In present value problems involving annuities, there are four variables: (1) present value of an ordinary annuity (PVA) or present value of an annuity due (PVAD), (2) the amount of each annuity payment, (3) the number of periods, n, and (4) the interest rate, i. If you know any three of these, the fourth can be determined.   ● LO8 
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ILLUSTRATION 611 Determining the Annuity Amount When Other Variables Are Known 
 Assume that you borrow $700 from a friend and intend to repay the amount in four equal annual installments beginning one year from today. Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. What is the required annual payment that must be made (the annuity amount), to repay the loan in four years? 
The following time diagram illustrates the situation:
  Determining the unknown annuity amount—ordinary annuity. 
The required payment is the annuity amount that will provide a present value of $700 when discounting that amount at a discount rate of 8%:
(K)   The unknown variable is the annuity amount. 
Rearranging algebraically, we find that the annuity amount is $211.34.
(K) You would have to make four annual payments of $211.34 to repay the loan. Total payments of $845.36 (4 × $211.34) would include $145.36 in interest ($845.36 − 700.00).
ILLUSTRATION 612 Determining n When Other Variables Are Known 
 Assume that you borrow $700 from a friend and intend to repay the amount in equal installments of $100 per year over a period of years. The payments will be made at the end of each year beginning one year from now. Your friend wishes to be reimbursed for the time value of money at a 7% annual rate. How many years would it take before you repaid the loan? 
Once again, this is an ordinary annuity situation because the first payment takes place one year from now. The following time diagram illustrates the situation:
  Determining the unknown number of periods—ordinary annuity. 
The number of years is the value of n that will provide a present value of $700 when discounting $100 at a discount rate of 7%:
(K)   The unknown variable is the number of periods. 
Rearranging algebraically, we find that the PVA table factor is 7.0.
(K)
p. 318 When you consult the PVA table, Table 4, you search the 7% column (i = 7%) for this value and find 7.02358 in row 10. So it would take approximately 10 years to repay the loan in the situation described.
ILLUSTRATION 613 Determining i When Other Variables Are Known 
 Suppose that a friend asked to borrow $331 today (present value) and promised to repay you $100 (the annuity amount) at the end of each of the next four years. What is the annual interest rate implicit in this agreement? 
First of all, we are dealing with an ordinary annuity situation as the payments are at the end of each period. The following time diagram illustrates the situation:   Q2, p.299 
(K)   Determining the unknown interest rate—ordinary annuity. 
The interest rate is the discount rate that will provide a present value of $331 when discounting the $100 fouryear ordinary annuity:
(K)   The unknown variable is the interest rate. 
Rearranging algebraically, we find that the PVA table factor is 3.31.
(K) When you consult the PVA table, Table 4, you search row four (n = 4) for this value and find it in the 8% column. So the effective interest rate is 8%.
ILLUSTRATION 614 Determining i When Other Variables Are Known—Unequal Cash Flows 
 Suppose that you borrowed $400 from a friend and promised to repay the loan by making three annual payments of $100 at the end of each of the next three years plus a final payment of $200 at the end of year four. What is the interest rate implicit in this agreement? 
The following time diagram illustrates the situation:
  Determining the unknown interest rate—unequal cash flow. 
The interest rate is the discount rate that will provide a present value of $400 when discounting the $100 threeyear ordinary annuity plus the $200 to be received in four years:
(K)   The unknown variable is the interest rate. 
p. 319 This equation involves two unknowns and is not as easily solved as the two previous examples. One way to solve the problem is to trialanderror the answer. For example, if we assumed i to be 9%, the total PV of the payments would be calculated as follows:
(K) Because the present value computed is less than the $400 borrowed, using 9% removes too much interest. Recalculating PV with i = 8% results in a PV of $405. This indicates that the interest rate implicit in the agreement is between 8% and 9%.  ANNUITIES Using the appropriate table, answer each of the following independent questions. 1.   What is the future value of an annuity of $2,000 invested at the end of each of the next six periods at 8% interest?  2.   What is the future value of an annuity of $2,000 invested at the beginning of each of the next six periods at 8% interest?  3.   What is the present value of an annuity of $6,000 to be received at the end of each of the next eight periods assuming an interest rate of 10%?  4.   What is the present value of an annuity of $6,000 to be received at the beginning of each of the next eight periods assuming an interest rate of 10%?  5.   Jane bought a $3,000 audio system and agreed to pay for the purchase in 10 equal annual installments of $408 beginning one year from today. What is the interest rate implicit in this agreement?  6.   Jane bought a $3,000 audio system and agreed to pay for the purchase in 10 equal annual installments beginning one year from today. The interest rate is 12%. What is the amount of the annual installment?  7.   Jane bought a $3,000 audio system and agreed to pay for the purchase by making nine equal annual installments beginning one year from today plus a lumpsum payment of $1,000 at the end of 10 periods. The interest rate is 10%. What is the required annual installment?  8.   Jane bought an audio system and agreed to pay for the purchase by making four equal annual installments of $800 beginning one year from today plus a lumpsum payment of $1,000 at the end of five years. The interest rate is 12%. What was the cost of the audio system? (Hint: What is the present value of the cash payments?)  9.   Jane bought an audio system and agreed to pay for the purchase by making five equal annual installments of $1,100 beginning four years from today. The interest rate is 12%. What was the cost of the audio system? (Hint: What is the present value of the cash payments?) 
SOLUTION 1.   FVA = $2,000 × 7.3359^{*} = $14,672
^{*} Future value of an ordinary annuity of $1: n = 6, i = 8% (from Table 3)  2.   FVAD = $2,000 × 7.9228^{*} = $15,846
^{*} Future value of an annuity due of $1: n = 6, i = 8% (from Table 5)  3.   PVA = $6,000 × 5.33493^{*} = $32,010
^{*} Present value of ordinary annuity of $1: n = 8, i = 10% (from Table 4)  4.   PVAD = $6,000 × 5.86842^{*} = $35,211
^{*} Present value of an annuity due of $1: n = 8, i = 10% (from Table 6)  5.   (K)
^{*} Present value of an ordinary annuity of $1: n = 10, i = ? (from Table 4, i approximately 6%)  6.   (K)
^{*} Present value of an ordinary annuity of $1: n = 10, i = 12% (from Table 4)  7.   (K)
^{*} Present value of an ordinary annuity of $1: n = 9, i = 10% (from Table 4)
^{†} Present value of $1: n = 10, i = 10% (from Table 2)  8.   PV = ($800 × 3.03735^{*}) + ($1,000 × .56743^{†} = $2,997
^{*} Present value of an ordinary annuity of $1: n = 4, i = 12% (from Table 4)
^{†} Present value of $1: n = 5, i = 12% (from Table 2)  9.   PVA = $1,100 × 3.60478^{*} = $3,965
^{*} Present value of an ordinary annuity of $1: n = 5, i = 12% (from Table 4) This is the present value three years from today (the beginning of the fiveyear ordinary annuity). This single amount is then reduced to present value as of today by making the following calculation: PV = $3,965 × .71178^{†} = $2,822
^{†}Present value of $1: n = 3, i = 12%, (from Table 2) 


