Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Chapter Outline. Multiple Cash Flows Example 2 Continued

(1)

00

6 6

Calculators

Discounted Cash Flow Valuation Discounted Cash Flow Valuation

1 1

Key Concepts and Skills Key Concepts and Skills

¾¾Be able to compute the future value of multiple Be able to compute the future value of multiple cash flows

cash flows

¾¾Be able to compute the present value of multiple Be able to compute the present value of multiple cash flows

cash flows

¾¾Be able to compute loan paymentsBe able to compute loan payments

¾

¾Be able to find the interest rate on a loanBe able to find the interest rate on a loan

¾

¾Understand how interest rates are quotedUnderstand how interest rates are quoted

¾

¾Understand how loans are amortized or paid offUnderstand how loans are amortized or paid off

2 2

Chapter Outline Chapter Outline

¾

Future and Present Values of Multiple Future and Present Values of Multiple Cash Flows

Cash Flows

¾

Valuing Level Cash Flows: Annuities and Valuing Level Cash Flows: Annuities and Perpetuities

Perpetuities

¾

Comparing Rates: The Effect of Comparing Rates: The Effect of Compounding

Compounding

¾

Loan Types and Loan Amortization Loan Types and Loan Amortization

3 3

Multiple Cash Flows

Multiple Cash Flows – –Future Future Value Example 6.1 Value Example 6.1

¾¾ Find the value at year 3 of each cash flow and add Find the value at year 3 of each cash flow and add them together.

them together.

z

z TodayToday’’s (year 0) CF: 3 N; 8 I/Y; s (year 0) CF: 3 N; 8 I/Y; --7,000 PV; CPT FV = 8817.987,000 PV; CPT FV = 8817.98

zz Year 1 CF: 2 N; 8 I/Y; Year 1 CF: 2 N; 8 I/Y; --4,000 PV; CPT FV = 4,665.604,000 PV; CPT FV = 4,665.60

z

z Year 2 CF: 1 N; 8 I/Y; Year 2 CF: 1 N; 8 I/Y; --4,000 PV; CPT FV = 4,3204,000 PV; CPT FV = 4,320

z

z Year 3 CF: value = 4,000Year 3 CF: value = 4,000

z

z Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 + 4,000 = Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 + 4,000 = 21,803.58

21,803.58

¾

¾ Value at year 4: 1 N; 8 I/Y; Value at year 4: 1 N; 8 I/Y; --21,803.58 PV; CPT FV = 21,803.58 PV; CPT FV = 23,547.87

23,547.87

4 4

Multiple Cash Flows Multiple Cash Flows – – FV FV

Example 2 Example 2

¾¾Suppose you invest $500 in a mutual fund Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays today and $600 in one year. If the fund pays 9% annually, how much will you have in two 9% annually, how much will you have in two years?

years?

z

z Year 0 CF: 2 N; -Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV = 594.05500 PV; 9 I/Y; CPT FV = 594.05

z

zz Total FV = 594.05 + 654.00 = 1,248.05Total FV = 594.05 + 654.00 = 1,248.05

5 5

Multiple Cash Flows

Multiple Cash Flows – – Example 2 Example 2 Continued

Continued

¾¾ How much will you have in 5 years if you make no How much will you have in 5 years if you make no further deposits?

further deposits?

¾

¾ First way:First way:

z

z Year 0 CF: 5 N; Year 0 CF: 5 N; --500 PV; 9 I/Y; CPT FV = 769.31500 PV; 9 I/Y; CPT FV = 769.31

zz Year 1 CF: 4 N; Year 1 CF: 4 N; --600 PV; 9 I/Y; CPT FV = 846.95600 PV; 9 I/Y; CPT FV = 846.95

z

z Total FV = 769.31 + 846.95 = 1,616.26Total FV = 769.31 + 846.95 = 1,616.26

¾¾ Second way Second way ––use value at year 2:use value at year 2:

z

z 3 N; 3 N; --1,248.05 PV; 9 I/Y; CPT FV = 1,616.261,248.05 PV; 9 I/Y; CPT FV = 1,616.26

(2)

6 6

Multiple Cash Flows Multiple Cash Flows – – FV FV

Example 3 Example 3

¾

¾Suppose you plan to deposit $100 into an Suppose you plan to deposit $100 into an account in one year and $300 into the account account in one year and $300 into the account in three years. How much will be in the in three years. How much will be in the account in five years if the interest rate is 8%?

account in five years if the interest rate is 8%?

z

z Total FV = 136.05 + 349.92 = 485.97Total FV = 136.05 + 349.92 = 485.97

7 7

Multiple Cash Flows

Multiple Cash Flows – – Present Present Value Example 6.3

Value Example 6.3

¾

¾ Find the PV of each cash flows and add themFind the PV of each cash flows and add them

z

z Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = --178.57178.57

zz Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = --318.88318.88

z

z Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93

8 8

Example 6.3 Timeline Example 6.3 Timeline

0 1 2 3 4

200 400 600 800

178.57

318.88

427.07

508.41 1,432.93

9 9

Multiple Cash Flows Using a Multiple Cash Flows Using a

Spreadsheet Spreadsheet

¾

¾You can use the PV or FV functions in Excel to You can use the PV or FV functions in Excel to find the present value or future value of a set of find the present value or future value of a set of cash flows

cash flows

¾¾Setting the data up is half the battle Setting the data up is half the battle ––if it is set if it is set up properly, then you can just copy the up properly, then you can just copy the formulas

formulas

¾

¾Click on the Excel icon for an exampleClick on the Excel icon for an example

10 10

Multiple Cash Flows Multiple Cash Flows – – PV PV

Another Example Another Example

¾¾You are considering an investment that will pay You are considering an investment that will pay you $1,000 in one year, $2,000 in two years you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn and $3,000 in three years. If you want to earn 10% on your money, how much would you be 10% on your money, how much would you be willing to pay?

willing to pay?

z

z N = 1; I/Y = 10; FV = 1,000; CPT PV = -N = 1; I/Y = 10; FV = 1,000; CPT PV = -909.09909.09

z

z N = 2; I/Y = 10; FV = 2,000; CPT PV = -N = 2; I/Y = 10; FV = 2,000; CPT PV = -1,652.891,652.89

zz N = 3; I/Y = 10; FV = 3,000; CPT PV = -N = 3; I/Y = 10; FV = 3,000; CPT PV = -2,253.942,253.94

z

z PV = 909.09 + 1,652.89 + 2,253.94 = 4,815.93PV = 909.09 + 1,652.89 + 2,253.94 = 4,815.93

11 11

Multiple Uneven Cash Flows Multiple Uneven Cash Flows – –

Using the Calculator Using the Calculator

¾

¾ Another way to use the financial calculator for uneven cash flowAnother way to use the financial calculator for uneven cash flows s is to use the cash flow keys

is to use the cash flow keys

z

z Press CF and enter the cash flows beginning with year 0.Press CF and enter the cash flows beginning with year 0.

z

z You have to press the You have to press the ““EnterEnter””key for each cash flowkey for each cash flow

z

z Use the down arrow key to move to the next cash flowUse the down arrow key to move to the next cash flow

z

z The The ““FF””is the number of times a given cash flow occurs in is the number of times a given cash flow occurs in consecutive periods

consecutive periods

z

z Use the NPV key to compute the present value by entering the Use the NPV key to compute the present value by entering the interest rate for I, pressing the down arrow, and then interest rate for I, pressing the down arrow, and then computing the answer

computing the answer

z

z Clear the cash flow worksheet by pressing CF and then 2Clear the cash flow worksheet by pressing CF and then 2^nd^nd CLR Work

CLR Work

(3)

12 12

Decisions, Decisions Decisions, Decisions

¾

¾ Your broker calls you and tells you that he has this great investment Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in o opportunity. If you invest $100 today, you will receive $40 in one year ne year and $75 in two years. If you require a 15% return on investment and $75 in two years. If you require a 15% return on investments of s of this risk, should you take the investment?

this risk, should you take the investment?

z

z Use the CF keys to compute the value of the investmentUse the CF keys to compute the value of the investment

••CF; CFCF; CF₀₀= 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1= 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1

•

•NPV; I = 15; CPT NPV = 91.49NPV; I = 15; CPT NPV = 91.49

z

z No No ––the broker is charging more than you would be willing the broker is charging more than you would be willing to pay.

to pay.

13 13

Saving For Retirement Saving For Retirement

¾

¾You are offered the opportunity to put some You are offered the opportunity to put some money away for retirement. You will receive five money away for retirement. You will receive five annual payments of $25,000 each beginning in annual payments of $25,000 each beginning in 40 years. How much would you be willing to 40 years. How much would you be willing to invest today if you desire an interest rate of invest today if you desire an interest rate of 12%?12%?

z

z Use cash flow keys:Use cash flow keys:

•

• CF; CFCF; CF₀₀= 0; C01 = 0; F01 = 39; C02 = 25,000; F02 = 5; = 0; C01 = 0; F01 = 39; C02 = 25,000; F02 = 5;

NPV; I = 12; CPT NPV = 1,084.71 NPV; I = 12; CPT NPV = 1,084.71

14 14

Saving For Retirement Timeline Saving For Retirement Timeline

0 1 2 … 39 40 41 42 43 44

0 0 0 … 0 25K 25K 25K 25K 25K

Notice that the year 0 cash flow = 0 (CF₀= 0) The cash flows years 1 – 39 are 0 (C01 = 0; F01 = 39) The cash flows years 40 – 44 are 25,000 (C02 = 25,000;

F02 = 5)

15 15

Quick Quiz

Quick Quiz – – Part I Part I

¾¾ Suppose you are looking at the following possible cash Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3

flows: Year 1 CF = $100; Years 2 and 3 CFsCFs= $200; = $200;

Years 4 and 5

Years 4 and 5 CFsCFs= $300. The required discount rate = $300. The required discount rate is 7%

is 7%

¾

¾ What is the value of the cash flows at year 5?What is the value of the cash flows at year 5?

¾

¾ What is the value of the cash flows today?What is the value of the cash flows today?

¾

¾ What is the value of the cash flows at year 3?What is the value of the cash flows at year 3?

16 16

Annuities and Perpetuities Annuities and Perpetuities

Defined Defined

¾¾Annuity Annuity ––finite series of equal payments that finite series of equal payments that occur at regular intervals

occur at regular intervals

zz If the first payment occurs at the end of the period, it If the first payment occurs at the end of the period, it is called an ordinary annuity

is called an ordinary annuity

z

z If the first payment occurs at the beginning of the If the first payment occurs at the beginning of the period, it is called an annuity due

period, it is called an annuity due

¾

¾Perpetuity Perpetuity ––infinite series of equal paymentsinfinite series of equal payments

17 17

Annuities and Perpetuities Annuities and Perpetuities – –

Basic Formulas Basic Formulas

¾¾

Perpetuity: PV = C / r Perpetuity: PV = C / r

¾

Annuities: Annuities:

⎥⎦

⎢ ⎤

⎣

⎡ + −

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− +

=

r C r

FV

r C r

PV

t t

1 ) 1 (

) 1 ( 1 1

(4)

18 18

Annuities and the Calculator Annuities and the Calculator

¾

¾You can use the PMT key on the calculator for You can use the PMT key on the calculator for the equal payment

the equal payment

¾

¾The sign convention still holdsThe sign convention still holds

¾¾Ordinary annuity versus annuity dueOrdinary annuity versus annuity due

z

z You can switch your calculator between the two You can switch your calculator between the two types by using the 2

types by using the 2^nd^ndBGN 2BGN 2^nd^ndSet on the TI BASet on the TI BA--II II Plus

Plus

z

z If you see “If you see “BGNBGN””or or ““BeginBegin””in the display of your in the display of your calculator, you have it set for an annuity due calculator, you have it set for an annuity due

zz Most problems are ordinary annuitiesMost problems are ordinary annuities

19 19

Annuity

Annuity – – Example 6.5 Example 6.5

¾

You borrow money TODAY so you need to You borrow money TODAY so you need to compute the present value.

compute the present value.

zz48 N; 1 I/Y; -48 N; 1 I/Y; -632 PMT; CPT PV = 23,999.54 632 PMT; CPT PV = 23,999.54 ($24,000)

($24,000)

¾

Formula: Formula:

54 . 999 , 01 23 .

) 01 . 1 ( 1 1 632

48 =

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

= PV

20 20

Annuity

Annuity – – Sweepstakes Sweepstakes Example

Example

¾¾Suppose you win the Publishers Clearinghouse Suppose you win the Publishers Clearinghouse

$10 million sweepstakes. The money is paid in

$10 million sweepstakes. The money is paid in equal annual end

equal annual end--ofof--year installments of year installments of

$333,333.33 over 30 years. If the appropriate

$333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the

discount rate is 5%, how much is the sweepstakes actually worth today?

sweepstakes actually worth today?

z

z 30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.2930 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29

21 21

Buying a House Buying a House

¾¾ You are ready to buy a house and you have $20,000 You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5%

loan is 6% per year with monthly compounding (.5%

per month) for a 30

per month) for a 30--year fixed rate loan. How much year fixed rate loan. How much money will the bank loan you? How much can you offer money will the bank loan you? How much can you offer for the house?

for the house?

22 22

Buying a House

Buying a House - - Continued Continued

¾

¾ Bank loanBank loan

z

z Monthly income = 36,000 / 12 = 3,000Monthly income = 36,000 / 12 = 3,000

z

z Maximum payment = .28(3,000) = 840Maximum payment = .28(3,000) = 840

••30*12 = 360 N30*12 = 360 N

•

• .5 I/Y.5 I/Y

•

• --840 PMT840 PMT

•

• CPT PV = 140,105CPT PV = 140,105

¾

¾ Total PriceTotal Price

z

z Closing costs = .04(140,105) = 5,604Closing costs = .04(140,105) = 5,604

z

z Down payment = 20,000 –Down payment = 20,000 –5,604 = 14,3965,604 = 14,396

z

z Total Price = 140,105 + 14,396 = 154,501Total Price = 140,105 + 14,396 = 154,501

23 23

Annuities on the Spreadsheet Annuities on the Spreadsheet - -

Example Example

¾¾

The present value and future value The present value and future value formulas in a spreadsheet include a place formulas in a spreadsheet include a place for annuity payments

for annuity payments

¾

Click on the Excel icon to see an Click on the Excel icon to see an example

example

(5)

24 24

Quick Quiz

Quick Quiz – – Part II Part II

¾¾You know the payment amount for a loan and You know the payment amount for a loan and you want to know how much was borrowed.

you want to know how much was borrowed.

Do you compute a present value or a future Do you compute a present value or a future value?

value?

¾

¾You want to receive 5,000 per month in You want to receive 5,000 per month in retirement. If you can earn .75% per month retirement. If you can earn .75% per month and you expect to need the income for 25 and you expect to need the income for 25 years, how much do you need to have in your years, how much do you need to have in your account at retirement?

account at retirement?

25 25

Finding the Payment Finding the Payment

¾

Suppose you want to borrow $20,000 for a Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, new car. You can borrow at 8% per year, compounded monthly (8/12 = .66667% per compounded monthly (8/12 = .66667% per month). If you take a 4

month). If you take a 4- -year loan, what is year loan, what is your monthly payment?

your monthly payment?

z

z4(12) = 48 N; 20,000 PV; .66667 I/Y; CPT 4(12) = 48 N; 20,000 PV; .66667 I/Y; CPT PMT = 488.26

PMT = 488.26

26 26

Finding the Payment on a Finding the Payment on a

Spreadsheet Spreadsheet

¾

¾Another TVM formula that can be found in a Another TVM formula that can be found in a spreadsheet is the payment formula spreadsheet is the payment formula

z

z PMT(rate,nper,pv,fv)PMT(rate,nper,pv,fv)

z

z The same sign convention holds as for the PV and The same sign convention holds as for the PV and FV formulas

FV formulas

¾¾Click on the Excel icon for an exampleClick on the Excel icon for an example

27 27

Finding the Number of Payments Finding the Number of Payments

– – Example 6.6 Example 6.6

¾

The sign convention matters!!! The sign convention matters!!!

z z1.5 I/Y1.5 I/Y

zz1,000 PV1,000 PV

z

z-20 PMT-20 PMT

z

zCPT N = 93.111 MONTHS = 7.75 yearsCPT N = 93.111 MONTHS = 7.75 years

¾

And this is only if you don And this is only if you don’ ’t charge t charge anything more on the card!

anything more on the card!

28 28

Finding the Number of Payments Finding the Number of Payments

– – Another Example Another Example

¾¾Suppose you borrow $2,000 at 5% and you are Suppose you borrow $2,000 at 5% and you are going to make annual payments of $734.42.

going to make annual payments of $734.42.

How long before you pay off the loan?

z

z Sign convention matters!!!Sign convention matters!!!

z z 5 I/Y5 I/Y

z

z 2,000 PV2,000 PV

z

z -734.42 PMT -734.42 PMT

z

z CPT N = 3 yearsCPT N = 3 years

29 29

Finding the Rate Finding the Rate

¾¾Suppose you borrow $10,000 from your parents Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly month for 60 months. What is the monthly interest rate?

interest rate?

zz Sign convention matters!!!Sign convention matters!!!

zz 60 N60 N

zz 10,000 PV10,000 PV

zz -207.58 PMT-207.58 PMT

z

z CPT I/Y = .75%CPT I/Y = .75%

(6)

30 30

Annuity

Annuity – – Finding the Rate Finding the Rate Without a Financial Calculator Without a Financial Calculator

¾

¾ Trial and Error ProcessTrial and Error Process

z

z Choose an interest rate and compute the PV of the Choose an interest rate and compute the PV of the payments based on this rate

payments based on this rate

z

z Compare the computed PV with the actual loan amountCompare the computed PV with the actual loan amount

z

z If the computed PV > loan amount, then the interest rate If the computed PV > loan amount, then the interest rate is too low

is too low

zz If the computed PV < loan amount, then the interest rate If the computed PV < loan amount, then the interest rate is too high

is too high

z

z Adjust the rate and repeat the process until the Adjust the rate and repeat the process until the computed PV and the loan amount are equal computed PV and the loan amount are equal

31 31

Quick Quiz

Quick Quiz – – Part III Part III

¾

¾ You want to receive $5,000 per month for the next 5 You want to receive $5,000 per month for the next 5 years. How much would you need to deposit today if years. How much would you need to deposit today if you can earn .75% per month?

you can earn .75% per month?

¾¾ What monthly rate would you need to earn if you only What monthly rate would you need to earn if you only have $200,000 to deposit?

have $200,000 to deposit?

¾¾ Suppose you have $200,000 to deposit and can earn Suppose you have $200,000 to deposit and can earn .75% per month.

.75% per month.

z

z How many months could you receive the $5,000 How many months could you receive the $5,000 payment?

payment?

z

z How much could you receive every month for 5 years?How much could you receive every month for 5 years?

32 32

Future Values for Annuities Future Values for Annuities

¾¾Suppose you begin saving for your retirement by Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in interest rate is 7.5%, how much will you have in 40 years?

40 years?

z

z Remember the sign convention!!!Remember the sign convention!!!

z z 40 N40 N

z z 7.5 I/Y7.5 I/Y

zz -2,000 PMT-2,000 PMT

zz CPT FV = 454,513.04CPT FV = 454,513.04

33 33

Annuity Due Annuity Due

¾¾ You are saving for a new house and you put $10,000 You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 made today. How much will you have at the end of 3 years?

years?

z

z 22^nd^ndBGN 2BGN 2^nd^ndSet (you should see BGN in the display)Set (you should see BGN in the display)

z z 3 N3 N

z

z --10,000 PMT10,000 PMT

zz 8 I/Y8 I/Y

z

z CPT FV = 35,061.12CPT FV = 35,061.12

z

z 22^nd^ndBGN 2BGN 2^nd^ndSet (be sure to change it back to an ordinary Set (be sure to change it back to an ordinary annuity)

annuity)

34 34

Annuity Due Timeline Annuity Due Timeline

0 1 2 3

10000 10000 10000

32,464

35,016.12

35 35

Perpetuity

Perpetuity – – Example 6.7 Example 6.7

¾

Perpetuity formula: PV = C / r Perpetuity formula: PV = C / r

¾

Current required return: Current required return:

z

z40 = 1 / r40 = 1 / r

z

zr = .025 or 2.5% per quarterr = .025 or 2.5% per quarter

¾¾

Dividend for new preferred: Dividend for new preferred:

zz100 = C / .025100 = C / .025

z

zC = 2.50 per quarterC = 2.50 per quarter

(7)

36 36

Quick Quiz

Quick Quiz – – Part IV Part IV

¾¾You want to have $1 million to use for retirement You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis much do you need to deposit on a monthly basis if the first payment is made in one month?

if the first payment is made in one month?

¾¾What if the first payment is made today?What if the first payment is made today?

¾¾You are considering preferred stock that pays a You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be is 3% per quarter, how much would you be willing to pay?

willing to pay?

37 37

Work the Web Example Work the Web Example

¾

¾Another online financial calculator can be found Another online financial calculator can be found at

at MoneyChimpMoneyChimp

¾

¾Click on the web surfer and work the following Click on the web surfer and work the following example

example

z

z Choose calculator and then annuity Choose calculator and then annuity

z

z You just inherited $5 million. If you can earn 6% on You just inherited $5 million. If you can earn 6% on your money, how much can you withdraw each year your money, how much can you withdraw each year for the next 40 years?

for the next 40 years?

z

z Money chimp assumes annuity due!!!Money chimp assumes annuity due!!!

z

z Payment = $313,497.81Payment = $313,497.81

38 38

Table 6.2 Table 6.2

39 39

Effective Annual Rate (EAR) Effective Annual Rate (EAR)

¾¾This is the actual rate paid (or received) after This is the actual rate paid (or received) after accounting for compounding that occurs during accounting for compounding that occurs during the year

the year

¾¾If you want to compare two alternative If you want to compare two alternative

investments with different compounding periods investments with different compounding periods you need to compute the EAR and use that for you need to compute the EAR and use that for comparison.

comparison.

40 40

Annual Percentage Rate Annual Percentage Rate

¾

¾This is the annual rate that is quoted by lawThis is the annual rate that is quoted by law

¾

¾By definition APR = period rate times the By definition APR = period rate times the number of periods per year

number of periods per year

¾

¾Consequently, to get the period rate we Consequently, to get the period rate we rearrange the APR equation:

rearrange the APR equation:

z

z Period rate = APR / number of periods per yearPeriod rate = APR / number of periods per year

¾

¾You should NEVER divide the effective rate by You should NEVER divide the effective rate by the number of periods per year

the number of periods per year ––it will NOT it will NOT give you the period rate

give you the period rate

41 41

Computing APRs Computing APRs

¾¾What is the APR if the monthly rate is .5%?What is the APR if the monthly rate is .5%?

z

z .5(12) = 6%.5(12) = 6%

¾¾What is the APR if the semiannual rate is .5%?What is the APR if the semiannual rate is .5%?

z

z .5(2) = 1%.5(2) = 1%

¾¾What is the monthly rate if the APR is 12% with What is the monthly rate if the APR is 12% with monthly compounding?

monthly compounding?

z

z 12 / 12 = 1%12 / 12 = 1%

(8)

42 42

Things to Remember Things to Remember

¾¾ You ALWAYS need to make sure that the interest rate You ALWAYS need to make sure that the interest rate and the time period match.

and the time period match.

z

z If you are looking at annual periods, you need an annual If you are looking at annual periods, you need an annual rate.

rate.

z

z If you are looking at monthly periods, you need a monthly If you are looking at monthly periods, you need a monthly rate.

rate.

¾¾ If you have an APR based on monthly compounding, If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have adjust the interest rate appropriately if you have payments other than monthly

payments other than monthly

43 43

Computing EARs

Computing EARs - - Example Example

¾¾ Suppose you can earn 1% per month on $1 invested Suppose you can earn 1% per month on $1 invested today.

today.

zz What is the APR? 1(12) = 12%What is the APR? 1(12) = 12%

z

z How much are you effectively earning?How much are you effectively earning?

•

• FV = 1(1.01)FV = 1(1.01)¹²¹²= 1.1268= 1.1268

•

• Rate = (1.1268 Rate = (1.1268 ––1) / 1 = .1268 = 12.68%1) / 1 = .1268 = 12.68%

¾

¾ Suppose you put it in another account and earn 3% per Suppose you put it in another account and earn 3% per quarter.

quarter.

z

z What is the APR? 3(4) = 12%What is the APR? 3(4) = 12%

z

z How much are you effectively earning?How much are you effectively earning?

•

• FV = 1(1.03)FV = 1(1.03)⁴⁴= 1.1255= 1.1255

•

• Rate = (1.1255 Rate = (1.1255 ––1) / 1 = .1255 = 12.55%1) / 1 = .1255 = 12.55%

44 44

EAR EAR - - Formula Formula

m 1 1 APR EAR

m

⎥⎦ −

⎢⎣ ⎤

⎡ +

=

Remember that the APR is the quoted rate m is the number of compounding periods per year

45 45

Decisions, Decisions II Decisions, Decisions II

¾

¾ You are looking at two savings accounts. One pays You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3%

5.25%, with daily compounding. The other pays 5.3%

with semiannual compounding. Which account should with semiannual compounding. Which account should you use?

you use?

z

z First account:First account:

•

• EAR = (1 + .0525/365)EAR = (1 + .0525/365)³⁶⁵³⁶⁵––1 = 5.39%1 = 5.39%

z

z Second account:Second account:

•

• EAR = (1 + .053/2)EAR = (1 + .053/2)²²––1 = 5.37%1 = 5.37%

¾¾ Which account should you choose and why?Which account should you choose and why?

46 46

Decisions, Decisions II Decisions, Decisions II

Continued Continued

¾

¾LetLet’’s verify the choice. Suppose you invest s verify the choice. Suppose you invest

$100 in each account. How much will you have

$100 in each account. How much will you have in each account in one year?

in each account in one year?

zz First Account:First Account:

•

• 365 N; 5.25 / 365 = .014383562 I/Y; 100 PV; CPT FV = 365 N; 5.25 / 365 = .014383562 I/Y; 100 PV; CPT FV = 105.39

105.39

z

z Second Account:Second Account:

•

• 2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = 105.372 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = 105.37

¾

¾You have more money in the first account.You have more money in the first account.

47 47

Computing APRs from EARs Computing APRs from EARs

¾¾If you have an effective rate, how can you If you have an effective rate, how can you compute the APR? Rearrange the EAR compute the APR? Rearrange the EAR equation and you get:

equation and you get:

⎥⎦ ⎤

⎢⎣ ⎡ +

= m (1 EAR) - 1

APR

¹^m

(9)

48 48

APR - APR - Example Example

¾

¾Suppose you want to earn an effective rate of Suppose you want to earn an effective rate of 12% and you are looking at an account that 12% and you are looking at an account that compounds on a monthly basis. What APR must compounds on a monthly basis. What APR must they pay?

they pay?

[ ]

11.39%

or

8655152 113

. 1 ) 12 . 1 (

12 +

¹^/¹²

− =

= APR

49 49

Computing Payments with Computing Payments with

APRs APRs

¾¾Suppose you want to buy a new computer Suppose you want to buy a new computer system and the store is willing to sell it to allow system and the store is willing to sell it to allow you to make monthly payments. The entire you to make monthly payments. The entire computer system costs $3,500. The loan computer system costs $3,500. The loan period is for 2 years and the interest rate is period is for 2 years and the interest rate is 16.9% with monthly compounding. What is 16.9% with monthly compounding. What is your monthly payment?

your monthly payment?

z

z 2(12) = 24 N; 16.9 / 12 = 1.408333333 I/Y; 3,500 2(12) = 24 N; 16.9 / 12 = 1.408333333 I/Y; 3,500 PV; CPT PMT =

PV; CPT PMT = --172.88172.88

50 50

Future Values with Monthly Future Values with Monthly

Compounding Compounding

¾

¾Suppose you deposit $50 a month into an Suppose you deposit $50 a month into an account that has an APR of 9%, based on account that has an APR of 9%, based on monthly compounding. How much will you monthly compounding. How much will you have in the account in 35 years?

have in the account in 35 years?

z

z 35(12) = 420 N35(12) = 420 N

z

z 9 / 12 = .75 I/Y9 / 12 = .75 I/Y

z z 50 PMT50 PMT

z

z CPT FV = 147,089.22CPT FV = 147,089.22

51 51

Present Value with Daily Present Value with Daily

Compounding Compounding

¾¾You need $15,000 in 3 years for a new car. If You need $15,000 in 3 years for a new car. If you can deposit money into an account that you can deposit money into an account that pays an APR of 5.5% based on daily pays an APR of 5.5% based on daily compounding, how much would you need to compounding, how much would you need to deposit?

deposit?

z

z 3(365) = 1,095 N3(365) = 1,095 N

z

z 5.5 / 365 = .015068493 I/Y5.5 / 365 = .015068493 I/Y

z

z 15,000 FV15,000 FV

z

z CPT PV = -CPT PV = -12,718.5612,718.56

52 52

Continuous Compounding Continuous Compounding

¾

¾Sometimes investments or loans are figured Sometimes investments or loans are figured based on continuous compounding

based on continuous compounding

¾

¾EAR = EAR = ee^q^q––11

z

z The e is a special function on the calculator normally The e is a special function on the calculator normally denoted by e

denoted by e^x^x

¾

¾Example: What is the effective annual rate of Example: What is the effective annual rate of 7% compounded continuously?

7% compounded continuously?

z

z EAR = eEAR = e^.07^.07––1 = .0725 or 7.25%1 = .0725 or 7.25%

53 53

Quick Quiz

Quick Quiz – – Part V Part V

¾

What is the definition of an APR? What is the definition of an APR?

¾

What is the effective annual rate? What is the effective annual rate?

¾

Which rate should you use to compare Which rate should you use to compare alternative investments or loans?

alternative investments or loans?

¾

Which rate do you need to use in the time Which rate do you need to use in the time value of money calculations?

value of money calculations?

(10)

54 54

Pure Discount Loans

Pure Discount Loans – – Example Example 6.12 6.12

¾

¾Treasury bills are excellent examples of pure Treasury bills are excellent examples of pure discount loans. The principal amount is repaid discount loans. The principal amount is repaid at some future date, without any periodic at some future date, without any periodic interest payments.

interest payments.

¾

¾If a TIf a T--bill promises to repay $10,000 in 12 bill promises to repay $10,000 in 12 months and the market interest rate is 7 months and the market interest rate is 7 percent, how much will the bill sell for in the percent, how much will the bill sell for in the market?

market?

z

z 1 N; 10,000 FV; 7 I/Y; CPT PV = -1 N; 10,000 FV; 7 I/Y; CPT PV = -9,345.799,345.79

55 55

Interest

Interest- -Only Loan Only Loan - - Example Example

¾

¾Consider a 5Consider a 5--year, interestyear, interest--only loan with a 7% only loan with a 7%

interest rate. The principal amount is $10,000.

Interest is paid annually.

z

z What would the stream of cash flows be?What would the stream of cash flows be?

••Years 1 Years 1 ––4: Interest payments of .07(10,000) = 7004: Interest payments of .07(10,000) = 700

•

• Year 5: Interest + principal = 10,700Year 5: Interest + principal = 10,700

¾

¾This cash flow stream is similar to the cash flows This cash flow stream is similar to the cash flows on corporate bonds and we will talk about them on corporate bonds and we will talk about them in greater detail later.

in greater detail later.

56 56

Amortized Loan with Fixed Principal Amortized Loan with Fixed Principal

Payment

Payment - - Example Example

¾

¾Consider a $50,000, 10 year loan at 8% Consider a $50,000, 10 year loan at 8%

interest. The loan agreement requires the firm interest. The loan agreement requires the firm to pay $5,000 in principal each year plus to pay $5,000 in principal each year plus interest for that year.

interest for that year.

¾

¾Click on the Excel icon to see the amortization Click on the Excel icon to see the amortization table

table

57 57

Amortized Loan with Fixed Amortized Loan with Fixed

Payment

Payment - - Example Example

¾¾ Each payment covers the interest expense plus reduces Each payment covers the interest expense plus reduces principal

principal

¾¾ Consider a 4 year loan with annual payments. The Consider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5,000.

interest rate is 8% and the principal amount is $5,000.

z

z What is the annual payment?What is the annual payment?

•

• 4 N4 N

•

• 8 I/Y8 I/Y

•

• 5,000 PV5,000 PV

•

• CPT PMT = CPT PMT = --1,509.601,509.60

¾

¾ Click on the Excel icon to see the amortization tableClick on the Excel icon to see the amortization table

58 58

Work the Web Example Work the Web Example

¾

¾ There are web sites available that can easily prepare There are web sites available that can easily prepare amortization tables

amortization tables

¾

¾ Click on the web surfer to check out the Click on the web surfer to check out the Bankrate.comBankrate.com site and work the following example

site and work the following example

¾¾You have a loan of $25,000 and will repay the loan You have a loan of $25,000 and will repay the loan over 5 years at 8% interest.

over 5 years at 8% interest.

z

z What is your loan payment?What is your loan payment?

zz What does the amortization schedule look like?What does the amortization schedule look like?

59 59

Quick Quiz

Quick Quiz – – Part VI Part VI

¾¾What is a pure discount loan? What is a good What is a pure discount loan? What is a good example of a pure discount loan?

example of a pure discount loan?

¾

¾What is an interestWhat is an interest--only loan? What is a good only loan? What is a good example of an interest

example of an interest--only loan?only loan?

¾

¾What is an amortized loan? What is a good What is an amortized loan? What is a good example of an amortized loan?

example of an amortized loan?

(11)

6060

6 6

Calculators

End of Chapter End of Chapter

61 61

Comprehensive Problem Comprehensive Problem

¾

¾ An investment will provide you with $100 at the end of An investment will provide you with $100 at the end of each year for the next 10 years. What is the present each year for the next 10 years. What is the present value of that annuity if the discount rate is 8%

value of that annuity if the discount rate is 8%

annually?

¾¾ What is the present value of the above if the payments What is the present value of the above if the payments are received at the beginning of each year?

are received at the beginning of each year?

¾¾ If you deposit those payments into an account earning If you deposit those payments into an account earning 8%, what will the future value be in 10 years?

8%, what will the future value be in 10 years?

¾

¾ What will the future value be if you open the account What will the future value be if you open the account with $1,000 today, and then make the $100 deposits at with $1,000 today, and then make the $100 deposits at the end of each year?

the end of each year?