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Introduction to Valuation: The Time Value of Money

Chapter 4

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Prepare for Capital Budgeting

Part 2: Understand financial statement and cash flowC2-Identify cash flow from financial statementC3-Financial statement and comparison

Part 3: Valuation of future cash flowC4-Basic conceptsC5-More exercise

Part 4: Valuing stocks and bonds C6-BondC7-Stock

Part 5: Capital budgeting

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Chapter Outline

1. Future Values: Definitions and Formula

2. Present Values

3. PV – Important Relationship

4. Calculate Rates and Number of Periods

5. Calculator Keys

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1. Example Suppose you put $5000 in bank for one year at 3%

interest rate per year. What is the value of your money in one year? Interest = 5000(.03) = 150 Value in one year = principal + interest

= 5000+5000(.03) = 5000 + 150 = 5150= 5000(1 + .03) = 5150

Suppose you leave the money in for another year. How much will you have two years from now? FV = [5000(1.03)](1.03)

= 5000(1.03)2 = 5304.5

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Example (cont..) 1 year: Value = 5000(1+.03)

2 years: Value = [5000(1+.03)](1+.03)

3 years: Value = {[5000(1+.03)](1+.03)} (1+.03)

4 years: Value ={{[5000(1+.03)](1+.03)} (1+.03)} (1+.03)

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Example (cont..) 1 year: Value = 5000(1+.03)

= 5000(1+.03)1

2 years: Value = [5000(1+.03)](1+.03) = 5000(1+.03)2

3 years: Value = {[5000(1+.03)](1+.03)} (1+.03)

= 5000(1+.03)3

4 years: Value ={{[5000(1+.03)](1+.03)} (1+.03)} (1+.03)

= 5000(1+.03)4

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Basic Definitions

FV = PV(1 + r)t

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

Time value of money: A dollar in hand today is worth more than a dollar promised at some time in the future.

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Future Values: General Formula

FV = PV(1 + r)t

FV = future value PV = present value r = period interest rate T = number of periods

Future value factor = (1 + r)t

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Effects of Compounding

Compounding: the process of accumulating interest over time to earn more interest.

Compound interest: interest earned on both the initial principal and the interest earned from prior periods.

Compound interest (total interest)

=Simple interest+Interest on interest Simple interest: interest on principal

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Illustration on Compounding

2004 2005 2006 2007

$5000 $5150 $5304.5Interest earned (2004-2005)=$150

$150

Interest earned (2004-2006)=150+(150+4.5)=$304.5

$150 $4.5

$150

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Illustration on Compounding (cont..)

2004 2005 2006 2007

$5000 $5150 $5304.5 $5463.6

Interest earned (04-06)=150+(150+4.5)=$304.5

Interest earned(04-07)=150+(150+4.5)+(150+9.1)=$463.6

$150 $4.5 $9.1

$150 $150

Compounding effect=Total interest earned - simple interest =463.6 – 150 – 150 – 150 = 463.6 - 3(150) = 13.6

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Figure 4.1

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Future Values – Example 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 = 447,189.84What is the effect of compounding? Total interest = FV- PV = 447, 179.84 Simple interest = 200[(10)(.055)] = 110 Compounding effect = Total interest – simple interest

= 447,179.84 -110= 447,069.84Compounding has added $447,069.84 to the value of the

investment.

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2. Present Values

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

Present Value factor (Discount factor)= 1 / (1 + r)t

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

When we just say 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

(last year) vs. (this year)

PV= 5000/(1+3%) = $4854

If you want to have $5000 in your account this year, how much you should put in the bank last year given 3% interest rate per year?

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PV –Example

Suppose your grandmother know you need $2500 in one year for car down payment. If you can earn 3% quarterly interest when put the money in the bank, how much does she need to give you today?

PV = 2500 / (1.03)4 = 2221.2

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3. PV – Important Relationship I

For a given interest rate and future value – 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

Future Value=Present Value+Interest Earned

PV = FV / (1 + r)t

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PV – Important Relationship II

For a given time period and future value – 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 = 248.58

Future Value=Present Value+Interest Earned

PV = FV / (1 + r)t

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4. The Basic PV Equation - RefresherPV = 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

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Discount Rate

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 If you are using formulas, you will want to

make use of both the yx and the 1/x keys

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Discount Rate – Example

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? r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%

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6. Calculator Keys

Texas Instruments BA-II Plus FV = future value PV = present value I/Y = period interest rate

Interest is entered as a percent, not a decimal

N = number of periods Remember to clear the registers (CLR TVM) after

each problem Other calculators are similar in format

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Calculator Settings (Appendix D)

Compounding frequency Set P/Y=1: Press [2 nd] [I/Y] (P/Y), show {P/Y},

[1] [ENTER] Set C/Y=1: [ ] [1] [ENTER] [2 nd] [CPT] (QUIT)

End mode and annuities due Start with [2 nd] [PMT] (BGN) Switch between END and BGN using [2 nd]

[ENTER] (SET) End with [2 nd] [CPT] (QUIT)

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Future Values – Example 1

Suppose you invest $1000 for 5 years with 5% interest rate. How much would you have at the end of 5th year? FV = 1000(1.05)5 = 1276.28

Calculator:

N = 5; I/Y = 5; PV = 1000;

CPT FV = -1276.28

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Present Value – Example 1

Suppose your grandmother know you need $2500 in one year for car down payment. If you can earn 3% quarterly when put the money in the bank, how much does she need to give you today? PV = 2500 / (1.03)4 = 2221.2

Calculator N=4 I/Y=3 FV=2500 CPT PV = -2221.2

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Using Financial Calculator

When using a financial calculator, be sure and remember the sign convention or you will receive an error when solving for r or t

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Discount Rate – Example 1

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? r = (1200 / 1000)1/5 – 1 = .03714 = 3.714% Calculator – the sign convention matters!!!

N = 5 PV = -1000 (you pay 1000 today) FV = 1200 (you receive 1200 in 5 years) CPT I/Y = 3.714 (%)

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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?

Calculator:

I/Y = 10; FV = 20,000; PV = -15,000;

CPT N = 3.02 years

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Review Questions

1. Know how to calculate the future value, present

value, and rate of return of an investment. What is the difference between simple interest

and compound interest? How to calculate compounding effect?

What is a compounding process and what is a discounting process?

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Review Questions (cont..)

3. How will discount factor and future value factor

change with the interest rate and length of time? As you increase the length of time involved,

what happens to FV for a given PV and rate? What happens to PV for a given FV and rate? If you increase the rate, what happens to FV for a given PV and time length? What happens to PV for a given FV and time length?