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Chapter 4

Introduction to Valuation: The Time Value of Money

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Overview

Future Value and Compounding Present Value and Discounting More on Present and Future Values

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The Time Value of Money Concept

We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs

The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner

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

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Future Value for a Single Payment

Notice that1. $110.00 = $100 X (1 + 0.10)2. $121.00 = $110 X (1 + 0.10) = $100 X 1.10 X 1.10 = $100 X (1.10)2

3. $133.10 = $121 X (1 + 0.10) = $100 X 1.10 X 1.10 X 1.10

= $100 X (1.10)3

In general, the future value, FVt, of $1 invested today at r% for t periods is

FVt = $1 x (1 + r)t or FVt = PV x (1 + r)t The expression (1 + r)t is the future value interest factor.

This is the process of compounding – interest earns interest

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Example 1

Q. 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?

A. Multiply the $5,000 by the future value interest factor:

$5,000 X (1 + r)t = $5,000 X (1.12)6

= $5,000 X 1.9738227 = $9,869.11

At 12%, the simple interest is 0.12 X $5000 = $600 per year.

After 6 years, this is 6 X $600 = $3,600 The difference between compound and simple

interest is thus $4,869.11 – $3,600 = $1,269.11

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Future Value of $100 at 10 Percent

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Future Value, Time, and Interest Rate

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Using TI BA II PLUS

Calculator Keys (TI BA II PLUS)

N Number of periods

I/Y Interest per Year (Entered in percent form)

P/Y Payment per Year (2ND Access)

C/Y Compounding per Year (2ND Access)

PV Present Value

PMT PayMenT (Periodic and Fixed)

FV Future Value

MODE END for ending and BGN for beginning (2ND Access)

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Some Important Notes on TI BA II PLUS

We will first use keys located at 3rd row from top of your calculator To clear previous entries you can use

2ND CLR TVM I prefer entering “0” for irrelevant variables

For most cases, PV should be entered as a negative number Always keep in mind that “N” is number of periods not necessarily

years. Compounding per Year (C/Y) can be modified by changing 2ND P/Y.

This determines how frequently interest is earned. Therefore we will use C/Y and P/Y interchangeably. There are situations in which you would have to distinguish between

the two that we will see in the next chapter Amount of interest would be based on periodic rate

Periodic Rate = Annual Rate / Number of Periods in a Year Always make sure P/Y is consistent with the problem at hand – this

means you need to check its setting every time. P/Y also help us determine number of periods

N = P/Y X Number of Years

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Example 2Interest 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?

Answer: The future value of the CD is $1 million x (1.06)10 = $1,790,847.70.

The future value of the investment with your brother-in-law, on the other hand, is $1 million + [$1 million X (0.06) X (10)] = $1,600,000.

Thus, compounding (or the payment of “interest on interest”), results in incremental wealth of nearly $191,000.

N I/Y P/Y PV PMT FV MODE

10 6 1 -1,000,00

0

0 1,790,847.70

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Present Value of a Single Payment

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

FVt = PV x (1 + r)t

Rearrange to solve for PV = FVt / (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.

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Example 3

Want to be a millionaire? No problem! Suppose you are currently 21 years old, and can earn 10 percent on your money. How much must you invest today in order to accumulate $1 million by the time you reach age 65?

First define the variables: FV = $1 million, r = 10 percent and t = 65 - 21 =

44 years PV = $1 million/(1.10)44 = $15,091.

N I/Y P/Y PV PMT FV MODE

44 10 1 -15,091.1

3

0 1,000,000

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Example 4

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

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

PV = $20,000/(1.08)3 = $15,876.64

N I/Y P/Y PV PMT FV MODE

3 8 1 -15,876.6

4

0 20,000

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Present Value, Time, and Interest Rate

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Example 5Finding the Rate

Suppose you deposit $5,000 today in an account paying r percent per year. If you are promised to get $10,000 in 10 years, what rate of return are you being offered?

Set this up as present value equation: FV = $10,000, PV = $ 5,000, t = 10 years, r = ? PV = FVt / (1 + r)t

$5,000 = $10,000 / (1 + r)10

Now solve for r: (1 + r)10 = $10,000 / $5,000 = 2 (1 + r)10 = 2 then r = (2)1/10 - 1 = 0.0718 = 7.18%

N I/Y P/Y PV PMT FV MODE

10 7.18

1 -5,000 0 10,000

1

1

t

PV

FVr

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Example 6Finding the Number of Periods

Suppose you need $2,000 to buy a new stereo for your car. If you have $500 to invest at 14% compounded annually, how long will you have to wait to buy the stereo?

Set this up as future value equation: FV = $2,000, PV = $ 500, r = 14%, t = ? FVt = PV x (1 + r)t

$2,000 = $500 x (1 + 0.14)t

Now solve for t: (1 + 0.14)t = $2,000 / $500 = 4 (1 + 0.14)t = 4 and t = ln (4) / ln (1.14) = 10.58 years

rPVFV

t

1ln

ln

N I/Y P/Y PV PMT FV MODE

10.58

14 1 -500 0 2,000

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A Note on Non-Annual Compounding – Continued

Remember that FVt = PV0 x (1 + r)t

When dealing with non annual interest and payment you should make two adjustment

Interest rate should be periodic Number of periods should reflect the total number of periods

given number of years and frequency of interest earnings in a year

Formulations should be adjusted as follows If “m” is the number of compounding in a year then where

you see “r” divide it by “m” and multiply “t” by “m.” This assumes that “t” is the number of years

FVt = PV0 x (1 + r/m)t X m

Excel adjustment are same as formulations that can be made in a function dialog box or data cells

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A Note on Non-Annual Compounding

FVt = PV0 x (1 + r/m)t X m

Entering the above equation into TI BA II PLUS Method 1:

Keep P/Y = 1 (so is C/Y = 1) N = t X m (Number of periods) I/Y = r/m (Periodic interest rate) Rest of the information is entered as before

Method 2: Change P/Y = m (so is C/Y = m) N = t X m (Number of periods) I/Y = r (Annual interest rate) Rest of the information is entered as before Note that P/Y has to be checked for consistency every time

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Example: Spreadsheet Strategies

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

The default setting for “type” is “0” which sugests end-of-period payments (pmt). It should be set to “1” for beginning-of-period payments.

See Excel file for the solution of examples.