1. Discounted Cash Flow Valuation
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Discounted Cash Flow Valuation
Chapter 5
Finance Is Fun!

2. Topics Be able to compute the future value of multiple cash flows
Be able to compute the present value of multiple cash flows
Understand how interest rates are quoted
Be able to compute loan payments
Be able to find the interest rate on a loan
Understand how loans are amortized or paid off

3. Page 113
Students who learn this material well will find that life is much easier down the road
Getting it straight now will save you a lot of headaches later

4. Annuities Annuity Definition: Timing for annuities:
A level steam of cash flows for a fixed period of time
Each payment is for the same amount
The time between payments is always the same
Timing for annuities:
Ordinary Annuity (Mortgage contracts)
Payments are made at the end of each period
The day you sign the contract, you do not make a payment
Annuity due (Lease contracts)
Payments are made at the beginning of each period

5. Annuities Types of annuities: Savings plan:
If I put $50 in the bank each month for 35 years, how much will I have when I retire? What is the future value?
Future value of future cash flows valuation:
If I want to be a millionaire, how much do I have to put in the bank each period. What is the PMTFV?
Loan (DEBT) periodic payment:
If I take out a loan, what is the periodic repayment amount? What is the PMTPV?
Present value of future cash flows valuation
If I know the asset will give me $50 at the end of each month for the next 25 years, what should I pay for this asset today? What is the present value?

6. Annuities (Math)
All the cash flow associated with an annuity represent a geometric sequences
Geometric sequences:
A geometric sequence is one in which each successive term of the sequence is the same nonzero constant multiple of the preceding term
Constant multiple = (successive term)/(preceding term)
Every two successive terms have a common ratio
The total value of an annuity represents a finite geometric series
Geometric series:
The sum of the terms in a geometric sequence

7. How To Determine The Present Value Of Investments With Multiple Future Cash Flows

8. Annuity – Sweepstakes Example
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Annuity – Sweepstakes Example
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?
PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29
Calculator:
30 N; 5 I/Y; 333, PMT; CPT PV = 5,124,150.29
Finance Is Fun!

9. Buying a House
You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?

10. Buying a House - Continued
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Buying a House - Continued
Bank loan
Monthly income = 36,000 / 12 = 3,000
Maximum payment = .28(3,000) = 840
PV = 840[1 – 1/ ] / .005 = 140,105
Total Price
Closing costs = .04(140,105) = 5,604
Down payment = 20,000 – 5604 = 14,396
Total Price = 140, ,396 = 154,501
You might point out that you would probably not offer 154,501. The more likely scenario would be 154,500.
Calculator:
30*12 = 360 N
.5 I/Y
840 PMT
CPT PV = 140,105
Finance Is Fun!

11. Students Are Smart!
Quick Quiz: Part 2
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 5000 per month in retirement. If you can earn .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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12. Students Are Smart!
Finding the Rate
Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $ per month for 60 months. What is the monthly interest rate?
Sign convention matters!!!
60 N
10,000 PV
PMT
In EXCEL use the RATE function
The next slide talks about how to do this without a financial calculator.
Finance Is Fun!

13. How To Determine The Future Value Of Investments With Multiple Future Cash Flows

14. Future Values for Annuities
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Future Values for Annuities
Suppose you begin saving for your retirement by depositing $2000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years?
FV = 2000( – 1)/.075 = 454,513.04
FV = 2000( – 1)/.075 = 454,513.04
Remember the sign convention!!!
40 N
7.5 I/Y
-2000 PMT
CPT FV = 454,513.04
Finance Is Fun!

15. Students Are Smart!
Annuity Due
You are saving for a new house and you put $10,000 per year in an account paying 8%, compounded yearly. The first payment is made today. How much will you have at the end of 3 years?
FV = 10,000[(1.083 – 1) / .08](1.08) = 35,061.12
Annuity Due trick: Annuity due value = Ordinary annuity value*(1+i/n)
PV Annuity Due trick: Subtract one period, then add one payment to PV
Note that the procedure for changing the calculator to an annuity due is similar on other calculators.
Calculator
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)
What if it were an ordinary annuity? FV = 32,464 (so receive an additional by starting to save today.)
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16. Perpetuity (Consol)
An annuity in which the cash flow continues forever
Equal cash flow goes on forever (like most preferred stock pays dividend)
Preferred Stock is a Perpetuity
Capitalization of Income
As x gets large,
(1+i/n)
Approaches
zero

17. Perpetuity
If you buy preferred stock that pays out a contractual yearly dividend of $5.50 and the appropriate discount rate is 12%, what is the stock worth? (What is the present value of this perpetuity?)
$5.5/.12 = $45.83
If RAD Corp. wants to sell preferred stock for $125 per share with a contractual quarterly dividend, and a similar company that pays a quarterly dividend of $2 and has a stock price of $150, what should the RAD Corp.’s dividend be if it wants to sell its stock?
PMT/(i/n) = PV  2/(i/n) = $150 2/150 = .1333
Thus RAD Corp.’s quarterly dividend must be 1.33%, or *$125 = $1.67

18. How Interest Rates Are Quoted (And Misquoted)
Periodic Rate
Annual Percentage Rate
Effective Annual Rate (EAR)

19. Periodic Rate
Interest rate per period

20. Annual Percentage Rate
Stated Interest Rates
The interest rate expressed in terms of the interest payment made each period. Also: Quoted Interest Rate.
Usually listed as:
APR (Annual Percentage Rate)
10%, compounded quarterly
Annual Percentage Rate
The interest rate charged per period multiplied by the number of periods per year
Truth-in-Lending Act Requires

21. 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%
Can you divide the above APR by 2 to get the semiannual rate? NO!!! You need an APR based on semiannual compounding to find the semiannual rate.

22. 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

23. Computing EARs - Example
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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 if 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%
Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.
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24. Effective Annual Rate (EAR)
The interest rate expressed as if it were compounded once
You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate

25. Effective Annual Rate (EAR)
One compounding period per year:
APR = EAR
When number of compounding periods per year goes up, EAR goes up, but up to a limit
365 periods per year is near the limit
Limit of EAR = ei
EAR = (1+.12/365) =
e.12 =
e 

26. Effective Annual Rate (EAR)
If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.

27. Decisions, Decisions II
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Decisions, Decisions II
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?
Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates.
Calculator:
2nd I conv 5.25 NOM up arrow 365 C/Y up arrow CPT EFF = 5.39%
5.3 NOM up arrow 2 C/Y up arrow CPT EFF = 5.37%
Finance Is Fun!

28. Decisions, Decisions II Continued
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Decisions, Decisions II 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 = / 2 = .0265
FV = 100(1.0265)2 =
You have more money in the first account.
It is important to point out that the daily rate is NOT .014, it is
First Account:
365 N; 5.25 / 365 = I/Y; 100 PV; CPT FV =
Second Account:
2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV =
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29. Loans Interest Only Loans Amortized Loans Pure Discount Loans
Principal = Amount lent by lender = Amount received by borrower
Interest Only Loans  Principal stays the same until the end of the loan, then principal is paid back
Amortized Loans  A small amount of the principal is paid off each period and principal amount gets smaller as payments are made
Periodic Interest = Principal*Periodic Rate

30. Loans
Each type of loan has different combinations of payments of cash flows:
Amounts
Timing
Interest payments
Principal payments
Sign

31. Loans
How Loan Payments Are Calculated And How To Find The Interest Rate On A Loan
How Loans Are Amortized Or Paid Off

32. Interest Only Loans (Coupon)
Pay fixed interest amount each period
Principal* Periodic Rate
Pay the principal back (all at once) at the end of the loan period (plus the last fixed interest amount)
Example: Bonds

33. Interest Only Loans (Coupon)

34. Amortized Loans - Repay Part Interest And Part Principal Each Period
Medium-term business loans
Period payments:
Interest amount paid changes each period
Principal amount paid is fixed
Consumer/mortgage loans
Period payments:
Interest amount paid changes each period
Principal amount paid changes each period
Ordinary annuity

35. Amortized Loans - Medium-term Business Loans
Periodic Interest Amount:
Principal* Periodic Rate
Pay changing interest amount each period (amount gets smaller each period)
Principal amount paid:
Fixed Amount
Total Periodic payment gets smaller each period

36. Amortized Loans: Medium-term Business Loans

37. Amortized Loans - Consumer/mortgage loans & Effective Interest Rate Method for Bonds
Periodic Interest Amount:
Principal* Periodic Rate
Pay changing interest amount each period (amount gets smaller each period)
Principal amount paid:
Periodic Payment - Periodic Interest Amount
Total Periodic payment stays the same each period
Ordinary Annuity: Solve for PMT

38. Amortized Loans: Consumer/mortgage loans

39. Pay Off Loan Early (Balloon Payment)
The present value of all remaining future cash flows will give you the amount to pay off

40. Pure Discount Loans (Zero Coupon)
Borrow an amount today, then pay back principal and all interest at the end of the loan period
Example: US Government Treasury Bills, or T-bills (government loans < 1year)
Payback
Amount
Loan amount
received today

41. Pure Discount Loans (Zero Coupon)

42. Multiple Cash Flows – FV Example 1
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Multiple Cash Flows – FV Example 1
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?
FV = 500(1.09) (1.09) =
Calculator:
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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43. Students Are Smart!
Example 1 Continued
How much will you have in 5 years if you make no further deposits?
First way:
FV = 500(1.09) (1.09)4 =
Second way – use value at year 2:
FV = (1.09)3 =
Calculator:
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 = =
Second way – use value at year 2:
3 N; PV; 9 I/Y; CPT FV =
Finance Is Fun!

44. Multiple Cash Flows – Present Value Example 2
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Multiple Cash Flows – Present Value Example 2
Find the PV of each cash flow and add them
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 = =
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?”
Calculator:
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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45. Example 2 Timeline 1 2 3 4
200
400
600
800
178.57
318.88
427.07
508.41

46. Multiple Cash Flows – PV Another Example
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Multiple Cash Flows – PV Another Example
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?
PV = 1000 / (1.1)1 =
PV = 2000 / (1.1)2 =
PV = 3000 / (1.1)3 =
PV = =
Calculator:
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 =
Finance Is Fun!

47. Students Are Smart!
Decisions, Decisions
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?
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.
Finance Is Fun!

48. 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%?

49. Summary Slide Annuities
How To Determine The Present Value Of Investments With Multiple Future Cash Flows
Finding the Rate
How To Determine The Future Value Of Investments With Multiple Future Cash Flows
Annuity Due (BEGIN mode)
Perpetuity (Consol)
How Interest Rates Are Quoted (And Misquoted)
Loans
Multiple Cash Flows