___________________________________________________________________________ The steps in solving future value and present value problems (listed in Illustration A-3 are illustrated below and on the following pages:
1. If $10,000 is deposited in the bank today at 8% interest compounded annually, what
will be the balance in 5 years?
Step 1: This is a future value of a single sum problem. Step 2: n = 5; i = 8% $10,000 PV FV 0 1 2 3 4 5 ? n = 5; i = 8%
Step 3: The interest factor from Table 1 is 1.46933. Step 4: Future Value = Present Value x FVFn,i
Future Value = $10,000 x 1.46933 Future Value = $14,693.30
2. A company needs $100,000 to retire debt when the debt matures two years from
now. What amount must be deposited on January 1, 2014 at 8% interest compounded quarterly in order to accumulate the desired sum by January 1, 2016?
Step 1: This is a present value of a single sum problem.
Step 2: It is 2 years from 1/1/14 to 1/1/16. The annual interest rate is 8%. n = 2 x 4 = 8;
i = 8% ÷ 4 = 2%. ? $100,000 = 8; n 0 1 2 3 4 5 6 7 8 FV PV i = 2%
Step 3: The interest factor from Table 2 is .85349. Step 4: Present Value = Future Value x PVFn,i
Present Value = $100,000 x .85349 Present Value = $85,349.00
6-10 Problem Solving Survival Guide for Intermediate Accounting, 15th Edition
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ILLUSTRATION 6-4
(Continued)___________________________________________________________________________
3. If $71,178 can be invested now, what annual interest rate must be earned in order to
accumulate $100,000 three years from now?
Step 1: This can be solved either as a future value or as a present value of a single sum problem. This solution illustrates the present value approach.
Step 2: n = 3; i must be solved for.
0 1 2 3
$71,178 $100,000
FV
PV n = 3; i = ?
Step 3: i must be solved for.
Step 4: Present Value = Future Value x PVFn,i $71,178 = $100,000 x PVFn,i
$71,178 ÷ $100,000 = PVFn,i .71178 = PVFn,i
Refer to Table 2 in the 3 period row.
i = 12%
4. If $1,000 is deposited into an account at the end of every year for six years, what will
be the balance in the account after the sixth deposit if all amounts on deposit earn 6% interest?
Step 1: This is a future value of an ordinary annuity problem. Step 2: n = 6; i = 6% FV 0 1 2 3 4 5 ? n = 6; i = 6% 6 $1,000 $1,000 $1,000 $1,000 $1,000 $1,000
Step 3: The interest factor from Table 3 is 6.97532.
Step 4: Future Value of an Ordinary Annuity = Rent x FVF-OAn,i Future Value of an Ordinary Annuity = $1,000 x 6.97532 Future Value of an Ordinary Annuity = $6,975.32
Accounting and the Time Value of Money 6-11 ____________________________________________________________________________
ILLUSTRATION 6-4
(Continued)___________________________________________________________________________
5. What amount must be deposited at 10% in an account on January 1, 2014 if it is
desired to make equal annual withdrawals of $10,000 each, beginning on January 1, 2015 and ending on January 1, 2018?
Step 1: This is a present value of an ordinary annuity problem. Step 2: The time diagram shows 4 withdrawals. n = 4; i = 10%
0 1 2 3 4
1/1/14 1/1/15 1/1/16 1/1/17 1/1/18
$10,000 $10,000 $10,000 $10,000
?
PV n = 4; i = 10%
Step 3: The interest factor from Table 4 is 3.16986.
Step 4: Present Value of an Ordinary Annuity = Rent x PVF-OAn,i Present Value of an Ordinary Annuity = $10,000 x 3.16986 Present Value of an Ordinary Annuity = $31,698.60
6. Beginning today, six annual deposits of $1,000 each will be made into an account
paying 6%. What will be the balance in the account one year after the sixth deposit is made?
Step 1: This is a future value of an annuity due problem. Step 2: n = 6; i = 6%. FV 0 1 2 3 4 5 ? n = 6; i = 6% 6 $1,000 $1,000 $1,000 $1,000 $1,000 $1,000
Step 3: Table 3 with factors for future value of an ordinary annuity (FVF-OAn,i) can be used to derive the factor needed here for future value of an annuity due (FVF-ADn,i). The process is as follows:
FVF-OA for n = 6, i = 6% 6.97532
Multiplied by 1 + i 1.06 FVF-AD for n = 6, i = 6% 7.39384 Step 4: Future Value of an Annuity Due = Rent x FVF-ADn,i
Future Value of Annuity Due = $1,000 x 7.39384 Future Value of Annuity Due = $7,393.84
6-12 Problem Solving Survival Guide for Intermediate Accounting, 15th Edition
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ILLUSTRATION 6-4
(Continued)___________________________________________________________________________
TIP: Compare the results of this problem with those of Problem 4 above. The solution to problem
4 can be multiplied by (1 + i) to get the answer to number 6. Proof: $6,975.32 x 1.06 = $7,393.84.
Although both situations use the same number of equal rents and the same interest rate, the interest is earned on all of the deposits for one period more under the annuity due situation.
7. What is the present value of four annual payments of $10,000 each if interest is 10%
and the first payment is made today?
Step 1: This is a present value of an annuity due problem. Step 2: n = 4; i = 10%.
0 1 2 3 4
$10,000 $10,000 $10,000 $10,000 ?
PV n = 4; i = 10%
Step 3: The interest factor from Table 5 is 3.48685.
This factor can also be derived by using the present value of an ordinary annuity table (Table 4) as follows:
PVF-OA for n = 4, i = 10% 3.16986 Multiplied by 1 + i 1.10 PVF-AD for n = 4, i = 10% 3.48685 Step 4: Present Value of an Annuity Due = Rent x PVF-ADn,i
Present Value of an Annuity Due = $10,000 x 3.48685 Present Value of an Annuity Due = $34,868.50
TIP: Compare the results of this problem with those of Problem 5 above. The solution to problem
5 can be multiplied by (1 + i) to get the answer to number 7. Proof: $31,698.60 x 1.10 = $34,868.46
(Difference of $.04 is due to the rounding of the factors.)
Although both situations use the same number of equal rents and the same interest rate, the discounting is done on all of the deposits for one period less under the annuity due situation.
Accounting and the Time Value of Money 6-13 ____________________________________________________________________________
ILLUSTRATION 6-4
(Continued)___________________________________________________________________________
8. What amount must be deposited at the end of each year in an account paying 8%
interest if it is desired to have $10,000 at the end of the fifth year?
Step 1: This is a future value of an ordinary annuity problem. Step 2: n = 5; i = 8%. $10,000 0 1 2 3 4 5 FV = 5; n i = 8% R R R R R
Step 3: The interest factor from Table 3 is 5.86660.
Step 4: Future Value of an Ordinary Annuity = Rent x FVF-OAn,i $10,000 = Rent x 5.86660
$10,000 ÷ 5.86660 = Rent Rent = $1,704.56
TIP: You can prove this solution by: $1,704.56 x 5.86660 = $9,999.97
The difference of $.03 is due to rounding.
6-14 Problem Solving Survival Guide for Intermediate Accounting, 15th Edition
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