chapter 4 introduction to valuation: the time value of money

Chapter 4Introduction to Valuation: The Time Value of Money

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Introduction and Financial Statements

Time Value of Money and

Discounted Cash Flows

Capital Investment Criteria

Bond and Stock Valuation

Risk, Return and Cost of Capital

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Chapters 1-3 Chapters 4-5 Chapters 8-9

Chapters 6-7 Chapters 10-12

Done Where we are now

Midterm Exam #1 on 3/6 will cover Chapters 1-4

Understanding time value of money

Future value of a single lump-sum amount

Present value of a singlelump-sum future payment

Introduction to Valuation

Chapter 4

Future value of multiplecash flows

Present value of multiplecash flows

Discounted Cash Flow Valuation

Chapter 5

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$528.8 million over 30 years- or -

$327.8 million today?4-5

Which do you choose if you won this jackpot - $528.8MM over 30 years or $327.8MM today?

A. Obviously $528.8MM over 30 years since it is a bigger number.

B. Obviously $327.8MM today so I can immediately get a really big house plus a yacht plus a plane plus lots of other stuff.

C. I’d choose the $528.8MM over 30 years so there was no risk that I would spend it all too fast.

D. I’d choose the $327.8MM today because I would invest it so well that it would make me much more money than $528.8MM over 30 years.

E. I’m really not sure, but can’t wait to learn the answer from FINC 311!!!

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Mike’s Project Emily’s Project Robert’s Project• Invest $1MM in Year

0 and Year 1 and Year 2

• Receive $2MM in Year 3 and $2MM in Year 4

Net Cash of $1MM

• Invest $3MM in Year 0

• Receive $5MM in Year 5

Net Cash of $2MM

• Invest $3MM in Year 0

• Receive $3.4MM in Year 1

Net Cash of $0.4MM

Today

$100- or -

Year 1

$102

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Which do you choose?

A. B.

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Both how much money and when you get it, or pay it

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How the factor of time impacts the value of money

Money available today is worth more than the same amount of money in the future

Because you can earn money on your money Interest Return on investment

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Interest is the price paid to borrow money Expressed as a percentage rate over a

period of time Two ways to calculate

Simple interest Compound interest

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$1 difference represents “interest on interest”

(i.e., 10% of $10)

SimpleInterest

$10 $110 $10 $120

Today Year 1 Year 2

$100Compound

Interest$10

$110$11

$121$100

Assume a 10% interest rate on $100

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Simple interest Interest earned only on the original principal

Compound interest Interest earned on principal and on interest

received “Interest on interest” – interest earned on

reinvestment of previous interest payments

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When? How much?

Assumed rate of return?

Other termsInterest rateDiscount rateCost of capitalOpportunity cost of capitalRequired returnHurdle rateReturn on investment

Decision maker

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Today

$100- or -

If interest rate is 1%?If interest rate is 5%?

Year 1

$102

Assume annual compounding

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Today

$100- or -

Year 1

$102

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Which do you choose – if the interest rate is 1%?

A. B.

Today

$100- or -

Year 1

$102

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Which do you choose – if the interest rate is 5%?

A. B.

Today

$100- or -

If interest rate is 1%?

If interest rate is 5%?

Year 1

$102

Choose $102 in 1 year

Choose $100 today

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The future value (FV) of $100 in 1 year (t) at 1% interest (r) is $101.

The present value (PV) of $105 in 1 year (t) at 5% interest (r) is $100.

Interest Rate1%5%

Year 1$101$105

Present Value (PV)

Interest Rate or

Discount Rate (r)

Future Value (FV)

Time (t)

Today$100$100

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Future Value (FV) The amount an investment is worth after

one or more periods. “Later” money on a time line

Present Value (PV) The current value of future cash flows

discounted at the appropriate discount rate Value at t=0 on a time line


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Long Form Formula Tables Excel Financial Calculator

There are 5 options for solving time value of money problems

Covered in class

All else held constant, the future value of a lump sum investment will decrease if the:

A. Amount of the lump sum investment increases.

B. Time period is increased.C. Interest is left in the investment.D. Interest rate increases.E. Interest is changed to simple interest from

compound interest.

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The long-form approach begins with drawing a time line of the problem.

The future value of cash is manually calculated for each time period.

This approach is important to understand intuitively how to solve time value of money problems

However, it is not practical when numerous time periods are involved.

Today Year 1 Year 2 Year 3

$X$1000 $Y $Z* (1+r) * (1+r) * (1+r)

FV = PV * (1 + r)t FV = Future ValuePV = Present Valuer = Interest Ratet = # of periods

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Future Value Factor

Note: The yx function on many calculators is useful when using the formula for solving problems.

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Note: When using tables, pay extra attention to following the correct row and column to get the correct factor amount.

Future Value Factors

FV of $1= $1 * (1 + r)t

Future Value Factor

FV = $1 * (1+.10)3

= $1 * 1.3310= $1.33

For a given interest rate: The longer the time period, The higher the future value


For a given r, as t increases, FV increases

FV = PV(1 + r)t

For a given time period: The higher the interest rate, The larger the future value

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For a given t, as r increases, FV increases

FV = PV(1 + r)t

Copyright (c) 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.


Yellow represents principal plus simple interest

Blue represents “interest on interest”

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The “Rule of 72” is a quick approximation on doubling dollar amounts based on any combination of rate and years equaling ~72

Today Rate Years Future Value

Rate *Years

$1,000 9% 8 $1,993 72

$1,000 18% 4 $1,939 72

$1,000 7% 10 $1,967 70

$1,000 5% 15 $2,079 75

Jane is a 20 year-old college student who invests $10,000 today and expects to earn 8% per year. Approximately how much money will Jane have from this investment when she retires at the age of 65?

A. $50,000B. $90,000C. $160,000D. $320,000E. $450,000

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At 8% per year, your money will double in how many years?

Thus, how many “doublings” will take place over 45 years?

So, if you start with $10,000 and have 5 doublings over 45 years, how much do you end up with?

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$10,000 $20,000 (doubling #1) $40,000 (doubling #2) $80,000

(doubling #3) $160,000 (doubling #4) $320,000 (doubling #5)

9 years

5 doublings

Actual Answer is: FV = PV*(1+r)t = $10,000*(1.08)45 = $319,204

Future Value (FV) The amount an investment is worth after

one or more periods. “Later” money on a time line

Present Value (PV) The current value of future cash flows

discounted at the appropriate discount rate Value at t=0 on a time line


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Today Future Period

Future Value$1,000

Present Value $1,000

“Discounting” to today

“Compounding” to the future

Note: “Discounting” is the reverse of “compounding.”

Today

$100- or -

Year 1

$102

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Present value and future value math is all about getting to an apples-to-apples comparison or

an oranges-to-oranges comparison

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If you have 20 euros and $20, how much money do you have? You need to convert the 20

euros to the equivalent value in dollars.

Or you can convert the $20 to the equivalent value in euros.

Conversion factor is FX exchange rate.

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If you have $100 cash and a Promissory Note for $102 in 1 year, how much money do you have?

Today 1 Year

You need to convert the Promissory Note for $102 in 1 year to the equivalent value today. Conversion factor is the present value factor.

Promissory Note value today

“Discount” to today

4-39

John and Sally want to deposit enough money today to have $100,000

in 18 years for their child’s college expenses.

Sam wants to deposit enough money today to have a $50,000 down

payment on a house in 5 years.

What is the present value of $100,000 in

18 years?

What is the present value of $50,000 in 5

years?

FV = PV * (1 + r)t

PV =FV

(1 + r)t

FV = Future ValuePV = Present Valuer = Interest Ratet = # of periods

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= FV *1

(1 + r)t

Future Value Factor

Present Value Factor

Which one of the following will increase the present value of a lump sum future amount to be received in 15 years?

A. An increase in the time period.B. An increase in the interest rate.C. A decrease in the future value.D. A decrease in the interest rate.E. Changing to compound interest from

simple interest.

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The long-form approach begins with drawing a time line of the problem.

The present value of cash is manually calculated for each time period.

This approach is important to understand intuitively how to solve time value of money problems

However, it is not practical when numerous time periods are involved.

Today Year 1 Year 2 Year 3

$Y$X $Z $1000* 1/(1+r) * 1/(1+r) * 1/(1+r)

FV = Future ValuePV = Present Valuer = Interest Ratet = # of periods

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Note: The yx and 1/x functions on many calculators are useful when using the formula for solving problems.

PV =FV

(1 + r)t

= FV *1

(1 + r)tPresent Value Factor

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Note: When using tables, pay extra attention to following the correct row and column to get the correct factor amount.

Present Value Factors

PV of $1= $1 * 1/(1 + r)t

Present Value Factor

PV = $1 * 1/(1+.10)3

= $1 * .7513= $0.75

The current value of future cash flows discounted at the appropriate discount rate

Value at t=0 (or today) on a time line Answers the questions:

How much do I have to invest today to have some amount in the future?

What is the current value of an amount to be received in the future?

Copyright (c) 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 4-45

For a given interest rate: The longer the time period, The lower the present value


For a given r, as t increases, PV decreases

PV =FV

(1 + r)t

4-46

What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%

r=10%0 5 10

PV? $500

PV? $500

10 yrs: PV = $500/(1.10)10= $192.77

$310.46

$192.77


5 yrs: PV = $500/(1.10)5 = $310.46

For a given time period: The higher the interest rate, The smaller the present value


For a given t, as r increases, PV decreases

PV =FV

(1 + r)t

4-48

What is the present value of $500 received in 5 years if the interest rate is 10%? 15%?

Rate = 10%Formula Solution:

PV = $500 / (1.1)5PV = $310.46

Rate = 15%Formula Solution:

PV = $500 / (1.15)5PV = $248.59

4-49

You want to have $40,000 for a down payment on a house 4 years from now. If you can earn 5.6 percent, compounded annually on your savings, how much do you need to deposit today to reach your goal?

A. $32,166.54B. $34,420.73C. $27,880.69D. $28,211.17E. $30,886.40

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Computing Return on

Investment

4-52

You have been offered the following investment opportunity from a reputable investment management company:

Open an account today with a deposit of $5,000 and, in 12 years, you will receive $10,000.

Is this a good investment?A. Yes, this is a good investment.B. No, this is a bad investment.C. I don’t know, more information is needed.

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You have been offered the following investment opportunity from a reputable

investment management company:Open an account today with a deposit of $5,000 and,

in 12 years, you will receive $10,000.

Is this a good investment?

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To answer this question, we need to understand what is the return on the investment.

Using the Rule of 72, we can approximate the return. What is this approximation?

FV = PV * (1 + r)t

4-55

= (1 + r) tFV

PV

= 1 + r1/tFV

PV

1/t- 1 = rFV

PV

1/t

- 1FV

PVr =

If using formulas with a calculator, make use of both the yx and the 1/x keys


4-56

What is the exact return on investment (or implied interest rate)?

r =FV

PV

1/t

- 1 =$10,000

$5,000

1/12

- 1

=1/12

- 12 = 1.0595 -1

r = 5.95%

You have been offered the following investment opportunity from a reputable investment management company:


Is this a good investment?A. Yes, this is a good investment.B. No, this is a bad investment.C. I don’t know, more information is needed.

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You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest?

Formula:r = (FV / PV)1/t – 1r = ($1200 / $1000)1/5 – 1 = .03714 = 3.714%


Finding # of Periods

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FV = PV * (1 + r)t

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PV =FV

(1 + r)t

r =FVPV

1/t

- 1

t = ?

4-61

Rule of 72 Tables Formula Excel Financial Calculator


Approximation -covered in class

Robert wants to buy a boat that costs $20,000 and he currently has only $10,000 saved. If he invests his savings at a 12% annual interest rate, in about how many years can he buy his boat?

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Rule of 72Rate * Years = ~ 7212 * Years = ~ 72Years = ~ 72/12

Years = ~ 6

Robert wants to buy a boat that costs $20,000 and he currently has only $10,000 saved. If he invests his savings at a 12% annual interest rate, in about how many years can he buy his boat?

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Future Value FactorPV = FV / (1+r)t

$10k = $20k / (1+.12)t2 = (1.12)t

2 = Future Value Factor

A. The period of time she has to wait until she reaches her goal is unaffected by the compounding of interest.

B. The lower the rate of interest she earns, the shorter the time she will have to wait to reach her goal.

C. She will have to wait longer if she earns 6 percent compound interest instead of 6 percent simple interest.

D. The period of time she has to wait decreases as the amount she invests increases.

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Today, Charity wants to invest less than $3,000 with the goal of receiving $3,000 back some time in the future. Which one of the following statements is correct?

Summary and Review

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Long Form Formula Tables Excel Financial Calculator


Covered in class

chapter 4 introduction to valuation: the time value of money

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