0. Multiple Cash Flows 6.1 – FV Example 1
You currently have $7,000 in a bank account earning 8% interest. You think you will be able to deposit an additional $4,000 at the end of each of the next three years. How much will you have in three years?
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1. Multiple Cash Flows FV Example 1 continued
Find the value at year 3 of each cash flow and add them together.
Formula Approach
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
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2. Multiple Cash Flows FV Example 1 continued
Calculator Approach
Today (year 0 CF): 3 N; 8 I/Y; PV; CPT FV =
Year 1 CF: 2 N; 8 I/Y; PV; CPT FV =
Year 2 CF: 1 N; 8 I/Y; PV; CPT FV = 4320
Year 3 CF: value = 4,000
Total value in 3 years = = 21,803.58
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3. 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?
Formula Approach
FV = 500(1.09) (1.09) =
Calculator Approach
Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV =
Year 1 CF: 1 N; -600 PV; 9 I/Y; CPT FV =
Total FV = =
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4. Multiple Cash Flows – FV Example 2 Continued
How much will you have in 5 years if you make no further deposits?
Formula Approach
First way:
FV = 500(1.09) (1.09)4 =
Second way – use value at year 2:
FV = (1.09)3 =
Calculator Approach
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 = =
3 N; PV; 9 I/Y; CPT FV =
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5. 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%?
Formula Approach
FV = 100(1.08) (1.08)2 = =
Calculator Approach
Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV =
Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV =
Total FV = =
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6. Multiple Cash Flows – PV Example 1
You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the year after, and $800 at the end of the following year. You can earn 12% on similar investments. How much is this investment worth today?
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7. Multiple Cash Flows - PV Example 1 - Timeline1 2 3 4
200
400
600
800
178.57
318.88
427.07
508.41
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8. Multiple Cash Flows - PV Example 1 continued
Find the PV of each cash flow and add them
Formula Approach
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 =
Total PV = =
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9. Multiple Cash Flows - PV Example 1 continued
Calculator Approach
Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV =
Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV =
Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV =
Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV =
Total PV = =
Remember the sign convention. The negative numbers imply that we would have to pay today to receive the cash flows in the future.
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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.
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11. Multiple Cash Flows – PV Example 2
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 to pay?
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12. Multiple Cash Flows – PV Example 2 continued
Formula Approach
PV = 1000 / (1.1)1 =
PV = 2000 / (1.1)2 =
PV = 3000 / (1.1)3 =
PV = =
Calculator Approach
N = 1; I/Y = 10; FV = 1000; CPT PV =
N = 2; I/Y = 10; FV = 2000; CPT PV =
N = 3; I/Y = 10; FV = 3000; CPT PV =
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13. Multiple Uneven Cash Flows – Using the Calculator
Another way to use the financial calculator for uneven cash flows is to use the cash flow keys
Texas Instruments BA-II Plus
Press CF and enter the cash flows beginning with year 0
You have to press the “Enter” key for each cash flow
Use the down arrow key to move to the next cash flow
The “F” is the number of times a given cash flow occurs in consecutive years
Use the NPV key to compute the present value by entering the interest rate for I, pressing the down arrow and then compute
Clear the cash flow keys by pressing CF and then CLR Work
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.
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Decisions, Decisions
LO1
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
CF; CF0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1
NPV; I = 15; CPT NPV = 91.49
No – the broker is charging more than you would be willing to pay.
You can also use this as an introduction to NPV by having the students put –100 in for CF0. When they compute the NPV, they will get – You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV.
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Saving For Retirement
LO1
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%?
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16. Saving For Retirement Timeline
LO1 …
… 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)
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17. Saving For Retirement continued
LO1
Calculator Approach
Use cash flow keys:
CF; CF0 = 0; C01 = 0; F01 = 39; C02 = 25000; F02 = 5; NPV; I = 12; CPT NPV =
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Quick Quiz – Part I
LO1
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 =
Value in year 3: PV = ; N = 3; I/Y = 7; CPT FV =
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 = =
Present value today = = (difference due to rounding)
Value at year 3 = =
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19. Annuities and Perpetuities 6.2
LO1
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
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20. Annuities and Perpetuities – Basic Formulas
LO1
Perpetuity: PV = C / r
Annuities:
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21. Annuities and the Calculator
LO1
You can use the PMT key on the calculator 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 are ordinary annuities
Other calculators also have a key that allows you to switch between Beg/End.
Warn students to be careful with this function. Every term, it seems that there is always one student who, during the midterm exam, switches their calculator to beginning of period payments and forgets to switch it back for the next question. Another way of handling the annuity due calculation is to calculate it as if it were a regular annuity, and then simply multiply the answer given by (1+R).
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Annuity – Example 1
LO1
After carefully going over your budget, you have determined that you can afford to pay $632 per month towards a new sports car. Your bank will lend to you at 1% per month for 48 months. How much can you borrow?
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23. Annuity – Example 1 continued
LO1
You borrow money TODAY so you need to compute the present value.
Formula Approach
Calculator Approach
48 N; 1 I/Y; -632 PMT; CPT PV = 23, ($24,000)
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24. Annuity – Sweepstakes Example
LO1
Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333, over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?
Formula Approach
PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29
Calculator Approach
30 N; 5 I/Y; 333, PMT; CPT PV = -5,124,150.29
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25. Annuities on the Spreadsheet - Example
LO1
The present value and future value formulas in a spreadsheet include a place for annuity payments
Click on the Excel icon to see an example
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Quick Quiz – Part II
LO1
You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value?
You want to receive $5,000 per month in retirement. If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement?
Calculator
PMT = 5000; N = 25*12 = 300; I/Y = .75; CPT PV = 595,808
Formula
PV = 5000[1 – 1 / ] / = 595,808
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27. Future Values for Annuities – Example 1
LO1
Suppose you begin saving for your retirement by depositing $2000 per year in an RRSP. If the interest rate is 7.5%, how much will you have in 40 years?
Formula Approach
FV = 2000( – 1)/.075 = 454,513.04
Calculator Approach
Remember the sign convention!!!
40 N
7.5 I/Y
-2000 PMT
CPT FV = 454,513.04
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Annuity Due – Example 1
LO1
You are saving for a new house and you put $10,000 per year in an account paying 8% compounded annually. The first payment is made today. How much will you have at the end of 3 years?
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29. Annuity Due – Example 1 Timeline
LO1
32,464
If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period.
35,061.12
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30. Annuity Due – Example 1 continued
LO1
Formula Approach
FV = 10,000[(1.083 – 1) / .08](1.08) = 35,061.12
Calculator Approach
2nd BGN 2nd Set (you should see BGN in the display)
3 N
-10,000 PMT
8 I/Y
CPT FV = 35,061.12
2nd BGN 2nd Set (be sure to change it back to an ordinary annuity)
Note that the procedure for changing the calculator to an annuity due is similar on other calculators.
What if it were an ordinary annuity? FV = 32,464 (so receive an additional by starting to save today.)
Remind students to change their calculator back to the regular annuity setting before proceeding with other questions!
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Perpetuity – Example 1
LO1
The Home Bank of Canada want to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend would the Home Bank have to offer if its preferred stock is going to sell?
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32. Perpetuity – Example 1 continued
LO1
Perpetuity formula: PV = C / r
First, find the required return for the comparable issue:
40 = 1 / r
r = .025 or 2.5% per quarter
Then, using the required return found above, find the dividend for new preferred issue:
100 = C / .025
C = 2.50 per quarter
This is a good preview to the valuation issues discussed in future chapters. The price of an investment such as a stock is just the present value of expected future cash flows.
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Growing Perpetuity
LO1
The perpetuities discussed so far have constant payments
Growing perpetuities have cash flows that grow at a constant rate and continue forever
Growing perpetuity formula:
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34. Growing Perpetuity – Example 1
LO1
Hoffstein Corporation is expected to pay a dividend of $3 per share next year. Investors anticipate that the annual dividend will rise by 6% per year forever. The required rate of return is 11%. What is the price of the stock today?
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Growing Annuity
LO1
Growing annuities have a finite number of growing cash flows
Growing annuity formula:
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36. Growing Annuity – Example 1
LO1
Gilles Lebouder has just been offered a job at $50,000 a year. He anticipates his salary will increase by 5% a year until his retirement in 40 years. Given an interest rate of 8%, what is the present value of his lifetime salary?
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Quick Quiz – Part IV
LO1
You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month?
What if the first payment is made today?
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?
Q1: 35(12) = 420 N; 1,000,000 FV; 1 I/Y; CPT PMT =
1,000,000 = C ( – 1) / .01
C =
Q2: Set calculator to annuity due and use the same inputs as above. CPT PMT =
1,000,000 = C[( – 1) / .01] ( 1.01)
C =
Q3: PV = 1.50 / .03 = $50
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Work the Web Example
LO1
Another online financial calculator can be found at MoneyChimp
Click on the web surfer and work the following example
Choose calculator and then annuity
You just inherited $5 million. If you can earn 6% on your money, how much can you withdraw each year for the next 40 years?
Payment = $332,307.68
Using MoneyChimp, select “Calculator” at the top of the screen, then select “Annuity” from the menu on the top, right-hand side of the screen.
You can choose between a regular annuity or an annuity due by clicking on the appropriate button in the box. If you chose “annuity due”, then the amount will change to $313,
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39. Table 6.2 – Summary of Annuity and Perpetuity Calculations
LO1
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