Consider the following three cases case 1 compute the

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rates increase, and (2)present values decline as the time lengthens. Consider the following three cases: Case 1. Compute the present value of $100 to be received one year from today, discounted at 10% compounded semiannually: 7-29 Module 7 I Liability Recognition and Nonowner Financing Calculator N=2 l/Yr = 5 PMT = 0 FV = 100 Fo:703J Calculator N=4 l/Yr = 5 PMT = 0 FV = 100 PV = 82.270 Calculator N=4 l/Yr = 6 PMT = 0 FV = 100 [pv~-79-.2091 Number of periods (one year, semiannually) = 2 Rate per period (10%/2) = 5% Multiplier = 0.90703 Present value = $100.00 x 0.90703 = $90.70 (rounded) Case 2. Compute the present value of $100 to be received two years from today, discounted at 10% compounded semiannually: Number of periods (two years, semiannually) = 4 Rate per period (10%/2) = 5% Multiplier = 0.82270 Present value = $100 x 0.82270 = $82.27 (rounded) Case 3. Compute the present value of $100 to be received two years from today, discounted at 12% compounded semiannually: Number of periods (two years, semiannually) = 4 Rate per period (12%/2) = 6% Multiplier = 0.79209 Present value = $100 = 0.79209 = $79.21 (rounded) In Case 2, the present value of $82.27 is less than for Case I ($90.70) because the time increased from two to fo compounding periods-the longer we must wait for money, the lower its value to us today. Then in Case 3, present value of $79.21 was lower than in Case 2 because, while there were still four compounding periods, tbe interest rate per year was higher (12% annually instead of IO%)-the higher the interest rate the more interest tit could have been earned on the money and therefore the lower the value today. Present Value of an Annuity In the examples above, we computed the present value of a single amount (also called a lump sum) made received in the future. Often, future cash flows involve the same amount being paid or received each period. Ex- amples include semiannual interest payments on bonds, quarterly dividend receipts, or monthly insurance premi- ums. If the payment or the receipt (the cash flow) is equally spaced over time and each cash flow is the same doll amount, we have an annuity. One way to calculate the present value of the annuity would be to calculate the prese value of each future cash flow separately. However, there is a more convenient method. To illustrate, assume $100 is to be received at the end of each of the next three years as an annuity. When annu- ity amounts occur at the endojeachperiod, the annuity is called an ordinaryannuity. As shown below, the prese value of this ordinary annuity can be computed from Table I by computing the present value of each of the three individual receipts and summing them (assume a 5% annual rate).
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Module 7 I Liability Recognition and Nonowner Financing 7-30 Future Receipts (ordinary annuity) PV Multiplier Present Year 1 Year 2 Year 3 (Table 1) Value Calculator $100 x 0.95238 $ 95.24 N=3 $100 x 0.90703 90.70 IlYr = 5 $100 x 0.86384 86.38 PMT = 100 --- FV = 0 2.72325 $272.32 ipV = 272.32 I - 2 in Appendix A provides a single multiplier for computing the present value of an ordinary annuity. Refer- _ to Table 2 in the row for three periods and the column for 5%, we see that the multiplier is 2.72325. When ied to the $100 annuity amount, the multiplier gives a present value of $272.33. As shown above, the same nt value (with 1cent rounding error) is derived by summing the three separate multipliers from Table 1.Con-
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