1. Discounted Cash Flow Valuation
MBAC 6060 Chapter 4
Discounted Cash Flow Valuation

2. Chapter Outline: 4.1 One-Period Case 4.2 Multi-Period Case
4.3 Compounding Periods
4.4 Simplifications
4.5 Loan Amortization
4.6 What Is a Firm Worth?

3. Key Concepts and Skills:
Compute the future value and/or present value of a single cash flow or series of cash flows
Compute the return on an investment
Use a spreadsheet to solve time value problems
Understand Annuities and Perpetuities
Learn the “Rule of 72’s”

4. Here is the Idea: Get $100 today or Get $100 in one year.
Which is better?
Obviously getting the $100 today is better.
Why?
If you want to buy something today, you can.
If you want to buy something in 1 year instead:
You can lend the $100 today for one year
And have more than $100 in one year
So if I don’t get the money for one year,
I need to get more than $100
How much more? Talk about that in a soon!

5. FV = C0 x (1+r)t PV = Ct /(1+r)t
4.1 and 4.2 One Cash Flow
One- and Multi-Period Cases:
FV = C0 x (1+r)t
PV = Ct /(1+r)t
FV = Future Value
PV = Present Value
Ct = Cash Flow at time t
r = The interest rate
We will solve for each of these variables
If we have the other 3.
And talk about what the variables mean

6. Future Value and Compounding
Future Value: What will a payment made today be worth later?
Save $100 for 1 year at 10% interest. What will we have in 1 year?
t = 1 r = 10% PV = $100 FV = ?
In 1 year the FV = PV( 1 + r)1 = $100(1.1)1 = $110
Save $100 for 2 years at 10% interest
Leave $110 in bank for a second year:
$110(1.1) = $121
$100(1.1)(1.1) = $100(1.1)2
General Notation
$100(1 + r)t  (1 + r)t is sometimes called the
Future Value Interest Factor

7. Table A.3 (page 966 in the book)
Actually shows at TABLE of FVIFs
Use the table to look up FVIF for 10% in 5 years:
So $100 in 5 years is worth $100(1.6105) = $161.05
But nobody uses tables anymore!
My grandfather used tables!
Use your spreadsheet (or calculator)!
This is equal to $100(1.10)5 = $100(1.6105) = $161.05
Using Excel:
=FV(rate, nper, pmt, [pv], [type])
=FV(.10,5,0,100) =
Why is the Excel answer negative? 

8. This is the formula that is in Excel:
0 = PV + FV/(1+r)t
Solve for FV:
FV = -PV(1 + r)t
= -100(1.1)5 =
This formulation allows for “signing: cash flows:
Positive is an inflow
Negative is an outflow
The assumption is
If PV is positive (get money now)
Then FV must be negative (pay money later)
But by convention, we report positive values

9. Simple Interest vs. Compound Interest
$100 in Five Years at 10% per year:
Interest Factor: (1 + r)t = 1.15 =
Future Value: $100( ) = $161.05
10% Interest on the $100 in each year is $10
Over five years it is $50
The extra $ $50 = $11.05 is interest on interest
Also called COMPOUND INTEREST

10. Present Value
Present Value: What will a payment made later be worth today?
Receive $5,000 in 12 years discounted by 6% interest.
Calculate the Present Value:
PV = FV/( 1 + r)t = $5,000/(1.06)12 = $2,484.85
Using Excel:
=PV(rate, nper, pmt, [fv], [type])
=PV(.06,12,0,5000) =
$2, < $5,000
$2, is the PV of $5,000 (at 6% over 12 years)
$5,000 is the FV of $2, (at 6% over 12 years)

11. Review Question:
What is the PV of $10,000 if you receive the money in 5 years and it is discounted at 12% per year?
What is the FV in 8 years of $30,000 paid today if it earns 9% per year?
Note: Even though Excel will show negative numbers as FV and PV outputs, we still report positive values.
PV = $16,000 and FV = $49,200
PV = $10,000 and FV = $49,200
PV = $4,000 and FV = $49,200
PV = $4,000 and FV = $59,777
PV = $5,674 and FV = $59,777

12. Review Answer: The answer is E. PV of $10,000 paid in 5 years at 12%:
=PV(rate, nper, pmt, [fv], [type])
=PV(.12,5,0,10000) = -5,674
FV in 8 years of $30,000 paid today at 9%:
=FV(rate, nper, pmt, [pv], [type])
=FV(.09,8,0,30000) = -59,777
The answer is E.

13. Interpreting PV and FV Example:
Example: Your company can pay $800 for an asset it believes it can sell for $1,200 in 5 yrs. Similar investments pay 10%
What does similar mean?
Another investment with the same risk (a stock or bond issued by another similar company) pays 10%
So is paying $800 for something that can be sold in 5 yrs for $1,200 a good idea?
PV = FV/(1+r)t = $1,200/(1.1)5 = $745.11
In Excel: =PV(.10,5,0,1200) =
FV = PV(1+r)t = $800(1.1)5 = $1,288.41
In Excel: =FV(.10,5,0,800) =
It is a Bad Idea! You need to earn 10% so either:
You should pay less than $800 (pay $745.11) to get $1,200
You should pay $800 to get more than $1,200 (get $1,288.41)

14. Review Question:
An investment costs $20,000 and you expect to hold it for 10 years.
Investments with similar risk earn 12% per year.
If the projected sale price is $75,000, is this a good investment idea?
YES.
NO.
MAYBE.

15. Review Answer: =FV(rate, nper, pmt, [pv], [type])
If the projected sale price exceeds the calculated FV, then it is a good idea:
FV = PV(1+r)t = $20,000(1.12)10 = $20,000( )
= $62,117
Or using Excel:
=FV(rate, nper, pmt, [pv], [type])
=FV(.12,10,0,20000) = -62,117
$75,000 > $62,117 so invest.
Here’s the idea:
Over 10 years, the invest, which costs $20k and pays $75k, has a return greater than 12%
If it paid only $62,117, the return would be 12%
Since it pays more ($75k) it must have a higher return than the required return

16. Determine the Discount Rate: Solve for r
PV(1 + r)t = FV
(1 + r)t = FV/PV
1 + r = (FV/PV)(1/t)
r = (FV/PV)(1/t) – 1
What rate is need to increase $200 to $400 in 10 years?
r = ($400/$200)(1/10) – 1 = = 7.18%
What rate is need to increase $200 to $400 in 8 years?
r = ($400/$200)(1/8) – 1 = = 9.05%
In Excel:
=RATE(nper, pmt, pv, [fv], [type])
=RATE(8,0,200,400) = #NUM
=RATE(8,0,-200,400) = 9.05%
=RATE(8,0,200,-400) = 9.05%

17. Review Question:
What rate is needed to increase $20,000 to $80,000 in 10 years?
Rate not Found
7.18%
14.87%
20.00%
30.00%

18. Review Answer: r = (FV/PV)(1/t) – 1 = (80/20)(1/10) – 1
= (80/20)(1/10) – 1
= (4)(1/10) - 1 = = 14.87% Or
=RATE(nper, pmt, pv, [fv], [type])
=RATE(10,0,-20,80) = = 14.87%
The answer is C.
Note:
=RATE(10,0,-1,4) = = 14.87%
=RATE(10,0,1,-4) = = 14.87%

19. See page 99 for tips to calculating FV, PV, RATE & NPER in Excel
Determine the Number of Periods: Solve for t (or N)
PV(1 + r)t = FV
(1 + r)t = FV/PV
ln(1 + r)t = ln(FV/PV)
t[ln(1 + r)] = ln(FV/PV)
t = [ln(FV/PV)]/[ln(1 + r)]
But we’ll just use the machine:
How many years are needed to increase $200 to $400 at 7.18%
=NPER(rate, pmt, pv, [fv], [type])
=NPER(.0718,0,200,400) = #NUM!
=NPER(.0718,0,-200,400) = 10.00
=NPER(.0718,0,200,-400) = 10.00
(–PV and +FV or +PV and –FV both work)
See page 99 for tips to calculating FV, PV, RATE & NPER in Excel

20. Review Question:
How many years are needed to get $1,000,000 if you invest $22,095 and earn 10% per year?
20.26 22
22.26 40
Not Found

21. Review Answer: t = [ln(FV/PV)]/[ln(1 + r)]
= ln(1,000,000/22,095)/ln(1.1) = 40 Or
=NPER(rate, pmt, pv, [fv], [type])
=NPER(.10,0,-22095, ) = 40
=NPER(.10,0,22095, ) = 40
The answer is D.

22. Rule of 72’s If FV/PV = 2 (your money doubles) then (r)(t) ≈ 72
Example: Start with $200 and get $400 then FV/PV = 2
$200 to $400 in 10 years at 7.18%
(10)(7.18) = 71.8 ≈ 72
$200 to $400 in 8 years at 9.05%
(8)(9.05) = 72.4 ≈ 72
If the value of your stock doubles in 5 years, what is the approximate annualized compounded return?
(r)(t) ≈ 72  (r)(5) ≈ 72  (r) ≈ 72/5 = 14.4%

23. Review Question:
You own a house that you believe has doubled in value over the last 20 years.
Using the Rule of 72’s, estimate the approximate annual return on the house.
3.60%
5.25%
7.20%
10.00%
20.00%

24. Review Answer: (r)(t) ≈ 72 (r)(20) ≈ 72  (r)(20) ≈ 72/20 = 3.6
That answer is A
Check this answers using the calculator’s TVM function:
N= 20 PV = -1 FV = 2 I/YR = 3.53 ≈ 3.6
Bonus Question:
Assume you will earn 12%? How long to quadruple in price?
Quadruple is double twice: (r)(t) ≈ 72  72/r ≈ t  72/12 = 6 years
So double in 6, double twice in approximately 12 years
Check this: r = 12 PV = -1 FV = 4 N = ≈ 12

25. Recap: FV = PV(1+r)t FV PV r (also called RATE)
Solve for any of the four variables FV PV
r (also called RATE)
t (also called N or NPER)
Use the Excel functions to solve for the one variable not given
Be sure to understand the economic meaning of the values

26. The FV of Multiple CFs For any one CF: FV = PV(1+r)t
For multiple CFs, the FV is the sum of each FV
Example:
Receive $100 at t = 0 and t = 1.
Calc the FV at t = 2 if the rate is 8%
The first $100 increases twice.
The second $100 increases once.
FV = $100(1.08)2 + $100(1.08) = $224.64

27. Another Example: You currently have $7,000 in an account (at t = 0)
You will deposit $4,000 at the end of each of the next 3 years (at t = 1, t = 2 and t = 3)
How much will you have at time 3 at 8%?
$7,000 at t = 0 with 3 years of interest  $7,000(1.08)3 = $8,818
$4,000 at t = 1 with 2 years of interest  $4,000(1.08)2 = $4,666
$4,000 at t = 2 with 1 year of interest  $4,000(1.08)1 = $4,320
$4,000 at t =  $4, = $4,000

28. Example continued Same Example, but now…
How much will you have at time 4 at 8%?
$7,000 at t = 0 with 4 years of interest  $7,000(1.08)4 = $9,523
$4,000 at t = 1 with 3 years of interest  $4,000(1.08)3 = $5,039
$4,000 at t = 2 with 2 years of interest  $4,000(1.08)2 = $4,666
$4,000 at t = 3 with 1 year of interest  $4,000(1.08)1 = $4,320

29. Calculations: FV at t = 3: FV at t = 4: $7,000(1.08)3 = $8,818
$4,000(1.08)2 = $4,666
$4,000(1.08)1 = $4,320
$4,000(1.08)0 = $4,000
$21,804
FV at t = 4:
$7,000(1.08)4 = $9,523
$4,000(1.08)3 = $5,039
$4,000(1.08)2 = $4,666
$4,000(1.08)1 = $4,320
$23,548

30. Now Calculate the PV of Multiple CFs:
You need $1,000 at t = 1 and $2,000 at t = 2
How much do you need to invest today if you earn 9%?
Or what is the PV of these cash flows at 9%?
$1,000/(1.09) + $2,000/(1.09)2 = $2,600.79

31. Think about the PV this way:
Invest $2,601 at 9%.
Show that you can
withdraw $1,000 at t = 1 and
withdraw $2,000 at t = 2:
$2,600.79(1.09) = $2, (at t = 1)
$2, $1,000 = $1, (withdraw $1,000 at t = 1)
$1,834.86(1.09)2 = $2,000 (available at t = 2)
So if you invest $2,601 at 9%, you can withdraw $1,000 at time 1 and $2,000 at time 2
The PV of $1,000 at time 1 and $2,000 at time 2 is $2,601

32. Review Question:
If your investment earns 10%, how much do you need to invest now to be able to withdraw $500 in one year and $800 in two years?
$1,000
$1,116
$1,200
$1,226
$1,300

33. Review Answer: PV of $500 in one year = $500/(1.1) = $455
In order to withdraw $500 in one year and then $800 in two years, you must invest the sum of the PVs of these withdrawals.
PV of $500 in one year = $500/(1.1) = $455
PV of $800 in two years = $800/(1.1)2 = $661
Sum = $455 + $661 = $1,116
The answer is B.

34. 4.3 Compounding Periods
We will cover this
after section 4.5

35. 4.4 Multiple CFs Some Terms: Annuity Perpetuity Growing Perpetuity
A stream of constant cash flows that lasts for a fixed number of periods
Perpetuity
A constant stream of cash flows that lasts forever
Growing Perpetuity
A stream of cash flows that grows at a constant rate forever
Growing Annuity
A stream of cash flows that grows at a constant rate for a fixed number of periods

36. Word Annuity has two definitions
Economic Definition:
All CFs are the same
CFs occur at regular intervals (Annually, Semi-annually, Quarterly, Monthly…)
All CFs are discounted at the same rate
The Financial Product:
Pay an insurance company or a bank a lump sum today
Receive CFs at regular intervals for a fixed period or until you die
Sometimes you pay now (or make regular payments starting now) and then receive payments when you retire at 65
Same pattern of Cash Flow rules for:
Loans (you pay)
A Purchased Annuity (you are paid)

37. Formula for PV of an Annuity (PVA)
𝑃𝑉𝐴= 𝐶 𝑟 1− 𝑟 𝑡
PVA = (C/r)[1 - 1/(1 + r)t] (Same thing but typed)
Text Book’s Notation:
1/(1 + r)t = Present Value Factor (PVF)
PVA = C{[1 - PVF]/r}
Other Notation:
{[1 - 1/(1 + r)t]/r} = Present Value Annuity Factor (PVAF)
PVA = C{PVAF}

38. We’ll use Excel’s PV function:
PV of $1,000 per for 5 6%:
We’ll use Excel’s PV function:
=PV(rate, nper, pmt, [fv], [type])
=PV(.06,5,1000) = -4,212.36

39. Future Value of an Annuity
You will receive $50 per year for next 10 years
When you get the money, you will deposit it in a bank and earn 7%
Calculate the FV of a 10 yr, $50, 7% annuity?
Formula: FVA = C{[(1 + r)t -1]/r}
= C{FVAF}
Excel: =FV(rate, nper, pmt, [pv], [type])
=FV(0.07,10,50) =
Note: 10 x $50 = $500 < $690.82
$ $500 = $ is interest and interest-on-interest

40. PV of a Growing Annuity
A growing stream of cash flows with a fixed maturity: 1 C 2
C×(1+g) 3
C ×(1+g)2 T
C×(1+g)T-1

41. Growing Annuity Example
A defined-benefit retirement plan pays $20,000 per year for 40 years
Payments increase by 3% each year.
Calculate the PV at retirement if the discount rate is 10%? 1
$20 2
$20×(1.03) 40
$20×(1.03)39

42. Growing Annuity Example
What if you will not retire for five more years?
Calculate today’s value of the retirement plan
Recall:
A defined-benefit retirement plan pays $20,000 per year for 40 years. Payments increase by 3% each year and the discount rate is 10%
At retirement the plan is worth:
PV = C/(r-g){1 – [(1 + g)/(1 + r)]T}
= 20,000/(.1 – .03){1 – [1.03/1.1)]40} = $265,121.57
Five years earlier it is worth:
265,121.57/(1.1)5 = 164,619.64

43. PV Annuity vs. Growing Annuity
There is no good way (I think) to calculate the PV of a growing annuity in Excel
𝑃𝑉Annuity= 𝐶 𝑟 1− 𝑟 𝑡
𝑃𝑉 Growing Annuity= 𝐶 𝑟−𝑔 1− 1+𝑔 1+𝑟 𝑡

44. Annuities Due
An Annuity Due means the payments are made at the beginning of each period, not at the end of each period:
So the First payment is made immediately, not at the end of the first period.
The figure below shows the payment timing of a four year $100 Annuity and an Annuity Due:
In Excel, use the “Type” indicator
=PV(rate, nper, pmt, [fv], [type])
Type = 1 for payments at the end of the period
Example 

45. Compare an Annuity Due to an Annuity
Is the PV of a 4 yr Annuity Due greater than or less than the PV a regular 4 yr annuity?
Would you rather be paid the Annuity Due or the Annuity?
Assume a 10% discount rate
The 4 yr Annuity Due is the same as a 3 yr (regular) Annuity plus an extra $100 now (at time zero):
3 yr Annuity: =PV(.10,3,100) = 249
PV 3 yr Annuity + $100 = 349
In Excel
=PV(rate, nper, pmt, [fv], [type]) = PV(.10,4,100,,1) = 349
=PV(rate, nper, pmt, [fv], [type]) = PV(.10,4,100) = 317

46. Review Question:
You will pay $1,000 per month to rent apartment for a year
The lease requires monthly payments at the beginning of each month.
Assume a 12% APR-Monthly discount rate.
Note: 12% APR means 1% per month
Calculate the NET BENEFIT (in present value terms) to the landlord of receiving the rent payments at the beginning of each month as opposed to the end of each month.
$1,343
$1,000
$743
$500
$113

47. Review Answer:
PV of a 12 month, $1,000 (Regular) Annuity discounted at 12% APR.
12% APR means 1% per month
=PV(.01,12,1000) = 11,255
PV of a 12 month, $1,000 Annuity Due discounted at 12% APR:
=PV(.01,12,1000,,1) = 13,368
Net Benefit = $11,368 – 11,255 = $113
The Answer is E

48. PV of a Perpetuity
A perpetuity is a level stream of CFs that lasts forever
The PV of a Perpetuity equals:
The denominator in each successive term in the brackets has a larger exponent, so the value approaches zero
This is known as a “convergent sequence”
The value to which it converges is:
𝑃𝑉= 𝐶 1+𝑟 + 𝐶 1+𝑟 𝐶 1+𝑟 3 +⋯
𝑃𝑉=𝐶 𝑟 𝑟 𝑟 3 +⋯
𝑃𝑉=𝐶 1 𝑟 = 𝐶 𝑟

49. Perpetuity Example
A company’s preferred stock will pay $5 annual dividend forever
Preferred stock of similar risk has a 6% return
Calculate the price of the preferred stock
PV = C/r = $5/.06 = $83.33

50. PV of a Growing Perpetuity
CFs grow at a constant rate (g) forever
The PV of a Growing Perpetuity equals:
If g < r (and it pretty much has to be), then
𝑃𝑉= 𝐶 𝑔 𝑟 𝐶 𝑔 𝑟 𝐶 𝑔 𝑟 3 +⋯
𝑃𝑉= 𝐶 𝑔 𝑟 𝑔 𝑟 𝑔 𝑟 3 +⋯
𝑃𝑉= 𝐶 𝑔 1+𝑟 𝑔 1+𝑟 𝑔 1+𝑟 3 +⋯
1+𝑔 1+𝑟 <1

51. PV of a Growing Perpetuity
If g < r, then each fraction is less than one
So this is also a convergent sequence
If converges to:
𝑃𝑉= 𝐶 𝑔 1+𝑟 𝑔 1+𝑟 𝑔 1+𝑟 3 +⋯
𝑃𝑉= 𝐶 𝑔 𝑟−𝑔 = 𝐶 0 1+𝑔 𝑟−𝑔 = 𝐶 1 𝑟−𝑔

52. Growing Perpetuity Example
A company has a fixed finance and operating policy such that its common stock dividend will grow at a fixed rate of 4% forever.
It is expected to pay a dividend of $2 in one year
Similar stocks have a required return of 14%
Calculate the price of the common stock
PV = C1/(r – g) = $2/(0.14 – 0.04) = $20
Now assume increased efficiency will allow the company increase its growth rate to 6% forever
But the change will not affect the next dividend
Calculate the new price of the common stock
PV = C1/(r – g) = $2/(0.14 – 0.05) = $25

53. 4.5 Loan Amortization How will loan’s principal will be repaid?
Pure Discount Loan
Two CFs: Borrower receives money at beginning
Borrows makes a single payment at end
The single payment covers both interest and principal
Zero-coupon Bonds, CP, T-Bills and CDss
Interest-Only Loans
Borrower makes periodic payments which are just interest
Principal (and the final interest payment) paid at the end
Most bonds (Gov, Corp, Muni)
Amortizing (or Self-Amortizing) Loans
Periodic payments include principal and interest
Payments are calculated using Excel PMT function
Loan amount is PV of the “annuity” payments
Consumer loans

54. Self-Amortization Loans
Example:
Consider a 5 year, $3,000 self-amortizing at 10% annual interest.
Calculate the annual payments:
Use Excel PMT function:
=PMT(rate, nper, pv, [fv], [type])
=PMT(.10,5,3000) =
The Amortization Schedule shows the portion of each fixed payment that is interest the portion that repays the loan

55. Amortization Schedule
5 year, $3,000, 10% annual, self-amortizing loan
Annual Payments are $791.39
Beg Bal = Previous Period’s End Bal = 2,508.61
Interest = Beg Balance x Rate = 2, x 0.1 =
Principal = Payment – Interest = – =
End Bal = Beg Bal – Prin = 2, – = 1,968.08
At the end of year 2, $1, not yet repaid
Period
Beg Bal
PMT
Interest
Principal
End Bal 1
$3,000.00
$791.39
$300.00
$491.39
$2,508.61 2
$250.86
$540.53
$1,968.08 3
$196.81
$594.58
$1,373.50 4
$137.35
$654.04
$719.46 5
$71.95
$719.44
$0.01

56. Amortization Schedule
What is the ending balance at time 2?
It is the amount that still has to be paid
It is also the PV of remaining payments
At time 2, there are 3 more $ payments at 10%
=PV(rate, nper, pmt, [fv], [type])
=PV(.1,3,791.39) = 1,968.08
Also the value of a “balloon payment” due at time 2
Period
Beg Bal
PMT
Interest
Principal
End Bal 1
$3,000.00
$791.39
$300.00
$491.39
$2,508.61 2
$250.86
$540.53
$1,968.08 3
$196.81
$594.58
$1,373.50 4
$137.35
$654.04
$719.46 5
$71.95
$719.44
$0.01

57. 4.3 Compounding Periods Annual Percentage Rate (APR)
APR = Periodic Rate x # of Periods
Periodic Rate = APR / # of Periods
A credit charges 18% APR monthly
The rate is actually 18/12 = 1.5% per month
What annual rate is equivalent?
Called the Effective Annual Rate (EAR)
EAR = (1.015)(1.015)(1.015)…(1.015) – 1
= ( /12)12 – 1 = 19.56%
So indifferent between paying 18% APR Monthly and paying 19.56% per year

58. 4.3 Compounding Periods Example:
An investment pays 12% APR Semi-Annual
Calculate the FV of $1,000 invested for 3 years
EAR = (1 + APR/m)m
= ( /2)2 - 1 = %
FV = C(1 + EAR)T = $1,418.52
= $1,000(1.1236)3 = $1,418.52
Combine the calculations:
FV = C(1 + APR/m)(m x T)
= $1,000( /2)(2 x 3) = $1,418.52

59. Three thousand, six hundred and eighty-six percent!
4.3 Compounding Periods
Formulas:
EAR = (1 + APR/m)m – 1
APR = m[1 + EAR)1/m – 1]
Example:
Payday Loans cost $75 for a $500 two-week loan
Calculate the APR and EAR
Assume there are exactly 26 two-week periods per year.
Periodic rate = $75/$500 = 15%
APR = Periodic Rate x # of Periods = 0.15 x 26 = 390%
EAR = (1 + APR/m)m – 1 = (1.15)26 – 1 = 3686%
Yes, Really.
Three thousand, six hundred and eighty-six percent!

60. Compounding Periods in Excel
EAR is called EFFECT
APR is called NOMINAL
Assume 8% mortgage (monthly rate). Calculate the EAR:
EFFECT(nominal_rate, npery)
EFFECT(.08,12) = 8.30%
A one-year CD pays 5%. What is the equivalent quarterly rate?
NOMINAL(effective_rate, npery)
NOMINAL(.05,4) = 4.91%
The periodic rate is 4.91%/4 = 1.23%

61. Continuous Compounding
EAR = (1 + APR/m)m – 1
As m increases, the EAR for 10% increases
As m approaches infinity  EAR = er – 1
Called “Continuous Compounding”
10% Continuously Compounded
EAR = e.10 – 1 = 10.52% m
EAR 1
10.00% 2
10.25% 4
10.38% 12
10.47%

62. 4.6 A Company’s Value
A company is worth the present value of its cash flows
The problems are
Determining the size of the cash flows
Determining the timing of the cash flows
Determining the correct discount rate for the cash flows
But we will value stocks and bonds by discounting the cash flows paid to the owners