FIN202- PRINCIPLES OF CORPORATE FINANCE

Session 6
CHAPTER 6: DISCOUNTED CASH FLOWS AND
VALUATION

Taught by Que Anh Nguyen - FPT School of Business (FSB)

Original Slides by Robert Parrino, Ph.D. & David S. Kidwell, Ph.D.

Based on: Parrino, R., KidWell, D., 2019, Fundamentals of Corporate

Finance (4ed)

1
 OBJECTIVES

1. Explain why cash flows occurring at different times must be adjusted to reflect their value as of a common
date before they can be compared, and compute the present value and the future value for multiple
cash flows.

2. Explain the difference between an ordinary annuity and an annuity due, and calculate the present value
and the future value of an ordinary annuity and an annuity due.

3. Explain what a perpetuity is and where we see them in business, and calculate the value of a perpetuity.

4. Discuss growing annuities and perpetuities, as well as their application in business, and calculate their
values.

5. Discuss why the effective annual interest rate (EAR) is the appropriate way to annualize interest rates,
and calculate the EAR.

2
1. Multiple Cash Flows
1.1. Future Value of Multiple Cash Flows

𝐹𝑉𝑛 = 𝑃𝑉 𝑥 (1 + 𝑖)𝑛

Solving future value problems with multiple cash flows.

1. Draw timeline to ascertain each cash flow is placed in correct time period.
2. Calculate future value of each cash flow for its time period.
3. Add up the future values.
3
1. Multiple Cash Flows
1.2. Present Value of Multiple Cash Flows

Many business situations call for computing present value of a series of expected future
cash flows.
✓ Determining market value of security.
✓ Deciding whether to make capital investment.

𝑃𝑉 = 𝐹𝑉𝑛 /(1 + 𝑖)𝑛

PROCESS:
1. First, prepare timeline to identify magnitude and timing of cash flows.
2. Next, calculate present value of each cash flow using equation 5.4 from the previous
chapter.
3. Finally, add up all present values.
4. Sum of present values of stream of future cash flows is their current market price, or value.
4
1. Multiple Cash Flows
1.2. Present Value of Multiple Cash Flows

Suppose that your best friend needs cash and offers to pay you $1,000 at the end of each
of the next three years if you will give him $3,000 cash today. You realize, of course, that
because of the time value of money, the cash flows he has promised to pay are worth less
than $3,000. If the interest rate on similar loans is 7 percent, how much should you pay for
the cash flows your friend is offering?

𝑃𝑉 = 𝐹𝑉𝑛 /(1 + 𝑖)𝑛

5
1. Multiple Cash Flows
1.2. Present Value of Multiple Cash Flows

For a student—or anyone else—buying a used car can be a harrowing experience. Once
you find the car you want, the next difficult decision is choosing how to pay for it—cash or
a loan. Suppose the cash price you have negotiated for the car is $5,600, but that amount
will stretch your budget for the year. The dealer says, “No problem. The car is yours for
$4,000 down and payments of $1,000 per year for the next two years. Or you can put
$2,000 down and pay $2,000 per year for two years. The choice is yours.” Which offer is
the best deal if the interest rate you can earn on your money is 8 percent?

𝑃𝑉 = 𝐹𝑉𝑛 /(1 + 𝑖)𝑛

Down Payment CF1 CF2 Total PV
Option A
Option B
Option C
6
2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)

Many situations exist where businesses and individuals would face either receiving or
paying constant amount for a length of period.
E.g. Constant payments on home or car loans, investment of a fixed amount year after
year saving for retirement.

▪ Annuity: any financial contract calling for equally spaced level cash flows over
finite number of periods.
▪ Perpetuity: contract calling for level cash flow payments to continue forever.
▪ Ordinary annuities: constant cash flows occurring at end of each period.

7
 2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)
▪ Can calculate PVA same way present value of multiple cash flows is calculated.

Tedious !!! 𝑛
1
PVAn = ෍ 𝐶𝐹 × 𝑡
1+𝑖
𝑡=0

8
2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)

▪ Instead, simplify equation 5.4 in chapter 5 to obtain annuity factor.

▪ Results in equation 6.1 that can be used to calculate the annuity’s present value.

PVAn = CF × Present value factor for an annuity

(1 − Present value factor)
= CF ×
i
1
1−
(1 + i)n
= CF × (6.1)
i

Exhibit 6.4: Present Value Annuity Factors

9
2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)

A financial contract pays $2,000 at the end of each year for three years and the
appropriate discount rate is percent? What is the present value of these cash flows?

1 1
Present value factor = = = 0.7938
(1+i)n (1+0.08)3
1 - Present value factor 1 - 0.7938
PV annuity factor = = = 2.577
i 0.08
PVA 3 = CF  PV annuity factor = $2,000  2.577 = $5,154.00

10
2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)

Present Value of Annuity example - Calculator solution

3 8 2,000 0
Enter
N i PV PMT FV
Answer -5,154.19

11
2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)

You have just purchased a $450,000 condominium. You were able to put $50,000 down
and obtain a 30-year fixed rate mortgage at 6.125 percent for the balance. What are
your monthly payments?

Monthly interest rate = 6.125 % / 12 months = 0.51042 %

Present value factor =

1
=
1
= 0.1599589 PVAn = CF  PV annuity factor
(1+i)n (1.0051042)360
$400,000 = CF  164.578406
1 - Present value factor
PV annuity factor = $400,000
i CF = = $2,430.45
1 - 0.1599589 164.578406
= = 164.57840
0.0051042

12
2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)

Financial Calculator Solution

360 0.51042 400,000 0

Enter
N i PV PMT FV
Answer -2,430.44

13
2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)
Preparing a Loan Amortization Schedule
▪ Amortization: the way the borrowed amount (principal) is paid down over life of loan.
→ Monthly loan payment is structured so each month portion of principal is paid off; at
time loan matures, it is entirely paid off.
→ Amortized loan: Monthly loan payment contains some payment of principal and an
interest payment.

Loan amortization schedule is a table showing:

➢ loan balance at beginning and end of each period.
➢ payment made during that period.
➢ how much of payment represents interest.
➢ how much represents repayment of principal.

14
 2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)
Preparing a Loan Amortization Schedule
Amortization Table for a Five-Year, $10,000 Loan with an Interest Rate of 5%
Step 1: The annual loan payment (CF), as calculated earlier, is $2,309.75.

1 1
1− 1−
(1 + i)n (1 + 0.05)5
PVAn = CF × (6.1) $10,000 = CF × (6.1)
i 0.05

Step 2: The interest payment for the first year is $500 = i × 𝑃0 = 0.05 × $10,000

Step 3: The principal paid for the year = $2,309.75 − $500 = $1,809.75

Step 4: Ending principal balance = $10,000 − $1,809.75 = $8,190.25

Step 5: Interest payment 2nd year = i × 𝑃1 = 0.05 × $8,190.25 = $409.51

15
 2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)

With amortized loan,

▪ Larger proportion of each month’s
payment goes towards interest in early
periods.
▪ As loan is paid down, greater proportion
of each payment is used to pay down
principal.
▪ Amortization schedules are best done
on a spreadsheet.

16
 2. Level Cash Flows: Annuities and Perpetuities
2.1. Present Value of an Annuities (PVA)
Finding the Interest Rate
▪ The annuity equation can also be used to find interest rate or discount rate or the rate of return for
an annuity.
▪ To determine rate of return for the annuity, we need to solve equation for the unknown value i.
▪ Other than using trial and error approach, easier to solve using financial calculator.

Your parents are getting ready to retire and decide to convert some of
their retirement portfolio into an annuity. Their insurance agent asks for
$350,000 for an annuity that guarantees to pay them $50,000 a year for
10 years. What is the return on the annuity?

10 -350,000 50,000 0
Enter
N i PV PMT FV

Answer 7.073 17
2. Level Cash Flows: Annuities and Perpetuities
2.2. Future Value of an Annuity (FVA)

▪ FVA calculations usually involve finding what a savings or investment activity is worth
at some future point.
▪ E.g. saving periodically for vacation, car, house, or retirement.

FVAn = CF × FV factor for an annuity

Future value factor −1
= CF ×
i
n
(1 + i) − 1
= CF × (6.2)
i

18
2. Level Cash Flows: Annuities and Perpetuities
2.2. Future Value of an Annuity (FVA)

Example Orbea bike problem in Exhibit 6.6.

Future value factor=(1+i)n =(1.08)4 =1.36049

Future value factor - 1 1.36049 - 1
FV annuity factor= = =4.5061
i 0.08
FVA 4 =CF  FV annuity factor=$1,000  4.5061 =$4,506.10

4 8 0 1,000
Enter
N i PV PMT FV
Answer -4,506.11
19
2. Level Cash Flows: Annuities and Perpetuities
2.3. Perpetuities

▪ A perpetuity is constant stream of cash flows that goes on for infinite period.
▪ In stock markets, preferred stock issues are considered to be perpetuities, with issuer
paying a constant dividend to holders.
▪ Equation for present value of a perpetuity can be derived from present value of an
annuity equation with n tending to infinity.

PVA∞ = CF × Present value factor for an annuity

1
1− (1 − 0)
(1 + i)∞
= CF × = CF ×
i i
CF
= (6.3)
i

20
2. Level Cash Flows: Annuities and Perpetuities
2.3. Perpetuities
Perpetuities - Example

Suppose you decided to endow a chair in finance. The goal of the

chair is to provide the chair holder with $100,000 of additional
financial support per year forever. If the rate of interest is 8 percent,
how much money will you have to give the university foundation to
provide the desired level of support?

PVA CF $100,000
= = = $1,250,000
i 0.08
IMPORTANT:
Just as perpetuity equation was derived from present value annuity equation,
one can also derive present value of a very long annuity from the equation for a
perpetuity.

21
 2. Level Cash Flows: Annuities and Perpetuities
2.4. Annuity Due
▪ Annuity is called an annuity due when there is an annuity with payments being incurred
at beginning of each period rather than at end.
▪ Rent or lease payments typically made at beginning of each period rather than at end.

▪ Annuity transformation

Annuity due = Ordinary annuity value  (1+i) (6.4)

▪ Each period’s cash flow thus earns extra period of

interest compared to ordinary annuity.
▪ Present or future value of annuity due is always > that
of ordinary annuity.

Annuity due = $3,312  (1.08) = $3,577

22
Exhibit 6.7: Ordinary Annuity versus Annuity Due
3. Cash Flows That Grow at a Constant Rate
3.1. Growing Annuity
▪ Growing annuities or growing perpetuities: In addition to constant cash flow streams,
one may have to deal with cash flows that grow at a constant rate over time.
▪ Ex: multiyear product or service contracts with cash flows that increase each year at
constant rate.

▪ When the growth rate is less than discount rate.

n
CF1 1+g
PVAn = × 1− (6.5)
i−g 1+i

23
3. Cash Flows That Grow at a Constant Rate
3.1. Growing Annuity

A coffee shop will be in business for 50-years. It produced $300,000 this year and the
discount rate used by similar businesses is 15 percent. The cash flows will grow at 2.5
percent per year. What is the estimated value of the coffee shop?

CF1 = $300,000  (1 + 0.025) = $307,500

$307,500   1.025 50 

PVA 50 =  1 −   
0.15 − 0.025   1.15  
 
= $2,460,000  0.9968 = $2,452,128

24
3. Cash Flows That Grow at a Constant Rate
3.2. Growing Perpeptuity
▪ When cash flow stream features constant growing annuity forever.

CF1
PVA  = (6.6)
i-g

Your account reports that a firm’s cash flow last year was $450,000 and the
appropriate discount rate for the club is 18 percent. You expect the firm’s
cash flows to increase by 5 percent per year and that the business will
have no fixed life. What is the value of the firm?

CF1 CF0 ×(1+g)

PVA  = =
i-g i-g
$450,000  (1+0.05)
= = $3,634,615
0.18 − 0.05 25
4. Effective Annual Interest Rate

▪ Interest rates can be quoted in financial markets in variety of ways.

▪ Annual percentage rate (APR): represents simple interest accrued on loan or
investment in a single period; annualized over a year by multiplying it by appropriate
number of periods in a year.
▪ Effective annual rate (EAR): is defined as annual growth rate that takes
compounding into account.

EAR = (1 + Quoted rate/m)m – 1 (6.7)

m is the # of compounding periods during a year.

→ EAR, thus effectively adjusts annualized interest rate for time value of money.
→ EAR is the true cost of borrowing and lending.
26
 4. Effective Annual Interest Rate

EAR = (1 + 0.12/12)12 – 1
Your credit card has an APR of 12 percent (1 percent = (1.01)12 – 1
per month). What is the effective annual interest rate? = 1.1268 – 1
= 0.1286 or 12.68%

Suppose you are the CFO of a manufacturing

company. The company is planning a $1 billion plant
expansion and will finance it by borrowing money for
five years.
Lender A: 10.40% compound monthly
Lender B: 10.90% compounded annually
Lender C: 10.50% compounded quarterly

27
4. Effective Annual Interest Rate

Consumer Protection Acts and Interest Rate Disclosures

▪ Truth-in-Lending (1968) ensures that true cost of credit was disclosed to
consumers, so they could make sound financial decisions.
▪ Truth-in-Savings Act provides consumers accurate estimate of return they
would earn on investment.
▪ Require that APR be disclosed on all consumer loans and savings plans, and
prominently displayed on advertising and contractual documents.
▪ Note that EAR, not APR, is the appropriate rate to use in present and future
value calculations.

28
Homework

HOMEWORK:
Q6.2→6.4, 6.6, 6.8, 6.10, 6.12 pg 364; 6.15, 6.16, 6.18 pg. 365
6.20, 6.22→ 6.25 pg. 365
6.35, 6.36, 6.38, 6.40 pg 367
(Parrino, R., KidWell, D., 2019, 4ed)

29
Exhibit 6.4: Present Value Annuity Factors

30