Fahmi Ben Abdelkader HEC, Paris Fall Students version 9/11/2012 7:50 PM 1

Financial Economics Time Value of Money Fahmi Ben Abdelkader HEC, Paris Fall 2012 Students version 9/11/2012 7:50 PM 1

Chapter Outline Time Value of Money: introduction Time Value of money Financial Decision making Learning objectives The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price Applying The Rules of Time Travel Compounding Laws and Annual Effective Rate 2

Time Value of Money: introduction Time Value of money Financial Decision making Learning objectives Introduction «Time is money» 3

Time Value of Money: introduction Time Value of money Financial Decision making Learning objectives What is our 1 worth after today? January 1913 1 January 2012 1 A euro today is worth more than a euro in one year You are lending 10,000 today to your friend. He promised to pay you back 2,500 every quarter next year. Is he a good friend? 4

Time Value of Money: introduction Time Value of money Financial Decision making Learning objectives How time impacts the value of money? You can receive either 1,000 today or 1,000 in the future. What do you prefer? Why? Uncertainty: You do not know what will happen tomorrow Inflation: Purchase power of 1,000 decreases with time. Opportunity cost: 1,000 can be invested today and will pay interests in the future. Money received today is better than money received tomorrow (just save it and spend it tomorrow) 5

Time Value of Money: introduction Time Value of money Financial Decision making Learning objectives Financial decision making : Risk and Return David Choe, Graffiti Artist Facebook Headquarters by David Choe In 2005, David Choe opted for Facebook stock instead of $60,000 in cash for covering the walls of Facebook headquarters with spray-painted murals. Today, Facebook Graffiti Artist Could be Worth more than $200 Million Example: Today you invest 1,000 and in 4 years time you will receive Eu 4,000 or nothing with probability 50%. Should you invest in this project? 6

Time Value of Money: introduction Time Value of money Financial Decision making Learning objectives Financial decision making : analyzing costs and benefits A financial manager s job is to make decisions that increase the value of the firm For good decisions, the benefits value exceeds the costs Example: The world's largest passenger aircraft, the Airbus A380, made its debut commercial flight in October 2007 with Singapore Airlines from Singapore to Sydney Development work of the A380 began in earnest in 1993. How did Airbus managers decide that this was a good decision? The A380 project involves revenues and expenses that will occur at different points in time, may be in a different currencies and may have different risks associated with them To make valid comparison, we must use the tools of finance to express all costs and benefits in common terms 7

Time Value of Money: introduction Time Value of money Financial Decision making Learning objectives Learning objectives Evaluate an investment decision by answering this question: Does the cash value today of its benefits exceed the cash value today of its costs? The Net Present Value (NPV) Describe and apply the three rules of time travel: comparing values at the same point in time, compounding and discounting Calculate the Net Present Value of a cash flow stream 8

The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price Never forget that Time is money! Example 1: Consider an investment opportunity with the following certain cash flows. Cost: 100,000 today Benefit: 102,000 in one year Should you invest in this project? The project s net value = Only values.. can be compared Interest rate (risk free)= 3% Value today Value in one year Cost : 100,000 today 100 000 Benefit: 102,000 in one year 102 000 The project s net value 10

The Net Present Value Rule How to measure the time value of money? Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price If you deposit 10,000 today in a bank account paying 3% interest, you will have 10,300 at the end of one year The Time Value of Money = 10,300-10,000 The rate at which we can exchange money today for money in the future is determined by the current interest rate Risk Free Interest Rate, r f : The interest rate at which money can be borrowed or lent without risk. 11

The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price The Interest Rate: An Exchange Rate Across Time Converting Between Dollars Today and Gold, Euros, or Dollars in the Future Interest rate factor Source : Pearson Education 2011 Discount Factor 12

The Interest Rate: An Exchange Rate Across Time Example 1 (Cont d): The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price Interest rate (risk free)= 3% Value today Value in one year Cost : 100,000 today 100 000 100 000 x (1+3%) = 103 000 Benefit: 102,000 in one year 102 000 / (1+3%) = 99 029.13 102 000 The project s net value - 970.87-1000 Compounding Present Value (PV) 2012 PV Future Value (FV) 2013 FV Discounting 2012 PV 2013 FV Discount Factor 13

Comparing costs at different points in time Problem The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price The cost of rebuilding the San Francisco Bay Bridge to make it earthquake-safe was approximately $3 billion in Jan 2012. At the time, engineers estimated that if the project were delayed to 2013, the cost would rise by 10%. If the interest rate was 2%, what was the cost of a delay in terms of dollars in 2012? 2012 2013 14

Three Rules of Time Travel The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price Financial decisions often require combining cash flows or comparing values. Three rules govern these processes Rule 1. Only values at the same point in time can be compared or combined Rule 2. To move a cash flow forward in time, you must compound it 2012 2013 PV FV Rule 3. To move a cash flow backward in time, you must discount it 2012 2013 PV FV 15

The Net Present Value Rule The NPV : a measure of value creation Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price The net present value (NPV) of a project or investment is the difference between the present value of its benefits and the present value of its costs. NPV = PV (Benefits) PV (Costs) NPV = PV (All project cash flows) CF t : Project Cash Flows CF 0 : Initial investment r = Discounting rate The NPV of a project can be interpreted as the value today of the wealth that could be created by the project Projects with positive NPV are potentially value-creating projects 16

The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price The NPV rule : the golden rule of financial decision making When making an investment decision, take the alternative with the highest NPV. Choosing this alternative is equivalent to receiving its NPV in cash today. Accepting or Rejecting a Project Accept those projects with positive NPV because accepting them would create value and potentially increase the wealth of investors Reject those projects with negative NPV because accepting them would reduce the wealth of investors. 17

The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price The NPV rule : the golden rule of financial decision making Example: Evaluating a real Estate investment project The purchase price 2012 Estimated selling price 2013-360,000 + 400,000 If the interest rate was 3%, should you invest in this project? 18

The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price The NPV rule : the golden rule of financial decision making Problem Your firm needs to buy a new 9,500 copier. As part of a promotion, the manufacturer has offered to let you pay 10,000 in one year, rather than pay cash today. Suppose the risk-free interest rate is 7% per year. Is this offer a good deal? Show that its NPV represents cash in your pocket. The purchase price today ( 9,500) The purchase price in one year ( 10,000) 2012 2013 19

The Net Present Value Rule Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price Choosing among alternative plans Problem: suppose you started a Web site hosting business and then decided to return to school. Now that you are back in school, you are considering selling the business within the next year. An investor has offered to buy the business for 200,000 whenever you are ready. If the interest rate is 10%, which of the following three alternatives is the best? 1. Sell the business now. 2. Scale back the business and continue running it while you are in school for one more year, and then sell the business (requiring you to spent 30,000 on expenses now, but generating 50,000 in profit at the end of the year). 3. Hire someone to manage the business while you are in school for one more year, and then sell the business (requiring you to spend 50,000 on expenses now, but generating 100,000 in profit at the end of the year). Cash Flows and NPVs for Web Site Business Alternatives Source : Berk J. and DeMarzo P. (2011), Corporate Finance, Second Edition. Pearson Education. (Example 3.5 p.61-62.) 20

The Net Present Value Rule Introduction to the Law of One Price Interest rate and the Time Value of Money Present Value versus Future Value The NPV decision rule Arbitrage and the Law of One Price Arbitrage The practice of buying and selling equivalent goods in different markets to take advantage of a price difference. An arbitrage opportunity occurs when it is possible to make a profit without taking any risk or making any investment. Normal Market A competitive market in which there are no arbitrage opportunities. Law of One Price If equivalent investment opportunities trade simultaneously in different competitive markets, then they must trade for the same price in both markets. 21

Applying The Rules of Time Travel Compounding Laws and Annual Effective Rate : may seem like a trivial task but very useful A timeline is a linear representation of the timing of potential cash flows Example : Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments Differentiate between two types of cash flows: Inflows are positive cash flows. Outflows are negative cash flows, which are indicated with a (minus) sign. 24

Applying The Rules of Time Travel Compounding Laws and Annual Effective Rate : may seem like a trivial task but very useful Quick-Check Problem Suppose you must pay tuition of $10,000 per year for the next two years. Your tuition payments must be made in equal installments at the start of each semester. What is the timeline of your tuition payments? 25

Compounding Laws Applying The Rules of Time Travel Compounding Laws and Annual Effective Rate Problem Alain Tauxvabien, your bank advisor, suggests you invest 1,000 in an account paying 10% interest per year. How much will you have in the account in 2 years? In 3 years? 3 x 1.10 1331 3 times Future Value of a Cash Flow 26

Compounding Laws Applying The Rules of Time Travel Compounding Laws and Annual Effective Rate Quick-Check Problem Suppose you have a choice between receiving 5,000 today or 10,000 in five years. You believe you can invest the 5,000 in an account paying 10% interest per year. What would be your choice? 27

Applying The Rules of Time Travel Compounding Laws and Annual Effective Rate Earning interest on interest : compound interest The Composition of Interest Over Time : the effect of compounding Source : Berk J. and DeMarzo P. (2011), Corporate Finance, Second Edition. Pearson Education. (Figure 4.1 p.90) 28

Compounding Laws Applying The Rules of Time Travel Compounding Laws and Annual Effective Rate A compounding law is a function of time that tells how many Euros an investor will receive at some future date t for each Euro invested today until t. Future Value of a Cash Flow The interest rate r is typically stated annually The annual percentage rate - APR but, interests can be compounded several times per year Frequency of compounding k : how often in a year I will receive the interests. 29

Applying The Rules of Time Travel Applying the rule of time travel to a stream of cash flows Example Suppose we plan to save $1000 today, and $1000 at the end of each of the next two years. If we can earn a fixed 10% interest rate on our savings, how much will we have three years from today? Three ways 30

Applying The Rules of Time Travel Applying the rule of time travel to a stream of cash flows A general formula for valuing a stream of cash flows Present Value of a Cash Flow Stream 33

Applying The Rules of Time Travel Present Value of a Cash Flow Stream Problem You have just graduated and need money to buy a new car. Your father will lend you the money so long as you agree to pay him back within four years, and you offer to pay him the rate of interest that he would otherwise get by putting his money in a savings account. Based on your earnings and living expenses, you think you will be able to pay him 5000 in one year, and then 8000 each year for the next three years. If your father would otherwise earn 6% per year on his savings, how much can you borrow from him? Your father should be willing to lend you 24,890 in exchange for your promised payments. You will pay him 5000+3*8000= 29 000. Is this transaction a good deal for your father? 0 1 2 3 4 24,890? 35

Applying The Rules of Time Travel The Net Present Value of a Cash Flow Stream Problem 0 1 2 3 37

Applying The Rules of Time Travel The Net Present Value of a Cash Flow Stream Quick-Check Problem Would you be willing to pay $5,000 for the following stream of cash flows if the discount rate is 7%? 0 1 2 3 38

Perpetuities Applying The Rules of Time Travel A perpetuity is a stream of equal cash flows that occur at regular intervals and last for ever. It has no fixed maturity date. Example: The Consol (The British government perpetual bond) 0 1 2 3 C C C The present value of a perpetuity P with payment C and interest r is given by: Geometric progression n DIG DEEPER See derivation of perpetuity formulas, http://fahmi.ba.free.fr/courses.html#hec_students 39

Applying The Rules of Time Travel The present value of a Perpetuity by the Law of One Price Example Suppose you could invest $100 in a bank account paying 5% interest per year forever. Suppose also you withdraw the interest and reinvest the $100 every year. By doing this, you can create a perpetuity paying $5 per year. The Law of One Price: the value of the perpetuity must be the same as the cost we incurred to create the perpetuity Let s generalize: suppose we invest an amount P in the bank. Every year we can withdraw the interest, = r * p, leaving the principal P. the present value of receiving C in perpetuity is then the upfront cost: C = r * P C P = r Quick check question: if r = 5%, how much will you need to invest to create a perpetuity of 500? 40

Perpetuities Applying The Rules of Time Travel Problem: Endowing a Perpetuity You are the president of the alumni association. You want to endow an annual Master graduation party at your School. You want the event to be a memorable one, so you budget 30,000 per year forever for the party. If you could earn 8% interest per year on your investments, and if the first party is in one year s time, how much will you need to donate to endow the party? 41

Annuities Applying The Rules of Time Travel An ordinary annuity is a stream of N equal cash flows paid at regular intervals. It has a fixed maturity date. Example: car loans, mortgages, bonds are annuities 0 1 2 N C C C The present value of an annuity A with payment C and interest r is given by: Geometric progression DIG DEEPER See derivation of perpetuity formulas, http://fahmi.ba.free.fr/courses.html#hec_students 42

Applying The Rules of Time Travel The present value of an Annuity by the Law of One Price Suppose you invest $100 in a bank account paying 5% interest. As with the perpetuity, suppose you withdraw the interest each year. Instead of leaving the $100 in forever, you close the account and withdraw the principal in 20 years. The Law of One Price: The present value of $5 for 20 years is $62.3 43

Applying The Rules of Time Travel The present value of an Annuity by the Law of One Price Let s generalize: suppose we invest an amount P in the bank. Every period we can withdraw the interest, C=r*P, leaving the principal P. After N periods, we close the account and we get back the original investment P. According to law of one price, P is the present value of all future cash flows. 0 1 2 N - P C C C + P Recall: 44

Applying The Rules of Time Travel The present value of a Lottery Prize Annuity Problem You are the lucky winner of the Euromillion lottery. You can take your prize money either as: (a) 30 payments of 1 million per year (starting today), or (b) 15 million paid today If the interest rate is 8%, which option should you take? 45

Applying The Rules of Time Travel The future value of an ordinary annuity 1 2 1 2 46

Applying The Rules of Time Travel The future value of an ordinary annuity Problem - Retirement Savings Plan Annuity Bernadette is 35 year old, and she has decided it is time to plan seriously for her retirement. At the end of each year until she is 65, she will save 10,000 in a retirement account. If the account earns 10% per year, how much will Bernadette have saved at age 65? 47

Growing Perpetuities Applying The Rules of Time Travel A Growing Perpetuity is a stream of cash flows that occur at regular intervals and grow at a constant rate for ever. 0 1 2 3 4 C *( 1 ) C *( 1+ g) 2 C *( 1+ g) 3 C + g The present value of a Growing Perpetuity P with payment C and interest r is given by: Geometric progression n DIG DEEPER See derivation of perpetuity formulas, http://fahmi.ba.free.fr/courses.html#hec_students 48

Growing Perpetuities Applying The Rules of Time Travel Problem: Endowing a Growing Perpetuity As the president of the alumni association, you planned to donate money to fund an annual 30,000 Master Graduation party. Given an interest rate of 8% per year, the required donation was 375,000 today. However, the association board asked that you increase the donation to account for the effetct of inflation on the cost of the party in future years. Although 30,000 is adequate for next year s party, the board estimates that the party s cost will rise by 4% per year thereafter. To satisfy their request, how much do you need to donate now? 49

Growing Annuities Applying The Rules of Time Travel A Growing Annuity is a stream of N growing cash flows, paid at regular intervals. It has a fixed maturity date. The first cash flow does not grow. 0 1 2 3 N C ( 1 ) C * + g ( 1 ) 2 C * + g C * N 1 ( 1+ g) The present value of an N-period Growing Annuity GA with initial cash flow C, growth rate g and interest rate r is given by: DIG DEEPER See derivation of perpetuity formulas, http://fahmi.ba.free.fr/courses.html#hec_students 50

Growing Annuities Applying The Rules of Time Travel Problem - Retirement Savings Plan Annuity In the previous example, Bernadette considered saving 101,000 per year for her retirement. Although 10,000 is the most she can save in the first year, she expects her salary to increase each year so that she will be able to increase her savings by 5% per year. With this plan, if she earnes 10% per year on her savings, how much will Bernadette have saved at age 65? 35 36 37 38 65 0 1 2 3 30 51

Applying The Rules of Time Travel Special cases: Computing a Loan Payment Problem Your firm plans to buy a warehouse for 100,000. The bank offers you a 30-year loan with equal annual payments and un interest rate of 8% per year. The bank requires that your firm pay 20% of the purchase price as a down payment, so your can borrow only 80,000. What is the annual loan payment? The timeline from the bank s perspective 52

Applying The Rules of Time Travel Special cases: solving for the cash flows to save to accumulate a certain amount in the future Problem Germaine and Bernabé have just had a child. They decide to be prudent and start saving this year for her college education. They would like to have 60,000 saved by the time their daughter is 18 years old. If they can earn 7% per year on their savings, how much do Germaine and Bernabé need to save each year to meet their goal? 53

Applying The Rules of Time Travel Special cases: Solving for the number of periods Example Suppose we invest 10,000 in an account paying 10% interest. How long will it take for the amount to grow to 20,000? 54

Applying The Rules of Time Travel Special cases: Solving for the number of periods Problem You are saving for a down payment on a house. You have 10,050 saved already, and you can afford to save an additional 5,000 per year at the end of each year. If you earn 7.25% per year on your savings, how long will it take you to save 60,000? 55

Applying The Rules of Time Travel Special cases: The Internal Return Rate (IRR) In some situations, you know the present value and cash flows of an investment opportunity but you do not know the internal rate of return (IRR), the interest rate that sets the net present value of the cash flows equal to zero. Example 1 Suppose that you have an investment opportunity that requires a 1,000 investment today and will have a 2,000 payoff in 6 years. What is the internal rate of return? 0 1 2 3 6 What is the discount rate that sets NPV to zero? 56

Applying The Rules of Time Travel Special cases: The Internal Return Rate (IRR) Example 2 Suppose your firm needs to puchase a new forklift with a 40,000 cash price. The dealer offers you financing with no down payment and four annual payments of 15,000. to evaluate the loan that the dealer is offering you, you will want to compare the rate on the loan with the rate that your bank is willing to offer you. Given the loan payment that the dealer quotes, how do you compute the interest rate charged by the dealer? 0 1 2 3 4-40,000-15,000-15,000-15,000-15,000 r? With NPV = 0 With 3 or more periods, there is no general formula to solve for r ; trial and error is the only way to compute the IRR 57

Applying The Rules of Time Travel Special cases: The Internal Return Rate (IRR) What is r so that : r = 10% : r = 20% : r = 18.45% = IRR : The interest rate charged by the dealer is 18.45% 58

Applying The Rules of Time Travel Special cases: The Internal Return Rate (IRR) Problem Abdel-Baptiste has just graduated with his MBA. Rather than take the job he was offered at a prestigious investment bank Lazard he has decided to go into business for himself. However, Lazard was so impressed with Abdel-Baptiste that it has decided to fund his business. In return for an initial investment of 1 million, Abdel-Baptiste has agreed to pay the bank 125,000 at the end of each year for the next 30 years. What is the internal rate of return on Lazard s investment in Abdel s company, assuming he fulfills his commitment? The timeline from the bank s perspective 0 1 2 3 30 59

Applying The Rules of Time Travel Special cases: The Internal Return Rate (IRR) Problem (cont d) Lazard offers Abdel-Baptiste a second option for repayment of the loan. He can pay 100,000 the first year, increase the amount by 4% each year, and continue to make these payments forever, rather than 30 years. What is the IRR in this case? The timeline from the bank s perspective - 1,000,000 0 1 2 3 30 + 125,000 + 125,000 + 125,000 + 125,000 60