Lecture Essentials of corporate finance (2/e) – Chapter 5: Discounted cash flow valuation

In this chapter, students will be able to understand: Be able to compute the future value of multiple cash flows, be able to compute the present value of multiple cash flows, be able to compute loan payments, be able to find the interest rate on a loan, understand how loans are amortised or paid off, understand how interest rates are quoted.

Discounted cash flow

valuation

Chapter 5

Key concepts and skills

• Be able to compute the future value of

multiple cash flows

• Be able to compute the present value of

multiple cash flows

• Be able to compute loan payments

• Be able to find the interest rate on a loan

• Understand how loans are amortised or

paid off

• Understand how interest rates are quoted

• Loan types and loan amortisation

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Future value with multiple cash

• Suppose you deposit $100 today in an account

paying 8% In one year, you will deposit another

$100 How much will you have in two years?

– At the end of first year = 100* (1.08)+100=208

– At the end of second year = 208*(1.08)=224.64

Future value: Multiple cash flows

Example 5.1

• You think you will be able to deposit

$4000 at the end of each of the next 3 years in a bank account paying 8%

interest

• You currently have $7000 in the

account.

• How much will you have in 3 years?

• How much in 4 years?

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Future value: Multiple cash

• Find the value at year 3 of each cash

flow and add them together

Future value: Multiple cash

Future value: Multiple cash

Future value: Multiple cash

flows Example 5.2 (cont.)

• How much in 5 years if you don’t add

Future value: Multiple cash

Future value: Multiple cash

flows Example 5.2 (cont.)

• Calculator keys:

– Year 1 CF: 2 N; 7 I/Y; -100 PV; CPT FV = 114.49

– Year 2 CF: 1 N; 7 I/Y; -200 PV; CPT FV = 214.00

Future value: Multiple cash

flows:

Another example

• Suppose you plan to deposit $100 into an account in one year and $300 into the

account in 3 years How much will be in

the account in 5 years if the interest rate is 8%?

Present value: Multiple cash

– $400 the next year;

– $600 the following year; and

– $800 at the end of the 4th year

You can earn 12% on similar

investments What is the most you

Present value: Multiple cash

Present value: Multiple cash

Present value: Multiple cash

Time (years)

= 1/(1.12)2 x

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Present value: Multiple cash

• the first $5000 in four years from today

• the second $5000 in five years

• the third $5000 in six years

You can earn 11%

What is the future value of cash flows?

Present value: Multiple cash

Present value: Multiple cash

flows

Another example—Formula solution

• You are considering an investment that will pay you $1000 in one year, $2000

in two years and $3000 in three years

If you want to earn 10% on your

money, how much would you be willing

Present value: Multiple cash

Calculator hints

• Use the internal memory of the

calculator to store cash flows.

• Use of cash flow or CF key

– Clear all

• [CF]-[2nd]-[CLR WORK]

– Enter period 0 cash flow (use [+/-] to

change the sign)

– Press [ENTER] to enter the figure in cash flow register

Example: Spreadsheet

strategies

• You can use the PV or FV functions in

Excel to find the present value or future

value of a set of cash flows.

• Setting the data up is half the battle—

once it is set up properly, you can simply copy the formulas.

• Click on the Excel icon for an example.

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Decisions, decisions…

• Your broker calls you and tells you that he has a great

investment opportunity for you If you invest $100 today, you

will receive $40 in one year and $75 in two years If you require

a 15% return on investments of this risk, should you take the

Saving for retirement

• You are offered the opportunity to put

some money away for retirement You will receive 5 annual payments of $25

000 each, beginning in 40 years How

much would you be willing to invest

today if you desire an interest rate of

12%?

– Use cash flow keys:

• CF; CF 0 = 0; C01 = 0; F01 = 39; C02 = 25000; F02 = 5; NPV; I = 12; CPT NPV = $1084.71

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Saving for retirement time

line

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 (CF 0 = 0)

The cash flows for years 1 – 39 are 0 (C01 = 0;

F01 = 39)

The cash flows for years 40 – 44 are 25 000

(C02 = 25 000; F02 = 5)

Quick quiz: Part 1

• Suppose you are looking at the

following possible cash flows:

– Year 1: CF = $100

– Years 2 and 3: CFs = $200

– Years 4 and 5: CFs = $300

– The required discount rate is 7%.

• What is the value of the CFs at year 5?

• What is the value of the CFs today?

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Quick quiz: Part 1 Solution: Calculator

• Use the uneven cash flow keys and

find the present value first, then

compute the others based on that

Quick Quiz: Part 1

– Value at year 5 = 131.08 + 245.01 + 228.98 + 321 + 300 =

1226.07

– Present value today = 93.46 + 174.69 + 163.26 + 228.87 + Copyright © 2011 McGraw-Hill Australia Pty Ltd

Quick quiz: Part 1

Quick quiz: Part 1

$ 163.26

$ 174.69

$ 93.46

$ 228.98

$ 245.01

$ 131.08

$

FV = $ 1,226.07

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Valuing level cash flows

Annuities and perpetuities

• Annuity—finite series of equal

payments that occur at regular intervals

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

– If the first payment occurs at the beginning

of the period, it is called an annuity due

• Perpetuity—infinite series of equal

payments

Annuities and perpetuities

FV

r

r C

PV

t

t

1 )

1 (

) 1

(

1 1

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Annuities and the calculator

• The [PMT] key on the calculator is used

for the equal payment.

• The sign convention still holds.

• Ordinary annuity versus annuity due

– You can switch your calculator between the

two types by using the [2nd] [BGN] [2nd]

[SET] on the TI BA-II Plus

– If you see ‘BGN’ or ‘Begin’ in the display of

your calculator, you have it set for an annuity due.

– Most problems arise with ordinary annuities.

Annuities present value

Spreadsheet strategy

• The present value and future value

formulas in a spreadsheet include a

place for annuity payments.

• Double-click on the Excel icon to see

an example.

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Annuity Example 5.5

• You can afford $632

54 999 ,

23 01

.

) 01 1 (

1 1

PV

=PV(0.01,48,-632,0)

Annuity—Sweepstakes

example

• Suppose you win the Publishers

Clearinghouse $10 million

sweepstakes The money is paid in

equal annual instalments of $333

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

sweepstakes actually worth today?

$5 124 150.29

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Annuity—Sweepstakes

example Calculator and Excel solution

Buying a house

• You are ready to buy a house and you have $20

000 for a down payment and closing costs.

• Closing costs are estimated to be 4% of the loan

value

• You have an annual salary of $36 000

• The bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly

income

• The interest rate on the loan is 6% per year with

monthly compounding (.5% per month) for a

30-year fixed-rate loan

• How much money will the bank loan you?

• How much can you offer for the house?

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Buying a house (cont.)

Quick quiz: Part 2

• You know the payment amount for a

loan and you want to know how much

Quick quiz: Part 2 (cont.)

• You want to receive $5000 per month in retirement If you can earn 75% per

month and you expect to need the

income for 25 years, how much do you need to have in your account at

Finding the payment

• Suppose you want to

• If you take a 4-year

loan, what is your

=PMT(0.006667,48,20000,0)

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Example: Spreadsheet strategies— Annuity payment

• Another TVM formula that can be

found in a spreadsheet is the payment formula:

– PMT(rate, nper, pv, fv)

– The same sign convention holds as for

the PV and FV formulas

• Click on the Excel icon for an

example.

Finding the number of payments Example 5.6

• $1000 is due on a credit card

• Payment = $20 month minimum

• Rate = 1.5% per month

– How long would it take to pay off the $1000?

– And this is only if you don’t charge anything

more on the card!

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Finding the number of payments Example 5.6 (cont.)

= 7.75 years

=NPER(0.015,-20,1000,0)

Finding the number of payments— Another example

• Suppose you borrow $2000 at 5% and you are going to make annual

payments of $734.42 How long before you pay off the loan?

Finding the number of payments— Another example

Finding the rate

• Suppose you borrow $10 000 from your

parents to buy a car You agree to pay

$207.58 per month for 60 months What is

the monthly interest rate?

Annuity—Finding the rate

without a financial calculator

• Trial and error method:

– Choose an interest rate and compute the PV of

the payments based on this rate.

– Compare the computed PV with the actual loan

amount.

– If the computed PV > loan amount, then the

interest rate is too low.

– If the computed PV < loan amount, then the

interest rate is too high.

– Adjust the rate and repeat the process until the

computed PV and the loan amount are equal.

Quick quiz: Part 3

• You want to receive $5000 per month

for the next 5 years How much would

you need to deposit today if you can

earn 75% per month?

Quick quiz: Part 3 (cont.)

• You want to receive $5000 per month

for the next 5 years

• What monthly rate would you need to

earn if you only have $200 000 to

Quick quiz: Part 3 (cont.)

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

– How many months could you receive

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Quick quiz: Part 3 (cont.)

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

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

Future values for annuities

• Suppose you begin saving for your

retirement by depositing $2000 per year in a superannuation fund If the interest rate is

7.5%, how much will you have in 40 years?

454 075

.

1 )

075 1 ( 2000 FV

r

1 )

r 1 ( PMT FV

40 t

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Annuity due

• An annuity for which the cash flows occur at the

beginning of the period.

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

08 1 ( 08

.

1 ) 08 1 ( 10000 FV

) r 1 ( r

1 ) r 1 ( PMT FV

3 AD

t AD

Annuity due time line

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Summary of annuity and

perpetuity calculations

Table 5.2

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Example: Work the Web

that has a financial calculator

• Click on the information icon, and work out the following example using the website calculator.

• Suppose you retire with $1 000 000 The

growth rate is 9%.

• How much you can withdraw for next 30 years.

• Do the calculation with a calculator and

compare the results.

Quick quiz: Part 4

• You want to have $1 million to use for retirement in

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Quick quiz: Part 4 (cont.)

• Q2: If you can earn 1% per month, how much do you need to deposit on a

monthly basis if the first payment is

Quick quiz: Part 4 (cont.)

• You are considering preferred stock

that pays a quarterly dividend of $1.50

If your desired return is 3% per quarter, how much would you be willing to pay?

– $1.50/0.03 = $50

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Effective annual rate (EAR)

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

• If you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison.

Annual percentage rate

(APR)

• This is the annual rate that is quoted by law.

• By definition APR = period rate times

the number of periods per year.

• So, to get the period rate we rearrange the APR equation:

– Period rate = APR/number of periods per year

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

—it will NOT give you the period rate. Copyright © 2011 McGraw-Hill Australia Pty Ltd

• What is the monthly rate if the APR is 12%,

with monthly compounding?

– 12 / 12 = 1%

– Can you divide the above APR by 2 to get the

semi-annual rate? NO!!! You need an APR based

on annual compounding to find the

– If you are looking at monthly periods, you

need a monthly rate.

• If you have an APR based on monthly

compounding, you have to use monthly

periods for lump sums, or adjust the

interest rate appropriately if you have

payments other than monthly.

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EAR formula

• APR = the quoted rate

• m = number of compounds per year

1

m m

APR 1

EAR

Computing EARs—Example

• Suppose you can earn 1% per month on $1

invested today.

– What is the APR? 1(12) = 12%

– How much are you effectively earning?

• FV = 1(1.01)12 = 1.1268

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

• Suppose you put it in another account, where you earn 3% per quarter.

– What is the APR? 3(4) = 12%

– How much are you effectively earning?

• FV = 1(1.03)4 = 1.1255

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

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EAR and APR on calculator

• [2nd][ICONV]

• [2nd][CLR WORK]

• 3 fields in worksheet:

– NOM (Nominal rate-APR)

– EFF (Effective annual rate)

– C/Y (Compounding periods/yr)

– Enter any 2 values, move to the 3 rd and press

EAR and NOM (APR) in

Excel

• 2 functions:

=EFFECT(Nom, Nper)

=NOMINAL(Eff, Nper)

• All rates entered as decimals

• Nper = number of compounding

periods per year

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Decisions, decisions… II

• Which savings accounts should you

choose:

– 5.25%, with daily compounding

– 5.30%, with semiannual compounding

Decisions, decisions… II

(cont.)

• Let’s verify the choice Suppose you

invest $100 in each account How much

will you have in each account in one year?

Computing APRs from

EARs

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

1 

­  EAR)

(1  

m  

m = number of compounding periods per year

• Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis What APR

must they pay?

11.39% or

8655 113

1 )

12 1 ( 12 APR 1 / 12

Computing payments with

APRs

• Suppose you want to buy a new computer system

and the store is willing to allow you to make monthly payments The entire computer system costs $3500 The loan period is for 2 years and the interest rate is 16.9%, with monthly compounding What is your

Future values with monthly compounding

• Suppose you deposit $50 a month into

an account that has an APR of 9%,

based on monthly compounding How

much will you have in the account in 35 years?

Present value with daily

compounding

• You need $15 000 in 3 years for a new car If you can deposit money into an account that pays an APR of 5.5% based on daily

compounding, how much would you need to deposit?