at different

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

UNIT 6 2 The Mortgage Amortization Schedule

UNIT 6 2 The Mortgage Amortization Schedule UNIT 6 2 The Mortgage Amortization Schedule A home mortgage is a contract that requires the homeowner to make a fixed number of monthly payments over the life of the mortgage. The duration, or length of

More information

January 22. Interest Rates

January 22. Interest Rates January 22 Interest Rates Compound Interest If you put $100 in a bank account at 5% interest per year, you will have $105 after one year. You earn $5 in interest. How much do you have after two years if

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

Jade Education Award Story: Smart Loan Strategies Page 1. This is Jade.

Jade Education Award Story: Smart Loan Strategies Page 1. This is Jade. Jade Education Award Story: Smart Loan Strategies Page 1 This is Jade. Jade borrowed a lot of money to pay for her two degrees. She consolidated several smaller loans into one big one. She wants to use

More information

Lesson 1. Key Financial Concepts INTRODUCTION

Lesson 1. Key Financial Concepts INTRODUCTION Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have

More information

CHAPTER 4 DISCOUNTED CASH FLOW VALUATION

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

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

MAT116 Project 2 Chapters 8 & 9

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

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

Table of Contents Reverse Mortgage O verview... Reverse Mortgage K ey Questions... 5 Reverse Mortgage Pros and Cons... 7 Reverse Mortgage Borr

Table of Contents Reverse Mortgage O verview... Reverse Mortgage K ey Questions... 5 Reverse Mortgage Pros and Cons... 7 Reverse Mortgage Borr 1 Table of Contents Reverse Mortgage Overview... Reverse Mortgage Key Questions... Reverse Mortgage Pros and Cons... Reverse Mortgage Borrower Qualification... Reverse Mortgage Key Items... Reverse Mortgage

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

Applying Time Value Concepts

Applying Time Value Concepts Applying Time Value Concepts C H A P T E R 3 based on the value of two packs of cigarettes per day and a modest rate of return? Let s assume that Lou will save an amount equivalent to the cost of two packs

More information

Outstanding mortgage balance

Outstanding mortgage balance Using Home Equity There are numerous benefits to owning your own home. Not only does it provide a place to live, where you can decorate as you want, but it also provides a source of wealth. Over time,

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

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

Handbook: The Cost of Borrowing. Learn what you need to know quickly.

Handbook: The Cost of Borrowing. Learn what you need to know quickly. Handbook: The Cost of Borrowing Learn what you need to know quickly. Money Matters Handbook: The Cost of Borrowing Groceries cost. Clothes cost. Furniture costs. And it costs to borrow money. The amount

More information

Young Doctors and Debt: A Script for Success

Young Doctors and Debt: A Script for Success Young Doctors and Debt: A Script for Success Shirley M. Mueller Published in Oncology Fellows: Volume 1, Issue 1 (12/08); pgs. 18-20 As the expected income of medical residents and fellows goes down, their

More information

AUSTRALIA S TOP 30 HOME LOAN MYTHS BUSTED

AUSTRALIA S TOP 30 HOME LOAN MYTHS BUSTED AUSTRALIA S TOP 30 HOME LOAN MYTHS BUSTED Australia s Top 30 Home Loan Myths BUSTED! Fairer home loans for Australians Hi, I m Mark Bouris from Yellow Brick Road. Australia, it s time for a fairer deal

More information

Chapter 3 Equivalence A Factor Approach

Chapter 3 Equivalence A Factor Approach Chapter 3 Equivalence A Factor Approach 3-1 If you had $1,000 now and invested it at 6%, how much would it be worth 12 years from now? F = 1,000(F/P, 6%, 12) = $2,012.00 3-2 Mr. Ray deposited $200,000

More information

3 More on Accumulation and Discount Functions

3 More on Accumulation and Discount Functions 3 More on Accumulation and Discount Functions 3.1 Introduction In previous section, we used 1.03) # of years as the accumulation factor. This section looks at other accumulation factors, including various

More information

Learning Objectives. Learning Objectives. Learning Objectives. Principles Used in this Chapter. Simple Interest. Principle 2:

Learning Objectives. Learning Objectives. Learning Objectives. Principles Used in this Chapter. Simple Interest. Principle 2: Learning Objectives Chapter 5 The Time Value of Money Explain the mechanics of compounding, which is how money grows over a time when it is invested. Be able to move money through time using time value

More information

F V P V = F V = P (1 + r) n. n 1. FV n = C (1 + r) i. i=0. = C 1 r. (1 + r) n 1 ]

F V P V = F V = P (1 + r) n. n 1. FV n = C (1 + r) i. i=0. = C 1 r. (1 + r) n 1 ] 1 Week 2 1.1 Recap Week 1 P V = F V (1 + r) n F V = P (1 + r) n 1.2 FV of Annuity: oncept 1.2.1 Multiple Payments: Annuities Multiple payments over time. A special case of multiple payments: annuities

More information

Engineering Economy. Time Value of Money-3

Engineering Economy. Time Value of Money-3 Engineering Economy Time Value of Money-3 Prof. Kwang-Kyu Seo 1 Chapter 2 Time Value of Money Interest: The Cost of Money Economic Equivalence Interest Formulas Single Cash Flows Equal-Payment Series Dealing

More information

THE TIME VALUE OF MONEY

THE TIME VALUE OF MONEY 1 THE TIME VALUE OF MONEY A dollar today is worth more than a dollar in the future, because we can invest the dollar elsewhere and earn a return on it. Most people can grasp this argument without the use

More information

7 Facts You Need to Know About Reverse Mortgages...

7 Facts You Need to Know About Reverse Mortgages... SPECIAL REPORT 7 Facts You Need to Know About Reverse Mortgages... By Quinn Berry 7 Facts You Need to Know About Reverse Mortgages... to help you enjoy a comfortable, worry-free retirement! Chances are

More information

5 More on Annuities and Loans

5 More on Annuities and Loans 5 More on Annuities and Loans 5.1 Introduction This section introduces Annuities. Much of the mathematics of annuities is similar to that of loans. Indeed, we will see that a loan and an annuity are just

More information

Loan Lessons. The Low-Down on Loans, Interest and Keeping Your Head Above Water. Course Objectives Learn About:

Loan Lessons. The Low-Down on Loans, Interest and Keeping Your Head Above Water. Course Objectives Learn About: usbank.com/student financialgenius.usbank.com Course Objectives Learn About: Different Types of Loans How to Qualify for a Loan Different Types of Interest Loan Lessons The Low-Down on Loans, Interest

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

Introduction 4. What is Refinancing? 5. Changing Home Loans 5 Changing Needs 6 Identifying Better Opportunities 6 Additional Home Loan Features 6

Introduction 4. What is Refinancing? 5. Changing Home Loans 5 Changing Needs 6 Identifying Better Opportunities 6 Additional Home Loan Features 6 Contents Introduction 4 What is Refinancing? 5 Changing Home Loans 5 Changing Needs 6 Identifying Better Opportunities 6 Additional Home Loan Features 6 What are the Advantages of Refinancing? 7 1. Consolidating

More information

A GUIDE TO HOME EQUITY LINES OF CREDIT. Call or visit one of our offices today to see what products in this guide we have to offer you!

A GUIDE TO HOME EQUITY LINES OF CREDIT. Call or visit one of our offices today to see what products in this guide we have to offer you! A GUIDE TO HOME EQUITY LINES OF CREDIT Call or visit one of our offices today to see what products in this guide we have to offer you! TABLE OF CONTENTS Introduction What is a home equity line of credit

More information

Appendix C- 1. Time Value of Money. Appendix C- 2. Financial Accounting, Fifth Edition

Appendix C- 1. Time Value of Money. Appendix C- 2. Financial Accounting, Fifth Edition C- 1 Time Value of Money C- 2 Financial Accounting, Fifth Edition Study Objectives 1. Distinguish between simple and compound interest. 2. Solve for future value of a single amount. 3. Solve for future

More information

_ Retirement. Planning for the Stages of. Getting started Your 20s and early 30s

_ Retirement. Planning for the Stages of. Getting started Your 20s and early 30s Planning for the Stages of _ Retirement _ Retirement is being reinvented. It s no longer our parent s retirement. Social Security alone w o n t see us through retirement, especially for higher income earners.

More information

Planning for the Stages of Retirement

Planning for the Stages of Retirement Planning for the Stages of Retirement The Financial Planning Association (FPA ) connects those who need, support and deliver financial planning. We believe that everyone is entitled to objective advice

More information

1.2-1.3 Time Value of Money and Discounted Cash Flows

1.2-1.3 Time Value of Money and Discounted Cash Flows 1.-1.3 ime Value of Money and Discounted ash Flows ime Value of Money (VM) - the Intuition A cash flow today is worth more than a cash flow in the future since: Individuals prefer present consumption to

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

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

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

Refinancing. Refinancing WISCONSIN HOMEOWNERSHIP PRESERVATION EDUCATION. Section Overview

Refinancing. Refinancing WISCONSIN HOMEOWNERSHIP PRESERVATION EDUCATION. Section Overview People refinance their homes to take advantage of lower interest rates or to decrease their monthly payment. Sometimes it is done to create extra money for purchases (like a car) or for debt repayment.

More information

21.1 Arithmetic Growth and Simple Interest

21.1 Arithmetic Growth and Simple Interest 21.1 Arithmetic Growth and Simple Interest When you open a savings account, your primary concerns are the safety and growth of your savings. Suppose you deposit $1000 in an account that pays interest at

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

Using Credit to Your Advantage.

Using Credit to Your Advantage. Using Credit to Your Advantage. Topic Overview. The Using Credit To Your Advantage topic will provide participants with all the basic information they need to understand credit what it is and how to make

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

Guide for Homebuyers

Guide for Homebuyers Guide for Homebuyers Tips for Getting a Safe Mortgage You Can Afford Q u i c k S u m m a ry Figure out what you can afford. Contact at least 3 different lenders or brokers. When you call, say: I m buying

More information

$ # $ + $ $ % $ ¾ $ ~ $² Money Math Lessons for Life

$ # $ + $ $ % $ ¾ $ ~ $² Money Math Lessons for Life $ # $ + $ $ % $ ¾ $ ~ $² Money Math Lessons for Life Written by Mary C. Suiter Sarapage McCorkle Center for Entrepreneurship and Economic Education University of Missouri St. Louis Mathematics Consultant

More information

READY TO REFI? A SIMPLE GUIDE TO REFINANCING YOUR HOME

READY TO REFI? A SIMPLE GUIDE TO REFINANCING YOUR HOME READY TO REFI? A SIMPLE GUIDE TO REFINANCING YOUR HOME INTRODUCTION You ve had your home loan for a few years, but the rate you are currently paying is much higher than the near-historic low rates you

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