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Introduction to Valuation: The Time Value of Money

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◦Assume interest rates are 4.3884%.◦You have just won a lottery and must

choose between the following two options: Receive a cheque for $150,000 today. Receive $10,000 a year for the next 25

years.

KEY QUESTIONS FOR YOU:

Which option gives you the biggest “winnings”

How should you tackle this kind of problem?

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1. Be able to compute the future value of an investment made today

2. Be able to compute the present value of cash to be received at some future date

3. Be able to compute the return on an investment4. Be able to compute the number of periods that equates

a present value and a future value given an interest rate5. Be able to use a financial calculator and/or a

spreadsheet to solve time value of money problems

Key Concepts and Skills

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Consider the time line below:

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

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

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

Time Value Terminology0 1 2 3 t

PV FV

……..

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Present Value – earlier money on a time line

Future Value – later money on a time line Interest rate – “exchange rate” between

earlier money and later money◦ Discount rate◦ Cost of capital◦ Opportunity cost of capital◦ Required return

Basic Definitions

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Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year?◦ Interest = 1,000(.05) = 50◦ Value in one year = principal + interest =

1,000 + 50 = 1,050◦ Future Value (FV) = 1,000(1 + .05) = 1,050

Suppose you leave the money in for another year. How much will you have two years from now?◦ FV = 1,000(1.05)(1.05) = 1,000(1.05)2 =

1,102.50

1. Future Values

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Simple interest Compound interest Consider the previous example

◦ FV with simple interest = 1,000 + 50 + 50 = 1,100

◦ FV with compound interest = 1,102.50◦ The extra 2.50 comes from the interest

of .05(50) = 2.50 earned on the first interest payment

Effects of Compounding

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Future Value of $100 at 10 Percent Beginning Simple Compound Total Total

Year Amount Interest Interest Interest Amount

1 $100.00 $10.00 $ 0.00 $10.00 $110.00 2 110.00 10.00 1.00 11.00 121.00

3 121.00 10.00 2.10 12.10 133.10

4 133.10 10.00 3.31 13.31 146.41

5 146.41 10.00 4.64 14.64 161.05 Totals $50.00 $ 11.05 $ 61.05

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FV = PV(1 + r)t

◦ FV = future value◦ PV = present value◦ r = period interest rate, expressed as a decimal◦ T = number of periods

Future value interest factor = (1 + r)t

Future Values: General Formula

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Suppose you invest the $1,000 from the previous example for 5 years. How much would you have?◦ FV = 1,000(1.05)5 = 1,276.28

The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1,250, for a difference of $26.28.)

Future Values – Example 2

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Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?◦ FV = 10(1.055)200 = 447,189.84

What is the effect of compounding?◦ Simple interest = 10 + 200(10)(.055) = 120.00◦ Compounding added $447,069.84 to the value

of the investment

Future Values – Example 3

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Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?◦ FV = 3,000,000(1.15)5 = 6,034,072

Future Value as a General Growth Formula

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What is the difference between simple interest and compound interest?

Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years.◦ How much would you have at the end of 15

years using compound interest?◦ How much would you have using simple

interest?

Example 4

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Example 5:

Q. You have just won a $1 million jackpot in the provincial 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 8% simple interest annually over the ten year

period. Which alternative will provide you with more money at the

end of ten years?

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

Example 6:

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How much do I have to invest today to have some amount in the future?◦ FV = PV(1 + r)t

◦ Rearrange to solve for PV = FV / (1 + r)t

When we talk about discounting, we mean finding the present value of some future amount.

When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

2. Present Values

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Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?

PV = 10,000 / (1.07)1 = 9,345.79

Present Value – One Period Example

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You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?◦ PV = 150,000 / (1.08)17 = 40,540.34

Present Values – Example 2

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Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest?◦ PV = 19,671.51 / (1.07)10 = 10,000

Present Values – Example 3

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For a given interest rate – the longer the time period, the lower the present value◦ What is the present value of $500 to be received

in 5 years? 10 years? The discount rate is 10%◦ 5 years: PV = 500 / (1.1)5 = 310.46◦ 10 years: PV = 500 / (1.1)10 = 192.77

Present Value – Important Relationship I

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For a given time period – the higher the interest rate, the smaller the present value◦ What is the present value of $500 received in 5

years if the interest rate is 10%? 15%? Rate = 10%: PV = 500 / (1.1)5 = 310.46 Rate = 15%; PV = 500 / (1.15)5 = 248.59

Present Value – Important Relationship II

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What is the relationship between present value and future value?

Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today?

If you could invest the money at 8%, would you have to invest more or less than at 6%? How much?

Example 4

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Suppose you need $3,000 next year to buy a new computer. The interest rate is 8 percent per year. How much money should you set aside now in order to pay for the purchase?

What if you can wait 2 years till you purchase the computer, how much money should you set aside?

Example 5

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Present Value vs Future Value PV and FV are related!

◦ Have you noticed that $620,921 becomes $1 million (and that $1 million requires $620,921) if you have a time period of 5 years and a discount rate of 10%?

$620,921$1,000,000

FV at 10%

PV at 10%

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You should understand that a dollar received today is worth more than a dollar received tomorrow.

Lesson: The value of cash flows received at different times can never be directly compared. You must first discount all cash flows to a common date and then compare them.

 

Key Principle

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Kangaroo Autos is offering free credit on a $10,000 car. You pay $4,000 down and then the balance at the end of 2 years. Turtle Motors next door does not offer free credit buy will give you $500 off the list price. If the interest rate is 10 percent, which company is offering the better deal?

Suppose your friend wants to buy a car but does not like to be indebt to the car company. What would you recommend?

Example 6: To see if you really understand the concept of PV?

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PV = FV / (1 + r)t

There are four parts to this equation◦ PV, FV, r and t◦ If we know any three, we can solve for the

fourth

The Basic PV Equation - Refresher

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Often we will want to know what the implied interest rate is in an investment

Rearrange the basic PV equation and solve for r◦ FV = PV(1 + r)t

◦ r = (FV / PV)1/t – 1

3. Discount Rate

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You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest?◦ r = (1,200 / 1,000)1/5 – 1 = .03714 = 3.714%

Discount Rate – Example 1

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Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?◦ r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25%

Discount Rate – Example 2

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Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?◦ r = (75,000 / 5,000)1/17 – 1 = .172688 = 17.27%

Discount Rate – Example 3

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What are some situations in which you might want to know the implied interest rate?

You are offered the following investments:◦ You can invest $500 today and receive $600 in 5 years.

The investment is considered low risk.◦ You can invest the $500 in a bank account paying 4%.◦ What is the implied interest rate for the first choice and

which investment should you choose?

Example 4

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Benjamin Franklin died on April 17, 1790. In his will, he gave 1,000 dollars to the city of Boston and $1000 to the city of Philadelphia. It was agreed that the money would be paid out 200 years after Franklin’s death in 1990. By that time, the Philadelphia bequest had grown to about $2 million; the Boston bequest had grown to $4.5 million. What was the rate of return for the two states earn?

Example 5

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Suppose you deposit $5000 today in an account paying r percent per year. If you will get $10,000 in 10 years, what rate of return are you being offered?

Example 6

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Start with basic equation and solve for t (remember your logs)◦ FV = PV(1 + r)t

◦ t = ln(FV / PV) / ln(1 + r)

4. Finding the Number of Periods

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You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?◦ t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years

Number of Periods – Example 1

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Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?

Number of Periods – Example 2

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When might you want to compute the number of periods?

Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money?

Example 3

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Suppose you need to come up with $40,000 for a world cruise. You have only $20,000. You know you can earn 15% interest per year. How many years do you have to wait?

Example 4

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Many financial calculators are available online

Click on the web surfer to go to Investopedia’s web site and work the following example:◦ You need $50,000 in 10 years. If you can earn

6% interest, how much do you need to invest today?

◦ You should get $27,919.74

Work the Web Example

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Use the following formulas for TVM calculations◦ FV(rate,nper,pmt,pv)◦ PV(rate,nper,pmt,fv)◦ RATE(nper,pmt,pv,fv)◦ NPER(rate,pmt,pv,fv)

The formula icon is very useful when you can’t remember the exact formula

Click on the Excel icon to open a spreadsheet containing four different examples.

Spreadsheet Example

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Table 5.4

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You have $10,000 to invest for five years. How much additional interest will you

earn if the investment provides a 5% annual return, when compared to a 4.5% annual return?

How long will it take your $10,000 to double in value if it earns 5% annually?

What annual rate has been earned if $1,000 grows into $4,000 in 20 years?

Comprehensive Problem

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Discounted Cash Flow Valuation

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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 loan payments Be able to find the interest rate on a loan Understand how interest rates are quoted Understand how loans are amortized or

paid off

Key Concepts and Skills

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Multiple Cash Flows

Future Value Calculations◦So far, we have looked at problems involving

only a single cash flow.◦This is unrealistic – most business

investments will involve multiple cash flows over time.

◦We need a method for coping with such streams of cash flows!

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Suppose we receive $2,000 each year starting next year for 5 years – what is FV?Future value calculated by compounding forward one period at a time or each cash flow (to time 5) separately.

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Assume interest rates are 8%.

You make 3 deposits to your bank account: $1,200 today $1,400 one year later. $1,000 two years later.How much will you have in three years time?

Example 1

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Doing Future Value Calculations◦ Calculate what each cash flow will be worth at the

specified future date and add up these future values.

0 1 2 3

$1,200 $1,400 $1,000

$1,080.00 = $1,000 x 1.08

$1,632.96 = $1,400 x (1.08)2

$4,224.61

FV in Year 3:

$1,511.65 = $1,200 x (1.08)3

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Find the value at year 3 of the following cash flows: Year 0 cash flow is 7000, year 1, 2, and 3 cash flows are 4000 each. ◦ Today (year 0): FV = 7000(1.08)3 = 8,817.98◦ Year 1: FV = 4,000(1.08)2 = 4,665.60◦ Year 2: FV = 4,000(1.08) = 4,320◦ Year 3: value = 4,000◦ Total value in 3 years = 8,817.98 + 4,665.60 +

4,320 + 4,000 = 21,803.58 Value at year 4 = 21,803.58(1.08) =

23,547.87

Example 2

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Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?◦ FV = 500(1.09)2 + 600(1.09) = 1,248.05

Example 3

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How much will you have in 5 years if you make no further deposits?

First way:◦ FV = 500(1.09)5 + 600(1.09)4 = 1,616.26

Second way – use value at year 2:◦ FV = 1,248.05(1.09)3 = 1,616.26

Example 3 Continued

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Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%?

Example 4

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PV calculation - what would we do to calculate PV of a $1,000 payment in the next 5 years?

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You wish to buy a car making three installments:

$8,000 today$4,000 one year later.$4,000 two years later.How much money would you have to place in

an account today to generate this stream of cash flows?

PV- Example 1

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Present Value Calculations◦You would need to place $15,133.06 in an

account today to generate the desired cash flows:

0 1 2

-$8,000 -$4,000 -$4,000

$8,000.00

$4,000 / (1.08) = $3,703.30

$4,000 / (1.08)2 = $3,429.36

$15,133.06

PV today:

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Example 2: Find the PV of each cash flows and add them

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

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You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows

Click on the Excel icon for an example

Multiple Cash Flows Using a Spreadsheet

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You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?

Example 3

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Another way to use the financial calculator for uneven cash flows is to use the cash flow keys◦ Usually you press CF and enter the cash flows beginning

with year 0.◦ In Excel you can use the NPV function.

Multiple Uneven Cash Flows – Using the Calculator

Microsoft Office Excel Worksheet

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Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount rate is 7%

What is the value of the cash flows at year 5? What is the value of the cash flows today? What is the value of the cash flows at year 3?

Question