2. 5
Time Value of Money
5.1 Explain the importance of the time value of money and how it is related to an investor’s opportunity costs.
5.2 Define simple interest and explain how it works.
5.3 Define compound interest and explain how it works.
5.4 Differentiate between an ordinary annuity and an annuity due, and explain how special constant payment problems can be valued as annuities and, in special cases, as perpetuities.
5.5 Differentiate between quoted rates and effective rates, and explain how quoted rates can be converted to effective rates.
5.6 Apply annuity formulas to value loans and mortgages and set up an amortization schedule.
5.7 Solve a basic retirement problem.
5.8 Estimate the present value of growing perpetuities and annuities.

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5.1 OPPORTUNITY COST
Money is a medium of exchange.
Money has a time value because it can be invested today and be worth more tomorrow.
The opportunity cost of money is the interest rate that would be earned by investing it.
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5.1 OPPORTUNITY COST
Required rate of return (k) is also known as a discount rate.
To make time value of money decisions, you will need to identify the relevant discount rate you should use.
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5. © John Wiley & Sons Canada, Ltd.
5.2 SIMPLE INTEREST
Simple interest is interest paid or received only on the initial investment (principal).
The same amount of interest is earned in each year.
Equation 5-1:
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6. EXAMPLE: Simple Interest
The same amount of interest is earned in each year.
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5.3 COMPOUND INTEREST
Compound interest is interest that is earned on the principal amount and on the future interest payments.
The future value of a single cash flow at any time ‘n’ is calculated using Equation 5.2.
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USING EQUATION 5.2
Given three known values, you can solve for the one unknown in Equation 5.2
Solve for:
FV - given PV, k, n (finding a future value)
PV - given FV, k, n (finding a present value)
k - given PV, FV, n (finding a compound rate)
n - given PV, FV, k (find holding periods)
[5.2]
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9. COMPOUND VERSUS SIMPLE INTEREST
Simple interest grows principal in a linear manner.
Compound interest grows exponentially over time.
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10. EXAMPLE: Compounding (Computing Future Values)
5.3 COMPOUND INTEREST
EXAMPLE: Compounding (Computing Future Values)
[5.2]
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11. Compounding (Computing Future Values) © John Wiley & Sons Canada, Ltd.
5.3 COMPOUND INTEREST
Compounding (Computing Future Values)
Compound value interest factor (CVIF) represents the future value of an investment at a given rate of interest and for a stated number of periods.
The CVIF for 10 years at 8% would be:
$100 invested for 10 years at 8% would equal:
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12. EXAMPLE: Using the CVIF
5.3 COMPOUND INTEREST
EXAMPLE: Using the CVIF
Find the FV20 of $3,500 invested at 3.25%.
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13. 5.3 COMPOUND INTEREST Discounting (Computing Present Values)
The inverse of compounding is known as discounting.
You can find the present value of any future single cash flow using Equation 5.3.
[5.3]
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14. 5.3 COMPOUND INTEREST Discounting (Computing Present Values)
Present value interest factor (PVIF) is the inverse of the CVIF.
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15. 5.3 COMPOUND INTEREST Discounting (Computing Present Values)
EXAMPLE: Using the PVIF Find the PV0 of receiving $100,000 in 10 years time if the opportunity cost is 5%.
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16. Solving for Time or “Holding Periods”
5.3 COMPOUND INTEREST
Solving for Time or “Holding Periods”
Equation 5.3 is reorganized to solve for n:
[5.3]
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17. EXAMPLE: Solving for ‘n’
5.3 COMPOUND INTEREST
EXAMPLE: Solving for ‘n’
How many years will it take $8,500 to grow to $10,000 at a 7% rate of interest?
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18. Solving for Compound Rate of Return
5.3 COMPOUND INTEREST
Solving for Compound Rate of Return
Equation 5.3 is reorganized to solve for k:
[5.3]
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19. EXAMPLE: Solving for ‘k’
5.3 COMPOUND INTEREST
EXAMPLE: Solving for ‘k’
Your investment of $10,000 grew to $12,500 after 12 years. What compound rate of return (k) did you earn on your money?
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20. 5.4 ANNUITIES AND PERPETUITIES
An annuity is a finite series of equal and periodic cash flows.
A perpetuity is an infinite series of equal and periodic cash flows.
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21. 5.4 ANNUITIES AND PERPETUITIES
An ordinary annuity offers payments at the end of each period.
An annuity due offers payments at the beginning of each period.
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22. 5.4 ANNUITIES AND PERPETUITIES
The formula for the compound sum of an ordinary annuity is:
[5.4]
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23. EXAMPLE: Find the Future Value of an Ordinary Annuity
5.4 ANNUITIES AND PERPETUITIES
EXAMPLE: Find the Future Value of an Ordinary Annuity
You plan to save $1,000 each year for 10 years. At 11% how much will you have saved if you make your first deposit one year from today?
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24. 5.4 ANNUITIES AND PERPETUITIES
The formula for the compound sum of an annuity due is:
[5.6]
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25. EXAMPLE: Find the Future Value of an Annuity Due
5.4 ANNUITIES AND PERPETUITIES
EXAMPLE: Find the Future Value of an Annuity Due
You plan to save $1,000 each year for 10 years. At 11% how much will you have saved if you make your first deposit today?
Booth • Cleary – 3rd Edition
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26. 5.4 ANNUITIES AND PERPETUITIES
The formula for the present value of an annuity is:
[5.5]
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27. EXAMPLE: Find the Present Value of an Ordinary Annuity
5.4 ANNUITIES AND PERPETUITIES
EXAMPLE: Find the Present Value of an Ordinary Annuity
What is the present value of an investment that offers to pay you $12,000 each year for 20 years if the payments start one year from day? Your opportunity cost is 6%.
Booth • Cleary – 3rd Edition
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28. 5.4 ANNUITIES AND PERPETUITIES
The formula for the present value of an annuity is:
[5.7]
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29. EXAMPLE: Find the Present Value of an Annuity Due
5.4 ANNUITIES AND PERPETUITIES
EXAMPLE: Find the Present Value of an Annuity Due
What is the present value of an investment that offers to pay you $12,000 each year for 20 years if the payments start one today? Your opportunity cost is 6%.
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30. 5.4 ANNUITIES AND PERPETUITIES
A perpetuity is an infinite series of equal and periodic cash flows.
[5.8]
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31. EXAMPLE: Find the Present Value of a Perpetuity
5.4 ANNUITIES AND PERPETUITIES
EXAMPLE: Find the Present Value of a Perpetuity
What is the present value of a business that promises to offer you an after-tax profit of $100,000 for the foreseeable future if your opportunity cost is 10%?
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32. 5.5 QUOTED VERSUS EFFECTIVE RATES
A nominal rate of interest is a ‘stated rate’ or quoted rate (QR).
An effective annual rate (EAR) rate takes into account the frequency of compounding (m).
[5.9]
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33. EXAMPLE: Find an Effective Annual Rate
5.5 QUOTED VERSUS EFFECTIVE RATES
EXAMPLE: Find an Effective Annual Rate
Your personal banker has offered you a mortgage rate of 5.5 percent compounded semi-annually. What is the effective annual rate charged (EAR)on this loan?
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34. EXAMPLE: Effective Annual Rates
5.5 QUOTED VERSUS EFFECTIVE RATES
EXAMPLE: Effective Annual Rates
EARs increase as the frequency of compounding increase.
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35. 5.6 LOAN OR MORTGAGE ARRANGEMENTS
A mortgage loan is a borrowing arrangement where the principal amount of the loan borrowed is typically repaid (amortized) over a given period of time making equal and periodic payments.
A blended payment is one where both interest and principal are retired in each payment.
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36. EXAMPLE: Loan Amortization Table
5.6 LOAN OR MORTGAGE ARRANGEMENTS
EXAMPLE: Loan Amortization Table
Determine the annual blended payment on a five –year $10,000 loan at 8% compounded semi-annually.
[5.5]
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37. EXAMPLE: Loan Amortization Table
5.6 LOAN OR MORTGAGE ARRANGEMENTS
EXAMPLE: Loan Amortization Table
The loan is amortized over five years with annual payments beginning at the end of year 1.
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38. 5.6 LOAN OR MORTGAGE ARRANGEMENTS © John Wiley & Sons Canada, Ltd.
EXAMPLE: Mortgage
Determine the monthly blended payment on a $200,000 mortgage amortized over 25 years at a QR = 4.5% compounded semi-annually.
Number of monthly payments = 25 × 12 = 300
Find EAR:
Find EMR:
Determine monthly payment:
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39. EXAMPLE: Mortgage Amortization
5.6 LOAN OR MORTGAGE ARRANGEMENTS
EXAMPLE: Mortgage Amortization
The mortgage is amortized over 25 years with annual payments beginning at the end of the first month.
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40. 5.7 COMPREHENSIVE EXAMPLES
Time value of money (TMV) is a tool that can be applied whenever you analyze a cash flow series over time.
Because of the long time horizon, TMV is ideally suited to solve retirement problems.
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41. COMPREHENSIVE EXAMPLE: Retirement Problem
Kelly, age 40 wants to retire at age 65 and currently has no savings.
At age 65 Kelly wants enough money to purchase a 30 year annuity that will pay $5,000 per month.
Monthly payments should start one month after she reaches age 65.
Today Kelly has accumulated retirement savings of $230,000.
Assume a 4% annual rate of return on both the fixed term annuity and on her savings.
How much will she have to save each month starting one month from now to age 65 in order for her to reach her retirement goal?
*NOTE – these are ordinary annuities
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42. COMPREHENSIVE EXAMPLE: Retirement Problem
How much will the fixed term annuity cost at age 65?
Steps in Solving the Comprehensive Retirement Problem
Calculate the present value of the retirement annuity as at Kelly’s age 65.
Estimate the value at age 65 of her current accumulated savings.
Calculate gap between accumulated savings and required funds at age 65.
Calculate the monthly payment required to fill the gap.
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43. COMPREHENSIVE EXAMPLE: Retirement Problem
Example Solution – Preliminary Calculations
Preliminary calculations Required
Monthly rate of return when annual APR is 4%
Number of months during savings period
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44. COMPREHENSIVE EXAMPLE: Retirement Problem
Time Line & Analysis Required to Identify Savings Gap
30 year fixed-term retirement annuity = 30 ×12 =360 months
Additional monthly savings
Existing Savings
Age
25 year asset accumulation phase
30 year asset depletion phase (retirement)
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45. COMPREHENSIVE EXAMPLE: Retirement Problem
Monthly Savings Required to fill Gap
Monthly savings to fill gap?
Your Answer
Additional monthly savings
Existing Savings
Age
25 year asset accumulation phase
30 year asset depletion phase (retirement)
Booth • Cleary – 3rd Edition
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46. Appendix 5A GROWING ANNUITIES & PERPETUITIES
Growing Perpetuity
A growing perpetuity is an infinite series of periodic cash flows where each cash flow grows larger at a constant rate.
The present value of a growing perpetuity is calculated using the following formula:
[5A-2]
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47. Appendix 5A GROWING ANNUITIES & PERPETUITIES
Growing Annuity
An annuity is a finite series of periodic cash flows where each subsequent cash flow is greater than the previous by a constant growth rate.
The formula for a growing annuity is:
[5A-4]
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48. © John Wiley & Sons Canada, Ltd.
Booth • Cleary – 3rd Edition
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