PRESENT VALUE – COMPOUND INTEREST Accounting Theory Present value concepts are practical in accounting for they determine the accounting values and resulting entries for certain monetary transactions. Various present value uses were described under leases and bonds. Additional uses of present value concepts are described in this section. They involve the concept of compound interest. Compound interest is accrued interest that after a period of time such as monthly is added to the original amount and thereafter the interest is computed on both the original amount and the accrued interest. Compound interest involves the following: Amount of 1 Amount of 1 - Determines what the amount of a sum invested today will amount to in the future at a compound interest rate. Amount of Annuity of 1 Amount of 1 - Determines what the amount of a sum of equal periodic investments will amount to in the future at a compound interest rate. The table of values determines the value at the beginning of the period. The equal periodic payments are made in the early years to fund a future amount. The payments can be made at the beginning or end of the year. Present Value of 1 Present Value of 1 - Determines what the future amount desired is worth today when discounted at a compound interest rate. Present Value of Annuity of 1 Present Value of 1 - Determines what the future amount desired of a sum of equal periodic payments is worth today when discounted at a compound interest rate. The table of values determines the value at the end of the period. The equal periodic payments are made in the years after the funding (Amount of Annuity of 1). The payments can be made at the beginning or end of the year. This table is also used to determine mortgage payments and installment note payments. These payments are an annuity. An interest rate is usually stated as an annual rate. The time period interest is compounded however may be annually, semiannually, quarterly or monthly. For example interest at 12% compounded annually. Interest would be 6% for each interest period if compounded semiannually. Interest would be 3% for each interest period if compounded quarterly. Interest would be 1% for each interest period if compounded monthly. Compound Interest Tables PRESENT VALUE – COMPOUND INTEREST Amount of 1 - Compound Interest Rate Table Calculate Future Amount If $1,000 earns 5% annually what will it accumulate to in 10 years: Using the years column and interest rate column of an amount of 1 compound interest rate table. $1,000 * 1.62889463 = $1,628.89. If $1,000 earns 5% semiannually what will it accumulate to in 10 years: 2.5% for 20 periods. Using the years column and interest rate column of an amount of 1 compound interest rate table. $1,000 * 1.63861644 = $1,638.62. Calculate Interest Rate What interest rate will be necessary for $1,000 to accumulate to $1,628.89 in 10 years: Using the years column of an amount of 1 compound interest rate table. $1,628.89 / $1,000 = 1.62889 = 5%. Calculate Number of Periods How many periods will be necessary for $1,000 to accumulate to $1,628.89 at 5% compounded annually: Using the interest rate column of an amount of 1 compound interest rate table. $1,628.89 / $1,000 = 1.62889 = 10 periods. Calculate Present Value Amount How much must be invested now to accumulate to $1,628.89 at 5% compounded annually in 10 years: Using the years column and interest rate column of an amount of 1 compound interest rate table. $1,628.89 / 1.6289 = $1,000. Amount of Annuity of 1 - Amount of Annuity of 1 Table Calculate Future Amount If $1,000 invested annually for 5 years earns 5% annually what will it accumulate to at the beginning of year 5: Using the years column and interest rate column of an amount of annuity of 1 compound interest rate table. $1,000 * 5.5256313 = $5,525.63. Calculate Future Amount If $1,000 invested annually for 5 years earns 5% annually what will it accumulate to at the end of year 5: Using the years column and interest rate column of an amount of annuity of 1 compound interest rate table. $1,000 * 5.8019128 = $5,801.91. (6.8019128 - 1 = 5.8019128) Proof: $5,525.63 * 1.05 = $5,801.91. Calculate Interest Rate What interest rate will be necessary for $1,000 invested annually to accumulate to $5,525.63 at the beginning of year 5: Using the years column of an amount of annuity of 1 compound interest rate table. $5,525.63 / $1,000 = 5.2563 = 5%. Calculate Number of Periods How many periods will be necessary for $1,000 invested annually to accumulate to $5,525.63 at the beginning of the period at 5% compounded annually : Using the interest rate column of an amount of annuity of 1 compound interest rate table $5,525.63 / $1,000 = 5.52563 = 5 periods. PRESENT VALUE – COMPOUND INTEREST Calculate Present Value Amount How much must be invested now to accumulate to $5,525.63 at the beginning of year 5 at 5% compounded annually: Using the years column and interest rate column of an amount of annuity of 1 compound interest rate table. $5,525.63 / 5.52563 = $1,000. Calculate Present Value Amount How much must be invested now to accumulate to $5,801.91 at the end of year 5 at 5% compounded annually: Using the years column and interest rate column of an amount of annuity of 1 compound interest rate table. $5,801.91 / 5.80191 = $1,000. (6.8019128 - 1 = 5.8019128) Present Value of 1 Calculate Present Value Amount How much must be invested now to accumulate to $1,628.89 at 5% compounded annually in 10 years: Using the years column and interest rate column of a present value of 1 compound interest rate table. $1,628.89 * .61391325 = $1,000. The amount can also be determined by using the amount of 1 compound interest rate table as described above as follows: Calculate Present Value Amount How much must be invested now to accumulate to $1,628.89 at 5% compounded annually in 10 years: Using the years column and interest rate column of an amount of 1 compound interest rate table. $1,628.89 / 1.6289 = $1,000. 1 / PV .61391325 = 1.6289. Present Value of Annuity of 1 Calculate Present Value Amount How much must be invested now in order to receive $30,000 annually beginning in 1year over a 15 year period interest at 5% compounded annually: Using the years column and interest rate column of a present value of annuity of 1 compound interest rate table. $30,000 * 10.3796580 = $311,389.74. Illustration Calculate Present Value Amount How much must be invested now in order to receive $30,000 annually beginning immediately over a 15 year period interest at 5% compounded annually: Using the years column and interest rate column of a present value of annuity of 1 compound interest rate table. $30,000 * 10.8986409 = $326,959.23. (9.8986409 +1 = 10.8986409) Illustration Calculate Interest Rate What interest rate will be necessary for a present value amount of $311,389.74 to provide for annual payments of $30,000 beginning in one year over a period of 15 years: Using the years column of a present value of annuity of 1 compound interest rate table. $311,389.74 / $30,000 = 10.3796580 = 5%. PRESENT VALUE – COMPOUND INTEREST Calculate Number of Periods How many periods will be necessary for a present value amount of $311,389.74 to provide for annual payments of $30,000 beginning in one year interest at 5% compounded annually: Using the interest rate column of a present value of annuity of 1 compound interest rate table $311,389.74 / $30,000 = 10.3796580 = 15 periods. Calculate Future Value Amount What is the future value of $311,389.74 invested now to be received over 15 years beginning in one year interest at 5% compounded annually: Using the years column and interest rate column of a present value of annuity of 1 compound interest rate table. $311,389.74 / 10.3796580 = $30,000. Calculate Future Value Amount What is the future value of $326,959.23 invested now to be received over 15 years beginning immediately interest at 5% compounded annually: Using the years column and interest rate column of a present value of annuity of 1 compound interest rate table. $326,959.23 / 10.8986409 = $30,000. (9.8986409 +1 = 10.8986409) See Deferred Annuity Below when payments begin after one year (deferral period) Mortgages, Installment Notes and Leases Mortgage or Installment Loan - Calculation of the Future Value Amount What would be the annual mortgage payment for a 20 year mortgage for $250,000... interest at an annual rate of 5%...the first payment beginning in one year: Using the years column and interest rate column of a present value of annuity of 1 compound interest rate table. $250,000 / 12.4622103 = $20,060.65. Early Payoff of a Debt obligation - Mortgage, Installment Loan, Lease Using the mortgage example above the debtor wants to pay off the mortgage 5 years sooner than the term of the mortgage. The interest rate on the mortgage is an annual rate of 5%. The lender agrees to discount the mortgage to an annual rate of 4%. There are 5 annual payments remaining. 4 payments Interest Rate 4% 3.6298952 1 payment 1.000000 Total 4.6298952 Annual payment $20,060.65 Total present value $92,878.71 Illustration PRESENT VALUE – COMPOUND INTEREST Deferred Annuities A deferred annuity exists when the amount to be received annually for a period of time begins after a one year period. Thus the payments are deferred and less has to be funded immediately because of compound interest earned during this deferral period. Calculate Present Value Amount How much must be invested now in order to receive $30,000 annually beginning in year 6 over a 15 year period interest at 5% compounded annually: This is a 15 year annuity deferred 5 years. Using the years column and interest rate column of a present value of annuity of 1 compound interest rate table. $30,000 * 10.3796580 = $311,389.74 Discount of 5 years at 5% .78352617 Present Value $243,982.01 Illustration Calculate Interest Rate What interest rate will be necessary for a present value amount of $243,982.01 to provide for annual payments of $30,000 beginning in year 6 over a period of 15 years: Using the years column of a present value of annuity of 1 compound interest rate table. $243,982.01 / .78352617 = $311,389.74 / $30,000 = 10.3796580 = 5%. Calculate Number of Periods How many periods will be necessary for a present value amount of $243.982.01 to provide for annual payments of $30,000 beginning in year 6 interest at 5% compounded annually: Using the interest rate column of a present value of annuity of 1 compound interest rate table. $243,982.01 / .78352617 = $311,389.74 / $30,000 = 10.3796580 = 15 periods. Calculate Future Value Amount If $243,982.01 is invested now how much will be received annually beginning in year 6 over a 15 year period interest at 5% compounded annually: Using the years column and interest rate column of a present value of annuity of 1 compound interest rate table. $243,982.01 / .78352617 = $311,389.74 / 10.3796580 = $30,000.00. Small Business is the Engine that Drives our Economy. The Men and Women who Work to make our Country Great Should be Recognized for their Achievement and Courage in Very Difficult Economic Times

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