# 1 17 Interest makes a dollar to be

1 17 Interest makes a dollar to be received tomorrow less valuable than a dollar received today Time Value of Money In depth: 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 A payment stream can be: a single amount, a series of payments any combination of both O17.1 CourseCollege.com 3 Interest Compensation or rent paid to the owner of cash for its use by others over time O17.1 CourseCollege.com 4 Future value The value in the future of a payment stream with the effect of interest included and expressed as a single value O17.1 CourseCollege.com Future value (FV) What will this dollar be worth one year from today? 5 Computing future value assumes some rate of interest At 8% per year Interest FV = \$1 x 1.08 Principal FV = \$1.08 O17.1 CourseCollege.com Future value (FV) 6 Any amount times 1 + interest rate equals the FV after one time period EXAMPLES Interest \$500 x 1.08 = \$540 Principal or \$12 x 1.08 = \$12.96 or \$15,560 x 1.08 = \$16,805 O17.1

CourseCollege.com Future value (FV) What will this dollar be worth two years from today? 7 Repeat the process for each interest period FV = \$1 x 1.08 x 1.08 = \$1.17 O17.1 CourseCollege.com Present Value Single Amount Future Value Single Amount Interest rate is 8% per year Interest earned Principal 8 Year 5 Year 4 Year 3 Year 1 Year 2 Year 0 X 1.08 X 1.08 X 1.08 X 1.08 X 1.08 O17.1 CourseCollege.com 9 Present value The value today of a payment stream with the effect of interest removed and expressed as a single value O17.1 CourseCollege.com 10 Rule #1 A dollar dollar received received today today is is A always more more valuable valuable than than a a always dollar received received in in the the future future dollar and A A dollar dollar received received in in the the and future is is always always less less valuable valuable future than a a dollar

dollar received received today today than O17.1 CourseCollege.com Present value (PV) If this dollar is received in one year, what is it worth today? 11 To compute PV DIVIDE by 1+ the interest rate At 8% per year PV = \$1 / 1.08 PV = \$.93 O17.1 CourseCollege.com Present value (PV) 12 Any amount DIVIDED by 1 + interest rate equals the PV for one time period EXAMPLES \$500 / 1.08 = \$463 or \$12 / 1.08 = \$11 or \$15,560/ 1.08 = \$14,407 O17.1 CourseCollege.com Present Value 0 1 Present value 2 3 13 Future Value 4 \$ SINGLE AMOUNT Reducing payment streams to their present value is called discounting O17.1 CourseCollege.com 14 Rule #2 The more more time time periods, periods, the the The higher the the future future value value and and higher the more more time time periods, periods, the the the lower the the present present value value lower EXAMPLE Future Value Value after after 11 year year =

= \$50 \$50 xx 1.08 1.08 = = \$54 \$54 Value Value after after 22 years years = = \$50 \$50 xx 1.08 1.08 xx 1.08 1.08 Value = \$58 \$58 = O17.1 CourseCollege.com 15 Rule #2 The more more time time periods, periods, the the The higher the the future future value value and and higher the more more time time periods, periods, the the the lower the the present present value value lower EXAMPLE Present value Value today today if if received received in in 11 year year = = \$50 \$50 // 1.08 1.08 Value = \$46 \$46 = Value today today if if received received in in 22 years years = = \$50 \$50 // 1.08 1.08 // Value 1.08 = = \$43 \$43 1.08 O17.1 CourseCollege.com 16 Rule #3 The value value in in the the future future is is The always higher higher if if the the interest interest always rate is is higher higher and and the the value value rate today is is always always lower lower if if the

the today interest rate rate is is higher higher interest EXAMPLE Present value O17.1 CourseCollege.com 17 Present Value Interest rate is 8% per year Future Value Fifth Year Principal Discounted Principal Year 5 Year 4 Year 3 Year 1 Year 2 Year 0 1.08 1.08 1.08 1.08 1.08 CourseCollege.com 18 Present Value Interest rate is 25% per year Future Value Fifth Year Principal Discounted Principal Year 5 Year 4 Year 3 Year 1 Year 2 Year 0 1.25 Much smaller than at 8% 1.25 1.25 1.25 1.25 CourseCollege.com 19 Compounding During a time period (year), the computation of interest and the addition to or subtraction from principle. O17.1 CourseCollege.com 20 Rule #4 The more more compounding compounding The periods the the higher higher the the future future periods value and and The The more more value compounding periods periods the

the compounding lower the the present present value. value. lower O17.1 CourseCollege.com 21 Compounding EXAMPLE Future value A \$5,000 \$5,000 savings savings deposit deposit pays pays 4% 4% per per A year compounded compounded every every six six months. months. year More 4%/2 = = 2% 2% per per six six months months 4%/2 compoundin \$5,000 x 1.02 x 1.02 = \$5,202.00 g periods = \$5,000 x 1.02 x 1.02 = \$5,202.00 higher future value A \$5,000 \$5,000 savings savings deposit deposit pays pays 4% 4% per per A year compounded compounded every every quarter. quarter. year 4%/4 = = 1% 1% per per quarter quarter 4%/4 \$5,000 xx 1.01 1.01 xx 1.01 1.01 xx 1.01 1.01 xx 1.01 1.01 = = \$5,000 \$5,203.02 \$5,203.02 O17.1 CourseCollege.com 22 Objective 17.2: Compute the present and future value of single amounts Financial calculators also can be used Time value of money tables provide a short cut to the computation of present and future values. O17.2 CourseCollege.com 23 Future value tables At 3%, the value of \$1 in 4 years is \$1.1255 The future value of \$1 table below gives the future value of \$1 at various interest rates and time periods.

FutureValue Valueof of\$1 \$1 Future Time Time Periods Periods 00 O17.2 11 22 33 44 55 20 20 21 21 1% 1% 1.0000 1.0000 1.0100 1.0100 1.0201 1.0201 1.0303 1.0303 1.0406 1.0406 1.0510 1.0510 1.2202 1.2202 1.2324 1.2324 2% 2% 1.0000 1.0000 1.0200 1.0200 1.0404 1.0404 1.0612 1.0612 1.0824 1.0824 1.1041 1.1041 1.4859 1.4859 1.5157 1.5157 Interestrate rate Interest 3% 3% 1.0000 1.0000 1.0300 1.0300 1.0609 1.0609 1.0927 1.0927 1.1255 1.1255 1.1593 1.1593 1.8061 1.8061 1.8603 1.8603 4% 4% 1.0000 1.0000 1.0400 1.0400 1.0816 1.0816 1.1249 1.1249 1.1699 1.1699 1.2167 1.2167 2.1911 2.1911 2.2788 2.2788 5% 5% 1.0000 1.0000 1.0500

1.0500 1.1025 1.1025 1.1576 1.1576 1.2155 1.2155 1.2763 1.2763 2.6533 2.6533 2.7860 2.7860 6% 6% 1.0000 1.0000 1.0600 1.0600 1.1236 1.1236 1.1910 1.1910 1.2625 1.2625 1.3382 1.3382 3.2071 3.2071 3.3996 3.3996 CourseCollege.com 24 Compute future value of single amount EXAMPL E A \$10,000 \$10,000 savings savings deposit deposit pays pays 6% 6% A per year year (( no no compounding) compounding) per What will will the the value value of of the the deposit deposit be be What in 55 years? years? in FutureValue Valueof of\$1 \$1 Future Time Time Periods Periods 00 11 22 33 44 55 66 O17.2 1% 1% 1.0000 1.0000 1.0100 1.0100 1.0201 1.0201 1.0303 1.0303 1.0406 1.0406 1.0510 1.0510 1.0615 1.0615 2% 2% 1.0000 1.0000

1.0200 1.0200 1.0404 1.0404 1.0612 1.0612 1.0824 1.0824 1.1041 1.1041 1.1262 1.1262 Interestrate rate Interest 3% 3% 1.0000 1.0000 1.0300 1.0300 1.0609 1.0609 1.0927 1.0927 1.1255 1.1255 1.1593 1.1593 1.1941 1.1941 4% 4% 1.0000 1.0000 1.0400 1.0400 1.0816 1.0816 1.1249 1.1249 1.1699 1.1699 1.2167 1.2167 1.2653 1.2653 5% 5% 1.0000 1.0000 1.0500 1.0500 1.1025 1.1025 1.1576 1.1576 1.2155 1.2155 1.2763 1.2763 1.3401 1.3401 6% 6% 1.0000 1.0000 1.0600 1.0600 1.1236 1.1236 1.1910 1.1910 1.2625 1.2625 1.3382 1.3382 1.4185 1.4185 From the the table table at at 6%, 6%, \$1 \$1 would would be be worth worth From \$1.3382, therefore, therefore, \$10,000 \$10,000 xx 1.3382 1.3382 = = \$1.3382, \$13,382 CourseCollege.com \$13,382 25 Present value tables At 2%, the

value today of \$1 received in 5 years is \$.9057 The present value of \$1 table below gives the present value of \$1 at various interest rates and time periods. PresentValue Valueof of\$1 \$1 Present Time Time Periods Periods 11 22 33 44 55 10 10 11 11 O17.2 1% 1% 0.9901 0.9901 0.9803 0.9803 0.9706 0.9706 0.9610 0.9610 0.9515 0.9515 0.9053 0.9053 0.8963 0.8963 2% 2% 0.9804 0.9804 0.9612 0.9612 0.9423 0.9423 0.9238 0.9238 0.9057 0.9057 0.8203 0.8203 0.8043 0.8043 Interestrate rate Interest 3% 3% 0.9709 0.9709 0.9426 0.9426 0.9151 0.9151 0.8885 0.8885 0.8626 0.8626 0.7441 0.7441 0.7224 0.7224 4% 4% 0.9615 0.9615 0.9246 0.9246 0.8890 0.8890 0.8548 0.8548 0.8219 0.8219 0.6756 0.6756 0.6496 0.6496 5% 5% 0.9524

0.9524 0.9070 0.9070 0.8638 0.8638 0.8227 0.8227 0.7835 0.7835 0.6139 0.6139 0.5847 0.5847 6% 6% 0.9434 0.9434 0.8900 0.8900 0.8396 0.8396 0.7921 0.7921 0.7473 0.7473 0.5584 0.5584 0.5268 0.5268 CourseCollege.com 26 Compute present value of single amount EXAMPL E At 5%, 5%, what what would would aa promise promise to to At receive \$10,000 \$10,000 in in four four years years (( no no receive compounding) compounding) be worth worth today? today? be Valueof of\$1 \$1 PPresent resent Value Time Time Periods Periods 11 22 33 44 55 O17.2 1% 1% 0.9901 0.9901 0.9803 0.9803 0.9706 0.9706 0.9610 0.9610 0.9515 0.9515 2% 2% 0.9804 0.9804 0.9612 0.9612 0.9423 0.9423 0.9238 0.9238 0.9057 0.9057 Interestrate rate Interest

3% 3% 0.9709 0.9709 0.9426 0.9426 0.9151 0.9151 0.8885 0.8885 0.8626 0.8626 4% 4% 0.9615 0.9615 0.9246 0.9246 0.8890 0.8890 0.8548 0.8548 0.8219 0.8219 5% 5% 0.9524 0.9524 0.9070 0.9070 0.8638 0.8638 0.8227 0.8227 0.7835 0.7835 6% 6% 0.9434 0.9434 0.8900 0.8900 0.8396 0.8396 0.7921 0.7921 0.7473 0.7473 From the the table table at at 5%, 5%, \$1 \$1 would would be be worth worth From \$.8227, therefore, therefore, \$10,000 \$10,000 xx .8227 .8227 = = \$8,227 \$8,227 \$.8227, 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 common example example of of an an annuity annuity A is an an installment installment loan loan payment payment such such as as is an auto auto loan loan with with identical identical an payments every every month. month. payments O17.3 CourseCollege.com 28

Annuity vs Single amount PAYMENT STREAMS PRESENT VALUE FUTURE VALUE Single Amount Annuity Equal payments in each time period Year 1 O17.3 Year 2 Year 3 Year 4 Year 5 CourseCollege.com 29 Present Value Future Value Interest rate is 8% per year Annuity payment (annuity) (annuity) Discounted Principal Year 5 Year 4 Year 3 Year 2 Year 1 Year 0 1.08 O17.3 1.08 1.08 1.08 1.08 CourseCollege.com 30 Future value tables At 5%, the value of a \$1 annuity for 5 years is \$5.5256 The future value of an annuity of \$1 table below gives future values at various interest rates and time periods. FutureValue Valueof ofan anannuity annuityof of\$1 \$1 Future Time Time Periods Periods 11 22 33 44 55 66 O17.3 1% 1% 1.0000 1.0000

2.0100 2.0100 3.0301 3.0301 4.0604 4.0604 5.1010 5.1010 6.1520 6.1520 2% 2% 1.0000 1.0000 2.0200 2.0200 3.0604 3.0604 4.1216 4.1216 5.2040 5.2040 6.3081 6.3081 Interestrate rate Interest 3% 3% 1.0000 1.0000 2.0300 2.0300 3.0909 3.0909 4.1836 4.1836 5.3091 5.3091 6.4684 6.4684 4% 4% 1.0000 1.0000 2.0400 2.0400 3.1216 3.1216 4.2465 4.2465 5.4163 5.4163 6.6330 6.6330 5% 5% 1.0000 1.0000 2.0500 2.0500 3.1525 3.1525 4.3101 4.3101 5.5256 5.5256 6.8019 6.8019 6% 6% 1.0000 1.0000 2.0600 2.0600 3.1836 3.1836 4.3746 4.3746 5.6371 5.6371 6.9753 6.9753 CourseCollege.com 31 Compute future value of an annuity EXAMPL E Tony will will deposit deposit \$1,000 \$1,000 per per year year in in aa Tony savings account account that

that pays pays 4% 4% annually. annually. savings What will will the the value value of of the the deposit deposit be be in in What years? 55 years? FutureValue Valueof ofan anannuity annuityof of\$1 \$1 Future Time Time Periods Periods 11 22 33 44 55 66 1% 1% 1.0000 1.0000 2.0100 2.0100 3.0301 3.0301 4.0604 4.0604 5.1010 5.1010 6.1520 6.1520 2% 2% 1.0000 1.0000 2.0200 2.0200 3.0604 3.0604 4.1216 4.1216 5.2040 5.2040 6.3081 6.3081 Interestrate rate Interest 3% 3% 1.0000 1.0000 2.0300 2.0300 3.0909 3.0909 4.1836 4.1836 5.3091 5.3091 6.4684 6.4684 4% 4% 1.0000 1.0000 2.0400 2.0400 3.1216 3.1216 4.2465 4.2465 5.4163 5.4163 6.6330 6.6330 5% 5% 1.0000 1.0000 2.0500 2.0500 3.1525

3.1525 4.3101 4.3101 5.5256 5.5256 6.8019 6.8019 6% 6% 1.0000 1.0000 2.0600 2.0600 3.1836 3.1836 4.3746 4.3746 5.6371 5.6371 6.9753 6.9753 From the the table table at at 4%, 4%, aa \$1 \$1 annuity annuity would would be be From worth \$5.4163 \$5.4163 in in 55 years, years, therefore, therefore, \$1,000 \$1,000 xx worth 5.4163 = = \$5,416.30 \$5,416.30 5.4163 O17.3 CourseCollege.com 32 Present value tables 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. PresentValue Valueof ofan anannuity annuityof of\$1 \$1 Present Time Time Periods Periods 11 22 33 44 55 66 O17.3 1% 1% 0.9901 0.9901 1.9704 1.9704 2.9410 2.9410 3.9020 3.9020 4.8534 4.8534 5.7955 5.7955 2% 2% 0.9804 0.9804 1.9416 1.9416

2.8839 2.8839 3.8077 3.8077 4.7135 4.7135 5.6014 5.6014 Interestrate rate Interest 3% 3% 0.9709 0.9709 1.9135 1.9135 2.8286 2.8286 3.7171 3.7171 4.5797 4.5797 5.4172 5.4172 4% 4% 0.9615 0.9615 1.8861 1.8861 2.7751 2.7751 3.6299 3.6299 4.4518 4.4518 5.2421 5.2421 5% 5% 0.9524 0.9524 1.8594 1.8594 2.7232 2.7232 3.5460 3.5460 4.3295 4.3295 5.0757 5.0757 6% 6% 0.9434 0.9434 1.8334 1.8334 2.6730 2.6730 3.4651 3.4651 4.2124 4.2124 4.9173 4.9173 CourseCollege.com 33 Compute present value of an annuity EXAMPL E A wealthy wealthy friend friend agrees agrees to to offer offer aa A loan to to you you at at 6% 6% for for 55 years. years. You You can can loan promise to to repay repay \$1,000 \$1,000 per per year. year. promise How much much can can you you borrow?

borrow? How PresentValue Valueof ofan anannuity annuityof of\$1 \$1 Present Time Time Periods Periods 11 22 33 44 55 66 1% 1% 0.9901 0.9901 1.9704 1.9704 2.9410 2.9410 3.9020 3.9020 4.8534 4.8534 5.7955 5.7955 2% 2% 0.9804 0.9804 1.9416 1.9416 2.8839 2.8839 3.8077 3.8077 4.7135 4.7135 5.6014 5.6014 Interestrate rate Interest 3% 3% 0.9709 0.9709 1.9135 1.9135 2.8286 2.8286 3.7171 3.7171 4.5797 4.5797 5.4172 5.4172 4% 4% 0.9615 0.9615 1.8861 1.8861 2.7751 2.7751 3.6299 3.6299 4.4518 4.4518 5.2421 5.2421 5% 5% 0.9524 0.9524 1.8594 1.8594 2.7232 2.7232 3.5460 3.5460 4.3295 4.3295 5.0757 5.0757 6% 6% 0.9434 0.9434 1.8334 1.8334 2.6730

2.6730 3.4651 3.4651 4.2124 4.2124 4.9173 4.9173 From the the table table at at 6%, 6%, aa \$1 \$1 annuity annuity would would be be From worth \$4.2124 \$4.2124 today, today, therefore, therefore, \$1,000 \$1,000 xx 4.2124 4.2124 worth = \$4,212.40 \$4,212.40 = O17.3 CourseCollege.com 34 Reference: Time value of money tables For quick reference, the following slides are time value of money tables 17.4 CourseCollege.com 35 FutureValue Valueof of\$1 \$1 Future Time Time Periods Periods 00 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 20 20 17.4 1% 1% 2% 2% 3% 3% Interestrate rate Interest 4% 4% 5% 5% 6% 6% 7%

7% 8% 8% 9% 9% 1.0000 1.0000 1.0100 1.0100 1.0000 1.0000 1.0200 1.0200 1.0000 1.0000 1.0300 1.0300 1.0000 1.0000 1.0400 1.0400 1.0000 1.0000 1.0500 1.0500 1.0000 1.0000 1.0600 1.0600 1.0000 1.0000 1.0700 1.0700 1.0000 1.0000 1.0800 1.0800 1.0000 1.0000 1.0900 1.0900 1.0406 1.0406 1.0510 1.0510 1.0824 1.0824 1.1041 1.1041 1.1255 1.1255 1.1593 1.1593 1.1699 1.1699 1.2167 1.2167 1.2155 1.2155 1.2763 1.2763 1.2625 1.2625 1.3382 1.3382 1.3108 1.3108 1.4026 1.4026 1.3605 1.3605 1.4693 1.4693 1.4116 1.4116 1.5386 1.5386 1.0201 1.0201 1.0303 1.0303 1.0615 1.0615 1.0721 1.0721 1.0829 1.0829

1.0937 1.0937 1.1046 1.1046 1.1157 1.1157 1.1268 1.1268 1.1381 1.1381 1.1495 1.1495 1.1610 1.1610 1.2202 1.2202 1.0404 1.0404 1.0612 1.0612 1.1262 1.1262 1.1487 1.1487 1.1717 1.1717 1.1951 1.1951 1.2190 1.2190 1.2434 1.2434 1.2682 1.2682 1.2936 1.2936 1.3195 1.3195 1.3459 1.3459 1.4859 1.4859 1.0609 1.0609 1.0927 1.0927 1.1941 1.1941 1.2299 1.2299 1.2668 1.2668 1.3048 1.3048 1.3439 1.3439 1.3842 1.3842 1.4258 1.4258 1.4685 1.4685 1.5126 1.5126 1.5580 1.5580 1.8061 1.8061 1.0816 1.0816 1.1249 1.1249 1.2653 1.2653 1.3159 1.3159 1.3686 1.3686 1.4233 1.4233 1.4802 1.4802 1.5395 1.5395 1.6010 1.6010 1.6651 1.6651 1.7317 1.7317 1.8009 1.8009 2.1911 2.1911 1.1025 1.1025 1.1576 1.1576 1.3401 1.3401 1.4071

1.4071 1.4775 1.4775 1.5513 1.5513 1.6289 1.6289 1.7103 1.7103 1.7959 1.7959 1.8856 1.8856 1.9799 1.9799 2.0789 2.0789 2.6533 2.6533 1.1236 1.1236 1.1910 1.1910 1.4185 1.4185 1.5036 1.5036 1.5938 1.5938 1.6895 1.6895 1.7908 1.7908 1.8983 1.8983 2.0122 2.0122 2.1329 2.1329 2.2609 2.2609 2.3966 2.3966 3.2071 3.2071 1.1449 1.1449 1.2250 1.2250 1.5007 1.5007 1.6058 1.6058 1.7182 1.7182 1.8385 1.8385 1.9672 1.9672 2.1049 2.1049 2.2522 2.2522 2.4098 2.4098 2.5785 2.5785 2.7590 2.7590 3.8697 3.8697 1.1664 1.1664 1.2597 1.2597 1.5869 1.5869 1.7138 1.7138 1.8509 1.8509 1.9990 1.9990 2.1589 2.1589 2.3316 2.3316 2.5182 2.5182 2.7196 2.7196 2.9372 2.9372 3.1722 3.1722 4.6610 4.6610 1.1881 1.1881 1.2950 1.2950

1.6771 1.6771 1.8280 1.8280 1.9926 1.9926 2.1719 2.1719 2.3674 2.3674 2.5804 2.5804 2.8127 2.8127 3.0658 3.0658 3.3417 3.3417 3.6425 3.6425 5.6044 5.6044 CourseCollege.com 36 FutureValue Valueof of\$1 \$1 Future Time Time Periods Periods 00 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 20 20 17.4 10% 10% 11% 11% 12% 12% Interestrate rate Interest 13% 14% 14% 15% 15% 13% 16% 16% 18% 18% 20% 20% 1.0000 1.0000 1.1000 1.1000 1.0000 1.0000 1.1100 1.1100 1.0000 1.0000 1.1200 1.1200 1.0000 1.0000 1.1300 1.1300

1.0000 1.0000 1.1400 1.1400 1.0000 1.0000 1.1500 1.1500 1.0000 1.0000 1.1600 1.1600 1.0000 1.0000 1.1800 1.1800 1.0000 1.0000 1.2000 1.2000 1.4641 1.4641 1.6105 1.6105 1.5181 1.5181 1.6851 1.6851 1.5735 1.5735 1.7623 1.7623 1.6305 1.6305 1.8424 1.8424 1.6890 1.6890 1.9254 1.9254 1.7490 1.7490 2.0114 2.0114 1.8106 1.8106 2.1003 2.1003 1.9388 1.9388 2.2878 2.2878 2.0736 2.0736 2.4883 2.4883 1.2100 1.2100 1.3310 1.3310 1.7716 1.7716 1.9487 1.9487 2.1436 2.1436 2.3579 2.3579 2.5937 2.5937 2.8531 2.8531 3.1384 3.1384 3.4523 3.4523 3.7975 3.7975 4.1772 4.1772 6.7275 6.7275 1.2321 1.2321 1.3676 1.3676 1.8704 1.8704 2.0762 2.0762 2.3045

2.3045 2.5580 2.5580 2.8394 2.8394 3.1518 3.1518 3.4985 3.4985 3.8833 3.8833 4.3104 4.3104 4.7846 4.7846 8.0623 8.0623 1.2544 1.2544 1.4049 1.4049 1.9738 1.9738 2.2107 2.2107 2.4760 2.4760 2.7731 2.7731 3.1058 3.1058 3.4785 3.4785 3.8960 3.8960 4.3635 4.3635 4.8871 4.8871 5.4736 5.4736 9.6463 9.6463 1.2769 1.2769 1.4429 1.4429 2.0820 2.0820 2.3526 2.3526 2.6584 2.6584 3.0040 3.0040 3.3946 3.3946 3.8359 3.8359 4.3345 4.3345 4.8980 4.8980 5.5348 5.5348 6.2543 6.2543 11.523 11.523 1.2996 1.2996 1.4815 1.4815 2.1950 2.1950 2.5023 2.5023 2.8526 2.8526 3.2519 3.2519 3.7072 3.7072 4.2262 4.2262 4.8179 4.8179 5.4924 5.4924 6.2613 6.2613 7.1379 7.1379 13.743 13.743 1.3225 1.3225 1.5209 1.5209 2.3131 2.3131

2.6600 2.6600 3.0590 3.0590 3.5179 3.5179 4.0456 4.0456 4.6524 4.6524 5.3503 5.3503 6.1528 6.1528 7.0757 7.0757 8.1371 8.1371 16.367 16.367 1.3456 1.3456 1.5609 1.5609 2.4364 2.4364 2.8262 2.8262 3.2784 3.2784 3.8030 3.8030 4.4114 4.4114 5.1173 5.1173 5.9360 5.9360 6.8858 6.8858 7.9875 7.9875 9.2655 9.2655 19.461 19.461 1.3924 1.3924 1.6430 1.6430 2.6996 2.6996 3.1855 3.1855 3.7589 3.7589 4.4355 4.4355 5.2338 5.2338 6.1759 6.1759 7.2876 7.2876 8.5994 8.5994 10.147 10.147 11.974 11.974 27.393 27.393 1.4400 1.4400 1.7280 1.7280 2.9860 2.9860 3.5832 3.5832 4.2998 4.2998 5.1598 5.1598 6.1917 6.1917 7.4301 7.4301 8.9161 8.9161 10.699 10.699 12.839 12.839 15.407 15.407 38.338 38.338 CourseCollege.com

37 resentValue Valueof of\$1 \$1 PP resent Time Time Periods Periods 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 20 20 17.4 1% 1% 2% 2% 3% 3% Interestrate rate Interest 4% 4% 5% 5% 6% 6% 7% 7% 8% 8% 9% 9% 0.9901 0.9901 0.9803 0.9803 0.9804 0.9804 0.9612 0.9612 0.9709 0.9709 0.9426 0.9426 0.9615 0.9615 0.9246 0.9246 0.9524 0.9524 0.9070 0.9070 0.9434 0.9434 0.8900 0.8900 0.9346 0.9346 0.8734 0.8734 0.9259 0.9259 0.8573 0.8573

0.9174 0.9174 0.8417 0.8417 0.9515 0.9515 0.9420 0.9420 0.9057 0.9057 0.8880 0.8880 0.8626 0.8626 0.8375 0.8375 0.8219 0.8219 0.7903 0.7903 0.7835 0.7835 0.7462 0.7462 0.7473 0.7473 0.7050 0.7050 0.7130 0.7130 0.6663 0.6663 0.6806 0.6806 0.6302 0.6302 0.6499 0.6499 0.5963 0.5963 0.9706 0.9706 0.9610 0.9610 0.9327 0.9327 0.9235 0.9235 0.9143 0.9143 0.9053 0.9053 0.8963 0.8963 0.8874 0.8874 0.8787 0.8787 0.8700 0.8700 0.8613 0.8613 0.8195 0.8195 0.9423 0.9423 0.9238 0.9238 0.8706 0.8706 0.8535 0.8535 0.8368 0.8368 0.8203 0.8203 0.8043 0.8043 0.7885 0.7885 0.7730 0.7730 0.7579 0.7579 0.7430 0.7430 0.6730 0.6730 0.9151 0.9151 0.8885 0.8885 0.8131

0.8131 0.7894 0.7894 0.7664 0.7664 0.7441 0.7441 0.7224 0.7224 0.7014 0.7014 0.6810 0.6810 0.6611 0.6611 0.6419 0.6419 0.5537 0.5537 0.8890 0.8890 0.8548 0.8548 0.7599 0.7599 0.7307 0.7307 0.7026 0.7026 0.6756 0.6756 0.6496 0.6496 0.6246 0.6246 0.6006 0.6006 0.5775 0.5775 0.5553 0.5553 0.4564 0.4564 0.8638 0.8638 0.8227 0.8227 0.7107 0.7107 0.6768 0.6768 0.6446 0.6446 0.6139 0.6139 0.5847 0.5847 0.5568 0.5568 0.5303 0.5303 0.5051 0.5051 0.4810 0.4810 0.3769 0.3769 0.8396 0.8396 0.7921 0.7921 0.6651 0.6651 0.6274 0.6274 0.5919 0.5919 0.5584 0.5584 0.5268 0.5268 0.4970 0.4970 0.4688 0.4688 0.4423 0.4423 0.4173 0.4173 0.3118 0.3118 0.8163 0.8163 0.7629 0.7629 0.6227 0.6227 0.5820 0.5820 0.5439 0.5439

0.5083 0.5083 0.4751 0.4751 0.4440 0.4440 0.4150 0.4150 0.3878 0.3878 0.3624 0.3624 0.2584 0.2584 0.7938 0.7938 0.7350 0.7350 0.5835 0.5835 0.5403 0.5403 0.5002 0.5002 0.4632 0.4632 0.4289 0.4289 0.3971 0.3971 0.3677 0.3677 0.3405 0.3405 0.3152 0.3152 0.2145 0.2145 0.7722 0.7722 0.7084 0.7084 0.5470 0.5470 0.5019 0.5019 0.4604 0.4604 0.4224 0.4224 0.3875 0.3875 0.3555 0.3555 0.3262 0.3262 0.2992 0.2992 0.2745 0.2745 0.1784 0.1784 CourseCollege.com 38 Valueof of\$1 \$1 PPresent resent Value Time Time Periods Periods 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 20 20 17.4 10% 10%

11% 11% 12% 12% Interestrate rate Interest 13% 14% 14% 15% 15% 13% 16% 16% 18% 18% 20% 20% 0.9091 0.9091 0.8264 0.8264 0.9009 0.9009 0.8116 0.8116 0.8929 0.8929 0.7972 0.7972 0.8850 0.8850 0.7831 0.7831 0.8772 0.8772 0.7695 0.7695 0.8696 0.8696 0.7561 0.7561 0.8621 0.8621 0.7432 0.7432 0.8475 0.8475 0.7182 0.7182 0.8333 0.8333 0.6944 0.6944 0.6209 0.6209 0.5645 0.5645 0.5935 0.5935 0.5346 0.5346 0.5674 0.5674 0.5066 0.5066 0.5428 0.5428 0.4803 0.4803 0.5194 0.5194 0.4556 0.4556 0.4972 0.4972 0.4323 0.4323 0.4761 0.4761 0.4104 0.4104 0.4371 0.4371 0.3704 0.3704

0.4019 0.4019 0.3349 0.3349 0.7513 0.7513 0.6830 0.6830 0.5132 0.5132 0.4665 0.4665 0.4241 0.4241 0.3855 0.3855 0.3505 0.3505 0.3186 0.3186 0.2897 0.2897 0.2633 0.2633 0.2394 0.2394 0.1486 0.1486 0.7312 0.7312 0.6587 0.6587 0.4817 0.4817 0.4339 0.4339 0.3909 0.3909 0.3522 0.3522 0.3173 0.3173 0.2858 0.2858 0.2575 0.2575 0.2320 0.2320 0.2090 0.2090 0.1240 0.1240 0.7118 0.7118 0.6355 0.6355 0.4523 0.4523 0.4039 0.4039 0.3606 0.3606 0.3220 0.3220 0.2875 0.2875 0.2567 0.2567 0.2292 0.2292 0.2046 0.2046 0.1827 0.1827 0.1037 0.1037 0.6931 0.6931 0.6133 0.6133 0.4251 0.4251 0.3762 0.3762 0.3329 0.3329 0.2946 0.2946 0.2607 0.2607 0.2307 0.2307 0.2042 0.2042 0.1807 0.1807 0.1599 0.1599 0.0868 0.0868

0.6750 0.6750 0.5921 0.5921 0.3996 0.3996 0.3506 0.3506 0.3075 0.3075 0.2697 0.2697 0.2366 0.2366 0.2076 0.2076 0.1821 0.1821 0.1597 0.1597 0.1401 0.1401 0.0728 0.0728 0.6575 0.6575 0.5718 0.5718 0.3759 0.3759 0.3269 0.3269 0.2843 0.2843 0.2472 0.2472 0.2149 0.2149 0.1869 0.1869 0.1625 0.1625 0.1413 0.1413 0.1229 0.1229 0.0611 0.0611 0.6407 0.6407 0.5523 0.5523 0.3538 0.3538 0.3050 0.3050 0.2630 0.2630 0.2267 0.2267 0.1954 0.1954 0.1685 0.1685 0.1452 0.1452 0.1252 0.1252 0.1079 0.1079 0.0514 0.0514 0.6086 0.6086 0.5158 0.5158 0.3139 0.3139 0.2660 0.2660 0.2255 0.2255 0.1911 0.1911 0.1619 0.1619 0.1372 0.1372 0.1163 0.1163 0.0985 0.0985 0.0835 0.0835 0.0365 0.0365 0.5787 0.5787 0.4823 0.4823

0.2791 0.2791 0.2326 0.2326 0.1938 0.1938 0.1615 0.1615 0.1346 0.1346 0.1122 0.1122 0.0935 0.0935 0.0779 0.0779 0.0649 0.0649 0.0261 0.0261 CourseCollege.com 39 FutureValue Valueof ofan anannuity annuity of of\$1 \$1 Future Time Time Periods Periods 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 20 20 17.4 1% 1% 2% 2% 3% 3% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.0100 2.0200 2.0200 2.0300 2.0300 2.0100 3.0301 3.0604 3.0604 3.0909 3.0909 3.0301 4.0604 4.1216 4.1216 4.1836 4.1836 4.0604 5.1010 5.2040 5.2040 5.3091 5.3091 5.1010 6.1520 6.3081 6.3081 6.4684 6.4684 6.1520 7.2135 7.4343 7.4343 7.6625 7.6625 7.2135 8.2857 8.5830 8.5830 8.8923 8.8923 8.2857 Interestrate rate Interest

4% 4% 5% 5% 6% 6% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.0400 2.0500 2.0500 2.0600 2.0600 2.0400 3.1216 3.1525 3.1525 3.1836 3.1836 3.1216 4.2465 4.3101 4.3101 4.3746 4.3746 4.2465 7% 7% 8% 8% 9% 9% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.0700 2.0800 2.0800 2.0900 2.0900 2.0700 3.2149 3.2464 3.2464 3.2781 3.2781 3.2149 4.4399 4.5061 4.5061 4.5731 4.5731 4.4399 5.4163 5.5256 5.5256 5.6371 5.6371 5.7507 5.7507 5.8666 5.8666 5.9847 5.9847 5.4163 6.6330 6.8019 6.8019 6.9753 6.9753 7.1533 7.1533 7.3359 7.3359 7.5233 7.5233 6.6330 7.8983 8.1420 8.1420 8.3938 8.3938 8.6540 8.6540 8.9228 8.9228 9.2004 9.2004 7.8983 9.2142 9.5491 9.5491 9.8975 9.8975 10.2598 10.2598 10.6366 10.6366 11.0285 11.0285 9.2142 9.3685 9.7546 9.7546 10.1591 10.1591 10.5828 10.5828 11.0266 11.0266 11.4913 11.4913 11.9780 11.9780 12.4876 12.4876 13.0210 13.0210 9.3685 10.4622 10.9497 10.9497 11.4639 11.4639 12.0061 12.0061 12.5779 12.5779 13.1808 13.1808 13.8164 13.8164 14.4866 14.4866 15.1929 15.1929 10.4622 11.5668 12.1687 12.1687 12.8078 12.8078 13.4864 13.4864 14.2068 14.2068 14.9716 14.9716 15.7836 15.7836 16.6455

16.6455 17.5603 17.5603 11.5668 12.6825 13.4121 13.4121 14.1920 14.1920 15.0258 15.0258 15.9171 15.9171 16.8699 16.8699 17.8885 17.8885 18.9771 18.9771 20.1407 20.1407 12.6825 13.8093 14.6803 14.6803 15.6178 15.6178 16.6268 16.6268 17.7130 17.7130 18.8821 18.8821 20.1406 20.1406 21.4953 21.4953 22.9534 22.9534 13.8093 14.9474 15.9739 15.9739 17.0863 17.0863 18.2919 18.2919 19.5986 19.5986 21.0151 21.0151 22.5505 22.5505 24.2149 24.2149 26.0192 26.0192 14.9474 16.0969 17.2934 17.2934 18.5989 18.5989 20.0236 20.0236 21.5786 21.5786 23.2760 23.2760 25.1290 25.1290 27.1521 27.1521 29.3609 29.3609 16.0969 22.0190 24.2974 24.2974 26.8704 26.8704 29.7781 29.7781 33.0660 33.0660 36.7856 36.7856 40.9955 40.9955 45.7620 45.7620 51.1601 51.1601 22.0190 CourseCollege.com 40 FutureValue Valueof ofan anannuity annuity of of\$1 \$1 Future Time Time Periods Periods 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 20 20 17.4 10% 10% 11% 11% 12% 12% 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 2.1000 2.1100 2.1100 2.1200 2.1200 2.1000 3.3100 3.3421 3.3421 3.3744 3.3744 3.3100 4.6410 4.7097 4.7097 4.7793 4.7793 4.6410 Interestrate rate Interest 13% 14% 14% 15% 15% 13% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.1300 2.1400 2.1400 2.1500 2.1500 2.1300 3.4069 3.4396 3.4396 3.4725 3.4725 3.4069 4.8498 4.9211 4.9211 4.9934 4.9934 4.8498 16% 16% 18% 18% 20% 20% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.1600 2.1800 2.1800 2.2000 2.2000 2.1600 3.5056 3.5724 3.5724 3.6400 3.6400 3.5056 5.0665 5.2154 5.2154 5.3680 5.3680 5.0665 6.1051 6.2278 6.2278 6.3528 6.3528 6.4803 6.4803 6.6101 6.6101 6.7424 6.7424 6.8771 6.8771 7.1542 7.1542 7.4416 7.4416 6.1051 7.7156 7.9129 7.9129 8.1152 8.1152 8.3227 8.3227 8.5355 8.5355 8.7537 8.7537 8.9775 8.9775 9.4420 9.4420 9.9299 9.9299 7.7156 9.4872 9.7833 9.7833 10.0890 10.0890 10.4047 10.4047 10.7305 10.7305 11.0668 11.0668 11.4139 11.4139 12.1415 12.1415 12.9159 12.9159 9.4872 11.4359 11.8594 11.8594 12.2997 12.2997 12.7573 12.7573 13.2328 13.2328 13.7268 13.7268 14.2401 14.2401 15.3270 15.3270 16.4991 16.4991 11.4359 13.5795 14.1640 14.1640 14.7757 14.7757 15.4157

15.4157 16.0853 16.0853 16.7858 16.7858 17.5185 17.5185 19.0859 19.0859 20.7989 20.7989 13.5795 15.9374 16.7220 16.7220 17.5487 17.5487 18.4197 18.4197 19.3373 19.3373 20.3037 20.3037 21.3215 21.3215 23.5213 23.5213 25.9587 25.9587 15.9374 18.5312 19.5614 19.5614 20.6546 20.6546 21.8143 21.8143 23.0445 23.0445 24.3493 24.3493 25.7329 25.7329 28.7551 28.7551 32.1504 32.1504 18.5312 21.3843 22.7132 22.7132 24.1331 24.1331 25.6502 25.6502 27.2707 27.2707 29.0017 29.0017 30.8502 30.8502 34.9311 34.9311 39.5805 39.5805 21.3843 24.5227 26.2116 26.2116 28.0291 28.0291 29.9847 29.9847 32.0887 32.0887 34.3519 34.3519 36.7862 36.7862 42.2187 42.2187 48.4966 48.4966 24.5227 27.9750 30.0949 30.0949 32.3926 32.3926 34.8827 34.8827 37.5811 37.5811 40.5047 40.5047 43.6720 43.6720 50.8180 50.8180 59.1959 59.1959 27.9750 31.7725 34.4054 34.4054 37.2797 37.2797 40.4175 40.4175 43.8424 43.8424 47.5804 47.5804 51.6595 51.6595 60.9653 60.9653 72.0351 72.0351 31.7725 57.2750 64.2028 64.2028 72.0524 72.0524 80.9468 80.9468 91.0249 91.0249 102.444 102.444 115.380 115.380 146.628 146.628 186.688 186.688 57.2750 CourseCollege.com 41 PresentValue Valueof ofan anannuity annuity of of\$1 \$1 Present Time Time Periods Periods 11 22 33 44 55 66 77 88 99 10 10 11 11 12

12 13 13 14 14 15 15 20 20 17.4 1% 1% 2% 2% 3% 3% Interestrate rate Interest 4% 4% 5% 5% 6% 6% 7% 7% 8% 8% 9% 9% 0.9901 0.9804 0.9804 0.9709 0.9709 0.9901 1.9704 1.9416 1.9416 1.9135 1.9135 1.9704 2.9410 2.8839 2.8839 2.8286 2.8286 2.9410 3.9020 3.8077 3.8077 3.7171 3.7171 3.9020 0.9615 0.9524 0.9524 0.9434 0.9434 0.9615 1.8861 1.8594 1.8594 1.8334 1.8334 1.8861 2.7751 2.7232 2.7232 2.6730 2.6730 2.7751 3.6299 3.5460 3.5460 3.4651 3.4651 3.6299 0.9346 0.9259 0.9259 0.9174 0.9174 0.9346 1.8080 1.7833 1.7833 1.7591 1.7591 1.8080 2.6243 2.5771 2.5771 2.5313 2.5313 2.6243 3.3872 3.3121 3.3121 3.2397 3.2397 3.3872 8.5660 8.1622 8.1622 7.7861 7.7861 8.5660 9.4713 8.9826 8.9826 8.5302 8.5302 9.4713 10.3676 9.7868 9.7868 9.2526 9.2526 10.3676 11.2551 10.5753 10.5753 9.9540 9.9540

11.2551 7.4353 7.1078 7.1078 6.8017 6.8017 7.4353 8.1109 7.7217 7.7217 7.3601 7.3601 8.1109 8.7605 8.3064 8.3064 7.8869 7.8869 8.7605 9.3851 8.8633 8.8633 8.3838 8.3838 9.3851 6.5152 6.2469 6.2469 5.9952 5.9952 6.5152 7.0236 6.7101 6.7101 6.4177 6.4177 7.0236 7.4987 7.1390 7.1390 6.8052 6.8052 7.4987 7.9427 7.5361 7.5361 7.1607 7.1607 7.9427 4.8534 4.7135 4.7135 4.5797 4.5797 4.8534 5.7955 5.6014 5.6014 5.4172 5.4172 5.7955 6.7282 6.4720 6.4720 6.2303 6.2303 6.7282 7.6517 7.3255 7.3255 7.0197 7.0197 7.6517 4.4518 4.3295 4.3295 4.2124 4.2124 4.4518 5.2421 5.0757 5.0757 4.9173 4.9173 5.2421 6.0021 5.7864 5.7864 5.5824 5.5824 6.0021 6.7327 6.4632 6.4632 6.2098 6.2098 6.7327 4.1002 3.9927 3.9927 3.8897 3.8897 4.1002 4.7665 4.6229 4.6229 4.4859 4.4859 4.7665 5.3893 5.2064 5.2064 5.0330 5.0330 5.3893 5.9713 5.7466 5.7466 5.5348 5.5348 5.9713 12.1337 11.3484 11.3484 10.6350 10.6350 9.9856 9.9856 9.3936 9.3936 8.8527 8.8527 8.3577 8.3577 12.1337 13.0037 12.1062 12.1062 11.2961 11.2961 10.5631 10.5631 9.8986 9.8986 9.2950 9.2950 8.7455 8.7455 13.0037 13.8651 12.8493 12.8493 11.9379 11.9379 11.1184 11.1184 10.3797 10.3797 9.7122 9.7122 9.1079

9.1079 13.8651 18.0456 16.3514 16.3514 14.8775 14.8775 13.5903 13.5903 12.4622 12.4622 11.4699 11.4699 10.5940 10.5940 18.0456 7.9038 7.4869 7.4869 7.9038 8.2442 7.7862 7.7862 8.2442 8.5595 8.0607 8.0607 8.5595 9.8181 9.1285 9.1285 9.8181 CourseCollege.com 42 Valueof ofan anannuity annuity of of\$1 \$1 PPresent resent Value Time Time Periods Periods 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 20 20 17.4 10% 10% 11% 11% 12% 12% Interestrate rate Interest 13% 14% 14% 15% 15% 13% 16% 16% 18% 18% 20% 20% 0.9091 0.9009 0.9009 0.8929 0.8929 0.9091 1.7355 1.7125 1.7125 1.6901 1.6901 1.7355 2.4869 2.4437 2.4437 2.4018 2.4018 2.4869 3.1699 3.1024 3.1024 3.0373 3.0373 3.1699

0.8850 0.8772 0.8772 0.8696 0.8696 0.8850 1.6681 1.6467 1.6467 1.6257 1.6257 1.6681 2.3612 2.3216 2.3216 2.2832 2.2832 2.3612 2.9745 2.9137 2.9137 2.8550 2.8550 2.9745 0.8621 0.8475 0.8475 0.8333 0.8333 0.8621 1.6052 1.5656 1.5656 1.5278 1.5278 1.6052 2.2459 2.1743 2.1743 2.1065 2.1065 2.2459 2.7982 2.6901 2.6901 2.5887 2.5887 2.7982 5.7590 5.5370 5.5370 5.3282 5.3282 5.7590 6.1446 5.8892 5.8892 5.6502 5.6502 6.1446 6.4951 6.2065 6.2065 5.9377 5.9377 6.4951 6.8137 6.4924 6.4924 6.1944 6.1944 6.8137 5.1317 4.9464 4.9464 4.7716 4.7716 5.1317 5.4262 5.2161 5.2161 5.0188 5.0188 5.4262 5.6869 5.4527 5.4527 5.2337 5.2337 5.6869 5.9176 5.6603 5.6603 5.4206 5.4206 5.9176 4.6065 4.3030 4.3030 4.0310 4.0310 4.6065 4.8332 4.4941 4.4941 4.1925 4.1925 4.8332 5.0286 4.6560 4.6560 4.3271 4.3271 5.0286 5.1971 4.7932 4.7932 4.4392 4.4392 5.1971 3.7908 3.6959 3.6959 3.6048 3.6048 3.7908 4.3553 4.2305 4.2305 4.1114 4.1114 4.3553 4.8684 4.7122 4.7122 4.5638 4.5638 4.8684 5.3349 5.1461 5.1461 4.9676 4.9676 5.3349 7.1034 6.7499 6.7499 6.4235 6.4235 7.1034 7.3667 6.9819 6.9819 6.6282

6.6282 7.3667 7.6061 7.1909 7.1909 6.8109 6.8109 7.6061 8.5136 7.9633 7.9633 7.4694 7.4694 8.5136 3.5172 3.4331 3.4331 3.3522 3.3522 3.5172 3.9975 3.8887 3.8887 3.7845 3.7845 3.9975 4.4226 4.2883 4.2883 4.1604 4.1604 4.4226 4.7988 4.6389 4.6389 4.4873 4.4873 4.7988 6.1218 5.8424 5.8424 5.5831 5.5831 6.1218 6.3025 6.0021 6.0021 5.7245 5.7245 6.3025 6.4624 6.1422 6.1422 5.8474 5.8474 6.4624 7.0248 6.6231 6.6231 6.2593 6.2593 7.0248 3.2743 3.1272 3.1272 2.9906 2.9906 3.2743 3.6847 3.4976 3.4976 3.3255 3.3255 3.6847 4.0386 3.8115 3.8115 3.6046 3.6046 4.0386 4.3436 4.0776 4.0776 3.8372 3.8372 4.3436 5.3423 4.9095 4.9095 4.5327 4.5327 5.3423 5.4675 5.0081 5.0081 4.6106 4.6106 5.4675 5.5755 5.0916 5.0916 4.6755 4.6755 5.5755 5.9288 5.3527 5.3527 4.8696 4.8696 5.9288 CourseCollege.com 43 End Unit 17 CourseCollege.com

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• Jesus' teachings alarmed many people; some complained to the Romans that Jesus was teaching that God was greater than their emperor. The Romans feared that Jesus would lead an armed revolt against the govt., so the Roman governor condemned Jesus...