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Time Value of Money Module An electronic presentation by Norman Sunderman Angelo State University An electronic presentation by Norman Sunderman Angelo State University COPYRIGHT © 2007 Thomson South-Western, a part of The Thomson Corporation. Thomson, the Star logo, and South-Western are trademarks used herein under license. TVM Intermediate Accounting 10th edition Nikolai Bazley Jones

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2 Some of the accounting items to which these techniques maybe applied are: 1.Receivables and payables 2.Bonds 3.Leases 4.Pensions 5.Sinking funds 6.Asset valuations 7.Installment contracts Uses of Time Value of Money

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3 Simple interest is interest on the original principal regardless of the number of time periods that have passed. Interest = Principal x Rate x Time Simple Interest

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4 Compound interest is the interest that accrues on both the principal and the past unpaid accrued interest. Compound Interest

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5 Value at Beginning of Quarter Compound Interest x Time 1st qtr.$10,000.00x 0.12x 1/4$ 300.00$10,300.00 2nd qtr.10,300.00x 0.12x 1/4309.0010,609.00 3rd qtr.10,609.00x 0.12x 1/4318.2710,927.27 4th qtr.10,927.27x 0.12x 1/4327.8211,255.09 5th qtr.11,255.09x 0.12x 1/4 337.6511,592.74 Compound interest on $10,000 at 12% compounded quarterly for 5 quarters………………………...$1,592.74 1st qtr.$10,000.00x 0.12x 1/4$ 300.00$10,300.00 2nd qtr.10,300.00x 0.12x 1/4309.0010,609.00 3rd qtr.10,609.00x 0.12x 1/4318.2710,927.27 4th qtr.10,927.27x 0.12x 1/4327.8211,255.09 5th qtr.11,255.09x 0.12x 1/4 337.6511,592.74 Compound interest on $10,000 at 12% compounded quarterly for 5 quarters………………………...$1,592.74 Periodx Rate= Value at End of Quarter Quarterly Compounded Interest

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6 One thousand dollars is invested in a savings account on December 31, 2007. What will be the amount in the savings account on December 31, 2011 if interest at 6% is compounded annually each year? Dec. 31, 2007 Dec. 31, 2008 Dec. 31, 2009 Dec. 31, 2010 Dec. 31, 2011 $1,000 is invested on this date How much will be in the savings account (the future value) on this date? Future Value of a Single Sum at Compound Interest

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7 2008$1,000.00$ 60.00$1,060.00 20091,060.0063.601,123.60 20101,123.6067.421,191.02 20111,191.0271.461,262.48 2008$1,000.00$ 60.00$1,060.00 20091,060.0063.601,123.60 20101,123.6067.421,191.02 20111,191.0271.461,262.48 Annual Future Value Value at Compound at End Beginning of Interest of Year Year Year (Col. 2 x 0.14) (Col. 2 + Col. 3) (1) (2) (3) (4) Future Value of a Single Sum at Compound Interest The future value of $1,000 compounded at 6% for four years is shown below.

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8 Formula Approach ƒ = p(1 + i) n where ƒ = future value of a single sum at compound interest i and n periods p =principal sum (present value) i =interest rate for each of the stated time periods n =number of time periods Future Value of a Single Sum at Compound Interest

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9 Formula Approach f = p(1 + i) n f n = 4, i = 6 = (1.06) 4 f = $1,000(1.2624796) = $1,262.48 Future Value of a Single Sum at Compound Interest

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10 Table Approach Future Value of a Single Sum at Compound Interest This time we will use a table to determine how much $1,000 will accumulate to in four years at 6% compounded annually.

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11 Table Approach Future Value of a Single Sum at Compound Interest Using Table 1 (the future value of 1) at the end of the TVM Module, determine the future value interest factor for an annual interest rate of 6 percent and four periods.

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12 Table Approach n 6.0% 8.0% 9.0% 10.0% 12.0% 14.0% 11.0600001.0800001.0900001.1000001.1200001.140000 21.1236001.1664001.1881001.2100001.2544001.299600 31.1910161.2597121.2950291.3310001.4049281.481544 41.2624771.3604891.4115821.4641001.5735191.688960 51.3382261.4693281.5386241.6105101.7623421.925415 61.4185191.5868741.6771001.7715611.9738232.194973 1.262477 Future Value of a Single Sum at Compound Interest

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13 Table Approach One thousand dollars times 1.262477 equals the future value, or $1,262.48. Future Value of a Single Sum at Compound Interest

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14 If $1,000 is worth $1,262.48 when it earns 6% compounded annually for 4 years, then it follows that $1,262.48 to be received in 4 years from now is worth $1,000 now at time period zero. Dec. 31, 2007 Dec. 31, 2008 Dec. 31, 2009 Dec. 31, 2010 Dec. 31, 2011 $1,000 (the present value) must be invested on this date $1,262.48 will be received on this date Present Value of a Single Sum

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15 Interest Rate Unknown If $1,000 is invested on December 31, 2007, to earn compound interest and if the future value on December 31, 2014 is $2,998.70, what is the quarterly interest rate? Future Value Present Value = Future value factor 28 periods $2,998.70 $1,000 = 2.99870

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16 Table Approach n 1.5% 4.0% 4.5% 5.0% 5.5% 6.0% 11.0150001.0400001.0450001.0500001.0550001.060000 21.0302281.0816001.0920251.1025001.1130251.123600 31.0456781.1248641.1411661.1576251.1742411.191016 281.5172222.9987033.4297003.9201294.4778435.111687 291.5399813.1186513.5840364.116`364.7241245.418388 301.5630803.2433983.7453184.3219424.9839515.743491 2.998703 Future Value of a Single Sum at Compound Interest The quarterly rate is 4%, which makes the annual rate 16%.

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17 1 (1 + i) n p = f Formula Approach Wherep = present value of any given future value due in the future ƒ = future value i = interest rate for each of the stated time periods n =number of time periods Present Value of a Single Sum

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18 p = $1,262.48 ( 0.792094 ) = $1,000.00 p n = 4, i = 6 = 1 (1.06) 4 = 0.792094 Formula Approach Present Value of a Single Sum

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19 Table Approach Find Table 3, the present value of 1, at the end of the Time Value of Money Module. Use 6% and four periods to obtain the future value interest factor. Present Value of a Single Sum

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20 Table Approach n 6.0% 8.0% 9.0% 10.0% 12.0% 14.0% 10.9433960.9259260.9174310.9090910.8928570.877193 20.8899960.8573390.8416800.8264460.7971940.769468 30.8396190.7938320.7721830.7513150.7117800.674972 40.7920940.7350300.7084250.6830130.6355180.592080 50.7472580.6805830.6499310.6209210.5674270.519369 60.7049610.6301700.5962670.5644740.5066310.455587 0.792094 Present Value of a Single Sum

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21 Table Approach $1,262.48 times 0.792094 equals $1,000. Present Value of a Single Sum

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22 Debbi Whitten wants to calculate the future value of four cash flows of $1,000, each with interest compounded annually at 6%, where the first cash flow is made on December 31, 2007. $1,000 Dec. 31, 2007 Dec. 31, 2008 Dec. 31, 2009 Dec. 31, 2010 The future value of an ordinary annuity is determined immediately after the last cash flow Future Value of an Ordinary Annuity

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23 Formula Approach (1 + i) - 1 n F o = C i WhereF = future value of an ordinary annuity of a series of cash flows of any amount C =amount of each cash flow n =number of cash flows i=interest rate for each of the stated time periods o Future Value of an Ordinary Annuity

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24 Formula Approach F o = n = 4, i = 6 = (1.06) – 1 4 = 4.37462 0.06 F o = $1,000(4.37462) = $4,374.62 Future Value of an Ordinary Annuity

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25 Table Approach Using the same data—four equal annual cash flows of $1,000 beginning on December 31, 2007, and an interest rate of 6 percent. Go to Table 2, the future value of an ordinary annuity of 1. Read the table value for n equals 4 and i equals 6%. Future Value of an Ordinary Annuity

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26 Table Approach n 6.0% 8.0% 9.0% 10.0% 12.0% 14.0% 11.0000001.0000001.0000001.0000001.0000001.000000 22.0600002.0800002.0900002.1000002.1200002.140000 33.1836003.2464003.2781003.3100003.3744003.439600 44.3746164.5061124.5731294.6410004.7793284.921144 55.6370935.8666015.9847116.1051006.3528476.610104 66.9753197.3359297.5233357.7156108.1151898.535519 4.374616 Future Value of an Ordinary Annuity

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27 So, cash flows of $1,000 each at 6% at the end of 2007, 2008, 2009, and 2010 will accumulate to a future value of $4,374.62. $1,000 x 4.374616 = $4, 374.62 Future Value of an Ordinary Annuity

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28 Cash Flows Unknown At the beginning of 2007, the Rexson Company issued 10-year bonds with a face value of $1,000,000 due on December 31, 2016. The company will accumulate a fund to retire these bonds at maturity. It will make annual deposits to the fund beginning on December 31, 2007. How much must the company deposit each year, assuming that the fund will earn 12% interest?

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29 Cash Flows Unknown Maturity value$1,000,000 Periods10 years Interest rate12% Future Value FV Annuity factor = Annual Cash flows for 10 periods $1,000,000 17.548735 = $56,984.16

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30 Kyle Vasby wants to calculate the present value on January 1, 2007, (one period before the first cash flow) of four future withdrawals (cash flows) of $1,000 each, with the first withdrawal being made on December 31, 2010. Assume again an interest rate of 6%. $1,000 Present Value of an Ordinary Annuity Dec. 31, 2007 Dec. 31, 2008 Dec. 31, 2009 Dec. 31, 2010 Jan. 1, 2007

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31 Go to Table 4, the present value of an ordinary annuity of 1. Read the table value for n equals 4 and i equals 6%. Present Value of an Ordinary Annuity

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32 Table Approach n 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.9615380.9523810.9433960.9345790.9259260.917431 2 1.8860951.8594101.8333931.8080181.7832651.759111 32.7750912.7232482.6730122.6243162.5770972.531296 43.628953.5459513.4651063.3872113.3121273.239720 54.4518224.3294774.2123644.1001973.9927103.889651 65.2421375.0756924.9173244.7665404.6228804.485919 3.465106 Present Value of an Ordinary Annuity

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33 Table Approach One thousand dollars times 3.46511 equals $3,465.11 So, the present value of this ordinary annuity is $3,465.11. Present Value of an Ordinary Annuity

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34 Cash Flows Unknown On January 1, 2007, Rex Company borrows $100,000 at 12% interest to finance a plant expansion project. Ten equal payment are to be made starting on December 31, 2007. What are the annual payments? Jan. 1, 2007 Dec. 31, 2007 Dec. 31, 2008 Dec. 31, 2009 Dec. 31, 2016 ? ?? ? The present value of 10 payments with first payment made one period later.

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35 Table Approach n 4.0% 5.0% 6.0% 7.0% 8.0% 12.0% 10.9615380.9523810.9433960.9345790.9259260.892857 2 1.8860951.8594101.8333931.8080181.7832651.690051 32.7750912.7232482.6730122.6243162.5770972.401831 43.628953.5459513.4651063.3872113.3121273.037349 54.4518224.3294774.2123644.1001973.9927103.604776 108.1108967.7217357.3600877.0235826.7100815.650223 5.650223 Present Value of an Ordinary Annuity $100,000 = Cash flows X 5.65023 Present value = Cash flows X PVA factor

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36 The present value of $100,000 divided by the present value of an annuity factor of 5.605223 equals an annual payment of $17,698.42, which includes both principle and interest. Present Value of an Ordinary Annuity

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37 The present value of an annuity due is determined on the date of the first cash flow in the series. Present Value of an Ordinary Annuity

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38 Barbara Livingston wants to calculate the present value of an annuity on December 31, 2007, which will permit four annual future receipts of $1,000 each, the first to be received immediately on December 31, 2007. $1,000 Dec. 31, 2007 Dec. 31, 2008 Dec. 31, 2009 Dec. 31, 2010 Present Value of an Annuity Due

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39 Look up the factor in the present value of an annuity due table (Table 5), for four periods at 6% Table Approach Present Value of an Annuity Due

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40 Table Approach n 5.0% 6.0 7.0% 8.0% 9.0% 10.0% 11.0000001.0000001.0000001.0000001.0000001.000000 21.9523811.9433961.9345791.9259261.9174311.909091 32.8594102.8333932.8080182.7832652.7591112.735537 43.7232483.6730123.6243163.5770973.5312953.486852 54.5459514.4651064.3872114.3121274.2397204.169865 65.3294775.2123645.1001974.9927104.8896514.790787 3.673012 Present Value of an Annuity Due

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41 One thousand dollars times 3.673012 equals $3,673.01. Table Approach Present Value of an Annuity Due

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42 Cash Flow Unknown Suppose that on Jan. 1, 2007, Katherine Spruill purchases an item that costs $10,000 and agrees to make 10 annual installments with interest of 8% starting immediately. What are her payments? Present value = Annual cash flow X PVAD factor OR Present value / PVAD factor = annual cash flow

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43 Table Approach n 5.0% 6.0 7.0% 8.0% 9.0% 10.0% 11.0000001.0000001.0000001.0000001.0000001.000000 21.9523811.9433961.9345791.9259261.9174311.909091 32.8594102.8333932.8080182.7832652.7591112.735537 43.7232483.6730123.6243163.5770973.5312953.486852 54.5459514.4651064.3872114.3121274.2397204.169865 68.1078227.8016927.5152327.2468886.9952476.759024 7.246888 Present Value of an Annuity Due

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44 The cash flow can be calculated by dividing $10,000 by the PVAD factor of 7.246888. Therefore, the annual payments, starting immediately, are $1,379.90 Present Value of an Annuity Due

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45 Helen Swain buys an annuity on January 1, 2007, that yields her four annual receipts of $1,000 each, with the first receipt on January 1, 2011. The interest rate is 6% compounded annually. What is the cost of the annuity? Present Value of a Deferred Ordinary Annuity

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46 $1,000 $1,000 $1,000 $1,000 Jan. 1, 2011 Jan. 1, 2012 Jan. 1, 2013 Jan. 1, 2014 Jan. 1, 2010 Jan. 1, 2009 Jan. 1, 2008 Jan.1, 2007 The present value of the deferred annuity is determined on this date $1,000 x 3.465106 (n=4, i=6) = $3,465.11 Present Value of a Deferred Ordinary Annuity

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47 Jan. 1, 2010 Jan. 1, 2009 Jan. 1, 2008 Jan.1, 2007 The present value of the deferred annuity is determined on this date $3,465.11 $3,465.11 x 0.839619 = $2,909.37 Present Value of a Deferred Ordinary Annuity

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48 If Helen buys an annuity for $2,909.37 on January 1, 2007, she can make four equal annual $1,000 withdrawals (cash flows) beginning on January 1, 2011. Present Value of a Deferred Annuity

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49 TVM Task Force Image Gallery clip art included in this electronic presentation is used with the permission of NVTech Inc.

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