1. Discounted Cash Flow Valuation
Chapter 5
Summer 2008

2. Objectives Be able to compute the future value of multiple cash flows
Be able to compute the present value of multiple cash flows
Be able to compute loan payments
Be able to find the interest rate on a loan
Understand how loans are amortized or paid off
Understand how interest rates are quoted
Summer 2008
Yunling Chen

3. Discounted Cash Flow Valuation
Summer 2008
Yunling Chen

4. Multiple Cash Flows
An asset or a project typically embeds multiple cash flows. Since money has time value, the value of such an asset or project is not simply the sum of individual cash flows.
We need to “transform” all cash flows to one point in time before we can meaningfully add them together.
Summer 2008
Yunling Chen

5. Example 1 – Future Value
You plan to deposit $4000 at the end of each next three years in a bank account paying 8 percent interest. You currently have $7000 in the account.
How much will you have in three years?
In four years?
Summer 2008
Yunling Chen

6. Example 1 - Key 1 2 3 4
$7000
$4000
$4000
$4000
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 = = 21,803.58
Value at year 4 = 21,803.58(1.08) = 23,547.87
Summer 2008
Yunling Chen

7. Example 2 – Future Value
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) =
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 =
Summer 2008
Yunling Chen

8. Quick Check
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 = 100(1.08) (1.08)2 = =
Summer 2008
Yunling Chen

9. Example 3 – Present Value
You are offered an investment that will pay you $200 in (the end) 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?
Summer 2008
Yunling Chen

10. Example 3 - key 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 = =
Summer 2008
Yunling Chen

11. Example 4 – Present Value
You are considering an investment that will pay you $1,000 in year 1, $2,000 in year 2 and $3,000 in year 3. If you want to earn 10% on your money, how much is the investment worth today?
PV = 1000 / (1.1)1 =
PV = 2000 / (1.1)2 =
PV = 3000 / (1.1)3 =
Total PV = =
Summer 2008
Yunling Chen

12. Investment decisions Net present value (NPV)
NPV = PV(cash inflow) – PV(cash outflow)
Cash inflows are earnings from investment, sale of assets, etc, from the investment.
Cash outflows are initial investment, future maintenance cost, etc, due to the investment.
What does the sign of NPV mean?
NPV measures the value created (or destroyed) by undertaking an investment.
Positive NPV suggests value is created and negative NPV suggests the opposite
Summer 2008
Yunling Chen

13. Investment Decisions – Example
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 the second year. You require a 15% return on investments of this risk (discount rate).
Should you take the investment?
Step 1: Figure out how much should you invest
if you can receive $40 in year 1 and $75
in year 2 and your annual return is 15%.
=> PV = 40/(1+15%) + 75/(1+15%)2 = $91.49
Step 2: Compare with the offer ($100)
Use the formula of NPV
NPV = /(1+15%) + 75/(1+15%)2
= = < 0
No – the broker is charging more than you would be willing to pay.
If you don’t invest in this project, you can earn 15% of return in other projects (opportunity cost).
Summer 2008
Yunling Chen

14. Multiple Cash Flows - A Note
The cash flow timing is critically important
Without explicitly told otherwise, it is assumed that the cash flow occur at the end of each period
A example--Suppose you are told a project has a first-year cash flow of $100, a second cash-flow of $ 200, a third-year cash flow of $300.
Then, the time line should be:
$100
$200
$300
1. Find out when there’s cash inflow/outflow
Year = 0;
Today 1 2 3 4
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Yunling Chen

15. Quick Quiz 1
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%
a. What is the value of the cash flows at year 5?
b. What is the value of the cash flows today?
c. What is the value of the cash flows at year 3?
(there are three ways of solving part c)
Summer 2008
Yunling Chen

16. Quick Quiz 1 (cont’d)
a) What is the value of the cash flows at year 5?
r = 7%
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Yunling Chen

17. Quick Quiz 1 (cont’d) b) What is the value of the cash flows today?
Method 1:
Method 2:
PV0 = $1, /(1.07)5 = $874.17
PV0 = ? FV5 = 1,
PV0 = ?
Summer 2008
Yunling Chen

18. Quick Quiz 1 (cont’d)
c) What is the value of the cash flows at year 3?
Method 1: You can use FV5 from the first part and compute PV3
FV3 = $1, / (1.07)2 = $1,070.89
FV3 = ? FV5 = 1,
Summer 2008
Yunling Chen

19. Quick Quiz 1 (cont’d) c)
Method 2: You can use PV0 from the second part and compute FV3
FV3 = $ (1.07)3 = $1,070.89
PV0 =
FV3 = ?
Summer 2008
Yunling Chen

20. Quick Quiz 1 (cont’d) c) Method 3: 1 2 3 4 5 100 200 200 300 300
FV3 = ?
FV3 = $100(1.07)2 + $200(1.07) + $ $300/(1.07) + $300/(1.07)2
= $1,070.89
Summer 2008
Yunling Chen

21. Go Back to Quick Survey in Ch.4
Today, you deposit $10,000 in a bank, which of the following option is better? (Suppose your required return is 4%.)
You get back $11,000 two years later.
You get $500 one year later and $10,500 two year later.
NPV(A) = $11,000/(1+4%)2 – 10,000 = $170.1
NPV(B) = $500/(1+4%) + 10,500 /(1+4%)2 – 10,000
= $188.6
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Yunling Chen

22. Discounted Cash Flow Valuation
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23. Special Cases of Multiple Cash Flows: Annuities and Perpetuities
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
Unless stated otherwise, all annuities are assumed to be ordinary annuities
Perpetuity – infinite series of equal payments
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Yunling Chen

24. Annuities and Perpetuities
Perpetuity C
t=0 1 2 3 4
…….. T
T+1
T+2 
Annuity
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Yunling Chen

25. Annuities and Perpetuities
Applicable when:
the values of cash flow are the same in all periods;
the discount rates are the same in all periods, i.e.
Recall how to add a geometric progression (GP) series
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Yunling Chen

26. Annuities and Perpetuities
Annuities (t = T):
FV of annuity = PV of annuity * (1+r)^T
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Yunling Chen

27. Annuity – Example 1 – buying a car
You can afford to pay $ 632 per month towards a car. The bank can loan you with 1% per month for 48 months. How much can you borrow?
Borrow money TODAY, so you need to compute the present value.
Using formula:
Summer 2008
Yunling Chen

28. Annuity – Example 2 – winning a lottery
Suppose you win the Mark Six $10 million. 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?
Future money worth TODAY, so you need to compute the present value.
PV = 333,333.33[(1 – 1/1.0530) / .05] = 5,124,150.29
Excel
PV = PV(Rate, Nper,pmt,FV) =PV(0.05,30, ,0)
Calculator:
30 N; 5 I/Y; 333, PMT; CPT PV = 5,124,150.29
Summer 2008
Yunling Chen

29. Annuity – Example 3 –Buying a House
You are ready to buy a house and you have HK$200,000 for a down payment (定金) and closing costs (借款手續费). Closing costs are estimated to be 4% of the loan value. You have an annual salary of HK$360,000 and the bank is willing to allow your monthly mortgage (抵押) payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (0.5% 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?
Summer 2008
Yunling Chen

30. You need to pay interest monthly.
Buying a House
How much loan you can get from the Bank?
Monthly income = 360,000 / 12 = 30,000
Maximum payment = .28(30,000) = 8400
PV(total loan) = 8400[1 – 1/ ] / .005 = 1,401,050
How much can you afford to buy the house?
Closing costs = .04(1,401,050) = 56,040
Closing costs are payments made to the bank, and therefore not to be included in the value of the house
Down payment = 200,000 – 56,040 = 143,960
Total price = PV(total loan) + down payment
Total Price = 1,401, ,960 = $1,545,010
You need to pay interest monthly.
1 period = 1 month
Summer 2008
Yunling Chen

31. Annuities- Example- Finding the Payment
Suppose you want to borrow $20,000 for a new car now. You can borrow at 8% per year, compounded monthly (8/12 = % per month). If you take a 4 year loan, what is your monthly payment?
20,000 = C[1 – 1 / ] /
C =
Excel
C = PMT(rate,nper,pv,fv) =PMT(0.08/12,48,20000,0) =
Note if you do not round the monthly rate and actually use 8/12, then the payment will be
Summer 2008
Yunling Chen

32. Annuities – Example - Finding the Number of Payments
Suppose you borrow $2000 at 5% and you are going to make annual payments of $ How long before you pay off the loan?
2000 = (1 – 1/1.05t) / .05
= 1 – 1/1.05t
= 1.05t
t = ln( ) / ln(1.05) = 3 years
Excel
t = NPER(rate,pmt,pv,fv) =NPER(0.05, ,2000,0) = 3
Summer 2008
Yunling Chen

33. Annuities – Example - Finding the Rate
Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $500 per month for 30 months. What is the monthly interest rate?
Excel
r = RATE(nper,pmt,pv,fv) = RATE(30,-500,10000,0) = 2.84%
 Verify r = 2.84%
Summer 2008
Yunling Chen

34. Annuities– Finding the Rate Without a Financial Calculator
Trial and Error Process
Choose an interest rate and compute the PV of the payments based on this rate
Compare the computed PV with the actual loan amount
If the computed PV > loan amount, then the interest rate is too low
If the computed PV < loan amount, then the interest rate is too high
Adjust the rate and repeat the process until the computed PV and the loan amount are equal
Summer 2008
Yunling Chen

35. Solving the interest rate by trial and error
r=3%: RHS =
r=2%: RHS =
By interpolation:
r = 2% + {( )/( )} x 1%
 r = 2% % = 2.857%
To get more precision: r=2.5% RHS =
 r = 2.5% + {( )/( )} x 0.5% = %
…..
Summer 2008
Yunling Chen

36. Annuities – Example - Finding the FV
Suppose you begin saving for your retirement by depositing $2000 per year in an retirement account. If the interest rate is 7.5%, how much will you have in 40 years?
FV = 2000( – 1)/.075 = 454,513.04
Summer 2008
Yunling Chen

37. Annuities Due
You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years?
Timeline:
Annuity due value
= Ordinary annuity value * (1+r)
FV = 10,000[(1.083 – 1) / .08](1.08)
= 35,061.12
10k k k
32,464
35,061.12
What if it were an ordinary annuity? FV = 32,464 (so receive an additional by starting to save today.)
Summer 2008
Yunling Chen

38. Perpetuities Recall the annuity formula:
Let t  infinity with r > 0
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39. Perpetuity: Example
You are looking into an investment which will pay you $4 per year for the foreseeable future. If you require a 12% return, what is the most that you would pay for this investment?
PV = $4 / 0.12
= $33.33
Summer 2008
Yunling Chen

40. Table 5.2
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Yunling Chen

41. Answer to Exercises
1. Company A has identified an investment project with the following cash flows. If the discount rate is 10% what is the present value of these cash flows? What is the present value at 18%?
Year
Cash Flow 1
$ 800 2
500 3
1,300 4
1,480
Summer 2008
Yunling Chen

42. Answer to Exercises (cont’d)
PV = ∑[FVt / (1 + r)t ]
@10%:
PV = $800 / $500 / $1,300 / $1,480 / = $3,128.07
@18%:
PV = $800 / $500 / $1,300 / $1,480 / = $2,591.65
Summer 2008
Yunling Chen

43. Answer to Exercises (cont’d)
2. An investment offers $7,000 per year for 15 years, with the first payment occurring 1 year from now. If the required return is 9%, what is the value of the investment? What would the value be if the payments occurred for 40 years? For 75 years? Forever?
Summer 2008
Yunling Chen

44. Answer to Exercises (cont’d)
PV = C({1 – [1/(1 + r)]t } / r )
For 15 yrs:
PV= $7,000{[1 – (1/1.09)15 ] / .09} = $56,424.82
For 40 yrs:
PV = $7,000{[1 – (1/1.09)40 ] / .09} = $75,301.52
For 75 yrs:
PV = $7,000{[1 – (1/1.09)75 ] / .09} = $77,656.48
Forever
PV = C/r = $7,000 / .09 = $77,777.78
Summer 2008
Yunling Chen

45. Answer to Exercises (cont’d)
3. If you put up $20,000 today in exchange for a 8.5%, 12-year annuity, what will the annual cash flow be?
PV = $20,000 = $C{[1 – (1/1.085)12 ] / .085}
=> C = $2,723.06
Summer 2008
Yunling Chen

46. Discounted Cash Flow Valuation
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47. Annual Percentage Rate (APR)
Sometimes it is called stated annual interest rate. This is the annual rate that is quoted in industry.
APR = period rate * the number of periods per year
It is the annual interest rate without consideration of compounding.
Summer 2008
Yunling Chen

48. Effective Annual Rate (EAR)
EAR is the actual rate paid (or received) after accounting for compounding that occurs during a year
EAR & APR
m = number of compounding per year
If you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison.
You should NEVER divide the EAR by the number of periods per year – it will NOT give you the period rate
Summer 2008
Yunling Chen

49. The Frequency of Compounding
You have a credit card that carries a rate of interest of 18% per year compounded monthly. What is the effective annual interest rate ?
18% per year compounded monthly is just code for 18%/12 = 1.5% per month
Monthly-compounded annual interest rate
= ( %)12 – 1 = 19.56%
EAR = 19.56%; APR = 18% (Diff = 1.56%)
Summer 2008
Yunling Chen

50. EAR and APR
Two equal Annual Percentage Rates with different frequency of compounding have different Effective Annual Rates. An example of an APR of 18%:
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51. Continuous Compounding
What occurs as the frequency of compounding rises to infinity?
Example:
The effective annual rate that’s equivalent to an annual percentage rate of 18%
= e 0.18 – 1 = 19.72%
(almost the same value with daily compounding)
Summer 2008
Yunling Chen

52. Computing EAR and APR: Examples
Suppose you can earn 1% per month on $1 invested today.
What is the APR?
How much are you effectively earning?
FV = $1(1.01)12 = $1.1268
Rate = $( – 1) / $1 = = 12.68%
EAR = $( )12 – 1 = 12.68%
Suppose you can earn 3% per quarter if you put the money in another account.
How much are you effectively earning per annum?
FV = $1(1.03)4 = $1.1255
Rate = $( – 1) / $1 = = 12.55%
1%(12) = 12%
3%(4) = 12%
Summer 2008
Yunling Chen

53. 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 rate to discounting monthly payments.
If you have payments other than monthly you need to adjust the interest rate into EAR of corresponding period.
Summer 2008
Yunling Chen

54. Future Values with Monthly Compounding
Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years?
Monthly rate = .09 / 12 = .0075
Number of months = 35(12) = 420
FV = 50[ – 1] / = 147,089.22
Summer 2008
Yunling Chen

55. Example
Suppose you have $10,000. You are looking at two savings accounts. One pays 5.25% per year, with daily compounding. The other pays 5.28% per year with semiannual compounding. (Suppose everything else is equal.)
Which account should you put your money in?
How much more can you earn from one account than another?
Use APR or EAR?
We should use EAR! Because it takes compound interest into consideration.
Summer 2008
Yunling Chen

56. Example (cont’d) EAR = (1 + .0525/365)365 – 1 = 5.39%
Calculate EAR
First account:
EAR = ( /365)365 – 1 = 5.39%
Second account:
EAR = ( /2)2 – 1 = 5.35%
Suppose you invest $10000 in each account.
First Account: FV = $10000(1.0539) = $10,539
Second Account: FV = $10000(1.0535) = $10,535
You have $4 more money in the first account.
Summer 2008
Yunling Chen

57. To avoid rounding error, keep as many decimal digits as possible.
Present Value with Daily Compounding
You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit?
Daily rate = .055 / 365 =
Number of days = 3(365) = 1095
PV = 15,000 / ( )1095 = 12,718.56
To avoid rounding error, keep as many decimal digits as possible.
Summer 2008
Yunling Chen

58. Computing Payments: Example
Suppose you want to buy a new stereo system and the store is willing to allow you to make monthly payments. The stereo system costs $3500. The loan period is for 2 years and the interest rate is 16.9%. What is your monthly payment?
Monthly rate = .169 / 12 =
Number of months = 2(12) = 24
$3500 = C[1 – 1 / )24] /
C = $172.88
ALWAYS make sure that the interest rate and the time period match.
Summer 2008
Yunling Chen

59. How to Use EARs?
A bank is offering 12% compounded quarterly. You put $100 in an account.
What is the EAR?
12.55%
How much will you have at the end of two years?
Method 1: $100 x(1+12%/4)8 =$126.68
Method 2: $100 x ( %)2 =$126.68
Summer 2008
Yunling Chen

60. Quick Quiz 1 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 effective semiannual rate?
NO!!! It should be (1+1%)6 – 1 = 6.15%
Summer 2008
Yunling Chen

61. Quick Quiz 2 Here is an advertisement:
“We make getting credit easy with low monthly payments. For just $94.35* a month, you can have $3,000 today”
* 48 low monthly payments only.
What is the interest rate of the loan?
What is the APR of the loan?
Summer 2008
Yunling Chen

62. Quick Quiz 2 (cont’d) Use formula :
Monthly Rate (by trial and error) = %
[RATE(48,-94.35,3000,0) (Excel)]
Hence, APR = 12 * % = %
And, Effective annual interest rate is
( %) = %
Summer 2008
Yunling Chen

63. Answer to Exercises
4. First National Bank charges 12.4% compounded monthly on its business loans. First United Bank charges 12.7% compounded semiannually. As a potential borrower, which bank would you go to for a new loan?
Summer 2008
Yunling Chen

64. Answer to Exercises (cont’d)
EAR = [1 + (APR / m)]m – 1
First National:
EAR = [1 + (.124 / 12)]12 – 1 = or 13.13%
First United:
EAR = [1 + (.127 / 2)]2 – 1 = or 13.10%
For a borrower, First United would be preferred since the EAR of the loan is lower.
Summer 2008
Yunling Chen

65. Answer to Exercises (cont’d)
5. Find the EAR in each of the following cases:
Stated Rate (APR)
Number of periods compounded per year
Effective Rate (EAR)
6 % 1 6 2 4 12 52
365
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Yunling Chen

66. Answer to Exercises (cont’d)
Stated Rate (APR)
Number of periods compounded per year
Effective Rate (EAR)
6 % 1 6% 6 2
6.090% 4
6.136% 12
6.168% 52
6.180%
365
6.183%
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67. Loans and Loan Amortization
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68. Types of Loans Pure Discount Loans Interest-Only Loans Amortized Loans
No periodic interest payment
Principal repaid at end of period
Periodic payments are interest only
Principal paid at end of period.
Periodic payments include both interest & some principal repayment
Interest is paid on the declining balance of outstanding principle
Example:
Zero Coupon Bond;
Treasury bills
Coupon Bond
Examples:
Mortgage Loan
Car Loan
Summer 2008
Yunling Chen

69. Pure Discount Loans: Example 5.11
The loan is sold at a discount. The principal amount is repaid at some future date, without any periodic interest payments.
If a 1-year loan promises to repay $10,000 in 12 months and the market interest rate is 7%, how much will this loan sell for in the market?
Price = PV = $10,000 / 1.07 = $9,345.79
Interest = $(10,000 – 9,345.79) = $654.21
Summer 2008
Yunling Chen

70. Interest-Only Loan: Example
Consider a 4-year, $10,000, interest only loan with a 7% interest rate. Interest is paid annually. The loan amount is $10,000. Principle is repaid at the end of year 4.
What would be the stream of cash flows of this loan?
Years 1 – 3: Interest payments
= .07($10,000) = $700
Year 4: Interest + Principal = $10,700
Summer 2008
Yunling Chen

71. Amortized Loan
The borrower is required to repay part of the loan amount over time
The most common way of amortizing a loan is to have the borrower make a single, fixed payment every period. Each payment covers the interest plus part of the principal
Example
Consider a 4-year loan with annual repayments. The interest rate is 8% and the principal amount is $5,000.
What is the annual repayment?
$5,000 = C[1 – 1 / 1.084] / .08
C = $1,509.60
Summer 2008
Yunling Chen

72. Amortization Table (Example, cont’d)
Year
Beginning Balance
Regular
Payment
Interest Paid
Principal Paid
Ending Balance A B C
D = B  8%
E = C  D
F = B  E 1
5,000.00
400.00 2
311.22 3
215.36 4
111.82
.00
Totals
Summer 2008
Yunling Chen

73. Fixed payment over periods Outstanding Balance as a function of Time: A 30-Year Mortgage Loan (8%)
Find the excel file on our course website
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74. Fixed payment over periods Proportion of Payments in an Amortized Loan
At first, most of the payment is interest.
Find the excel file on our course website
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Yunling Chen

75. Summary The periodic payment is fixed (the same) each year.
The principal balance is reduced by the different amount each period.
Principal paid will increase each year
Interest paid is decreasing over time
At the end of the last period, the principal is reduce to zero.
Summer 2008
Yunling Chen

76. Takeaways Be able to compute the future value of multiple cash flows
Be able to compute the present value of multiple cash flows
Be able to compute loan payments
Be able to find the interest rate on a loan
Understand the effect of compounding periods and how interest rates are quoted
EAR
APR
Understand how loans are amortized or paid off
Summer 2008
Yunling Chen

77. Individual Homework Critical Thinking and Concepts Review
5.1, 5.2
Questions and Problems
6, 10, 12, 20, 21, 34, 45, 55, 56
Summer 2008
Yunling Chen