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1 Appendix D Compound Interest 2 Objectives 1. Understand simple interest and compound interest. 2. Compute and use the future value of a single sum. 3. Compute and use the present value of a single sum. 4. Compute and use the future value of an ordinary annuity. 5. Compute and use the future value of an annuity due. Continued 3 Objectives 6. Compute and use the present value of an ordinary annuity. 7. Compute and use the present value of an annuity due. 8. Compute and use the present value of a deferred ordinary annuity. 9. Explain the conceptual issues regarding the use of present value in financial reporting. 4 Simple Interest Simple interest is interest on the original principal regardless of the number of time periods that have passed. Interest = Principal x Rate x Time 5 Compound Interest Compound interest is the interest that accrues on both the principal and the past unpaid accrued interest. 6 Compound Interest Value at Value at Beginning Compound End of Period of Quarter x Rate x Time = Interest Quarter 1st qtr. $10,000.00 x 0.12 x 1/4 $ 300.00 $10,300.00 2nd qtr. 10,300.00 x 0.12 x 1/4 309.00 10,609.00 3rd qtr. 10,609.00 x 0.12 x 1/4 318.27 10,927.27 4th qtr. 10,927.27 x 0.12 x 1/4 327.82 11,255.09 5th qtr. 11,255.09 x 0.12 x 1/4 337.65 11,592.74 Compound interest on $10,000 at 12% compounded quarterly for 5 quarters………………………... $1,592.74 7 Future Value of a Single Sum at Compound Interest One thousand dollars is invested in a savings account on December 31, 2004. What will be the amount in the savings account on December 31, 2008 if interest at 14% is compounded annually each year? How much will be in the $1,000 is invested savings account (the future on this date value) on this date? Dec. 31, Dec. 31, Dec. 31, Dec. 31, Dec. 31, 2004 2005 2006 2007 2008 8 Future Value of a Single Sum at Compound Interest (1) (2) (3) (4) Annual Future Value Value at Compound at End Beginning of Interest of Year Year Year (Col. 2 x 0.14) (Col. 2 + Col. 3) 2005 $1,000.00 $140.00 $1,140.00 2006 1,140.00 159.60 1,299.60 2007 1,299.60 181.94 1,481.54 2008 1,481.54 207.42 1,688.96 9 Future Value of a Single Sum at Compound Interest Formula Approach n ƒ = p(1 + i) 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 10 Future Value of a Single Sum at Compound Interest Formula Approach f = p(1 + i) n 4 fn=4, i=14 = (1.14) f = $1,000(1.688960) = $1,688.96 11 Future Value of a Single Sum at Compound Interest Table Approach This time we will use a table to determine how much $1,000 will accumulate to in four years at 14% compounded annually. 12 Future Value of a Single Sum at Compound Interest Table Approach Using Table 1 (the future value of 1) at the end of Appendix D, determine the table value for an annual interest rate of 14 percent and four periods. 13 Future Value of a Single Sum at Compound Interest Table Approach n 8.0% 9.0% 10.0% 12.0% 14.0% 16.0% 1 1.080000 1.090000 1.100000 1.120000 1.140000 1.160000 2 1.166400 1.188100 1.210000 1.254400 1.299600 1.345600 3 1.259712 1.295029 1.331000 1.404928 1.481544 1.560896 4 1.360489 1.411582 1.464100 1.573519 1.688960 1.688960 1.810639 5 1.469328 1.538624 1.610510 1.762342 1.925415 2.100342 6 1.586874 1.677100 1.771561 1.973823 2.194973 2.436396 14 Future Value of a Single Sum at Compound Interest Table Approach One thousand dollars times 1.688960 equals the future value, or $1,688.96. 15 Present Value of a Single Sum If $1,000 is worth $1,688.96 when it earns 14% compounded annually for 4 years, then it follows that $1,688.96 to be received in 4 years from now is worth $1,000 now at time period zero. $1,000 (the present value) For $1,688.96 to be must be invested received on this date on this date Dec. 31, Dec. 31, Dec. 31, Dec. 31, Dec. 31, 2004 2005 2006 2007 2008 16 Present Value of a Single Sum Formula Approach 1 p = f (1 + i) n Where p = 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 17 Present Value of a Single Sum Formula Approach 1 p n=4, i=14 = (1 .14) 4 = 0.592080 p = $1,688.96(0.592080) = $1,000.00 18 Present Value of a Single Sum Table Approach Use 14% 3, the present Find Tableand four value of obtain end periods to1, at thethe of Appendix table value. D. 19 Present Value of a Single Sum Table Approach n 8.0% 9.0% 10.0% 12.0% 14.0% 16.0% 1 0.925926 0.917431 0.909091 0.892857 0.877193 0.862069 2 0.857339 0.841680 0.826446 0.797194 0.769468 0.743163 3 0.793832 0.772183 0.751315 0.711780 0.674972 0.640658 4 0.735030 0.708425 0.683013 0.635518 0.592080 0.592080 0.552291 5 0.680583 0.649931 0.620921 0.567427 0.519369 0.476113 6 0.630170 0.596267 0.564474 0.506631 0.455587 0.410442 20 Present Value of a Single Sum Table Approach $1,688.96 times 0.592080 equals $1,000. 21 Future Value of an Ordinary Annuity The future value Debbi Whitten wants to calculate the future value of of an ordinary four cash flows of $1,000, each with interest annuity compounded annually at 14%, where the first cash is flow is made on December 31, 2004. determined immediately after the last cash flow $1,000 $1,000 $1,000 $1,000 Dec. 31, Dec. 31, Dec. 31, Dec. 31, 2004 2005 2006 2007 22 Future Value of an Ordinary Annuity Formula Approach n (1 + i) - 1 Fo = C i Where Fo= 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 23 Future Value of an Ordinary Annuity Formula Approach (1 .14)4 – 1 Fo = n=4, i=14 = = 4.921144 0.14 Fo = $1,000(4.921144) = $4,921.14 24 Future Value of an Ordinary Annuity Table Approach Using the same data—four Go to Table 2, the future value of equal annual cash flows of an ordinary annuity of 1. Read $1,000 beginning on December 31, 2004 and an interest rate and the table value for n equals 4 of i equals 14%. 14 percent. 25 Future Value of an Ordinary Annuity Table Approach n 8.0% 9.0% 10.0% 12.0% 14.0% 16.0% 1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 2 2.080000 2.090000 2.100000 2.120000 2.140000 2.160000 3 3.246400 3.278100 3.310000 3.374400 3.439600 3.505600 4 4.506112 4.573129 4.641000 4.779328 4.921144 4.921144 5.066496 5 5.866601 5.984711 6.105100 6.352847 6.610104 6.877135 6 7.335929 7.523335 7.715610 8.115189 8.535519 8.977477 26 Future Value of an Ordinary Annuity So, cash flow of $1,000 each at 14% at the end of 2004, 2005, 2006, and 2007 will accumulate to a future value of $4,921.14. $1,000 x 4.921144 = $4,921.14 27 Future Value of an Annuity Due Solutions Approach How much will be in the fund on this date, which is 1 period after the last cash flow in the series? $1,000 $1,000 $1,000 $1,000 Dec. 31, Dec. 31, Dec. 31, Dec. 31, 2004 2005 2006 2007 28 Future Value of an Annuity Due Solutions Approach Step 1: In the ordinary annuity table (Table 2), look up the value of n + 1 cash flows at 14% or the value of 5 cash flows at 14%. 29 Future Value of an Annuity Due Solutions Approach n 8.0% 9.0% 10.0% 12.0% 14.0% 16.0% 1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 2 2.080000 2.090000 2.100000 2.120000 2.140000 2.160000 3 3.246400 3.278100 3.310000 3.374400 3.439600 3.505600 4 4.506112 4.573129 4.641000 4.779328 4.921144 5.066496 5 5.866601 5.984711 6.105100 6.352847 6.610104 6.610104 6.877135 6 7.335929 7.523335 7.715610 8.115189 8.535519 8.977477 30 Future Value of an Annuity Due Solutions Approach Step 1: In the ordinary annuity table (Table 2), look up the value of n + 1 cash flows at 14% or the value of 5 cash flows at 14%. 6.610104 Step 2: Subtract 1 without interest. (1.000000) Table value 5.610104 31 Future Value of an Annuity Due Solutions Approach Step 3: Multiply the amount of each cash flow ($1,000) by the table value from Step 2. Fd = $1,000(5.610104) = $5,610.10 32 Future Value of an Annuity Due So, if $1,000 is deposited …a cumulative total of annually for four years $5,610 can be withdrawn beginning on December on December 31, 2008. 31, 2004… 33 Present Value of an Ordinary Annuity Table Approach Kyle Vasby wants to calculate the present value on January 1, 2004 (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, 2004. Assume again an interest rate of 14%. $1,000 $1,000 $1,000 $1,000 Dec. 31, Dec. 31, Dec. 31, Dec. 31, 2004 2005 2006 2007 34 Present Value of an Ordinary Annuity Go to Table 4, the present value of an ordinary annuity of 1. Read the table value for n equals 4 and i equals 14%. 35 Present Value of an Ordinary Annuity Table Approach n 8.0% 9.0% 10.0% 12.0% 14.0% 16.0% 1 0.925926 0.917431 0.909091 0.892857 0.877193 0.862069 2 1.783265 1.759111 1.735537 1.690051 1.646661 1.605232 3 2.577097 2.531295 2.486852 2.401831 2.321632 2.245890 4 3.312127 3.239720 3.169865 3.037349 2.913712 2.913712 2.798181 5 3.992710 3.889651 3.790787 3.604776 3.433081 3.274294 6 4.622880 4.485919 4.355261 4.111407 3.888668 3.684736 36 Present Value of an Ordinary Annuity Table Approach One thousand dollars times 2.913713 equals $2,913.71. So, the present value of this ordinary annuity is $2,913.71. 37 Present Value of an Annuity Due Table Approach Barbara Livingston wants to calculate the present value of an annuity on December 31, 2004, which will permit four annual future receipts of $1,004 each, the first to be received on December 31, 2004. $1,000 $1,000 $1,000 $1,000 Dec. 31, Dec. 31, Dec. 31, Dec. 31, 2004 2005 2006 2007 38 Present Value of an Annuity Due Table Approach Step 1: In the ordinary annuity table (Table 4), look up the value of n – 1 cash flows at 14% or the value of 3 cash flows at 14%. 39 Present Value of an Annuity Due Table Approach n 8.0% 9.0% 10.0% 12.0% 14.0% 16.0% 1 0.925926 0.917431 0.909091 0.892857 0.877193 0.862069 2 1.783265 1.759111 1,735537 1.690051 1.546661 1.605232 3 2.577097 2.531295 2.485852 2.402831 2.321632 2.321632 2.245890 4 3.312127 3.329720 3.159865 3.037349 2.913712 2.798181 5 3.992710 3.889651 3.790787 3.604776 3.443081 3.274294 6 4.622880 4.485919 4.355261 4.111407 3.888668 3.684736 40 Present Value of an Annuity Due Table Approach Step 1: In the ordinary annuity table (Table 4), look up the value of n – 1 cash flows at 14% or the value of 3 cash flows at 14%. 2.321632 Step 2: Add 1 without interest. 1.000000 3.321632 41 Present Value of an Annuity Due Table Approach One thousand dollars times 3.321632 equals $3,321.63. So, this is the present value of an ordinary annuity due. 42 Present Value of a Deferred Ordinary Annuity Table Approach Helen Swain buys an annuity on January 1, 2004 that yields her four annual payments of $1,000 each, with the first payment on January 1, 2008. The interest rate is 14% compounded annually. What is the cost of the annuity? 43 Present Value of a Deferred Ordinary Annuity The present Table Approach value of the deferred $1,000 $1,000 $1,000 $1,000 annuity is determined Jan. 1, Jan. 1, Jan. 1, Jan. 1, on this date 2008 2009 2011 2010 Jan.1, Jan. 1, Jan. 1, Jan. 1, 2004 2005 2006 2007 $1,000 x 2.913712(n=4, i=14) = $2,913.71 44 Present Value of a Deferred Ordinary Annuity The present Table Approach value of the deferred annuity is $2,913.71 determined on this date Jan.1, Jan. 1, Jan. 1, Jan. 1, 2004 2005 2006 2007 $2,913.71 x 0.674972 = $1,966.67 45 Present Value of a Deferred Ordinary Annuity If Helen buys an annuity for $1,966.67 on January 1, 2004, she can make four equal annual $1,000 withdrawals (cash flows) beginning on January 1, 2008. 46 Appendix D The End 47