# ©CourseCollege.com 1 17 In depth: Time Value of Money Interest makes a dollar to be received tomorrow less valuable than a dollar received today Learning.

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©CourseCollege.com 1 17 In depth: Time Value of Money Interest makes a dollar to be received tomorrow less valuable than a dollar received today Learning Objectives 1.Explain the effect of interest on payment streams 2.Compute the future and present value of single amounts 3.Compute the future and present value of annuities 4.Reference: Time value of money tables

©CourseCollege.com 2 Objective 17.1: Explain the effect of interest on payment streams O17.1 A payment stream can be: a single amount, a series of payments any combination of both

©CourseCollege.com 3 Compensation or rent paid to the owner of cash for its use by others over time Interest O17.1

©CourseCollege.com 4 The value in the future of a payment stream with the effect of interest included and expressed as a single value Future value O17.1

©CourseCollege.com 5 Computing future value assumes some rate of interest Future value (FV) O17.1 What will this dollar be worth one year from today? At 8% per year FV = \$1 x 1.08 FV = \$1.08 Interest Principal

©CourseCollege.com 6 Any amount times 1 + interest rate equals the FV after one time period Future value (FV) O17.1 \$500 x 1.08 = \$540 or \$12 x 1.08 = \$12.96 or \$15,560 x 1.08 = \$16,805 Interest Principal EXAMPLES

©CourseCollege.com 7 Repeat the process for each interest period Future value (FV) O17.1 What will this dollar be worth two years from today? FV = \$1 x 1.08 x 1.08 = \$1.17

©CourseCollege.com 8 Future Value Single Amount Present Value Single Amount Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 X 1.08 Interest rate is 8% per year Interest earned Principal X 1.08 O17.1

©CourseCollege.com 9 The value today of a payment stream with the effect of interest removed and expressed as a single value Present value O17.1

©CourseCollege.com 10 Rule #1 O17.1 A dollar received today is always more valuable than a dollar received in the future and A dollar received in the future is always less valuable than a dollar received today

©CourseCollege.com 11 To compute PV DIVIDE by 1+ the interest rate Present value (PV) O17.1 If this dollar is received in one year, what is it worth today? At 8% per year PV = \$1 / 1.08 PV = \$.93

©CourseCollege.com 12 Any amount DIVIDED by 1 + interest rate equals the PV for one time period Present value (PV) O17.1 \$500 / 1.08 = \$463 or \$12 / 1.08 = \$11 or \$15,560/ 1.08 = \$14,407 EXAMPLES

©CourseCollege.com 13 123 4 0 \$ SINGLE AMOUNT Present Value Present value Reducing payment streams to their present value is called discounting Future Value O17.1

©CourseCollege.com 14 Rule #2 O17.1 The more time periods, the higher the future value and the more time periods, the lower the present value Value after 1 year = \$50 x 1.08 = \$54 Value after 2 years = \$50 x 1.08 x 1.08 = \$58 Value after 1 year = \$50 x 1.08 = \$54 Value after 2 years = \$50 x 1.08 x 1.08 = \$58 EXAMPLE –Future Value

©CourseCollege.com 15 Rule #2 O17.1 The more time periods, the higher the future value and the more time periods, the lower the present value Value today if received in 1 year = \$50 / 1.08 = \$46 Value today if received in 2 years = \$50 / 1.08 /1.08 = \$43 Value today if received in 1 year = \$50 / 1.08 = \$46 Value today if received in 2 years = \$50 / 1.08 /1.08 = \$43 EXAMPLE –Present value

©CourseCollege.com 16 Rule #3 O17.1 The value in the future is always higher if the interest rate is higher and the value today is always lower if the interest rate is higher EXAMPLE –Present value

©CourseCollege.com 17 Present Value Future Value Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 ÷ 1.08 Interest rate is 8% per year Fifth Year Principal ÷ 1.08 Discounted Principal

©CourseCollege.com 18 Present Value Future Value Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 ÷ 1.25 Interest rate is 25% per year Fifth Year Principal ÷ 1.25 Discounted Principal Much smaller than at 8%

©CourseCollege.com 19 During a time period (year), the computation of interest and the addition to or subtraction from principle. Compounding O17.1

©CourseCollege.com 20 Rule #4 O17.1 The more compounding periods the higher the future value and The more compounding periods the lower the present value.

©CourseCollege.com 21 Compounding O17.1 A \$5,000 savings deposit pays 4% per year compounded every six months. 4%/2 = 2% per six months \$5,000 x 1.02 x 1.02 = \$5,202.00 A \$5,000 savings deposit pays 4% per year compounded every quarter. 4%/4 = 1% per quarter \$5,000 x 1.01 x 1.01 x 1.01 x 1.01 = \$5,203.02 A \$5,000 savings deposit pays 4% per year compounded every six months. 4%/2 = 2% per six months \$5,000 x 1.02 x 1.02 = \$5,202.00 A \$5,000 savings deposit pays 4% per year compounded every quarter. 4%/4 = 1% per quarter \$5,000 x 1.01 x 1.01 x 1.01 x 1.01 = \$5,203.02 EXAMPLE –Future value More compounding periods = higher future value

©CourseCollege.com 22 Financial calculators also can be used Objective 17.2: Compute the present and future value of single amounts O17.2 Time value of money tables provide a short cut to the computation of present and future values.

©CourseCollege.com 23 Future value tables O17.2 The future value of \$1 table below gives the future value of \$1 at various interest rates and time periods. At 3%, the value of \$1 in 4 years is \$1.1255

©CourseCollege.com 24 Compute future value of single amount O17.2 A \$10,000 savings deposit pays 6% per year ( no compounding) What will the value of the deposit be in 5 years? A \$10,000 savings deposit pays 6% per year ( no compounding) What will the value of the deposit be in 5 years? EXAMPLE From the table at 6%, \$1 would be worth \$1.3382, therefore, \$10,000 x 1.3382 = \$13,382

©CourseCollege.com 25 Present value tables O17.2 The present value of \$1 table below gives the present value of \$1 at various interest rates and time periods. At 2%, the value today of \$1 received in 5 years is \$.9057

©CourseCollege.com 26 Compute present value of single amount O17.2 At 5%, what would a promise to receive \$10,000 in four years ( no compounding) be worth today? At 5%, what would a promise to receive \$10,000 in four years ( no compounding) be worth today? EXAMPLE From the table at 5%, \$1 would be worth \$.8227, therefore, \$10,000 x.8227 = \$8,227

©CourseCollege.com 27 Objective 17.3: Compute the present and future value of annuities An annuity is series of equal payments, paid or received, each time period. A common example of an annuity is an installment loan payment such as an auto loan with identical payments every month. A common example of an annuity is an installment loan payment such as an auto loan with identical payments every month. O17.3

©CourseCollege.com 28 Annuity vs Single amount O17.3

©CourseCollege.com 29 Present Value (annuity) Future Value (annuity) Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Interest rate is 8% per year Annuity payment Discounted Principal ÷ 1.08 O17.3

©CourseCollege.com 30 Future value tables O17.3 The future value of an annuity of \$1 table below gives future values at various interest rates and time periods. At 5%, the value of a \$1 annuity for 5 years is \$5.5256

©CourseCollege.com 31 Compute future value of an annuity O17.3 Tony will deposit \$1,000 per year in a savings account that pays 4% annually. What will the value of the deposit be in 5 years? EXAMPLE From the table at 4%, a \$1 annuity would be worth \$5.4163 in 5 years, therefore, \$1,000 x 5.4163 = \$5,416.30

©CourseCollege.com 32 Present value tables O17.3 At 4%, the value today of \$1 annuity for 4 years is \$3.6299 The present value of an annuity of \$1 table below gives present values at various interest rates and time periods.

©CourseCollege.com 33 Compute present value of an annuity O17.3 A wealthy friend agrees to offer a loan to you at 6% for 5 years. You can promise to repay \$1,000 per year. How much can you borrow? EXAMPLE From the table at 6%, a \$1 annuity would be worth \$4.2124 today, therefore, \$1,000 x 4.2124 = \$4,212.40

©CourseCollege.com 34 Reference: Time value of money tables 17.4 For quick reference, the following slides are time value of money tables