Solution manual intermediate accounting 9e 2

The present value of 1 is the amount that must be invested today in order to grow to 1 in a given number of periods at a given compound interest rate.. The present value of 1 tells how m

APPENDIX D

COMPOUND INTEREST

CONTENT ANALYSIS OF EXERCISES AND PROBLEMS

(minutes) ED-1 Future Value Single investment, compound interest 5-10 ED-2 Future Value Single investment, compound interest 5-10

ED-4 Future Value Ordinary annuity, interest compounded

ED-7 Amount of an Annuity Various annual withdrawal dates,

ED-8 Amount of Each Cash Flow Present value, calculate monthly

ED-9 Amount of Each Cash Flow Present value and future value,

ED-10 Amount of an Annuity Different future value dates, amount of

ED-11 Compound Interest Future value and present value, ordinary

annuity and annuity due, withdrawal determination 15-20 ED-12 Amount of an Annuity Ordinary annuity Deposits,

ED-13 Present Value of Leased Asset Lease payments Annuity due

ED-14 Number of Cash Flows Future value, interest compounded

Number Content Time Range (minutes) PD-2 Present Value Various single sums, compound interest 15-30 PD-3 Future Value Annuity due, ordinary annuity, compound interest 15-30 PD-4 Amount of Each Cash Flow Different present value dates, interest

PD-5 Present Value Ordinary annuity, annuity due, deferred annuity

PD-6 Present Value Ordinary annuity, annuity due, compound interest 20-30 PD-7 Compound Interest Issues Future value, installment determinations 30-40 PD-8 Amount of an Annuity Ordinary annuity, present value, withdrawal

PD-9 Amount of Each Cash Flow Numerous first withdrawal dates, future

PD-10 Number of Cash Flows Present value, future value, compound interest 20-30 PD-11 Serial Installments Future value, amounts applicable to interest and

PD-12 Determining Loan Repayments Present value, recalculation of cash

PD-13 Purchase of Asset Alternative financing plans to acquire asset

PD-14 Fund to Retire Bonds Ordinary annuity, future value, interest

PD-15 Asset Purchase Price Given future cash inflows, compute purchase

PD-16 Acquisition of Asset Compute cost of asset, record purchase, and

PD-17 Present Value Issues Four different payment plans, determine smallest

PD-18 (AICPA adapted) Comprehensive: Compound Interest Issues 30-45

ANSWERS TO QUESTIONS

QD-1 Interest is the cost of the use of money over time Interest and the price of any

merchandise item are similar because both are costs associated with items acquired by

a company

QD-2 Simple interest is interest only on the principal amount There is no compounding of

interest on "previously earned" interest when computations are based on simple interest Compound interest is interest that accrues on past unpaid accrued interest, as well as on the principal

The time value of money is interest This term indicates that a dollar held today is worth more than a dollar to be received a year from now because a dollar today can be invested to earn a return (interest), whereas a dollar received a year from now yields no return during the year This future dollar must have the interest element removed from it

to determine its value today

Discount Discounting involves finding out what a sum or sums of money in the future is worth today by removing the time value of money Dollars in the future are brought back to the present at some interest rate The higher the interest rate, the lower the present value

QD-3 The future amount of 1 tells how much one single monetary unit will accrue to in a given

number of periods at a given interest rate The future value of an ordinary annuity of 1 tells how much a series of end-of-the-period deposits of one monetary unit will accrue to

at a given periodic interest rate

QD-4 Interest Rate Frequency of Compounding

Per Period Per Year

a 9% 2 times

b 4% 4 times

c 1¼% 12 times

QD-5 The future value of 1 is 1 plus the interest compounded at a given interest rate for a given

number of periods The present value of 1 is the amount that must be invested today in order to grow to 1 in a given number of periods at a given compound interest rate The present value of 1 tells how much one monetary unit in the future is worth today, given the interest rate and the number of periods The present value of an ordinary annuity of 1 tells how much a series of payments of one monetary unit at the end of each period is worth today, given the interest rate

QD-6 The only difference between the future value of an ordinary annuity and the future value

of an annuity due is the number of time periods over which interest accrues With the future value of an annuity due, interest accrues for one period after the last cash flow in the series With the future value of an ordinary annuity, interest compounding ends on the date of the last payment

QD-6 (continued)

Future value of an {ordinary annuity of 4 cash flows is determined

immediately after the last cash flow is made

$ $ $ $

* _ * * *

Dec 31 Dec 31 Dec 31 Dec 31

Year 1 Year 2 Year 3 Year 4

Future value of an annuity due of

4 cash flows is determined one period after the last { cash flow is made

$ $ $ $

* _ * * *

Dec 31 Dec 31 Dec 31 Dec 31 Dec 31

Year 1 Year 2 Year 3 Year 4 Year 5

Arrows indicate date to which computation applies

QD-7 The present value of an annuity due is based on cash payments made at the

beginning of each period, and is determined on the date of the first payment The present value of a deferred annuity refers to an annuity where the first payment in the series is postponed for two or more periods in the future

Present value of an annuity due of four cash flows

$ $ $ $ * _* * _*

Jan 1 Jan 1 Jan 1 Jan 1 Year 1 Year 2 Year 3 Year 4

QD-7 (continued)

Present value of an annuity of

four cash flows deferred three periods

$ $ $ $

_* _ _ _ _* * *

Jan 1 Jan 1 Jan 1 Jan 1

Year 5 Year 6 Year 7 Year 8

Jan 1 Jan 1 Jan 1 Jan 1

Year 1 Year 2 Year 3 Year 4

Arrows indicate date to which computation applies

QD-8 a Step 1: Compute the present value of 1 at 10% for 4 years, as follows:

4 0.10) (1 1

Step 2: Multiply $10,000 by the answer to step 1

b Step 1: To convert the factor obtained in step 1 above from four periods to five periods, simply divide by 1.10, as follows:

1.10 0.10) (1

1 4

Step 2: Multiply $5,000 by the answer to step 1

c Step 1: Compute the future amount of an ordinary annuity of 1 for five cash flows,

at 10%, as follows:

0.10

1 0.10)

Step 2: Multiply $3,000 by the answer to step 1

QD-9 First, the two desired withdrawals are discounted back to the present at 12%

compounded semiannually The sum of the two present values of the withdrawals equals the required deposit

Required deposit = [ $40,000(pn 8,i 6%)] [ $50,000(pn 20,i 6%)]

QD-9 (continued)

Required deposit = ($40,000 x 0.627412) + ($50,000 x 0.311805)

= $25,096.48 + $15,590.25

= $40,686.73

QD-10 All of the factors have two things in common: a 14% interest rate, and 16 periods

(cash flows) If the factors given have the same number of time periods and/or cash flows for the same interest rate, the table value classification can be determined without using the table The number given for e is the only table value given less than 1 It must therefore be the present value of 1 The reciprocal of the present value of 1 is the future value of 1 Therefore, a is the future value of 1 (1 ¸ 0.122892 = 8.137249) Of the answers remaining, b c and d., the largest is the future value of an ordinary annuity of 1 and the smallest is the present value of an ordinary annuity of 1, again assuming the same number of cash flows and same interest rate The present value of an annuity due is d because it is equal to the present value of an ordinary annuity of one less period with 1 added to the factor

Table Value Classification

a 8.137249 Future value of 1

b 50.980352 Future value of an ordinary annuity of 1

c 6.265060 Present value of an ordinary annuity of 1

d 7.142168 Present value of an annuity due of 1

e 0.122892 Present value of 1

QD-11 There are two approaches to the determination of the converted factor for a

deferred annuity:

1 Converted factor for present value of a deferred annuity of 1 = (Factor for

present value of an ordinary annuity of n cash flows of 1) x (Factor for present value

of 1 for period of deferment) The ultimate present value of the deferred annuity is determined by multiplying the above factor by the value of each cash flow

2 Converted factor for present value of a deferred annuity of 1 = (Factor for

present value of an ordinary annuity of n + k cash flows of 1) - (Factor for present value of an ordinary annuity of k rents of 1) The ultimate present value of the

deferred annuity is determined by multiplying the above factor by the value of each cash flow

QD-12 (continued)

Table value: P n 3,i 14% = 2.321632

Equal installments = $20,000 2.321632 QD-13

a Invert the given value, or

4.411435 1

b Square the given value, or (4.411435) 2

c Use the following equation:

0.16

1 value Given

or

0.16

1 4.411435

d Use the following equation:

0.16

1

value Given

1

or

0.16 4.411435

1 1

e Use the following equation:

0.16

1 value)

or

0.16

1 (4.411435)2

ANSWERS TO CASES

CD-1

Annual cost of the 1-year plan: $4,480.00

Annual cost of the 3-year plan:

) C(P

Pd dn 3,i 12%

$11,200 = C (2.690051)

$4,163.49 2.690051

$11,200 C

Annual cost of the 5-year plan:

) C(P

Pd dn 5,i 12%

CD-1 (continued)

$4,438.56 4.037349

$17,920 C

The 3-year plan is the least expensive plan given the 12% rate The savings over the other two plans are computed as follows:

Yearly savings over the 1-year plan

$4,480.00 - $4,163.49 = $316.51

Yearly savings over the 5-year plan

$4,438.56 - $4,163.49 = $275.07

CD-2

Plan 1 Purchase the equipment

The present value of the purchase alternative equals the sum of the initial cash payment, less the present value of the resale value to be received in 5 years, computed as follows:

Present value of the resale value:

P of $5,500 for 5 years at 12%:

Plan 2 Lease the equipment

The present value of leasing the equipment equals the present value of $9,100 per year for 5 years, discounted at 12% Since the payments are made at the beginning of each year, this is an annuity due situation

) i n, d C(P d P

CD-3

1 If White takes the discount, it must pay $396,000 By not taking the discount, White can

use the $396,000 for 10 days (assuming that White follows its usual policy of paying after

30 days) For waiting the extra 10 days, the company must pay an additional $4,000 The effective annual interest cost is

$36.36%

0.3636 10days

360days x

$396,000

$4,000

The 36.36% rate is compared with the effective annual interest cost of borrowing the money from a bank to see if the discount should be taken The effective annual interest cost of borrowing the funds is computed as shown in the following

Since White has to keep a 15% compensating balance in the bank, $396,000 equals only 85% of the funds that must be borrowed, therefore,

2 The effective annual interest cost of not taking the discount is lower in this case, since

White could use the $396,000 for 40 days instead of just 10 days The effective annual interest cost of waiting the entire 60-day period is

9.09%

0.0909 40days

360days x

$396,000

$4,000

Since this rate is lower than the effective rate of borrowing from the bank (16.47%) White should not take the discount, and pay at the end of the 60-day period.

CD-3 (continued)

3 It has become less desirable for White to borrow from the bank By waiting one day, or

40 days past the discount period, White must pay $4,000 more than if the discount were taken The longer White waits to pay, the lower the effective interest cost since White pays the same charge ($4,000), but gets a longer use of the $396,000 Increasing the amount of time funds are used, while keeping the interest charge constant, always lowers the effective interest cost

CD-4

1 The amount of interest earned equals the future value minus the present value,

computed as follows:

23.049803 (4.801021)

) (f

) (fn 80,i 4% n 40,i 4%2 2

) 16%

i 20, n o

$10,000(F o

F o

1 Either argument may be correct depending on the circumstances If the note was given

solely in exchange for cash, then the president is correct However, the requirements of FASB Statement No 57, "Related Party Disclosures," must be considered

CD-5 (continued)

2 Since the interest amount at 4% is compounded annually, the future value due in 5 years

is:

f = $300,000 x 1.216653 = $364,995.90 Present value of note at January 1, 2004:

p = $364,995.90 pn 5,i 16%

= $364,995.90 x 0.476113 = $173,779.29

CD-6

Whether or not Perry would pay back the loan would depend on her reinvestment rate

If she could earn more than 12% on some investment, then she would be earning more than the interest charge on the $5,000 she borrowed In this case, she would be better off not paying back the principal amount

On the other hand, if Perry could not earn 12% on some investment, she would be better off if she paid back the money, and avoided the interest charges on the $5,000 debt (This solution ignores the fact that, should Perry die, the policy proceeds to the

beneficiary would be reduced by $5,000 It also ignores income taxes.)

ANSWERS TO MULTIPLE CHOICE

SOLUTIONS TO EXERCISES

ED-1

1 f = $ 40,000 fn 7,i 12%

= $ 40,000 (2.210681) = $ 88,427.24

2 f = $ 10,000 fn 22,i 4%

= $ 10,000 (2.369919) = $ 23,699.19

3 The compound interest equals the total interest for the 5 years, therefore,

Future Value - Present Value = Compound Interest

ED-3

1 p = $ 30,000 pn 5,i 12%

= $ 30,000 (0.567427) = $ 17,022.81

= $ 8,000 - $4,967.37 = $ 3,032.63

ED-4

1 Fo $10,000(Fon 7,i 12%)

= $ 10,000 (10.089012) = $100,890.12

2 The future value determined in (1) accumulates interest for 1 more year ( dF )

d

F = $100,890.12 (1.12) = $112,996.93

or $10,000 (12.299693* - 1) = $112,996.93

*Fon 8,i 12%

i6,no(FCoF

$30,000 = C (7.715610)

7.715610

$30,000C

= $3,888.22 ED-7

1 Since the $25,000 is invested one year before the first withdrawal, the calculation is based on the Po formula:

)in,o(PCoP

$25,000 = C(Pon 5,i 12%)

$25,000 = C (3.604776)

3.604776

$25,000C

ED-7 (continued)

2 Since the deposit is made on the date of the first withdrawal, the

computation is based on the Pd formula:

12%

i5,ndPCdP

$25,000 = C (4.037349)

4.037349

$25,000C

i8,knoPCdeferred

C = $9,743.52

or

12%

i3,kP12%

i5,noPCdeferred

C = $9,743.52

ED-8

)in,o(PCoP

% 1i24,no(PC

2 1

$10,000 = C (20.030405)

20.030405

$10,000C

C = $499.24 ED-9

Step 1: Find the present value of the trust

p = $25,000 pn 10,i 12%

= $25,000 (0.321973) = $8,049.33

Step 2: The present value of the annual payments must equal the amount of the

loan minus the present value of the trust

o

P = $40,000 - $8,049.33 = $31,950.67

Step 3: With the known present value of the annuity, compute the amount of the equal payments

$31,950.67 = C(Pon 9,i 12%)

$31,950.67 = C (5.328250)

5.328250

$31,950.67C

C = $5,996.47

ED-10

1 Since the future value is determined immediately after the last cash payment

is made (January 1, 2009), the calculation is based on the future value of an ordinary annuity

)in,oC(FoF

$200,000 = C(Fon 5,i 10%)

$200,000 = C (6.105100)

6.105100

$200,000C

C = $32,759.50

2 Since the future value is determined one period after the last cash payment, the calculation is based on the future value of an annuity due

)10%

i5,ndC(FdF

C = $29,781.36

P = $50,303.06

$30,000 = C(Fon 1 6,i 10% -1) $30,000 = C (7.715610 - 1) $30,000 = C (6.715610)

6.715610

$30,000C

C = $4,467.20

The present value of the six annual withdrawals of $3,000 is first calculated This present value is then used as the known future amount of the 10 annual unknown cash flows beginning on January 1, 2004

Step 1: The Po value of six cash flows of $3,000 at 10%:

o

P = C(Pon 6,i 10%)o

P = $ 3,000 (4.355261) o

P = $13,065.78

Step 2: The $13,065.78 is the known future value to which the 10 payments

beginning on January 1, 2004 must accumulate To find the amount

of these payments, we use the Fo formula

o

F = C(Fon 10,i 10%)13,065.78 = C (15.937425)

15.937425

$13,065.78C

C = $819.82

ED-13

d

P = C(Pdn 5,i 12%) = $3,000 (4.037349) = $12,112.05

?Looking down the 10% column of the present value of an annuity due table, we find that 4.790788 is the factor for six cash flows The Boston Company must make six payments

p = $30,000 pn 4,i 10%

p = $30,000 (0.683013)

p = $20,490.39