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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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**© John Wiley & Sons Canada, Ltd.**

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 Edition © John Wiley & Sons Canada, Ltd.

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**© John Wiley & Sons Canada, Ltd.**

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 Edition © John Wiley & Sons Canada, Ltd.

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**© 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: Booth • Cleary – 3rd Edition © 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 Edition © John Wiley & Sons Canada, Ltd.

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**© John Wiley & Sons Canada, Ltd.**

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 Edition © John Wiley & Sons Canada, Ltd.

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**© John Wiley & Sons Canada, Ltd.**

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] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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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 Edition © John Wiley & Sons Canada, Ltd.

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**EXAMPLE: Compounding (Computing Future Values)**

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

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**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: Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**EXAMPLE: Using the CVIF**

5.3 COMPOUND INTEREST EXAMPLE: Using the CVIF Find the FV20 of $3,500 invested at 3.25%. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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

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**5.3 COMPOUND INTEREST Discounting (Computing Present Values)**

Present value interest factor (PVIF) is the inverse of the CVIF. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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%. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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? Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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? Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**5.4 ANNUITIES AND PERPETUITIES**

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

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**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? Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**5.4 ANNUITIES AND PERPETUITIES**

The formula for the compound sum of an annuity due is: [5.6] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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 © John Wiley & Sons Canada, Ltd.

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**5.4 ANNUITIES AND PERPETUITIES**

The formula for the present value of an annuity is: [5.5] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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 © John Wiley & Sons Canada, Ltd.

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**5.4 ANNUITIES AND PERPETUITIES**

The formula for the present value of an annuity is: [5.7] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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%. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**5.4 ANNUITIES AND PERPETUITIES**

A perpetuity is an infinite series of equal and periodic cash flows. [5.8] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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%? Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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? Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**EXAMPLE: Effective Annual Rates**

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

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**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. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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: Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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. Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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. Booth • Cleary – 3rd Edition © 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 Edition © 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 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. Booth • Cleary – 3rd Edition © 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 Edition © John Wiley & Sons Canada, Ltd.

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**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) Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**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 © John Wiley & Sons Canada, Ltd.

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

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**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] Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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**© John Wiley & Sons Canada, Ltd.**

Booth • Cleary – 3rd Edition © John Wiley & Sons Canada, Ltd.

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