1. Chapter 4 The Time Value of Money Chapter Outline
Future Values and Compound Interest
Present Values
Multiple Cash Flows
Level Cash Flows: Perpetuities and Annuities
Inflation and the Time Value of Money
Effective Annual Interest Rates
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2. Introduction Money Problems … Two basic types of cash flow problems:
As a person or a financial manager you will often have to compare cash payments which occur at different dates: Cash flows now, versus cash flows later.
To make optimal decisions, you must understand the relationship between a dollar received (paid) today and a dollar received (paid) in the future.
Two basic types of cash flow problems:
Present Value (PV) problems.
Future Value (FV) problems.
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3. Introduction Present Value (PV) Problems
PV problems involve calculating the value today of future cash flow(s).
For example:
Interest rates are 7%. If I need to have $100,000 saved in 10 years, how much money must I put aside today to create that cash flow?
Interest rates are 12%. If I need to create an income of $5,000 per year for 10 years, how much money must I put aside today to create that cash flow?
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4. Introduction Future Value (FV) Problems
FV problems involve calculating the value an investment will grow to after earning interest.
For example:
Interest rates are 5%. If I invest $1,000 today, how much will it be worth in 8 years?
Interest rates are 10%. If I open an account and invest $2,500 per year, how much will it be worth in 12 years?
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5. Future Values 1. Simple Interest Compound Interest vs Simple Interest
There are two types of interest you may receive:
Compound interest.
Simple interest.
1. Simple Interest
Simple interest means that interest is earned only on your original investment:
No interest is earned on the interest.
Example:
Assume interest rates are 6%.
You invest $100 in an account paying simple interest.
How much will the account be worth in 5 years?
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6. Future Values Simple Interest
You earn interest only on the amount invested.
Therefore you would earn:
$100 x 6% = $6.00 per year for 5 years.
Answer – you would have $130 after 5 years:
Balance in your account:
$100
$106
$112
$118
$124
$130
Period (t) 1 2 3 4 5
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7. Future Values 2. Compound Interest
Compound interest means that interest is earned on interest.
The result: the income you earn would be higher than it would be with simple interest. Why?
Because you earn interest on both the original investment and the interest earned in previous years.
Try the example again using compound interest:
Interest rates are 6%. You invest $100 in an account paying compound interest. How much will the account be worth in 5 years?
Most financial problems you will deal with will involve compound interest.
Compound interest means that interest is earned on interest.
The result: the income you earn would be higher than it would be with simple interest. Can you see why?
Your income would be higher than it would be with simple interest because you earn interest on both the original investment and the interest earned in previous years.
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8. Future Values Compound Interest You earn interest on your interest:
$100 x 6% = $6.00 the first year.
$106 x 6% = $6.36 the second year.
$ x 6% = $6.74 the third year … etc.
After 5 years you would have $ :
Balance in your account:
$100
$106
$112.36
$119.10
$126.25
$133.82
Period (t) 1 2 3 4 5
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9. Future Values Formula for Calculating FV FV = Investment x (1 + r)t
Where r is the interest rate and t is the number of periods.
Try the example again using the formula above:
Interest rates are 6%. You invest $100 in an account paying compound interest. How much will the account be worth in 5 years?
FV = $100 x ( )5
= $100 x
= $133.82
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10. Why Compound Interest?
Future Value (Dollars)
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11. Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in Was this a good deal?
To answer, determine $24 is worth in the year 2000, compounded at 8%.
FYI - The value of Manhattan Island land is well below this figure.
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12. Present Value
Assume interest rates are 10%.
You have just won a lottery and must choose between the following two options:
Receive $1,000,000 today.
Receive $1,000,000 five years from now.
Which option gives you the biggest “winnings”
The above example demonstrates a basic financial principle:
A dollar received today is worth more than a dollar received tomorrow.
The key question is:
How much less valuable is a dollar received tomorrow as versus a dollar received today?
That question is answered by using the interest rate (also known as the discount rate) to calculate the PV of the second option.
More Money Problems …
This is an example of a present value problem.
You shouldn’t even have to do a calculation to get the correct answer.
Obviously the first option is the better choice!
You would want to take the money today so that you could immediately start earning interest on your winnings.
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13. PV = Future Value x 1/(1 + r)t
Present Value
Formula for Calculating PV
PV = Future Value x 1/(1 + r)t
You have been offered $1 million five years from now. Interest rates are 10%.
What is that worth to you in today’s dollars?
PV = $1.0 million x 1/ ( )5
= $1.0 million x
= $620,921
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14. Present Value More Money Problems …
Thus, you could have $1 million today.
Or you could have the second option, which equates to $620,921 in today’s dollars.
$1 million now
The equivalent of $620,921 now vs
You knew before that the first option was better, but now you can calculate exactly how much better off you are:
$379,079 better off!
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15. Present Value vs Future Value
PV and FV are related!
Have you noticed that $620,921 becomes $1 million (and that $1 million requires $620,921) if you have a time period of 5 years and a discount rate of 10%?
PV at 10%
$1,000,000
$620,921
FV at 10%
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16. Present Value vs Future Value
PV and FV are related!
$620,921 invested for 5 years at 10% grows to $1 million.
Or, working it in reverse, if rates are 10%, and you need $1 million in 5 years, you must put aside $620,921 right now.
FV = PV x (1 + r)t
= $620,921 x ( )5
= $620,921 x
= $1 million
PV = FV x 1/(1 + r)t
= $1 million x 1/ ( )5
= $1 million x
= $620,921
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17. Present Value vs Future Value
PV and FV are related!
To calculate the FV of money which is available now (PV) to be invested for t years at an interest rate r, multiply the PV by (1+r)t. To calculate the PV of a future payment, run the process in reverse and divide the FV by (1+r)t.
(1+r)t is called the future value factor.
r is called the discount rate
Finding the PV is often called discounting
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18. Present Value vs Future Value
Two Key Principles for Financial Calculations
Think of the example in which we compared receiving $1 million today against $1 million received 5 years from now.
You should see from that example that:
A dollar received today is worth more than a dollar received tomorrow.
Lesson: The value of cash flows received at different times can never be directly compared.
You must first discount all cash flows to a common date and then compare them.
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19. Present Value vs Future Value
Finding the Unknown …
FV = PV x (1 + r)t
PV = FV x 1/(1 + r)t
The FV and PV formulas have many applications.
Note that the variables used in these two equations are: FV PV r t
Given any three variables in the equation, you can always solve for the remaining variable!
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20. Present Value vs Future Value
Example
You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?
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21. Present Value vs Future Value
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22. Present Value vs Future Value
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23. Multiple Cash Flows Future Value Calculations
So far, we have looked at problems involving only a single cash flow.
This is unrealistic – most business investments will involve multiple cash flows over time.
We need a method for coping with such streams of cash flows!
To calculate the FV of multiple cash flow stream, you need to calculate the FV of each CF and then add up these FVs.
To calculate the PV of multiple cash flow stream, you need to calculate the PV of each CF and then add up these PVs
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24. FV of Multiple Cash Flows
Doing Future Value Calculations
Assume interest rates are 8%. You make 3 deposits to your bank account: $1,200 today, $1,400 one year later, $1,000 two years later.
How much money will you have in your account 3 years from now?
Calculate what each cash flow will be worth at the specified future date and add up these future values. 1 2 3
$1,200
$1,400
$1,000
FV in Year 3:
$1, = $1,000 x 1.08
$1, = $1,400 x (1.08)2
$1, = $1,200 x (1.08)3
$4,224.61
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25. PV of Multiple Cash Flows
EXAMPLE
Assume interest rates are 8%.
You wish to buy a car making three installments:
$8,000 today
$4,000 one year later.
$4,000 two years later.
How much money would you have to place in an account today to generate this stream of cash flows?
If Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments, which do you prefer?
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26. Multiple Cash Flows Present Value Calculations
You would need to place $15, in an account today to generate the desired cash flows: 1 2
$8,000
$4,000
PV today:
$8,000.00
$4,000 / (1.08) = $3,703.30
$4,000 / (1.08)2 = $3,429.36
$15,133.06
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27. Example: FV & PV of Mixed Stream (4% Interest Rate)
Compounding
- $12,166.5
$3,509.6
FV $6,413.8
$5,624.3
$4,326.4
$3,120.0
-$10, $3, $5, $4, $3, $2,000.0
End of Year
$2,884.6
PV $5,271.7
$4,622.8
$3,556.0
$2,564.4
$1,643.9
Discounting
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28. Multiple Cash Flows Special Situations
In the previous examples, we worked with multiple cash flows of different sizes.
Sometimes we have a situation in which a series of equal cash flows is involved:
Any sequence of equally spaced, level cash flows is called an Annuity.
If the payment Stream lasts forever, it is called a perpetuity.
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29. PV of an Annuity: the Long Method
In previous examples, we have worked with multiple cash flows of different sizes.
Suppose we now need to calculate the PV of a stream of level future cash flows.
We could use the same procedure as before:
Calculate what each cash flow would be worth today, i.e. get its PV.
Add up these present values.
EXAMPLE
Assume interest rates are 10%. You wish to buy a car making three installments of $4000, first installment will be a year from today
How much money would you have to place in an account today to generate this stream of cash flows?
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30. PV of an Annuity: the Long Method
You would need to place $9, in an account today to generate the desired cash flows: 1 2 3
$4,000
4,000
PV today:
$4,000 / (1.10) = $3,636.36
$4,000 / (1.10)2 = $3,305.79
$4,000 / (1.10)3 = $3,005.26
$9,947.41
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31. PV of Annuity Formula or
C = cash payment
r = interest rate
t = Number of periods cash payment is received
PVannuity = $4,000 x [1/0.10 – 1/(0.10 ( )3)]
= $4,000 x
= $9,947.41
Using the PV of an annuity calculation, we get the same answer as before:
Put aside $9, to fund the cash flows.
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32. Calculating the FV of an Annuity
Suppose interest rates are 10% and you decide to save $4,000 per year for 20 years. How much will you have saved for your retirement?
This is a FV problem. We could use the same procedure as we used for multiple cash flows of different sizes:
Calculate what each cash flow would be worth in, 20 years, i.e. get its FV and Add up these future values.
Can you see the problem with using this method?
Calculating the FV this way would mean working out the FV for 20 separate cash flows ... 
Is there an easier way?
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33. FV of Annuity Formula
C = cash payment
r = interest rate
t = Number of periods cash payment is received
FVannuity = $4,000 x [ (( )20 – 1) / 0.10 ]
= $4,000 x
= $229,100
Using the FV of an annuity calculation, we see that you will have $229,100 in your account when you retire in 20 years.
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34. Examples of Special Annuities
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35. Annuities Due
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36. Annuities Due
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37. Annuities Due
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38. Examples of Special Annuities
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39. Annuities with missing CFs
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40. Annuities with missing CFs
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41. PV of Perpetuity Perpetuities
The PV of a perpetuity is calculated by dividing the cash payment by the interest rate: C r
Cash Payment
Interest rate
PV of a perpetuity = =
The interest rate on a perpetuity is calculated by dividing the cash payment by the PV: C PV
Cash Payment
Present Value
Interest rate on a perpetuity = =
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42. PV of Perpetuity Example
What would you pay to own a guaranteed income of $1,000 per year to be received forever, if interest rates are 4%? C r
Cash Payment
Interest rate
PV of a perpetuity = =
$1,000 4% = =
$25,000
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43. What is the maximum price you should pay for this condo?
Growing Cash Flows
Cash Flows Growing at a Constant Rate
What if the cash flows in a financial problem are not equal, but are instead growing at a constant rate?
For example:
Assume the discount rate is 8%. You are thinking of buying a condo which generates $12,000 per year in net cash flow in perpetuity. These cash flows grow at 3% per year.
What is the maximum price you should pay for this condo?
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44. Discount Rate – Growth Rate
Growing Cash Flows
Valuing Growing Perpetuities
The PV of a growing perpetuity is calculated by dividing the cash payment by the discount rate less the growth rate: C
r-g
Cash Payment
Discount Rate – Growth Rate
PV of a perpetuity = =
$12,000
0.08 – 0.03 = =
$240,000
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45. Now, what is the maximum price you should pay for this condo?
Growing Cash Flows
Cash Flows Growing at a Constant Rate
In the previous problem we assumed that the cash flows grew at a constant rate forever.
It may be more reasonable to assume a constant growth rate for a limited time period.
For example:
Assume the condo in the previous problem generates $12,000 per year in net cash flow for 20 years. These cash flows grow at 3% per year.
Now, what is the maximum price you should pay for this condo?
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46. Growing Cash Flows Valuing Finite Growing Cash Flows
The PV of cash flows which grow at a constant rate for a limited time period (T) is calculated by:
PV of cash flow = = =
$240,000 * =
$147,000
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47. Inflation, Real and Nominal Rates
Inflation - Rate at which prices as a whole are increasing.
Nominal Interest Rate - Rate at which money invested grows.
Real Interest Rate - Rate at which the purchasing power of an investment increases.
The relation between nominal and real rates and the (expected) future inflation rate is
approximation formula
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48. Inflation, Real and Nominal Rates
Example
If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate?
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49. Frequency of Compounding
General Formula: FVt= PV0(1 + [r/m])mt
t: Number of Years
m: Compounding Periods per Year
r: Annual Interest Rate
FVt,m: FV at the end of Year t
PV0: PV of the Cash Flow today
Impact of Frequency
Julie Miller has $1,000 to invest for 2 years at an annual interest rate of 12%.
Annual FV2 = 1,000(1+ [.12/1])(1)(2) = 1,254.40
Semi FV2 = 1,000(1+ [.12/2])(2)(2) = 1,262.48
Qrtly FV2 = 1,000(1+ [.12/4])(4)(2) = 1,266.77
Monthly FV2 = 1,000(1+ [.12/12])(12)(2) = 1,269.73
Daily FV2 = 1,000(1+[.12/365])(365)(2)= 1,271.20
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50. Effective Interest Rates
EAR vs APR
So far, we have used annual interest rates applied to annual cash flows.
But interest can be applied daily, weekly, monthly, semi-annually – or for any other convenient time period.
The simplest way to deal with this situation is to convert to annual rates.
There are two ways to convert a periodical interest rate to an annual basis:
Calculate the Effective Annual Rate (EAR).
Calculate the Annual Percentage Rate (APR).
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51. Effective Interest Rates
Annual Percentage Rate (APR)
The annual percentage rate (APR) is an interest rate that is annualized using simple interest.
For example:
Your credit card charges 1.5% per month. What is the annual charge?
APR = Quoted rate x Number of Periods Per Year
= 1.5% x 12
= 18%
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52. Effective Interest Rates
Effective Annual Interest Rate (EAR)
The effective annual interest rate (EAR) is an interest rate that is annualized using compound interest.
For example:
Your credit card charges 1.5% per month. What is the annual charge?
EAR = (1 + period rate) # of Periods Per Year-1
= ( )12-1
= 19.56%
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53. Effective Interest Rates
Calculating the EAR
Convert the APR to a period rate and then apply the equation:
(1 + Period Rate) m-1 (m = Number of periods per year)
Quoted APR: 12% (a)
(b)
(c)
Compounding
Period
Periods per Year (m)
Period Rate
(= a/b)
EAR
= (1+ c)m - 1
1 year 1 %
Semiannually 2
6.0000% %
Quarterly 4
3.0000% %
Monthly 12
1.0000% %
Daily
365
0.0329% %
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54. Home Mortgage Home Mortgage Monthly Payment
Assume we are on October 1st and you negotiate a home mortgage of $250,000 to pay during 25 years with equal monthly installments starting at the end of October with a monthly interest rate of 0.5%. What would be your monthly payment ? C 1 2
298
299
300
PV = $250,000
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55. Home Mortgage C 1 2
298
299
300
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56. Home Mortgage in Canada
(See Example 4.19 pages 113)
Mortgage rates in Canada are APRs for month interest rates
They must be converted for use in calculations
Assuming interest rate is 6% (compounded semi-annually) , we first calculate the 6-month interest rate: 6% /2 = 3% per 6-month
The EAR is then given by (1+3%)2 – 1 = 6.09%
The monthly rate is then given by (1+6.09%)1/12 – 1 = %
This monthly rate will be used in calculations
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57. Home Mortgage in Canada
(See Example 4.19 pages 113) C 1 2
298
299
300
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