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Laurence Booth Sean Cleary

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LEARNING OBJECTIVES Time Value of Money Explain the importance of the time value of money and how it is related to an investors 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. Booth Cleary – 3rd EditionPage 3© John Wiley & Sons Canada, Ltd.

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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. Booth Cleary – 3rd EditionPage 4© John Wiley & Sons Canada, Ltd.

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5.2SIMPLE 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: Booth Cleary – 3rd EditionPage 5© John Wiley & Sons Canada, Ltd.

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EXAMPLE: Simple Interest The same amount of interest is earned in each year. Booth Cleary – 3rd EditionPage 6© John Wiley & Sons Canada, Ltd. 5.2SIMPLE INTEREST

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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. Booth Cleary – 3rd EditionPage 7© John Wiley & Sons Canada, Ltd.

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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) Booth Cleary – 3rd EditionPage 8© John Wiley & Sons Canada, Ltd. [5.2]

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COMPOUND VERSUS SIMPLE INTEREST Simple interest grows principal in a linear manner. Compound interest grows exponentially over time. Booth Cleary – 3rd EditionPage 9© John Wiley & Sons Canada, Ltd.

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EXAMPLE:Compounding (Computing Future Values) Booth Cleary – 3rd EditionPage 10© John Wiley & Sons Canada, Ltd. [5.2] 5.3 COMPOUND INTEREST

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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: Booth Cleary – 3rd EditionPage 11© John Wiley & Sons Canada, Ltd. 5.3 COMPOUND INTEREST Compounding (Computing Future Values)

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EXAMPLE: Using the CVIF Find the FV 20 of $3,500 invested at 3.25%. Booth Cleary – 3rd EditionPage 12© John Wiley & Sons Canada, Ltd. 5.3 COMPOUND INTEREST

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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. Booth Cleary – 3rd EditionPage 13© John Wiley & Sons Canada, Ltd. [5.3]

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Present value interest factor (PVIF) is the inverse of the CVIF. Booth Cleary – 3rd EditionPage 14© John Wiley & Sons Canada, Ltd. 5.3 COMPOUND INTEREST Discounting (Computing Present Values)

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EXAMPLE: Using the PVIF Find the PV 0 of receiving $100,000 in 10 years time if the opportunity cost is 5%. Booth Cleary – 3rd EditionPage 15© John Wiley & Sons Canada, Ltd. 5.3 COMPOUND INTEREST Discounting (Computing Present Values)

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Solving for Time or Holding Periods Equation 5.3 is reorganized to solve for n: Booth Cleary – 3rd EditionPage 16© John Wiley & Sons Canada, Ltd. [5.3] 5.3 COMPOUND INTEREST

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EXAMPLE: Solving for n How many years will it take $8,500 to grow to $10,000 at a 7% rate of interest? Booth Cleary – 3rd EditionPage 17© John Wiley & Sons Canada, Ltd. 5.3 COMPOUND INTEREST

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Solving for Compound Rate of Return Equation 5.3 is reorganized to solve for k: Booth Cleary – 3rd EditionPage 18© John Wiley & Sons Canada, Ltd. [5.3] 5.3 COMPOUND INTEREST

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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? Booth Cleary – 3rd EditionPage 19© John Wiley & Sons Canada, Ltd. 5.3 COMPOUND INTEREST

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5.4ANNUITIES 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. Booth Cleary – 3rd EditionPage 20© John Wiley & Sons Canada, Ltd.

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An ordinary annuity offers payments at the end of each period. An annuity due offers payments at the beginning of each period. Booth Cleary – 3rd EditionPage 21© John Wiley & Sons Canada, Ltd. 5.4ANNUITIES AND PERPETUITIES

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The formula for the compound sum of an ordinary annuity is: Booth Cleary – 3rd EditionPage 22© John Wiley & Sons Canada, Ltd. [5.4] 5.4ANNUITIES AND PERPETUITIES

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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? Booth Cleary – 3rd EditionPage 23© John Wiley & Sons Canada, Ltd. 5.4ANNUITIES AND PERPETUITIES

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The formula for the compound sum of an annuity due is: Booth Cleary – 3rd EditionPage 24© John Wiley & Sons Canada, Ltd. [5.6] 5.4ANNUITIES AND PERPETUITIES

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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 EditionPage 25© John Wiley & Sons Canada, Ltd. 5.4ANNUITIES AND PERPETUITIES

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The formula for the present value of an annuity is: Booth Cleary – 3rd EditionPage 26© John Wiley & Sons Canada, Ltd. [5.5] 5.4ANNUITIES AND PERPETUITIES

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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 EditionPage 27© John Wiley & Sons Canada, Ltd. 5.4ANNUITIES AND PERPETUITIES

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The formula for the present value of an annuity is: Booth Cleary – 3rd EditionPage 28© John Wiley & Sons Canada, Ltd. [5.7] 5.4ANNUITIES AND PERPETUITIES

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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%. Booth Cleary – 3rd EditionPage 29© John Wiley & Sons Canada, Ltd. 5.4ANNUITIES AND PERPETUITIES

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A perpetuity is an infinite series of equal and periodic cash flows. Booth Cleary – 3rd EditionPage 30© John Wiley & Sons Canada, Ltd. [5.8] 5.4ANNUITIES AND PERPETUITIES

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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%? Booth Cleary – 3rd EditionPage 31© John Wiley & Sons Canada, Ltd. 5.4ANNUITIES AND PERPETUITIES

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5.5QUOTED 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). Booth Cleary – 3rd EditionPage 32© John Wiley & Sons Canada, Ltd. [5.9]

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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? Booth Cleary – 3rd EditionPage 33© John Wiley & Sons Canada, Ltd. 5.5QUOTED VERSUS EFFECTIVE RATES

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EXAMPLE: Effective Annual Rates EARs increase as the frequency of compounding increase. Booth Cleary – 3rd EditionPage 34© John Wiley & Sons Canada, Ltd. 5.5QUOTED VERSUS EFFECTIVE RATES

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5.6LOAN 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. Booth Cleary – 3rd EditionPage 35© John Wiley & Sons Canada, Ltd.

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EXAMPLE: Loan Amortization Table Determine the annual blended payment on a five –year $10,000 loan at 8% compounded semi-annually. Booth Cleary – 3rd EditionPage 36© John Wiley & Sons Canada, Ltd. [5.5] 5.6LOAN OR MORTGAGE ARRANGEMENTS

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EXAMPLE: Loan Amortization Table The loan is amortized over five years with annual payments beginning at the end of year 1. Booth Cleary – 3rd EditionPage 37© John Wiley & Sons Canada, Ltd. 5.6LOAN OR MORTGAGE ARRANGEMENTS

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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: Booth Cleary – 3rd EditionPage 38© John Wiley & Sons Canada, Ltd. 5.6LOAN OR MORTGAGE ARRANGEMENTS

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EXAMPLE: Mortgage Amortization The mortgage is amortized over 25 years with annual payments beginning at the end of the first month. Booth Cleary – 3rd EditionPage 39© John Wiley & Sons Canada, Ltd. 5.6LOAN OR MORTGAGE ARRANGEMENTS

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5.7COMPREHENSIVE 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. Booth Cleary – 3rd EditionPage 40© John Wiley & Sons Canada, Ltd.

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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 Booth Cleary – 3rd EditionPage 41© John Wiley & Sons Canada, Ltd.

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COMPREHENSIVE EXAMPLE: Retirement Problem How much will the fixed term annuity cost at age 65? Steps in Solving the Comprehensive Retirement Problem 1.Calculate the present value of the retirement annuity as at Kellys age Estimate the value at age 65 of her current accumulated savings. 3.Calculate gap between accumulated savings and required funds at age Calculate the monthly payment required to fill the gap. Booth Cleary – 3rd EditionPage 42© John Wiley & Sons Canada, Ltd.

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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 Booth Cleary – 3rd EditionPage 43© John Wiley & Sons Canada, Ltd.

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COMPREHENSIVE EXAMPLE: Retirement Problem Time Line & Analysis Required to Identify Savings Gap Booth Cleary – 3rd EditionPage 44© John Wiley & Sons Canada, Ltd. Age year asset accumulation phase 30 year asset depletion phase (retirement) 30 year fixed-term retirement annuity = 30 ×12 =360 months Existing Savings Additional monthly savings

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COMPREHENSIVE EXAMPLE: Retirement Problem Monthly Savings Required to fill Gap Booth Cleary – 3rd EditionPage 45© John Wiley & Sons Canada, Ltd. Age year asset accumulation phase Existing Savings Additional monthly savings Monthly savings to fill gap? Your Answer 30 year asset depletion phase (retirement)

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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: Booth Cleary – 3rd EditionPage 46© John Wiley & Sons Canada, Ltd. [5A-2]

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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: Booth Cleary – 3rd EditionPage 47© John Wiley & Sons Canada, Ltd. [5A-4] Appendix 5A GROWING ANNUITIES & PERPETUITIES

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WEB LINKS Wiley Weekly Finance Updates site (weekly news updates): Textbook Companion Website (resources for students and instructors): Booth Cleary – 3rd Edition© John Wiley & Sons Canada, Ltd. Page 48

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Copyright © 2013 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (the Canadian copyright licensing agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these files or programs or from the use of the information contained herein. COPYRIGHT Copyright © 2013 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (the Canadian copyright licensing agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these files or programs or from the use of the information contained herein. COPYRIGHT

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