time value of money - freefahmi.ba.free.fr/docs/courses/2012 hec/fba_fe_chap1_time_value.pdf ·...

Financial Financial EconomicsEconomics

Time Value of MoneyTime Value of MoneyTime Value of MoneyTime Value of Money

Fahmi Ben Abdelkader ©

HEC, ParisFall 2012

9/11/2012 7:50 PM 1

Students version

Chapter Outline

Time Value of Money: introduction Time Value of moneyFinancial Decision makingLearning objectives

The Net Present Value Rule Interest rate and the Time Value of MoneyPresent Value versus Future ValueThe NPV decision ruleArbitrage and the Law of One Price

Applying The Rules of Time Travel The Timeline

Fahmi Ben Abdelkader © Financial Economics – Time Value of Money9/11/2012 7:50 PM 2

Applying The Rules of Time Travel The TimelineCompounding Laws and Annual Effective RateValuing a stream of cash flowsPerpetuities, Annuities, and Other Special Cases

Introduction



« Time is money»

What is our 1€ worth after today?

Time Value of Money: introduction

January 1913 1€

Time Value of moneyFinancial Decision makingLearning objectives

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?

How time impacts the value of money?

Time Value of Money: introduction Time Value of mone yFinancial Decision makingLearning objectives

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


Money received today is better than money received tomorrow(just save it and spend it tomorrow)

Financial decision making : Risk and Return


In 2005, David Choe opted for Facebook stock instead of $60,000 in cash for covering the walls of Facebookheadquarters with spray-painted murals.

Facebook Headquarters by David ChoeDavid Choe, Graffiti Artist


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 ?

A financial manager’s job is to make decisions that increase the value of the firm

Financial decision making : analyzing costs and ben efits


For good decisions, the benefits value exceeds the costs


How did Airbus managers decide that this was a good decision?

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 19 93.

Example:

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


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

Evaluate an investment decision by answering this question:

Does the cash value today of its benefits exceed the cash value today of its costs ?

Learning objectives


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


Chapter Outline





Example 1:Consider an investment opportunity with the following certain cash flows.

•Cost: €100,000 today•Benefit: €102,000 in one year

Never forget that “Time is money” !


•Benefit: €102,000 in one yearShould you invest in this project ?

The project’s net value =

Interest rate (risk free)= 3% Value today Value in one year

Only values …………………………………… ………….. can be compared



Cost : €100,000 today 100 000€

Benefit: €102,000 in one year 102 000€

The project’s net value

If you deposit €10,000 today in a bank account paying 3% interest, you will have €10,300 at the end of one year

How to measure the time value of money?

The Net Present Value Rule

The Time Value of Money = €10,300 - €10,000

Interest rate and the Time Value of MoneyPresent Value versus Future ValueThe NPV decision ruleArbitrage and the Law of One Price

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, rf : The interest rate at which money can be borrowed or lent without risk.Risk–Free Interest Rate, rf : The interest rate at which money can be borrowed or lent without risk.


Converting Between Dollars Today and Gold, Euros, or Dollars in the Future

The Interest Rate: An Exchange Rate Across Time



Source : Pearson Education 2011

Interest rate factor

Discount Factor

Example 1 (Cont’d):

The Interest Rate: An Exchange Rate Across Time



Cost : €100,000 today 100 000€

Benefit: €102,000 in one year 102 000€

100 000 x (1+3%) = 103 000€

102 000 / (1+3%) = 99 029.13€Benefit: €102,000 in one year 102 000€

The project’s net value

102 000 / (1+3%) = 99 029.13€

- 970.87€ - 1000€

Present Value (PV) Future Value (FV)

Compounding

2013

FV

2012

PV


Discount Factor

Discounting

2013

FV

2012

PV

Problem

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?

Comparing costs at different points in time


20132012


Financial decisions often require combining cash flows or comparing values. Three rules govern these processes

Three Rules of Time Travel


Rule 1. Only values at the same point in time can be compar ed or combinedRule 1. Only values at the same point in time can be compar ed or combined

Rule 2. To move a cash flow forward in time, you must compo und it

2013

FV

2012

PV


Rule 3. To move a cash flow backward in time, you must disc ount it

2013

FV

2012

PV

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)

The NPV : a measure of value creation


NPV = PV (All project cash flows)

CFt : Project Cash FlowsCF0 : Initial investmentr = 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

When making an investment decision, take the alternative with the highest NPV . Choosing this alternative is equivalent to receiving its NPV in cash today.

The NPV rule : the “golden rule” of financial decis ion making


Accepting or Rejecting a Project

– Accept those projects with positive NPV because accepting them would createvalue and potentially increase the wealth of investors

– Reject those projects with negative NPV because accepting them would


– Reject those projects with negative NPV because accepting them would reduce the wealth of investors.

Example: Evaluating a real Estate investment project



20132012

Estimated selling price

The purchase price

2013

+ €400,000

2012

- €360,000

If the interest rate was 3%, should you invest in this project?


ProblemYour 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.



20132012

The purchase price in one year

(€10,000)

The purchaseprice today

(€9,500)


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.

Choosing among alternative plans


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 Alternati ves


Source : Berk J. and DeMarzo P. (2011), Corporate Finance, Second Edition. Pearson Education. (Example 3.5 p.61-62.)

Introduction to 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 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


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.

Introduction to the Law of One Price


No free lunch


Chapter Outline





A timeline is a linear representation of the timing of potential cash flows

The Timeline: …may seem like a trivial task but very useful …


Example : Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 paymentsAssume 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:


Differentiate between two types of cash flows:

• Inflows are positive cash flows.

• Outflows are negative cash flows, which are indicated with a – (minus) sign.

The Timeline: …may seem like a trivial task but very useful …


Quick-Check ProblemSuppose 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?


Compounding Laws


ProblemAlain 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?

33

1331 €x 1.10

3 times


Future Value of a Cash Flow

Compounding Laws


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?


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)

Compounding Laws


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


Frequency of compounding k : how often in a year I will receive the interests.

but, interests can be compounded several times per year

The annual percentage rate - APR

Applying the rule of time travel to a stream of cas h flows

Applying The Rules of Time Travel The TimelineCompounding Laws and Effective Annual RateValuing a stream of cash flowsPerpetuities, Annuities, and Other Special Cases

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 …



Example





A general formula for valuing a stream of cash flows


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 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?




Problem


Your father should be willing to lend you €24,890 i n exchange for your promised payments. You will pay him 5000+3*8000= 29 000. Is this transaction a good deal for your father?

210 4

?€24,890

3




Problem


Your father should be willing to lend you €24,890 i n exchange for your promised payments. You will pay him 5000+3*8000= 29 000. Is this transaction a good deal for your father?How to make sure your father won’t lose money?

4

8000 €

2

8000 €

1

5000 €

0 3

8000 €


x 1.065300 €

13 300 € 14 098€x 1.06

22 089 €x 1.06

23 423.8€

31 423.8€

The Net Present Value of a Cash Flow Stream


Problem

210 3


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%?

210 3210 3


Perpetuities


A perpetuity is a stream of equal cash flows that occur at regular intervals and last for ever.

210 3

It has no fixed maturity date. Example: The Consol (The British government perpetual bond)

The present value of a perpetuity P with payment C and interest r is given by:

Geometric progression

CC C

…


See derivation of perpetuity formulas, http://fahmi.ba.free.fr/courses.html#HEC_Students

DIG DEEPER

∞→n

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 youwithdraw 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: r

CP =

PrC *=

Quick check question : if , how much will you need to invest to create a perpetuity of €500? %5=r

Perpetuities


Problem: Endowing a Perpetuity

You are the president of the alumni association. You want to endow an annual Master graduation party at yourSchool. You want the event to be a memorable one, so you budget €30,000 per year forever for the party. If youcould earn 8% interest per year on your investments, and if the first party is in one year’s time, how much willyou need to donate to endow the party?you need to donate to endow the party?


Annuities


An ordinary annuity is a stream of N equal cash flows paid at regular intervals.

210 N

…

It has a fixed maturity date. Example: car loans, mortgages, bonds … are annuities

The present value of an annuity A with payment C and interest r is given by:


CC C

…



DIG DEEPER

The present value of an Annuity by the Law of One P rice


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

The present value of an Annuity by the Law of One P rice


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.

210 N

…

CC C + P

…

- P


Recall:

The present value of a Lottery Prize Annuity


ProblemYou 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 todayIf the interest rate is 8%, which option should you take?If the interest rate is 8%, which option should you take?


The future value of an ordinary annuity


1

2

1 2


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 eachyear until she is 65, she will save €10,000 in a retirement account. If the account earns 10% per year, how muchwill Bernadette have saved at age 65?

…


Growing Perpetuities


A Growing Perpetuity is a stream of cash flows that occur at regular intervals and grow at a constant rate for ever.

…210 3 4


…

( )gC +1* ( )21* gC + ( )31* gC +C

The present value of a Growing Perpetuity P with payment C and interest r is given by:



DIG DEEPER

∞→n

Growing Perpetuities


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’sthe party in future years. Although €30,000 is adequate for next year’s party, the board estimates that the party’scost will rise by 4% per year thereafter. To satisfy their request, how much do you need to donate now?


Growing Annuities


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.

210 3 N…

210

( )gC +1*

3

( )21* gC +

N

( ) 11* −+ NgCC

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

Growing Annuities


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?will Bernadette have saved at age 65?

373635 65

…210 3 30

38


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


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? goal?


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?


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?


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 1Example 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?

…210 3 6

What is the discount rate that sets NPV to zero?




Example 2

Suppose your firm needs to puchase a new forklift with a €40,000 cash price. The dealer offers you financingwith 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 loanpayment that the dealer quotes, how do you compute the interest rate charged by the dealer?payment that the dealer quotes, how do you compute the interest rate charged by the dealer?

r ? With NPV = 0

210 3 4

- €15,000- €15,000 - €15,000- €15,000- €40,000


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



What is r so that :

r = 10% :

r = 20% :

r = 18.45% = IRR :


r = 18.45% = IRR :

The interest rate charged by the dealer is 18.45%



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 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?

210 3 30…





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?

210 3 30

+ €125,000+ €125,000 + €125,000+ €125,000- €1,000,000

…



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