1. Discounted cash flow valuation
Chapter 4
Discounted cash flow valuation

3. 4.1 Valuation: the one-period case
10,000 (return of principle)
+ (0.12 x 10,000) (interest) = 11,200
Future value / compound value
Present value
PV = C1 / (1+r)

4. NPV = - cost + PV
Is the present value of future cash flows minus the present value of the cost of the investment
The selection of the discounted rate (example 4.2)

5. 4.2 The multi-period case 1 x (1+r)
Compounding (the process of leaving the money in the financial market and lending it for another year)
1 x (1+r) x (1+r) = 1 + 2r (simple interest) + r2 (interest on interest)

6. What is the effect of compounding?1
FV = C0 x (1 + r)t
(example )
Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?
FV = 10(1.055)200 = 447,189.84
What is the effect of compounding?
Simple interest = (10)(.055) =
Compounding added $447, to the value of the investment. 6

7. FUTURE VALUE AS A GENERAL GROWTH FORMULA1
FUTURE VALUE AS A GENERAL GROWTH FORMULA
Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in year 5?
FV = 3,000,000(1.15)5 = 6,034,072 7

8. Present value & discounting
PV x (1.09)2 = 1
Discounting - the process of calculating the present value of a future cash flow

9. 1
Present Values
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. 9

11. Present Value – One Period Example1
Present Value – One Period Example
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 11

12. Present Values – Example 21
Present Values – Example 2
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?
PV = 150,000 / (1.08)17 = 40,540.34 12

13. Present Values – Example 31
Present Values – Example 3
Your parents set up a trust fund for you 10 years ago that is now worth $19, If the fund earned 7% per year, how much did your parents invest?
PV = 19, / (1.07)10 = 10,000 13

17. 1
Discount Rate
Often we will want to know what the implied interest rate is in an investment
Rearrange the basic PV equation and solve for r
FV = PV(1 + r)t
r = (FV / PV)1/t – 1 17

18. Discount Rate – Example 1
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 = = 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% 18

19. Discount Rate – Example 21
Discount Rate – Example 2
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?
r = (20,000 / 10,000)1/6 – 1 = = 12.25% 19

20. Discount Rate – Example 31
Discount Rate – Example 3
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?
r = (75,000 / 5,000)1/17 – 1 = = 17.27% 20

21. Finding the Number of Periods1
Finding the Number of Periods
Start with 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. 21

22. Number of Periods – Example 1
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?
t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years
I/Y = 10; PV = -15,000; FV = 20,000
CPT N = 3.02 years 22

23. Number of Periods – Example 21
Number of Periods – Example 2
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? 23

24. Number of Periods – Example 2 Continued1
Number of Periods – Example 2 Continued
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, ,750 = 21,750
Compute the number of periods
Using the formula
t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years
Per a financial calculator:
PV = -15,000, FV = 21,750, I/Y = 7.5, CPT N = 5.14 years 24

25. Spreadsheet Example Use the following formulas for calculations1
Spreadsheet Example
Use the following formulas for 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 25

26. 1 26 26

27. FUTURE VALUE OF MULTIPLE CASH FLOWS1
FUTURE VALUE OF MULTIPLE CASH FLOWS
The time line:
Calculating the future value: 1 2
Time
(years)
$100 CF 1 2
Time
(years)
$100 CF
Future values
+108
x1.08
$208
$224.64 27

28. Multiple Cash Flows –Future Value Example1
Multiple Cash Flows –Future Value Example
Suppose you deposit $4000 at end of next 3 years in a bank account paying 8% interest. You currently have $7000 in the account. How much you will have in year 3, in year 4? 1
Time
(years)
$4000
$7000 CF 2 3 4 28

29. Multiple Cash Flows –Future Value Example1
Multiple Cash Flows –Future Value Example
Find the value at year 3 of each cash flow and add them together.
Today (year 0): FV = 7000(1.08)3 = 8,817.98
Year 1: FV = 4,000(1.08)2 = 4,665.60
Year 2: FV = 4,000(1.08) = 4,320
Year 3: value = 4,000
Total value in 3 years = 8, , , ,000 = 21,803.58
Value at year 4 = 21,803.58(1.08) = 23,547.87 29

30. Multiple Cash Flows – FV Example1
Multiple Cash Flows – FV Example
Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in year 2?
FV = 500(1.09) (1.09) = 1,248.05 1 2
Time
(years)
$600
$500 30

31. Multiple Cash Flows – Example Continued1
Multiple Cash Flows – Example Continued
How much will you have in year 5 if you make no further deposits?
First way:
FV = 500(1.09) (1.09)4 = 1,616.26
Second way – use value at year 2:
FV = 1,248.05(1.09)3 = 1,616.26 1 2
Time
(years)
$600
$500 3 4 5 31

32. Multiple Cash Flows – FV Example1
Multiple Cash Flows – FV Example
Suppose you plan to deposit $100 into an account in year 1 and $300 into the account in year 3. How much will be in the account in year 5 if the interest rate is 8%?
FV = 100(1.08) (1.08)2 = = 1 2 3 4 5
Time
(years)
$100
$300 32

33. 4.3 Compounding periods
Stated annual interest rate/annual percentage rate
By definition APR = period rate times the number of periods per year
Consequently, to get the period rate we rearrange the APR equation:
Period rate = APR / number of periods per year
Effective annual rate (EAR): (1+r/m)m - 1

35. Computing APRs What is the APR if the monthly rate is .5%?1
Computing APRs
What is the APR if the monthly rate is .5%?
.5(12) = 6%
What is the APR if the semiannual rate is .5%?
.5(2) = 1%
What is the monthly rate if the APR is 12% with monthly compounding?
12 / 12 = 1% 35

37. 1
Things to Remember
You ALWAYS need to make sure that the interest rate and the time period match.
If you are looking at annual periods, you need an annual rate.
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 37

38. Computing EARs - Example1
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 =
Rate = ( – 1) / 1 = = 12.68%
Suppose you put it in another account, you earn 3% per quarter.
What is the APR? 3(4) = 12%
FV = 1(1.03)4 =
Rate = ( – 1) / 1 = = 12.55% 38

39. 1
EAR - Formula 39

40. 1
Decisions, Decisions
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?
First account:
EAR = ( /365)365 – 1 = 5.39%
Second account:
EAR = ( /2)2 – 1 = 5.37%
Which account should you choose and why? 40

41. Decisions, Decisions Continued1
Decisions, Decisions Continued
Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year?
First Account:
Daily rate = / 365 =
FV = 100( )365 =
Second Account:
Semiannual rate = .053 / 2 = .0265
FV = 100(1.0265)2 =
You have more money in the first account. 41

42. Computing APRs from EARs1
Computing APRs from EARs
If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: 42

43. 1
APR - Example
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? 43

44. 4.4 Simplifications Perpetuity Growing perpetuity Annuity
Growing annuity

45. Annuities and Perpetuities Defined1
Annuities and Perpetuities Defined
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 45

46. Annuities and Perpetuities – Basic Formulas1
Annuities and Perpetuities – Basic Formulas
Perpetuity: PV = C / r
Annuities: 46