Future Value and Compounding Present Value and Discounting

Future Value and CompoundingFuture Value and Compounding

Present Value and DiscountingPresent Value and Discounting

More on Present and Future ValuesMore on Present and Future Values

Summary and ConclusionsSummary and Conclusions

Chapter 4 Introduction to Valuation : The Time Value of Money

Chapter OrganizationChapter Organization

Corporate Finance 1

Consider the time line below:Consider the time line below:

PV is the Present Value, that is, the value PV is the Present Value, that is, the value todaytoday.. FV is the Future Value, or the value at a future date.FV is the Future Value, or the value at a future date. The number of time periods between the Present Value and The number of time periods between the Present Value and

the Future Value is represented by “t”.the Future Value is represented by “t”. The rate of interest is called “r”. The rate of interest is called “r”. All time value questions involve the four values above: PV, All time value questions involve the four values above: PV,

FV, r, and t. Given three of them, it is always possible to FV, r, and t. Given three of them, it is always possible to calculate the fourth.calculate the fourth.

Time Value TerminologyTime Value Terminology

. . .0 1 2 3 t

PV FV

Corporate Finance 2

Notice thatNotice that

1.1. $110 $110 = = $100 $100 (1 + .10) (1 + .10) 2.2. $121 $121 = = $110 $110 (1 + .10) = $100 (1 + .10) = $100 1.10 1.10 1.10 = $100 1.10 = $100

1.101.1022

3.3. $133.10 $133.10 = = $121 $121 (1 + .10) = $100 (1 + .10) = $100 1.10 1.10 1.10 1.10 1.10 1.10 == $100 $100 ________ ________

Future Value for a Lump SumFuture Value for a Lump Sum

Corporate Finance 3

Quick QuizQuick Quiz

Q.Q. Deposit $5,000 today in an account paying 12%. Deposit $5,000 today in an account paying 12%. How How much will you have in 6 years? How much is simple much will you have in 6 years? How much is simple

interest? How much is compound interest?interest? How much is compound interest?

A.A. Multiply the $5000 by the future value interest Multiply the $5000 by the future value interest

factor:factor: $5000 $5000 (1 + (1 + r r ))tt = = $5000 $5000 ___________ ___________ = = $5000 $5000 1.9738227 1.9738227 = = $9869.11$9869.11 At 12%, the simple interest is .12 At 12%, the simple interest is .12 $5000 = $_____ per $5000 = $_____ per

year. After 6 years, this is 6 year. After 6 years, this is 6 $600 = $_____ ; the difference $600 = $_____ ; the difference between compound and simple interest is thus $_____ - between compound and simple interest is thus $_____ - $3600 = $_____$3600 = $_____

Corporate Finance 4

Interest on Interest IllustrationInterest on Interest Illustration

Q. You have just won a $1 million jackpot in the state lottery. You can buy a ten year certificate of deposit which pays 6% compounded annually. Alternatively, you can give the $1 million to your brother-in-law, who promises to pay you 6% simple interest annually over the ten year period. Which alternative will provide you with more money at the end of ten years?

Corporate Finance 5

Future Value of $100 at 10 Percent Future Value of $100 at 10 Percent

Beginning Simple Compound TotalBeginning Simple Compound Total Ending EndingYear Year Amount Interest Interest Interest Earned Amount Amount Interest Interest Interest Earned Amount

1 $100.001 $100.00 $10.00 $ 0.00 $10.00 $110.00 $10.00 $ 0.00 $10.00 $110.00 2 110.002 110.00 10.00 1.00 11.00 121.00 10.00 1.00 11.00 121.00 3 121.003 121.00 10.00 2.10 12.10 133.10 10.00 2.10 12.10 133.10 4 133.104 133.10 10.00 3.31 13.31 146.41 10.00 3.31 13.31 146.41 5 146.415 146.41 10.00 4.64 14.64 161.05 10.00 4.64 14.64 161.05 Totals $50.00 $ 11.05 $ 61.05Totals $50.00 $ 11.05 $ 61.05

Corporate Finance 6

Quick QuizQuick Quiz Want to be a millionaire? No problem! Suppose you are currently Want to be a millionaire? No problem! Suppose you are currently

21 years old, and can earn 10 percent on your money. How much 21 years old, and can earn 10 percent on your money. How much must you invest must you invest today today in order to accumulate in order to accumulate $1 million$1 million by the time by the time you reach age 65?you reach age 65?

First define the variables:First define the variables:FVFV = $1 million= $1 million rr = 10 percent = 10 percenttt = 65 - 21 = 44 years = 65 - 21 = 44 years PV = ?PV = ?

Set this up as a future value equation and solve for the present value:Set this up as a future value equation and solve for the present value:$1 million = PV $1 million = PV (1.10) (1.10)4444 PV = $1 million/(1.10)PV = $1 million/(1.10) 44 44 == $15,091.$15,091.

Of course, we’ve ignored taxes and other complications, but stay tuned - right now you need Of course, we’ve ignored taxes and other complications, but stay tuned - right now you need to figure out where to get $15,000!to figure out where to get $15,000!

7

Q.Q. Suppose you need $20,000 in three years to pay Suppose you need $20,000 in three years to pay your college tuition. If you can earn 8% on your money, your college tuition. If you can earn 8% on your money, how much do you need today?how much do you need today?

A.A. Here we know the future value is $20,000, the rate Here we know the future value is $20,000, the rate (8%), (8%), and the number of periods (3). What is and the number of periods (3). What is the unknown the unknown present amount (i.e., the present amount (i.e., the present present valuevalue)? From before:)? From before:

FVFVtt = = PV PV (1 + (1 + r r ))tt

$20,000$20,000 = = PV PV __________ __________

Rearranging:Rearranging: PVPV = $20,000/(1.08)= $20,000/(1.08)33

= = $$ ________________

Present Value for a Lump SumPresent Value for a Lump Sum

Corporate Finance 8

Present Value of $1 for Different Periods and Rates Present Value of $1 for Different Periods and Rates PresentPresentvaluevalueof $1 ($)of $1 ($)

Time( years)

r = 0%

r = 5%

r = 10%

r = 15%

r = 20%

1 2 3 4 5 6 7 8 9 10

1.00

.90

.80

.70

.60

.50

.40

.30

.20

.10

Corporate Finance 9

Example: Finding the RateExample: Finding the Rate

Benjamin Franklin died on April 17, 1790. In his will, he gave Benjamin Franklin died on April 17, 1790. In his will, he gave 1,000 pounds sterling to Massachusetts and the city of Boston. He 1,000 pounds sterling to Massachusetts and the city of Boston. He gave a like amount to Pennsylvania and the city of Philadelphia. gave a like amount to Pennsylvania and the city of Philadelphia. The money was paid to Franklin when he held political office, but The money was paid to Franklin when he held political office, but he believed that politicians should not be paid for their service(!).he believed that politicians should not be paid for their service(!).

Franklin originally specified that the money should be paid out 100 Franklin originally specified that the money should be paid out 100 years after his death and used to train young people. Later, years after his death and used to train young people. Later, however, after some legal wrangling, it was agreed that the money however, after some legal wrangling, it was agreed that the money would be paid out 200 years after Franklin’s death in 1990. By that would be paid out 200 years after Franklin’s death in 1990. By that time, the Pennsylvania bequest had grown to about $2 million; the time, the Pennsylvania bequest had grown to about $2 million; the Massachusetts bequest had grown to $4.5 million. The money was Massachusetts bequest had grown to $4.5 million. The money was used to fund the Franklin Institutes in Boston and Philadelphia.used to fund the Franklin Institutes in Boston and Philadelphia.

Assuming that 1,000 pounds sterling was equivalent to 1,000 Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn? (Note: the dollar didn’t dollars, what rate did the two states earn? (Note: the dollar didn’t become the official U.S. currency until 1792.)become the official U.S. currency until 1792.)

Corporate Finance 10

Example: Finding the Rate (continued)Example: Finding the Rate (continued)

Q.Q. Assuming that 1,000 pounds sterling was equivalent Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn?to 1,000 dollars, what rate did the two states earn?

A.A. For Pennsylvania, the future value is $________ and For Pennsylvania, the future value is $________ and the present value is $______ . There are 200 years the present value is $______ . There are 200 years involved, so we need to solve for involved, so we need to solve for rr in the following: in the following:

________ = _____________/(1 + ________ = _____________/(1 + rr ))200200

(1 + (1 + rr ))200200 = ________ = ________

Solving for Solving for rr, the Pennsylvania money grew at about , the Pennsylvania money grew at about 3.87% per year. The Massachusetts money did better; 3.87% per year. The Massachusetts money did better; check that the rate of return in this case was 4.3%. check that the rate of return in this case was 4.3%. Small differences can add up!Small differences can add up!


The Rule of 72The Rule of 72

The “Rule of 72” is a handy rule of thumb that states the The “Rule of 72” is a handy rule of thumb that states the following: following: If you earn If you earn r r % per year, your money will double in % per year, your money will double in about 72/about 72/r r % years.% years.So, for example, if you invest at 6%, your money will So, for example, if you invest at 6%, your money will double in 12 years. double in 12 years.

Why do we say “about?” Because at higher-than-normal Why do we say “about?” Because at higher-than-normal rates, the rule breaks down.rates, the rule breaks down.

What if What if rr = 72%? = 72%? FVIF(72,1) = 1.72, not 2.00 FVIF(72,1) = 1.72, not 2.00 And if And if rr = 36%? = 36%? FVIF(36,2) = 1.8496 FVIF(36,2) = 1.8496 The lesson? The Rule of 72 is a useful rule of thumb, but The lesson? The Rule of 72 is a useful rule of thumb, but

it is it is onlyonly a rule of thumb! a rule of thumb!


Quick Quiz Quick Quiz Suppose you deposit $5000 today in an account paying Suppose you deposit $5000 today in an account paying r r percent percent

per year. If you will get $10,000 in 10 years, what per year. If you will get $10,000 in 10 years, what rate of returnrate of return are you being offered?are you being offered?

Set this up as present value equation:Set this up as present value equation:

FV = $10,000FV = $10,000 PV = $ 5,000PV = $ 5,000 tt = 10 years = 10 years PV PV = = FVFVtt/(1 + /(1 + r r ))tt

$5000 $5000 = = $10,000/(1 + $10,000/(1 + rr))1010

Now solve for Now solve for rr::

((1 + 1 + rr))10 10 = $10,000/$5,000 = 2.00= $10,000/$5,000 = 2.00

rr = (2.00) = (2.00)1/10 1/10 - 1 = .0718 = - 1 = .0718 = 7.18 percent7.18 percent


Example: The (Really) Long-Run Example: The (Really) Long-Run Return on Common StocksReturn on Common Stocks

According to Stocks for the Long Run, by Jeremy Siegel, the average annual compound rate of return on common stocks was 8.4% over the period from 1802-1997. Suppose a distant ancestor of yours had invested $1000 in a diversified common stock portfolio in 1802. Assuming the portfolio remained untouched, how large would that portfolio be at the end of 1997? (Hint: if you owned this portfolio, you would never have to work for the rest of your life!)Common stock values increased by 28.59% in 1998 (as proxied by the growth of the S&P 500). How much would the above portfolio be worth at the end of 1998?


Summary of Time Value Calculations Summary of Time Value Calculations

I. Symbols:I. Symbols:PV = Present value, what future cash flows are worth todayPV = Present value, what future cash flows are worth todayFVFVtt = Future value, what cash flows are worth in the future = Future value, what cash flows are worth in the future

r r = Interest rate, rate of return, or discount rate per period = Interest rate, rate of return, or discount rate per periodt t = number of periods = number of periods C = cash amountC = cash amount

II. Future value of II. Future value of CC invested at invested at rr percent per period for percent per period for tt periods:periods:FVFVtt = = CC (1 + (1 + r r ))tt

The term (1 + The term (1 + r r ))tt is called the is called the future value factorfuture value factor..


Summary of Time Value Calculations (cont)Summary of Time Value Calculations (cont)

III. Present value of III. Present value of CC to be received in to be received in tt periods at periods at rr percent percent per period:per period:PV = PV = CC/(1 + /(1 + r r ))tt

The term 1/(1 + The term 1/(1 + r r ))tt is called the is called the present value factorpresent value factor. .

IV. The basic present value equation giving the relationship IV. The basic present value equation giving the relationship between present and future value is: between present and future value is: PV = FVPV = FVtt/(1 + /(1 + r r ))tt


Quick Quiz Quick Quiz

Now let’s see what you remember! Now let’s see what you remember! 1. Which of the following statements is/are true?1. Which of the following statements is/are true?

Given Given rr and and tt greater than zero, future value interest factors FVIF( greater than zero, future value interest factors FVIF(rr,,t t ) are ) are always greater than 1.00always greater than 1.00..

Given r and t greater than zero, present value interest factors PVIF(Given r and t greater than zero, present value interest factors PVIF(rr,,tt ) are ) are always less than 1.00always less than 1.00..

2. True or False: For given levels of2. True or False: For given levels of r r and and tt, PVIF(, PVIF(rr,,tt ) is the reciprocal of FVIF( ) is the reciprocal of FVIF(rr,,tt ). ).3. All else equal, the higher the discount rate, the (lower/higher) the present value of 3. All else equal, the higher the discount rate, the (lower/higher) the present value of

a set of cash flows.a set of cash flows.

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Quick Quiz (cont.)Quick Quiz (cont.)

1. Both statements are true. If you use time value tables, use this 1. Both statements are true. If you use time value tables, use this information to be sure that you are looking at the correct information to be sure that you are looking at the correct table. table.

2. This statement is also true. PVIF(2. This statement is also true. PVIF(rr,,t t ) = 1/FVIF() = 1/FVIF(rr,,tt ). ).

3. The answer is 3. The answer is lower - lower - discountingdiscounting cash flows at higher rates cash flows at higher rates results in lower present values. And compounding cash flows results in lower present values. And compounding cash flows at higher rates results in higher future valuesat higher rates results in higher future values . .


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Future Value and Compounding Present Value and Discounting