© 2003 The McGraw-Hill Companies, Inc. All rights reserved.

Introduction to Valuation: The Time

Value of Money

Chapter

Five

5.2 Time Value of Money

Time value of money: $1 received today is not the same as $1 received in the future.

How do we equate cash flows received or paid at different points in time?

Time value of money uses compounding of interest (i.e., interest is reinvested and receives interest).

5.3 Basic Definitions

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

money received today and money to be received in the future Discount rate Cost of capital Opportunity cost of capital Required return

5.4 Future Value – Example 1 – 5.1

Suppose you invest $1000 for one year at 5% per year. What is the future value in one year? The future value is determined by two components: Interest = Principal = $1,000 Future Value (in one year) = principal + interest Future Value (FV) = Suppose you leave the money in for another year.

How much will you have two years from now? FV =

5.5 Future Value: General Formula

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

Future value interest factor = FVIF= (1 + r)t

tFV = PV 1 + r

5.6 Effects of Compounding

Simple interest – earn interest on principal only Compound interest – earn interest on principal and

reinvested interest Consider the previous example

FV with simple interest =

FV with compound interest =

The extra $2.50 comes from the interest of 5% earned on the $50 of interest paid in year one.

5.7 Calculator Keys

Financial calculator function keys FV = future value (amount at which a cash flow or series

of cash flows will grow over a given period of time when compounded at a given interest rate).

PV = present value (value today of a cash flow or series of cash flows)

PMT = the periodic payment for an annuity or perpetuity I/Y = period interest rate Interest is entered as a percent, not a decimal N = number of interest periods in investment horizon,

number of times interest is paid

Most calculators are similar in format

5.8

Calculator Approach: PV

PMTN

I/YFV

Future Value – Example 2

Suppose you invest $1000 for 5 years at 5% interest. How much will you have at the end of the five years?

Formula Approach:

5.9 Future Value – Example 2 continued

The effect of compounding is small for a small number of periods, but increases as the number of periods increases.

For example, if were to invest $1,000 for five years at 5% simple interest, the future value would be $1,250 versus $1,276.28 when the interest is compounded (see calculation on the previous page).

The compounding effect makes a difference of $26.28.

5.10 Future Value

Note 1: The longer the investment horizon, the greater the FV of a present amount.

The reason for that is there is more interest and interest on interest (compounded interest).

FV is PV discounted at an appropriate rate

5.11

Calculator Approach

PV

PMT

N

I/YFV

Future Value – Example 3

Compounding over long periods of time makes a huge difference Suppose you had a relative who deposited $10 at 5.5%

interest 200 years ago. How much would the investment be worth today?

Formula Approach

5.12 Future Value – Example 3 continued

What is the effect of compounding? Using simple interest

Using compound interest

10 10 .055 200

$120

FV PV PV I t

200

1

10 1.055

$447,189.84

tFV PV r

5.13 Future Value as a General Growth Formula

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?

Formula Approach Calculator Approach

PV

PMT

N

I/YFV units

5.14 Present Values – 5.2

How much do I have to invest today to have some specified amount in the future? Start with the formula for FV and rearrange

Rearrange to solve for PV:

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.

1t

FVPV

r

1t

FV PV r

5.15 Present Value – One Period Example

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?

Formula Approach Calculator ApproachFVPMTN

I/YPV

5.16 Present Value – Example 2

You want to begin saving for you 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?

Formula Approach Calculator Approach

FV

PMT

N

IPV

5.17 Present Value – Example 3

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?

Formula Approach Calculator ApproachFVPMT

N

I/YPV

5.18 Present Value – Important Relationships

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

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

What is the present value of $500 To be received in 5 years? Discount rate 10% and 15%

FV=500, N=5, I/Y=10 » PV=

FV=500, N=5, I/Y=15 » PV= To be received in 10 years? Discount rate 10% and 15%

FV=500, N=10, I/Y=10 » PV=

FV=500, N=10, I/Y=15 » PV=

5.19 The Basic PV Equation - Refresher

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

If you are using a financial calculator, the calculator views cash inflows as positive numbers and cash outflows as negative numbers.

When solving for time (N) or rate (I/Y) using a financial calculator, you must have both an inflow (positive number) and an outflow (negative number), or you will receive an error message

1t

FV PV r

5.20 Discount Rate – 5.3

We often want to know the implied interest rate for an investment

Rearrange the basic PV equation and solve for r

t

1

t

FV= PV 1+r

FVr = -1

PV

5.21 Discount Rate – Example 1

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

Formula Approach Calculator Approach PV FV

PMTN

I/Y %

5.22 Discount Rate – Example 2

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?

Formula Approach Calculator Approach

FV

PV

PMTN

I/Y %

5.23 Discount Rate – Example 3

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?

Formula Approach Calculator ApproachFVPV

PMTN

I/Y %

5.24 Finding the Number of Periods

Start with basic equation and solve for t

.

t

t

FV = PV 1+r

FV= 1+r

PVFV

ln = t ln 1+rPV

FVln

PVt =

ln 1+r

5.25 Number of Periods – Example 1

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?

Formula Approach Calculator Approach

FV

PV

PMT

I/YN years

5.26 Number of Periods – Example 2

Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment. If the type of house you want costs about $20,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment?

Formula Approach Calculator Approach

FV

PVPMT

I/YN years

5.27 Table 5.4