CLA 2 Paper & PPT - Financial Management

voyage

Ross_12e_PPT_Ch05_Calculator2.pptx

Home >Business & Finance homework help >CLA 2 Paper & PPT - Financial Management

CHAPTER 5

INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY (CALCULATOR)

5C-‹#›

4.1

This version relies primarily on the financial calculator with a brief presentation of formulas. The calculator discussed is the TI BA-II+. The slides are easy to modify for whatever calculator you prefer.

Determine the future value of an investment made today

Determine the present value of cash to be received at a future date

Find the return on an investment

Calculate how long it takes for an investment to reach a desired value

Key Concepts and Skills

5C-‹#›

Future Value and Compounding

Present Value and Discounting

More about Present and Future Values

Chapter Outline

5C-‹#›

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

5C-‹#›

4.4

Section 5.1

It’s important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an “interest rate” only in terms of loans and savings accounts.

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

Future Value – Example 1

5C-‹#›

4.5

Section 5.1 (A)

Point out that we are just using algebra when deriving the FV formula. We have 1,000(1) + 1,000(.05) = 1,000(1+.05)

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 Value: General Formula

5C-‹#›

Section 5.1 (A)

4.6

Simple interest vs. 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

5C-‹#›

4.7

Section 5.1 (B)

Lecture Tip: Slide 5.7 distinguishes between simple interest and compound interest and can be used to emphasize the effects of compounding and earning interest on interest. It is important that students understand the impact of compounding now, or they will have more difficulty distinguishing when it is appropriate to use the APR and when it is appropriate to use the effective annual rate.

Texas Instruments BA-II Plus

FV = future value

PV = present value

I/Y = period interest rate

P/Y must equal 1 for the I/Y to be the period 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.

Calculator Keys

5C-‹#›

4.8

Section 5.1 (B)

We are providing information on the Texas Instruments BA-II Plus – other calculators are similar. If you recommend or require a specific calculator other than this one, you may want to make the appropriate changes.

Note: the more information students have to remember to enter, the more likely they are to make a mistake. For this reason, I normally tell my students to set P/Y = 1 and leave it that way. Then I teach them to work on a period basis, which is consistent with using the formulas. If you want them to use the P/Y function, remind them that they will need to set it every time they work a new problem and that CLR TVM does not affect P/Y.

If students are having difficulty getting the correct answer, make sure they have done the following:

Set decimal places to floating point (2nd Format, Dec = 9 enter) or show 4 to 5 decimal places if using an HP

Double check and make sure P/Y = 1

Make sure to clear the TVM registers after finishing a problem (or before starting a problem) It is important to point out that CLR TVM clears the FV, PV, N, I/Y and PMT registers. C/CE and CLR Work DO NOT affect the TVM keys

The remaining slides will work the problems using the notation provided above for calculator keys. The formulas are presented in the notes section.

Suppose you invest the $1,000 from the previous example for 5 years. How much would you have?

5 N; 5 I/Y; 1,000 PV

CPT FV = -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 Value – Example 2

5C-‹#›

4.9

Section 5.1 (B)

It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay $1,276.28 in 5 years. Show the students that if they enter the 1,000 as negative, the FV will compute as a positive number.

Also, you may want to point out the change sign key on the calculator. There seems to be a few students each semester that have never had to use it before.

Formula: FV = 1,000(1.05)5 = 1,000(1.27628) = 1,276.28

Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?

200 N; 5.5 I/Y; 10 PV

CPT FV = -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 Value – Example 3

5C-‹#›

4.10

Section 5.1 (B)

You might also want to point out that it doesn’t matter what order you enter the information into the calculator.

Formula: FV = 10(1.055)200 = 10(44,718.9838) = 447,189.84

Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you sell 3 million widgets in the current year, how many widgets do you expect to sell in the fifth year?

5 N;15 I/Y; 3,000,000 PV

CPT FV = -6,034,072 units (remember the sign convention)

Future Value as a General Growth Formula

5C-‹#›

4.11

Section 5.1 (C)

Formula: FV = 3,000,000(1.15)5 = 3,000,000(2.011357187) = 6,034,072

This example also presents a good illustration of the Rule of 72, which approximates the number of years it will take to double an initial amount at a given rate. In this example, 72/15 = 4.8, or approximately 5 years.

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?

Quick Quiz – Part I

5C-‹#›

4.12

Section 5.1

N = 15; I/Y = 8; PV = 500; CPT FV = -1,586.08

Formula: 500(1.08)15 = 500(3.172169) = 1,586.08

500 + 15(500)(.08) = 1,100

Lecture Tip: You may wish to take this opportunity to remind students that, since compound growth rates are found using only the beginning and ending values of a series, they convey nothing about the values in between. For example, a firm may state that “EPS has grown at a 10% annually compounded rate over the last decade” in an attempt to impress investors of the quality of earnings. However, this just depends on EPS in year 1 and year 11. For example, if EPS in year 1 = $1, then a “10% annually compounded rate” implies that EPS in year 11 is (1.10)10 = 2.5937. So, the firm could have earned $1 per share 10 years ago, suffered a string of losses, and then earned $2.59 per share this year. Clearly, this is not what is implied by management’s statement above.

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.

Present Value

5C-‹#›

4.13

Section 5.2

Point out that the PV interest factor = 1 / (1 + r)t

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

Calculator

1 N

7 I/Y

10,000 FV

CPT PV = -9,345.79

Present Value –Example 1

5C-‹#›

4.14

Section 5.2 (A)

The remaining examples will just use the calculator keys.

Lecture Tip: It may be helpful to utilize the example of $100 compounded at 10 percent to emphasize the present value concept. Start with the basic formula: FV = PV(1 + r)t and rearrange to find PV = FV / (1 + r)t. Students should recognize that the discount factor is the inverse of the compounding factor. Ask the class to determine the present value of $110 and $121 if the amounts are received in one year and two years, respectively, and the interest rate is 10%. Then demonstrate the mechanics: $100 = $110 (1 / 1.1) = 110 (.9091) $100 = $121 (1 / 1.12) = 121(.8264)

The students should recognize that it was an initial investment of $100 invested at 10% that created these two future values.

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?

N = 17; I/Y = 8; FV = 150,000

CPT PV = -40,540.34 (remember the sign convention)

Present Value – Example 2

5C-‹#›

4.15

Section 5.2 (B)

Formula: 150,000 / (1.08)17 = 150,000(.270268951) = 40,540.34

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?

N = 10; I/Y = 7; FV = 19,671.51

CPT PV = -10,000

Present Value – Example 3

5C-‹#›

4.16

Section 5.2 (B)

The actual number computes to –9999.998. This is a good place to remind the students to pay attention to what the question asked, and to be reasonable in their answers. A little common sense should tell them that the original amount was 10,000 and that the calculation doesn’t come out exactly because the future value is rounded to the nearest cent.

Formula: 19,671.51 / (1.07)10 = 19,671.51(.508349292) = 9999.998 = 10,000

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: N = 5; I/Y = 10; FV = 500 CPT PV = -310.46

10 years: N = 10; I/Y = 10; FV = 500 CPT PV = -192.77

Present Value – Important Relationship I

5C-‹#›

4.17

Section 5.2 (B)

Remember the sign convention.

Formulas: PV = 500 / (1.1)5 = 500(.620921323) = 310.46

PV = 500 / (1.1)10 = 500(.385543289) = 192.77

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%: N = 5; I/Y = 10; FV = 500 CPT PV = -310.46

Rate = 15%; N = 5; I/Y = 15; FV = 500 CPT PV = -248.59

Present Value – Important Relationship II

5C-‹#›

4.18

Section 5.2 (B)

Formulas: PV = 500 / (1.1)5 = 500(.620921323) = 310.46

PV = 500 / (1.15)5 = 500(.497176735) = 248.59

Since there is a reciprocal relationship between PVIFs and FVIFs, you should also point out that future values increase as the interest rate increases.

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?

Quick Quiz – Part II

5C-‹#›

4.19

Section 5.2

Relationship: The mathematical relationship is PV = FV / (1 + r)t. One of the important things for them to take away from this discussion is that the present value is always less than the future value when we have positive rates of interest.

N = 3; I/Y = 6; FV = 15,000; CPT PV = -12,594.29

PV = 15,000 / (1.06)3 = 15,000(.839619283) = 12,594.29

N = 3; I/Y = 8; FV = 15,000; CPT PV = -11,907.48 (Difference = 686.81)

PV = 15,000 / (1.08)3 = 15,000(.793832241) = 11,907.48

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.

If you are using a financial calculator, be sure to remember the sign convention or you will receive an error (or a nonsense answer) when solving for r or t.

The Basic PV Equation - Refresher

5C-‹#›

4.20

Section 5.3

Lecture Tip: Students who fail to grasp the concept of time value often do so because it is never really clear to them that given a 10% opportunity rate, $110 to be received in one year is equivalent to having $100 today (or $90.90 one year ago, or $82.64 two years ago, etc.). At its most fundamental level, compounding and discounting are nothing more than using a set of formulas to find equivalent values at any two points in time. In economic terms, one might stress that equivalence just means that a rational person will be indifferent between $100 today and $110 in one year, given a 10% opportunity. This is true because she could (a) take the $100 today and invest it to have $110 in one year or (b) she could borrow $100 today and repay the loan with $110 in one year. A corollary to this concept is that one can’t (or shouldn’t) add, subtract, multiply or divide money values in different time periods unless those values are expressed in equivalent terms, i.e., at a single point in time.

Often we will want to know what the implied interest rate is on 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

Discount Rate

5C-‹#›

Section 5.3 (B)

4.21

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%

Calculator – the sign convention matters!!!

N = 5

PV = -1,000 (you pay 1,000 today)

FV = 1,200 (you receive 1,200 in 5 years)

CPT I/Y = 3.714%

Discount Rate – Example 1

5C-‹#›

4.22

Section 5.3 (B)

It is very important at this point to make sure that the students have more than 2 decimal places visible on their calculator.

Efficient key strokes for formula: 1,200 / 1,000 = yx 5 1/x = - 1 = .03714

If they receive an error when they try to use the financial keys, they probably forgot to enter one of the numbers as a negative.

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?

N = 6

PV = -10,000

FV = 20,000

CPT I/Y = 12.25%

Discount Rate – Example 2

5C-‹#›

4.23

Section 5.3 (B)

Formula: r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25%

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?

N = 17; PV = -5,000; FV = 75,000

CPT I/Y = 17.27%

Discount Rate – Example 3

5C-‹#›

4.24

Section 5.3 (B)

Formula: r = (75,000 / 5,000)1/17 – 1 = .172686 = 17.27%

This is a great problem to illustrate how TVM can help you set realistic financial goals and possibly adjust your expectations based on what you can currently afford to save.

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?

Quick Quiz – Part III

5C-‹#›

4.25

Section 5.3

Implied rate: N = 5; PV = -500; FV = 600; CPT I/Y = 3.714%

r = (600 / 500)1/5 – 1 = 3.714%

Choose the bank account because it pays a higher rate of interest (assuming tax rates and other issues are consistent across both investments).

How would the decision be different if you were looking at borrowing $500 today and either repaying at 4%, or repaying $600? In this case, you would choose to repay $600 because you would be paying a lower rate.

Start with the basic equation and solve for t (remember your logs).

FV = PV(1 + r)t

t = ln(FV / PV) / ln(1 + r)

You can use the financial keys on the calculator as well; just remember the sign convention.

Finding the Number of Periods

5C-‹#›

4.26

Section 5.3 (C)

Remind the students that ln is the natural logarithm and can be found on the calculator.

The rule of 72 is a quick way to estimate how long it will take to double your money: # years to double = 72 / r, where r is number of percent.

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?

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

CPT N = 3.02 years

Number of Periods – Example 1

5C-‹#›

4.27

Section 5.3 (C)

Formula: t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years

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

5C-‹#›

Section 5.3 (C)

4.28

How much do you need to have in the future?

Down payment = .1(150,000) = 15,000

Closing costs = .05(150,000 – 15,000) = 6,750

Total needed = 15,000 + 6,750 = 21,750

Compute the number of periods.

Using a financial calculator:

PV = -15,000; FV = 21,750; I/Y = 7.5

CPT N = 5.14 years

Using the formula:

t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years

Number of Periods – Example 2 (ctd.)

5C-‹#›

4.29

Section 5.3 (C)

Loan amount = 150,000 – down payment = 150,000 – 15,000 = 135,000

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?

Quick Quiz – Part IV

5C-‹#›

4.30

Section 5.3

Calculator: PV = -500; FV = 600; I/Y = 6; CPT N = 3.13 years

Formula: t = ln(600/500) / ln(1.06) = 3.13 years

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

5C-‹#›

4.31

Section 5.4

Click on the tabs at the bottom of the worksheet to move between examples.

Many financial calculators are available online.

Go to Investopedia’s website 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

5C-‹#›

Section 5.4

4.32

Table 5.4

5C-‹#›

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

5C-‹#›

4.34

Section 5.4

N = 5

PV = -10,000

At I/Y = 5, the FV = 12,762.82

At I/Y = 4.5, the FV = 12,461.82

The difference is attributable to interest. That difference is 12,762.82 – 12,461.82 = 301

To double the 10,000:

I/Y = 5

PV = -10,000

FV = 20,000

CPT N = 14.2 years

Note, the rule of 72 indicates 72/5 = 14 years, approximately.

N = 20

PV = -1,000

FV = 4,000

CPT I/Y = 7.18%

End of Chapter

Chapter 5 - Calculator

5C-‹#›

4.35

Future Value

You have $10,000 to invest. You will need the money in 5 years and you expect to earn 8% per year. How much will you have in 5 years.
What are you looking for?		PV =	10,000
		NPER =	5
Use the FV formula:		RATE =	8%	(Same as .08)
FV(rate,nper,pmt,pv)
	Compute	FV =	$14,693.28
(Notice that the spreadsheet has the same sign convention as the calculators with positive inflows and negative outflows. A negative sign was placed before the FV formula to make the result positive.)
Note that this problem does not include a payment, so it was entered as 0.

Present Value

You need $150,000 in 18 years for your daughter's eductation. If you can earn 6% per year, how much do you need to invest today?
What are you looking for?		FV =	150,000
		NPER =	18
Use the PV formula:		RATE =	6%	(Same as .06)
PV(rate,nper,pmt,fv)
	Compute	PV =	$52,551.57

Rate

You have $30,000 to invest and you need $45,000 for a down payment and closing costs on a house. If you want to buy the house in 2 years, what rate of interest do you need to earn?
What are you looking for?		PV =	30,000
		FV =	45,000
Use the RATE formula: RATE(nper,pmt,pv,fv)		NPER =	2
	Compute	RATE =	22.47%
(Note that the rate will display as a whole percent, you need to format the cell to see the decimal places.)
Note a negative sign was entered before the cell reference for the FV to maintain the sign convention.

Number of Periods

You have $15,000 to invest right now and you figure you will need $25,000 to buy a new car. If you can earn 9% per year, how long before you can buy the car?
What are you looking for?		PV =	15,000
		FV =	25,000
Use the NPER formula:		RATE =	9%	(Same as .09)
NPER(rate,pmt,pv,fv)
	Compute	NPER =	5.9275850487	years