1. How to calculate present values
CORPORATE FINANCE COURSE
University LUISS Guido Carli, Rome
Academic year
I Semester, September – December 2014
Jacopo Carmassi
Massimo Arnone
Rome, 26 September 2014

2. Outline Future values and Present values: significance and methods
-The relationship among present value and rates of return
-The relationship among present value and risk factors
-The cost of capital
The present values and future values of annuities:
-The Annuities Due
-The perpetuities
-The growing perpetuities
Real interest, Nominal Interest
Continuous compounding

3. The future value formula
Future Value = $C0 * (1+r)t
C0 is the cash flow at first stage and generally assume negative value (that is the cost of investment)
r is compound rate
(1+r)t is the gain of interests and called compound interest

4. Multiple test 1
If you invest $100,000 today at 12% interest rate for one year, what is the amount you will have at the end of the year?
$90,909
$112,000
$100,000
$102,100

5. The future value: exercise
At an interest rate of 12%, the six-year discount factor is How many dollars is $.507 worth in six year if invested at 12%?
Solution:
$ % $x?
The discount factor: 1/(1+r)6= 1/(1+0.12)6= 0.507
The future value (FV) is equal to:
FV= $C0* (1+r)t.
=.507*(1.12)6=$1
Source: Problem Sets within Brealey et al. (2014) 6
(Year) T
Temporal horizontal line

6. The future value: exercise
If you invest $100 an interest rate of 15% how much will you have at the end of eight years?
Solution:
15%
$ $x? t
(Today) (Year) Temporal Horizontal Line
Future Value = FV = $C0* (1+r)t
Future Value = 100* (1+0.15)8=$305.90
The spread among the initial cash flow ($100) and the future value ($305.90) equal to is called compound interest
Source: Problem Sets within Brealey et al. (2014)

7. The present value formula
We can try to answer to following questions:
How much we need to invest today to produce $ at the end of second year?
What’s the present value (PV) of this $ pay-off
The present value of this future payment is:
Present Value =PV= Ct/(1+r)t
Where
Ct is cash flow amount received at the end of year t, r is the discount rate, 1/(1+r)t is called the discount factor

8. The discount factor DF = 1/(1+r)t
It measures the present value of one dollar received in year t
For longer time we attend that it will be lower and even the present value (PV)
Small variatons in the interest rate can have a powerful impact on the present value of future cash flow

9. The discount factor: exercise
If the PV of $139 is $125, what is the discount factor?
Solution:
DF= 1/(1+r)
It can be obtained trough the ratio among the present value (PV) and the initial cash flow. In our example:
DF = PV /C0= $125/$139=.899
Source: Problem Sets within Brealey et al. (2014)

10. Multiple test 2
A two-year discount factor at a discount rate of 10% per year is:
0.826
1.000
0.909
0.814

11. Multiple test 3
The present value of $115,000 expected to be received one year from today at an interest rate (discount rate) of 10% per year is:
$121,000
$100,500
$110,000
$104,545

12. Multiple test 4
If the present value of $444 to be paid at the end of one year is $400, what is the one-year discount factor?
0.9009
1.11
0.11
None of these options

13. The Present Value: exercise
If the cost of capital is 9% what is the PV of $374 paid in year 9?
Solution:
$x? $374 t
Today Year Temporal Horizontal Line
Present Value=PV= C1/(1+r)t
In our example: the opportunity cost of capital (r) is 9%, t=9 years and C1= $374. Thus:
PV= $374 * 1/(1,09)9=$172.20
Source: Problem Sets within Brealey et al. (2014)

14. The Present Value: exercise
A project produce a cash flow of $432 in year 1, $137 in year 2, and $797 in year 3, if the cost of capital is 15% what is the project’s PV? t
Today Year Year Year Temporal Horizontal Line
Present Value=PV= C1/(1+r) + C2/(1+r)2+ C3/(1+r)3
The each cah flow are discounted by the opportunity cost of capital (15%)
PV = 432/1,15+137/(1.15)2+797/(1.15)3= =$1003
Source: Problem Sets within Brealey et al. (2014)

15. The Present Value: exercise
What is the PV of $100 received in:
Year 10 (at a discount rate of 1%)
Step 1: 1%
$x? $100 t
(Today) (Year) Temporal Horizontal Line
Present Value = PV =C1/(1+r)t
PV=100 /(1,01)10=$90.53
Source: Problem Sets within Brealey et al. (2014)

16. The Present Value: exercise
What is the PV of $100 received in:
Year 10 (a discount rate of 13%)
Step 2:
13%
$x? $100 t
(Today) (Year) Temporal Horizontal Line
Present Value = PV =C1/(1+r)t
PV=100 /(1,13)10=$29.46
Source: Problem Sets within Brealey et al. (2014)

17. The Present Value: exercise
What is the PV of $100 received in:
Year 15 (at a discount rate of 25%)
Step 3:
$x? % $100 t
(Today) (Year) Temporal Horizontal Line
Present Value = PV =C1/(1+r)t
PV=100 /(1,25)15=$3.52
Source: Problem Sets within Brealey et al. (2014)

18. The Present Value: exercise
What is the PV of $100 received in:
Each of years 1 trough 3 (at a discount rate of 12%)
Step 4: t
Today Year Year Year Temporal Horizontal Line
Present Value=PV= C1/(1+r) + C2/(1+r)2+ C3/(1+r)3
The each cah flow are discounted by the opportunity cost of capital (12%)
PV = 100/1,12+100/(1.12)2+100/(1.12)3= $240.18
Source: Problem Sets within Brealey et al. (2014)

19. The discount factor and present value: exercise
It the one-year discount factor is .905, what is the one-year interest rate?
If the two-year interest rate is 10.5%, what is the two-year discount factor?
Given these one and the two year discount factors, calculate the two-year annuity factor?
If the PV 0f $10 a year for three year is $24.65, what is the three-year annuity factor?
From uour answers to last questions calculate the three-year discount factor?
Solution:
DF=1/(1+r)=.905
1=.905(1+r)
1= r
1-.905=.905r
0.095=.905r
r=10.50%
Source: Problem Sets within Brealey et al. (2014)
DF = 1/(1+r)t

20. The discount factor and present value: exercise
If the two-year interest rate is 10.5%, what is the two-year discount factor?
Solution:
DF2 = 1/(1+r)t
DF2=1/(1+r)2=1/(1.105)2=0.819
Given these one and the two year discount factors, calculate the two-year annuity factor?
Two-year annuity factor is sum of two discount factors:
AF=DF1+DF2= =1.724
Source: Problem Sets within Brealey et al. (2014)

21. The discount factor and present value: exercise
If the PV 0f $10 a year for three year is $24.65, what is the three-year annuity factor?
Solution:
Present Value = PV =C1/(1+r)3
PV=10/(1+r)3=24.65
Three-year annuity factor = AF3= 1/(1+r)3=24.65/10=2.465
From our answers to last questions calculate the three-year discount factor?
AF3=DF1+DF2+DF3
DF3=AF3-DF1-DF2
DF3= =0.747
Source: Problem Sets within Brealey et al. (2014)

22. Net Present Value formula
Net Present Value = NPV= C0+C1/(1+r)
Where C0 is the cah flow at time 0 (or today) and is negative number (that is an investment, a cash outflow)
C0=-$

23. The Net Present value: exercise
Halcyon Lines is considering the purchase of a new bulk carrier for $8 million. The forecasted revenues are $5 million a year and operating costs are $4 million. A major refit costing $2 million will be required after both the fifth and tenth years. After 15 years, the ship is expected to be sold for scrap at $1.5 million. If the discount rate is 8%, what is the ship’s NPV?
Solution:
Cost of the ship is $8 million, PV = $8 million, Revenue is $5 million per year, and operating expenses are $4 million. Thus, operating cash flow is $1 million per year for 15 years.
Major cost $2 million each and will occur at times t = 5 and t = 10. PV = ($2 million)/ ($2 million)/ = $2.288 million
Sale for scrap brings in revenue of $1.5 million at t = 15. PV = $1.5 million/ = $0.473 million
We can break this down into several different cash flows, such that the sum of these separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project
NPV = $8 million + $8.559 million  $2.288 million + $0.473 million= $1.256 million

24. The Net Present value: exercise
Recalculate the NPV of the office building at interest rate of 5, 10 and 15%. Plot the point on graph with NPV on vertical axis and the discount rate on the horizontal axis. At what discount rate (approximately) would the project have zero NPV?
Solution:
The cost of land and building: $
The rents for each two next years =$30.000
The sale: $
Interest rate 5%
Interest rate 10%
Interest rate 15%

25. The Net Present value: exercise (2)
NPV at 13.5% is:

26. Net Present Value: exercise
An investment costs $1.548 and pays $138 in perpetuity. If the interest rate is 9% what is the NPV?
Solution:
Net Present Value =NPV= -C0 + Present Value of Perpetuity
NPV= /0.09=-14.67
The negative NPV suggests that the investment cost more than its value. In other words this investment doesn’t increase the wealth
Source: Problem Sets within Brealey et al. (2014)

27. The Net Present Value: exercise
A factory costs $ You thought that it will produce an inflow after operating costs of $ a year for 10 years. If the opportunity cost of capital is 14%, what is the net present value of the factory? What will the factory be worth at the end of five years?
Solution:
The factory will produce inflow equal to $ a year for next ten years in perpetuity. That is we to find the present value of a annuity 1 trough 10 year we must calculale the difference between a perpetuity starting from now and perpetuity starting from 10 year:
Present Value 1 trough 10 year= PV=C1 [1/r-1/(r*(1+r)10]=
= *[1/0.14-1/(0.14(1.14)10]=$886,739.66
Net Present Value = NPV =PV-C0= 886, =86,739.66
Present Value 1 trough 5 year = PV=C1 [1/r-1/(r*(1+r)5]=
= *[1/0.14-1/(0.14(1.14)5]=$583,623.76
Source: Problem Sets within Brealey et al. (2014)

28. The Net Present Value: exercise
A machine cost $ , and is expected to produce the following cash flows. If the cost of capital is 12%, what is the machine’s NPV?
Year 1 2 3 4 5 6 7 8 9 10
Cash Flows (000s)
$50
$57
$75
$80
$85
$92
$68
Solution
We calculate the present value trough the sum of discounted cash flow. The first cash flow is negative and measure the cost of investment (an outflow). To find this result we can use the function sum of Excel 2007 t
Temporal Horizontal Line
Source: Problem Sets within Brealey et al. (2014)

29. The discounted cash flows
Quality of present value: the cash flow are expressed in some currency so you can add them up. The cash flow of A+B is equal to the present value of cash flow A plus the present value of cash flow B
Suppose that you wish to value a series of cash flows extending over a number of years.
PV=C1/(1+r)+C2/(1+r)2+C3/(1+r) Ct/(1+r)t
This is called discounted cash flow (DCF) formula
A short way to write it is:

30. Multiple test 5
If the present value of cash flow X is $200, and the present value of cash flow Y is $150, then the present value of the combined cash flow is:
$200
$150
$50
$350

31. The discounted cash flows: example
Your real estate advisor suggests that you rent out the building for two years at $30000 a year and predicts that at the end of that time you will be able to sell the building for
First Cash Flow (at the enf of one year): C1=$30000
Second Cash Flow (further): C2= =$870000
The present value of first cash flow: C1/(1+r) = 30000/1.12 = $26.78
The present value of second cash flow: C2/(1+r)2 = /1.122 = $69.559
The our rule for calculate the present value tell us the total present value of your investment is:
PV=C1/(1+r) + C2/(1+r)2 = =$

32. Multiple test 6
The opportunity cost of capital for a risky project is:
the expected rate of return on a government security having the same maturity as the project
the expected rate of return on a well-diversified portfolio of common stocks
the expected rate of return on a portfolio of securities of similar risks as the project
none of these options

33. The value of perpetuities
They are bond that government is under no obligation to repay but that offer a fixed income for each year to perpetuity
The annual rate of return on a perpetuity is equal to the promised annual payment divided by the present value
Return = Cash Flow /Present Value 0 r= C/PV
We can certanly twist this ratio and find the present value of a perpetuity given the discount rate r and the cash payment C:
PV=C/r
Two warnings about this formula: first, we must no confuse this formula with the present value of a single payment. A payment of $1 at the end of one year has a present value of 1/(1+r) while the perpetuity has value of 1/r. Second the perpetuity formula tell us the value of regular stream of payment starting one period from now

34. The Perpetuities: exercise
The interest rate is 10%
What is the PV of an asset that pays $1 a year in perpetuity?
The value of an asset that appreciates at 10% per annum approximately doubles in seven years. What is the aproximate PV of an asset that pays $1 a year in perpetuity beginning in year 8?
What is the approximate PV of an asset that pays $1 a year for each of next six years?
A piece of land produces an income that grow by 5% per annum. If the first year’s income is $ what is the value of the land?
Solution:
PV=C/r
PV= 1/0.10= $10
Source: Problem Sets within Brealey et al. (2014)

35. The Perpetuities: exercise
The value of an asset that appreciates at 10% per annum approximately doubles in seven years. What is the aproximate PV of an asset that pays $1 a year in perpetuity beginning in year 8?
Solution:
To find the value of this perpetuity we must multiplicate the present value of perpetuity starting now and the seven-year discount factor
PV = (C/r) * 1/(1+r)t
PV=(1/0.10)*1/(1.08)7=$5
Source: Problem Sets within Brealey et al. (2014)

36. The Perpetuities: exercise
What is the approximate PV of an asset that pays $1 a year for each of next six years?
Solution:
A perpetuity paying $1 starting now would be worth $10, whereas a perpetuity starting in year 8 would be worth roughly $5. The difference betwwen these cash flows is therefore approximately $5
Present Value = PV= $10-$5=$5
Source: Problem Sets within Brealey et al. (2014)

37. The Growing Perpetuities: exercise
A piece of land produces an income that grow by 5% per annum. If the first year’s income is $ what is the value of the land?
Solution:
This case is must considered a growing perpetuity. That is we value the stream of cash flow that grows at a constant rate
This is a geometric serie with rate r higher than rate of growth of salaries g. Then this formula can be simplified:
PV=10.000/( )=$
Source: Problem Sets within Brealey et al. (2014)

38. Multiple test 7
You are a charitable organization that plans to provide $100,000 per year in perpetuity to needy children. How much would a donor need to provide today to fund this goal? Assume that the first payment out of the charitable organization will start one year from today. The interest rate is 10%.
$1,000,000
$10,000,000
$100,000
$111,000

39. Multiple test 9
Find the present value of a perpetuity that pays $3.40 one year from now and is growing at a constant rate of 3%. The discount rate is 14%:
$3.40
$24.29
$30.91
$113.33

40. The Growing Perpetuities: exercise
A common stock will pay a cash dividend of $4 next year. After that the dividend are expected to increase indefinitely at 4% per year. If the discount rate is 14%, what is the PV of the stream of dividend payments
Solution:
This case is must considered a growing perpetuity. That is we value the stream of cash flow that grows at a constant rate
This is a geometric serie with rate r higher than rate of growth of salaries g. Then this formula can be simplified:
PV=4/( )=$40
Source: Problem Sets within Brealey et al. (2014)

41. The delayed perpetuities
Sometimes you may need to calculate the value of perpetuity that does not to start to make payments for several years
Example: you decide to provide $1 billion a year with the first payment four years from now
The present value: to find it we need to multiply by the three-year discount factor 1/(1+r)3= 1/(1+0.10)3=.751.
Thus the delayed perpetuity is worth 10 billion *.751 =7.51 billion
PV =1 billion * 1/r * 1/(1+r)3= 1 billion * 1/0.10 * 1/(1,10)3=$7.51

42. The Annuities
It is an asset that pays a fixed sum each year for a specified number of years
Examples: house mortgages, installment credit agreements, interest payments on bonds
The present value:
You can calculate the value of each cash flow and funding the total
To use a simple formula that states that if the interest rate is r, then the present value of an annuity that pays $C a period for each t period is:
The expression in brackets shows the present value of $1 a year for each t years. It is generally called ad the t-year annuity factor . This formula comes from three investments (figure 2.7)

43. Present Value of 3 year annuity of $1 a year = 1/r -1/(1+r)3
The Annuities
First investment: provides a perpetual stream of $1 starting at the end of the first year. The present value is equal to 1/r
Second investment: provides a perpetual stream of $1 payments but these doesn’t start untili year 4. This stream of payments is identical to the first investment, except that they are delayed for an additional three years. The present value is equal to 1/r * 1/(1+r)3
Third investment: provides a level payment of $1 a year for each of three years. In other words is a three-year annuity. Thus the value of annuity must be equal to the value of first investment less the value of delayed second perpetuity
Present Value of 3 year annuity of $1 a year = 1/r -1/(1+r)3

44. The Annuities
Brealey et al. (2014)

45. The Annuities: exercise
The cost of a new automobile is $ If the interest rate is 5%, how much would you have to set aside now to provide this sum in five years?
You have to pay $ a year in school fees at the end of each the next six years. If the interest rate is 8%, how much do you need to set aside today to cover these bills?
You have invested $ at 8%. After paying the above school fees, how much would remain at the end of six years?
Solution:
Presente Value =PV=C0 (1+r)t
Present Value = (1.05)5=$7,835.26
The our hypothesis is that the cost of new automobile remain the same for next five years and it isn’t subject to growth or declining
Source: Problem Sets within Brealey et al. (2014)

46. The Annuities: exercise
You have to pay $ a year in school fees at the end of each the next six years. If the interest rate is 8%, how much do you need to set aside today to cover these bills?
Solution:
We must value the annuity that pays cash flows 1 trough 6 year. This value is the difference among the perpetuity with uniform payments starting now and the delayed perpetuity starting the six year
In our example is
PV = C*six-year annuity factor
The six-year annuity factor= [1/0.08-1/(0.08)(1.08)6]=4.623
PV=12.000*4.623=$55.475
Source: Problem Sets within Brealey et al. (2014)

47. The Annuities: exercise
You have invested $ at 8%. After paying the above school fees, how much would remain at the end of six years?
Solution:
The remaining balance is after paying the school fees is:
(1.08)6 * ( )=$7.935
That is the future value of remaining balance.
Their investment value is positive. That is you have been able to cover these education fees and to have a savings
Source: Problem Sets within Brealey et al. (2014)

48. The Growing Annuities: exercise
Yasuo Obuchi is 30 years old and his salary next year will be $ He forecasts that his salary will increase at a costante rate of 5% per annum until his retirement at age 60
If the discount rate is 8%, what is the PV of these future salary payments?
If Mr Obuchi saves 5% of his salary each year and invests these savings at an interest rate of 8%, how much will he has saved by age 60?
If the plans to spend these savings in even amounts over the subsequent 20 years, how much can he spend each year?
Solution:
To value this 30-year growing annuity we can calculate as difference among the growing perpetuity starting from now and the delayed growing perpetuity starting from year 30
Source: Problem Sets within Brealey et al. (2014)

49. The Growing Annuities: exercise
If Mr Obuchi saves 5% of his salary each year and invests these savings at an interest rate of 8%, how much will ha have saved by age 60?
Solution:
The interest earning on PV salary is this:
Interest earning = PV salary * (0.05)= *0.05=$
The future value of the interest earning (saving) is this:
FV=Interest earning *(1+i)t
FV=$ *(1.08)30=382714,30
Source: Problem Sets within Brealey et al. (2014)

50. The Growing Annuities: exercise
If the plans to spend these savings in even amounts over the subsequent 20 years, how much can he spend each year?
Solution
In our case:
$382714,30= C* [1/0.08-1/(0.08*(1.08)20)]
C=$382714,30 * / [1/0.08-1/(0.08*(1.08)20)]=$38980,30
Source: Problem Sets within Brealey et al. (2014)

51. The Annuities Due
A level stream of payments starting immediately is called an annuity due. An annuity due is worth (1+r) times the value of an ordinary annuity. It’s possible to reinvest the earning interest
Example “Paying the bank loan”
Bank loan are paid off in equal installments. Suppose that you take out a four-year loan of $1000. The bank requires to repay the loan during the four years. It must therefore set the four annual payments so that they have a present value of $1000
PV= annual loan payment * 4-year annuity factor =$1000
Annual loan payment =$1000/4-year annuity factor
Suppose that the interest rate is 10% a year. Then:
4-year annuity factor = [1/0.10-1/(0.10)*(1+0.10)4]= 3.17
Annual loan payment=1000/3.17=$ > $1000
the annual payment are major that cost of bank loan today and so we are able to repay the loan (Table 2.1)

52. Multiple test 8
What is the present value annuity due factor of $1 at a discount rate of 15% for 15 years?
5.8474
8.5143
7.1324
6.7245

53. The Amortizing Loans
At the end of first year the interest amount is 10%of r 100
First Year: 100 is absorbed by interest and the remaining $ is used to reduce the loan balance to $ ( )
Second Year: the outstanding balance is lower and then interest amount is only $ Therefore $ (annual payments) = can be applied to paying of the loan.
The loan is progressively paid off and so:
The fraction of each payment devoted to interest steadily falls over time,
The fraction used to reduce the balance increases
This series of level payments are known as amortizing loans. That is a part of regular payments is used to pay interest on the loan and part is used to reduce the amount of the loan

54. The Amortizing Loans
Brealey et al. (2014)

55. Multiple test 10
Mr. Hopper is expected to retire in 28 years and he wishes accumulate $750,000 in his retirement fund by that time. If the interest rate is 10% per year, how much should Mr. Hopper put into the retirement fund at the end of each year in order to achieve this goal?
$4,559.44
$5,588.26
$9,118.88
$10,018.67

56. The amortizing loans: exercise
A mortgage requires you to pay $70,000 at the end of each of the next eight years. The interest rate is 8%. What is the present value of these payments? Calculate for each year the loan balance that remains outstanding, the interest payment on the loan, and the reduction in the loan balance
Solution:
PV = * 8-year annuity factor =[1/.08 – 1/.08(1.08)8]
Amortizing Loan =Annual Payment-Year-end interest on Balance
Year-end interest on Balance=0.08 * Beginning of-Year Balance
Amortizing Loan increase (+)
Year-end Interest on Balance falls (-)

57. The amortizing loans: exercise
Suppose that you take out a $200,000, 20-year mortgage loan to buy a condo. The interest rate on the loan is 6%, and payments on the loan are made annually at the end of each year.
What is your annual payment on the loan?
Construct a mortgage amortization table in Excel similar, showing the interest payment, the amortization of the loan, and the loan balance for each year
What fraction of your initial loan payment is interest? What about the last payment? What fraction of the loan has been paid off after 10 years? Why is the fraction less than half?
Solution:
20-year annuity factor = [1/.06 – 1/.06(1.06)20] =
Mortgage payment or Annual Payments= PV / 20-year annuity factor =$200,000/ = $17,436.91
Amortizing Loan =Annual Payment-Year-end interest on Balance
Year-end interest on Balance=0.08 * Beginning of-Year Balance
Amortizing Loan +
Year-end Interest on Balance -

58. The Annuity: future value
Sometimes you need to calculate the future value of a level stream of payments
You find it first we must calculate its presente value and then multiplying by (1+r)t
Future Value at the end of year t =PV * (1+r)t
Future Value at the end of year t = [1/r – 1/r(1+r)t] * (1+r)t= [(1+r)t-1]/r
Example “Saving to buy a sailboat”
Perhaps your ambition is to buy a sailboat, something like a 40-foot Benetou would fit the bill very well. You estimate that, once you start to work you coul save $20000 out of your income and earn a return of 8% on these savings. How much will you be able to spend after five years?
Solutions:
Present Value =PV=20000* [1/.08 – 1/.08(1.08)5] =$79.854
Future Value at the end of year 5 =PV * (1+r)t
Future Value at the end of year 5 =PV * (1+0.08)5= $117.32
You after five years will able to buy yourself a nice boat for $117.32

59. Growing Perpetuities Stream of cash flows that grows at a costant rate
For example you plan to donate $10 billion to flight malaria and other infectious diseases
Unfortunately you made no allowance for the growth in salaries and other costs, which probably average about 4% a year starting in year. Therefore instead of providing $1 billion in perpetuity, you must provide $1 billion in year 1, 1.04*1 billion in year 2 and so on
G is growth rate in cost, the present value of growing perpetuity is:
With r>g this formula can be simplified is the following:

60. Growing Annuities
The formula to find the present value of growing annuitiy is:
It’s sometimes the case that the annuity pays cah flows at the end of each years. For example in France and Germany the government pays interest on its bonds annually. In the United States and Britain government bonds pay interest semiannually
You invest $100 in a bond that pays interest of 10% compounded semiannually your wealth will grow to 1.05*100=105 by the end of six months and to 1.05* 105 = by the end of the year. In other words the interest rate of 10% compounded semiannually is equivalent to 10.25% compounded annually. The effective annual interest rate is 10.25%
In general, if you invest $1 at a rate of r per year compounded m time a year, your investment at the end of the year will be worth [1+(r(m)]m and the effective interest rate is [1+r/m]m-1

61. Continuous Compounding
A situation where the payments are spread evenly and continuosly throught the year define the interest rate continuously compounded
The rate could be compounded weakly (m=52) or daily (m=365). There is no limit to how frequently interest could be paid.
To find the present value you recall m approaches infinity [1+(r/m)]m approaches (2.718)r. This figura it is called the base of natural logarithms.
Therefore $1 invested at a continuously compounded rate of r will grow to er =(2.718)r by the end of the first year. By the end of the t years it will grow to ert =(2.718)rt
If you invest $1 at a continuously compounded rate of 11% for one year (t=1). The end-year value is e11 =(2.718)11=$ in other words investing at a 11% a year continuously compounded rate is exactly the same as investing 1.116% a year annually compounded

62. Continuous Compounding: exercise
The continuously compounded interest rate is 12%
You invest $1000 at this rate. What is the investment worth after five years?
What is the PV of $ 5 million to be received in eight years?
What is the PV of a continuous stream of cash flow, amounting to $ 2000 per year, starting immediately and continuing for 15 years?
Solutions:
the futre value and the present value of annuity with pay cash flows that grow at interest rate continuously compounded are:
FV=C(ert)
FV=1000*( *5)=1000 e0.6=$1822,12
PV=C(e-rt)
PV=1000*( *8)=1000 e0.96=$1914

63. Continuous Compounding: exercise
What is the PV of a continuous stream of cash flow, amounting to $ 2000 per year, starting immediately and continuing for 15 years?
Solutions:
To answer to third question we can think to difference among perpetuity starting now and the perpetuity at continuously rate that starts from 15 year
=2000 (1/0.12-1/0.12*1/e0.12*15)=$13.912

64. Key Formula Meaning Formula Discount factor DF=1/1+r Present Value
PV=DF x C1=C1/(1+r)
Where:
C1 cash flow at the end of year 1,
r opportunity cost of capital
Net Present Value
NPV =C0 + C1/(1+r)
C0 is the cost of investment (negative)
C1 first cash flow,
Perpetuity
PV=1/r
Where :

65. Compounding Capitalization
Key Formula
Meaning
Formula
Annuity
PV=1/r-1/r(1+r)t
Where:
r opportunity cost of capital
Growing Perpetuities
PV=1/(r-g)
g growth rate of cash payments
r >g
Compounding Capitalization
PV=1/ert
e = (Nepero Number)