1 Chapter 5 Discounted Cash Flow Valuation. 2 Overview Important Definitions Finding Future Value of an Ordinary Annuity Finding Future Value of Uneven

1

Chapter 5

Discounted Cash Flow Valuation

2

Overview

Important Definitions Finding Future Value of an Ordinary Annuity Finding Future Value of Uneven Cash Flows Finding Present Value of an Ordinary Annuity Finding Present Value of Uneven Cash Flows Valuing Level Cash Flows: Annuities and

Perpetuities Comparing Rates: The Effect of

Compounding Periods Loan Types and Loan Amortization

3

Important Definitions

Annuity: A level stream of cash flows for a fixed period of time.

Ordinary Annuity: Cash flows take place at the end of each period.

Annuity Due: Cash flows takes place at the beginning of each period.

Perpetuity: A level stream of cash flows forever. Note: For now computations will be shown using

formulations and calculator entries. I recommend focusing on calculator entries because it is much more efficient in getting answers. Formulations are provided for demonstration purposes. You should also review Excel file to see how excel functions can be used.

4

Finding Future Value of an Ordinary Annuity

20.210,12

10.0

110.01000,2

11 5

r

rCFV

t

t

N I/Y P/Y PV PMT FV MODE

5 10 1 0 2,000 -12,210.20

5

Finding Future Value of Uneven Cash Flows

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

FV can be computed for each cash flow at a common future period (in this case 5 years from today)

FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97

100

0 1 2 3 4 5

300

136.05

349.92

485.97

6

Finding Future Value of Uneven Cash Flows – Continued

Calculator solution of the previous problem would involve determining FV of each payment in five years from today and then adding them

Total FV = 136.05 + 349.92 = 485.97


4 8 1 -100 0 -136.05

2 8 1 -300 0 -349.92

7

Finding Present Value of an Ordinary Annuity

37.212,4

06.006.01

11

000,11

11 5

rr

CPVt

t


5 6 1 -4,212.37

1,000 0

8

Finding Present Value of Uneven Cash Flows

Find the PV of each cash flow and add them Year 1 CF: 200 / (1.12)1 = 178.57 Year 2 CF: 400 / (1.12)2 = 318.88 Year 3 CF: 600 / (1.12)3 = 427.07 Year 4 CF: 800 / (1.12)4 = 508.41 Total PV = 178.57 + 318.88 + 427.07 +

508.41 Total PV = 1,432.93

Years 0 1 2 3 4Cash Flow 200 400 600 800

9

Finding Present Value of Uneven Cash Flows – Continued

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

10

Finding Present Value of Uneven Cash Flows – TI BA II PLUS

CF

CF0 0 ENTER

C01 200 ENTER F01 1.00 ENTER




NPV 12 ENTER CPT 1,432.93

When using CF make sure to clear previous entries CF 2ND CLR WORK

11

Example You are offered the opportunity to put some

money away for retirement. You will receive five 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%?

This cash flow set has two parts PV of annuity payments 39 years from today PV of a single payment in 39 years See the timeline below

0 1 2 … 39 40 41 42 43 44

0 0 0 … 0 25K 25K 25K 25K 25K

12

Example – Continued

PV of annuity 39 years from today

PV of single payment in 39 years

41.119,90

12.012.01

11

000,251

11 5

rr

CPVt

t

71.084,1

12.01

41.119,9039

tPV

13

PV of annuity 39 years from today

PV of single payment in 39 years

Example – Continued – TI BA II PLUS


5 12 1 -90,119.4

1

25,000

0


39 12 1 -1,084.7

1

0 90,119.41

14

Example – Continued – TI BA II PLUS

CF

CF0 0 ENTER


C02 25,000 ENTER F02 5.00 ENTER

NPV 12 ENTER CPT 1,084.71

15

Finding Future or Present Value of an Annuity Due

Annuity Due cash flows take place at the beginning of each period – this means that there is a payment now

The immediate payment requires that we account its impact on FV and PV correctly

The basic adjustments Treat the annuity as if it were an ordinary annuity and determine

FV or PV Then multiply the answer by (1 + r) FV or PV of Annuity Due = (FV or PV of Ordinary Annuity) x (1 + r)

If you are computing PMT then the adjustment is slightly different

PMT of Annuity Due = (PMT of Ordinary Annuity) / (1 + r) This is because immediate payment with annuity due reduces

amount borrowed leading to lower overall payment I suggest using calculator settings for PMT, N, and I/Y

computations

16

Finding Future or Present Value of an Annuity Due – Continued

Consider the above annuity due. This is a five year annuity due. When you compute FV it is computed at the end of year 5. Assume 10% rate.

95.667,110.0110.0

10.01

11

40011

11 5

r

rr

CPVt

t

24.686,210.0110.0

110.014001

11 5

r

r

rCFV

t

t

17

Finding Future or Present Value of an Annuity Due – TI BA II PLUS

To compute FV and PV of an annuity due you need to change MODE to BGN

2ND BGN 2ND SET 2ND QUIT Once complete you will see BGN on the display To return to END follow the same steps END mode is not displayed


5 10 1 -1,667.9

5

400 0 BGN


5 10 1 0 400 -2,686.24 BGN

18

Example

Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?


30 5 1 -5,124,150.2

9

333,333.33

0

19

Non Annual Interest Earnings (Compounding of Interest Rate) and Payments

Remember that FVt = PV0 x (1 + r)t

When dealing with non annual interest and payment you should make two adjustment

Interest rate should be periodic Number of periods should reflect the total number of periods

given number of years and frequency of interest earnings in a year

Formulations should be adjusted as follows If “m” is the number of compounding in a year then where

you see “r” divide it by “m” and multiply “t” by “m” This assumes that “t” is the number of years

FVt = PV0 x (1 + r/m)t X m

Excel adjustment are same as formulations that can be made in a function dialog box or data cells

20

Non Annual Interest Earnings (Compounding of Interest Rate) and Payments – Continued

FVt = PV0 x (1 + r/m)t X m

Entering the above equation into TI BA II PLUS Method 1:

Keep P/Y = 1 (so is C/Y = 1) N = t X m (Number of periods) I/Y = r/m (Periodic interest rate) Rest of the information is entered as before

Method 2: Change P/Y = m (so is C/Y = m) N = t X m (Number of periods) I/Y = r (Annual interest rate) Rest of the information is entered as before Note that P/Y has to be checked for consistency every time

21

ExampleBuying a Car

Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly. If you take a 4-year loan, what is your monthly payment?

26.488

12

08.012

08.01

11

000,20

1

11

48124

CC

m

rm

r

CPV

mt

t

22

ExampleBuying a Car – TI BA II PLUS

Note the adjustment


48 8 12 -20,000 488.26 0

2ND P/Y12 ENTER2ND QUIT

23

ExampleCredit Card

Note: Finding interest rate and number of periods in annuity formulations are relatively time consuming. By this point you should be more comfortable with your calculator. From now on, solutions will be shown based on calculator entries.

You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month (18% per year compounded monthly). How long will you need to pay off the $1,000.


93.11

18 12 -1,000 20 0

24

ExampleBorrowing Money from Unlikely Resources

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 annual interest rate compounded monthly?N I/Y P/Y PV PMT FV MODE

60 9.00 12 -10,000 207.58 0

25

Present Value of a Perpetuity Suppose you will receive a fixed payment every

period (month, year, etc.) forever. This is an example of a perpetuity

You can think of a perpetuity as an annuity that goes on

r

C PV

r

rr

CPV

t

t

toreducesequation PV theTherefore

zero. approaches 1

1 term then thelarge gets t If

1

11

:as PVfor equation theknow We

26

ExamplePerpetuity

What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

PV = $10,000 / 0.08 = $125,000

27

Effective Annual Rate (EAR) vs. Annual Percentage Rate (APR)

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.

Sometimes it is also called Annual Percentage Yield (APY) or Effective Rate

Annual Percentage Rate (APR) This is the annual rate that is quoted by law By definition APR = periodic rate times the number of periods per

year Consequently, to get the periodic rate we rearrange the APR

equation: Periodic rate = APR / number of periods per year Sometimes it is also called Stated Rate, Quoted Rate or Nominal

Rate You should NEVER divide the effective rate by the number of

periods per year – it will NOT give you the periodic rate

28

Rate Conversions

APR to EAR

EAR to APR

Calculator convention NOM = APR EFF = EAR C/Y = m

11 m

m

APR EAR

1

11 - m EAR m APR

29

ExampleRate Conversion

You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?

Compare EAR and choose the larger one since you are investing. If you were borrowing then you would choose the lower EAR alternative

5.25% compounded daily 2nd ICONV NOM 5.25 ENTER (EFF) C/Y 365 ENTER (EFF) CPT 5.39

5.30% compounded semiannually 2nd ICONV NOM 5.30 ENTER (EFF) C/Y 2 ENTER (EFF) CPT 5.37

Choose 5.25% compounded daily

30

ExampleRate Conversion – Continued

Let’s verify the choice. Suppose you invest $100,000 in each account. How much will you have in each account after 5 years? Daily Compounded Account:

Semiannually Compounded Account:

You have more money in the first account.


1,825 5.25 365 -100,000 0 130,015.19


10 5.30 2 -100,000 0 129,894.13

31

Amortized Loan with Fixed Payment

Each payment covers the interest expense plus reduces principal

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


4 8 1 -5,000 1,509.60 0

32

Amortization Table

Computations: Interest Paid = Beginning Balance X Interest Rate Principal Paid = Total Payment – Interest Paid Ending Balance = Beginning Balance – Principal Paid

Year Beginning Balance

Total Payment

Interest Paid

Principal Paid

Ending Balance

1 5,000.00 1,509.60 400.00 1,109.60 3,890.40

2 3,890.40 1,509.60 311.23 1,198.37 2,692.02

3 2,692.02 1,509.60 215.36 1,294.24 1,397.78

4 1,397.78 1,509.60 111.82 1,397.78 0.00

Totals 6,038.42 1,038.41 4,999.99

33

Amortization Table – Continued

If you want to determine the outstanding balance of the loan after 2 years: 2nd AMORT 2 ENTER 2 ENTER

This will allow you to see loan information at that point in time.

You can change P1 and P2 to get the data for the specified payment range

Documents

1 Chapter 5 Discounted Cash Flow Valuation. 2 Overview Important Definitions Finding Future Value of an Ordinary Annuity Finding Future Value of Uneven