1. Discounted Cash Flow Valuation: Part I

2. Chapter Outline Part I: Part II:
Future and Present Values of Multiple Cash Flows
Part II:
Valuing Level Cash Flows: Annuities and Perpetuities
Comparing Rates: The Effect of Compounding
Loan Types and Loan Amortization

3. Multiple Cash Flows –Future Value Example 6.1
Find the value at year 3 of each cash flow and add them together.
Calculate FV of each CF
Today’s (year 0) CF: 3 N; 8 I/YR; -7,000 PV; FV =
Year 1 CF: 2 N; 8 I/YR; -4,000 PV; FV = 4,665.60
Year 2 CF: 1 N; 8 I/YR; -4,000 PV; FV = 4,320
Year 3 CF: value = 4,000
Then, total future value in 3 years = 8, , , ,000 = 21,803.58
The book discusses that there are two ways to work this problem. The first method, computing the FV one year at a time and adding the cash flows as you go along, is illustrated in Example 6.1 in the book. The slides illustrate the other method, finding the future value at the end for each cash flow and then adding.
Point out that you can find the value of a set of cash flows at any point in time, all you have to do is get the value of each cash flow at that point in time and then add them together.
The students can read the example in the book. It is also provided here.
You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in three years? In four years?
Point out that there are several ways that this can be worked. The book works this example by rolling the value forward each year. The presentation will show the second way to work the problem.
I entered the PV as negative for two reasons. (1) It is a cash outflow since it is an investment. (2) The FV is computed as positive and the students can then just store each calculation and then add from the memory registers, instead of writing down all of the numbers and taking the risk of keying something back into the calculator incorrectly.
Formula:
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 = = 21,803.58
Value at year 4 = 21,803.58(1.08) = 23,547.87

4. Multiple Cash Flows – FV Example 2
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 two years?
Year 0 CF: 2 N; -500 PV; 9 I/YR; FV =
Year 1 CF: 1 N; -600 PV; 9 I/YR; FV =
Total FV = = 1,248.05
Formula: FV = 500(1.09) (1.09) = 1,248.05

5. Multiple Cash Flows – Example 2 Continued
How much will you have in 5 years if you make no further deposits?
First way:
Year 0 CF: 5 N; -500 PV; 9 I/Y; CPT FV =
Year 1 CF: 4 N; -600 PV; 9 I/Y; CPT FV =
Total FV = = 1,616.26
Second way – use value at year 2:
3 N; -1, PV; 9 I/Y; CPT FV = 1,616.26
Formula:
First way: FV = 500(1.09) (1.09)4 = 1,616.26
Second way: FV = (1.09)3 = 1,616.26

6. Multiple Cash Flows – FV Example 3
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%?
Year 1 CF: 4 N; -100 PV; 8 I/YR; FV =
Year 3 CF: 2 N; -300 PV; 8 I/YR; FV =
Total FV = =
FV = 100(1.08) (1.08)2 = =

7. Multiple Cash Flows – Present Value Example 6.3
Find the PV of each cash flows and add them
Year 1 CF: N = 1; I/YR = 12; FV = 200; PV =
Year 2 CF: N = 2; I/YR = 12; FV = 400; PV =
Year 3 CF: N = 3; I/YR = 12; FV = 600; PV =
Year 4 CF: N = 4; I/YR = 12; FV = 800; PV =
Total PV = = 1,432.93
The students can read the example in the book.
You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one?
Point out that the question could also be phrased as “How much is this investment worth?”
Remember the sign convention. The negative numbers imply that we would have to pay 1, today to receive the cash flows in the future.
Formula:
Year 1 CF: 200 / (1.12)1 =
Year 2 CF: 400 / (1.12)2 =
Year 3 CF: 600 / (1.12)3 =
Year 4 CF: 800 / (1.12)4 =

8. Multiple Cash Flows – Present Value Example 6.3
Alternatively
4 N; 12 I/YR;
0 CFj; 200 CFj; 400 CFj; 600 CFj; 800 CFj
SHIFT NPV = 1,432.93
NPV = net present value = total present value
The students can read the example in the book.
You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one?
Point out that the question could also be phrased as “How much is this investment worth?”
Remember the sign convention. The negative numbers imply that we would have to pay 1, today to receive the cash flows in the future.
Formula:
Year 1 CF: 200 / (1.12)1 =
Year 2 CF: 400 / (1.12)2 =
Year 3 CF: 600 / (1.12)3 =
Year 4 CF: 800 / (1.12)4 =

9. Example 6.3 Timeline 1 2 3 4
200
400
600
800
178.57
318.88
427.07
508.41
1,432.93

10. Multiple Cash Flows Using a Spreadsheet
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 – if it is set up properly, then you can just copy the formulas
Click on the Excel icon for an example
Click on the tabs at the bottom of the worksheet to move from a future value example to a present value example.

11. Multiple Cash Flows – PV Another Example
You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?
N = 1; I/YR = 10; FV = 1,000; PV =
N = 2; I/YR = 10; FV = 2,000; PV = -1,652.89
N = 3; I/YR = 10; FV = 3,000; PV = -2,253.94
PV = , , = 4,815.93
Alternatively;
3 N; 10 I/YR; 0 CFj; 1,000 CFj; 2,000 CFj; 3,000 CFj
SHIFT NPV = 4,815.93
Formula:
PV = 1000 / (1.1)1 =
PV = 2000 / (1.1)2 = 1,652.89
PV = 3000 / (1.1)3 = 2,253.94

12. Multiple Uneven Cash Flows – Using CF Key
Your broker calls you and tells you that he has this great investment opportunity. 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 investment?
Use the CF keys to compute the value of the investment
3 N; 15 I/YR; 0 CF0; 40 CF1; 75 CF2;
NPV = 91.49
No – the broker is charging more (100) than the worth of that investment (91.49).
The next example will be worked using the cash flow keys.
Note that with the BA-II Plus, the students can double check the numbers they have entered by pressing the up and down arrows. It is similar to entering the cash flows into spreadsheet cells.
Other calculators also have cash flow keys. You enter the information by putting in the cash flow and then pressing CF. You have to always start with the year 0 cash flow, even if it is zero.
Remind the students that the cash flows have to occur at even intervals, so if you skip a year, you still have to enter a 0 cash flow for that year.

13. Saving For Retirement
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%?
Since the cash outflows are equal each period, you can either use CF or PMT functions.

14. Saving For Retirement Timeline…
… K 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. Saving For Retirement: Calculator
Two steps: First, calculate PV of total 5 withdrawals from Year 40 to Year 44
Using CF Function
6 N, 12 I/YR; 0 CFj; 25,000 CFj; 25,000 CFj; 25,000 CFj ; 25,000 CFj; 25,000 CFj ;
SHIFT NPV = 90,119.41
Or Using PMT Function (only if CF is equal each year)
5 N; 12 I/YR; 25,000 PMT; PV = -90,119.41
This value is actually PV at Year 39 => FV
Then, calculate PV of this value (FV)
39 N, 12 I/YR; 90, FV; PV = -1,084.71

16. Quick Quiz – Part I
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 cash flows at year 5?
What is the value of the cash flows today?
What is the value of the cash flows at year 3?
The easiest way to work this problem is to use the uneven cash flow keys and find the present value first and then compute the others based on that.
CF0 = 0; C01 = 100; F01 = 1; C02 = 200; F02 = 2; C03 = 300; F03 = 2; I = 7; CPT NPV =
Value in year 5: PV = ; N = 5; I/Y = 7; CPT FV = 1,226.07
Value in year 3: PV = ; N = 3; I/Y = 7; CPT FV = 1,070.90
Using formulas and one CF at a time:
Year 1 CF: FV5 = 100(1.07)4 = ; PV0 = 100 / 1.07 = 93.46; FV3 = 100(1.07)2 =
Year 2 CF: FV5 = 200(1.07)3 = ; PV0 = 200 / (1.07)2 = ; FV3 = 200(1.07) = 214
Year 3 CF: FV5 = 200(1.07)2 = ; PV0 = 200 / (1.07)3 = ; FV3 = 200
Year 4 CF: FV5 = 300(1.07) = 321; PV0 = 300 / (1.07)4 = ; PV3 = 300 / 1.07 =
Year 5 CF: FV5 = 300; PV0 = 300 / (1.07)5 = ; PV3 = 300 / (1.07)2 =
Value at year 5 = = 1,226.07
Present value today = = (difference due to rounding)
Value at year 3 = = 1,070.89

17. Quick Quiz – Part VI
What is a pure discount loan? What is a good example of a pure discount loan?
What is an interest-only loan? What is a good example of an interest-only loan?
What is an amortized loan? What is a good example of an amortized loan?

18. End of Chapter