APPENDIX 3 TIME VALUE OF MONEY -...

APPENDIX 3 TIME VALUE OF MONEY

DISCUSSION QUESTIONS

1. The timing of the receipt (or payment) of money affects the value of that receipt (or payment). Money received now can earn interest. Money received in the future forgoes an opportunity to earn interest and therefore is less valuable than money received now.

2. Future value of a single amount―An amount is put on deposit and earns compound interest until maturity.

Present value of a single amount―An amount is to be paid or received at some future time. The value today of that future flow is the pre-sent value.

Future value of an annuity―A series of pay-ments at equal intervals of time of the same amount are put into a fund and earn compound interest until maturity at which time the interest plus the annuity payments are returned.

Present value of an annuity―A series of regu-lar payments of the same amount are to be re-ceived or paid. The value today of those pay-ments is the present value of the annuity.

3. Compound interest is computed by multiplying the interest rate times the balance of the account at the beginning of the interest period

and adding the product to the account. A future value is the amount to which a single deposit or a series of deposits will grow by a specified fu-ture time when interest is compounded. A pre-sent value is the value today of a single pay-ment or a series of payments to be received in the future.

4. An annuity is a series of payments or receipts (cash flows) of the same amount with payments exactly one period apart. From the viewpoint of a lender, the present value of the annuity is the amount lent, and the repeating cash flows re-pay the loan with interest.

5. The value of a business is the present value of the estimated future cash flows generated by the business. Since there is usually great un-certainty about the various inputs to this calcu-lation (including interest rates, future cash flows, and the life of the business), the time-value-of-money calculation is usually performed for various combinations of the possible input amounts.

© 2010 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different

from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part.

A3-1

CORNERSTONE EXERCISES

Cornerstone Exercise A3–1

Tables: Future Value = $6,000 × (FV of a Single Amount, 20 Periods, 3%) = $6,000 × 1.80611 = $10,836.66 Formula: Future Value = $6,000 × 1.0320 = $10,836.67

Calculator* N I/Y PV PMT FV Solution 20 3 –6,000 0 Compute 10,836.67

*The columns in this table have been put in the order of the keys on most, if not

all, financial calculators. The key with “compute” underneath it is what you are solving for in this problem. The order you press the keys does not matter; how-ever, you should do the compute key last. The last column shows the solution to the problem.


Tables: Future Value of Annuity = $500 × (FV of an Annuity, 3 Periods, 6%) = $500 × 3.18360 = $1,591.80 Formula: Future Value of Annuity = $500 × [(1.063 – 1)/0.06] = $1,591.80

Calculator* N I/Y PV PMT FV Solution 3 6 0 –500 Compute 1,591.80

*See footnote explanation for Cornerstone Exercise A3–1.

© 2010 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in

part.

A3-2


Tables: Future Value = $140,000 × (FV of a Single Amount, 30 Periods, 4%) = $140,000 × 3.24340 = $454,076.00 Formula: Future Value = $140,000 × 1.0430 = $454,075.65 Calculator* N I/Y PV PMT FV Solution 30 4 –140,000 0 Compute 454,075.65



Tables: $3,000 = Investment × (FV of a Single Amount, 4 Periods, 3%) Investment = $3,000/1.12551 = $2,665.46 Formula: $3,000 = Investment × 1.034 Investment = $2,665.46 Calculator* N I/Y PV PMT FV Solution 4 3 Compute 0 –3,000 2,665.46



part.

A3-3


Tables: Future Value of Annuity = $500 × (FV of an Annuity, 8 Periods, 4%) = $500 × 9.21423 = $4,607.12 Formula: Future Value of Annuity = $500 × [(1.048 – 1)/0.04] = $4,607.11 Calculator* N I/Y PV PMT FV Solution 8 4 0 –500 Compute 4,607.11



Tables: Future Value = $2,000 × (FV of Single Amount, 20 Periods, 7%) = $2,000 × 40.99549 = $81,990.98 Formula: Future Value = $2,000 × [(1.0720 – 1)/0.07] = $81,990.98 Calculator* N I/Y PV PMT FV Solution 20 7 0 –2,000 Compute 81,990.98



part.

A3-4


Tables: Present Value = $7,000 × (PV of Single Amount, 7 Periods, 7%) = $7,000 × 0.62275 = $4,359.25 Formula: Present Value = $7,000 × (1/1.077) = $4,359.25 Calculator* N I/Y PV PMT FV Solution 7 7 Compute 0 –7,000 4,359.25



Tables: Present Value = $600 × (PV of a Single Amount, 4 Periods, 6%) = $600 × 0.79209 = $475.25 Formula: Future Value = $600 × (1/1.064) = $475.26 Calculator* N I/Y PV PMT FV Solution 4 6 Compute 0 –600 475.26



part.

A3-5


Tables: Present Value of Annuity = $200 × (PV of an Annuity, 10 Periods, 6%) = $200 × 7.36009 = $1,472.02 Formula: Present Value of Annuity = $200 × {[1 – (1/1.0610)]/0.06} = $1,472.02 Calculator* N I/Y PV PMT FV Solution 10 6 Compute –200 0 1,472.02



Tables: Present Value of Annuity = $20,000 × (PV of an Annuity, 10 Periods, 6%) = $20,000 × 7.36009 = $147,201.80 Formula: Present Value of Annuity = $20,000 × {[1 – (1/1.0610)]/0.06} = $147,201.74

Calculator* N I/Y PV PMT FV Solution 10 6 Compute –20,000 0 147,201.74



part.

A3-6

EXERCISES

Exercise A3–11

a. Tables: Future Value = $5,000 × (FV of a Single Amount, 10 Periods, 7%) = $5,000 × 1.96715 = $9,835.75

Formula: Future Value = $5,000 × 1.0710 = $9,835.76



b. Tables: Future Value of Annuity = $500 × (FV of an Annuity, 10 Periods, 7%) = $500 × 13.81645 = $6,908.23

Formula: Future Value of Annuity = $500 × [(1.0710 – 1)/0.07] = $6,908.22

Calculator* N I/Y PV PMT FV Solution 10 7 0 500 Compute 6,908.23



part.

A3-7

Exercise A3–11 (Concluded)

c. Tables: Present Value = $5,000 × (PV of a Single Amount, 10 Periods, 7%) = $5,000 × 0.50835 = $2,541.75

Formula: Present Value = $5,000 × (1/1.0710) = $2,541.75

Calculator* N I/Y PV PMT FV Solution 10 7 Compute 0 –5,000 2,541.75


d. Tables: Present Value of Annuity = $500 × (PV of an Annuity, 10 Periods, 7%) = $500 × 7.02358 = $3,511.79

Formula: Present Value of Annuity = $500 × {[1 – (1/1.0710)]/0.07} = $3,511.79

Calculator* N I/Y PV PMT FV Solution 10 7 Compute –500 0 3,511.79



part.

A3-8

Exercise A3–12

a. Tables: Present Value = $1,200 × (PV of a Single Amount, 7 Periods, 8%) = $1,200 × 0.58349 = $700.19

Formula: Present Value = $1,200 × (1/1.087) = $700.19

Calculator* N I/Y PV PMT FV Solution 7 8 Compute 0 –1,200 700.19


b. Tables: Present Value of Annuity = $1,200 × (PV of an Annuity, 7 Periods, 8%) = $1,200 × 5.20637 = $6,247.64

Formula: Present Value of Annuity = $1,200 × {[1 – (1/1.087)]/0.08} = $6,247.64

Calculator* N I/Y PV PMT FV Solution 7 8 Compute –1,200 0 6,247.64



part.

A3-9


c. Tables: Future Value = $1,200 × (FV of a Single Amount, 7 Periods, 8%) = $1,200 × 1.71382 = $2,056.58

Formula: Future Value = $1,200 × 1.087 = $2,056.59*

Calculator** N I/Y PV PMT FV Solution 7 8 –1,200 0 Compute 2,056.59*

*Differs from value calculated with the tables due to rounding. **See footnote explanation for Cornerstone Exercise A3–1.

d. Tables: Future Value of Annuity = $1,200 × (FV of an Annuity, 7 Periods, 8%) = $1,200 × 8.92280 = $10,707.36

Formula: Future Value of Annuity = $1,200 × [(1.087 – 1)/0.08]

= $10,707.36

Calculator* N I/Y PV PMT FV Solution 7 8 0 –1,200 Compute 10,707.36



part.

A3-10

Exercise A3–13

a. Tables: Future Value = $15,000 × (FV of a Single Amount, 4 Periods, 10%) = $15,000 × 1.46410 = $21,961.50




b. Tables: $13,416.80 = $8,000 × (FV of a Single Amount, 6 Periods, Unknown Inter-

est Rate) $13,416.80 = $8,000 × Table Factor Table Factor = 1.67710; Interest Rate = 9%

Formula: $13,416.80 = $8,000 × (1 + Unknown Interest Rate)6 1.67710 = (1 + i)6

= 1 + i 1.09 = 1 + i i = 0.09 = 9%

Calculator* N I/Y PV PMT FV Solution 6 Compute –8,000 0 13,416.80 9



part.

A3-11


c. Tables: Unknown Cash Flow × (FV of an Annuity, 9 Periods, 9%) = $79,428.10 Unknown Cash Flow × 13.02104 = $79,428.10 Unknown Cash Flow = $6,099.98

Formula: Unknown Cash Flow × [(1.099 – 1)/0.09] = $79,428.10 Unknown Cash Flow × 13.02104 = $79,428.10 Unknown Cash Flow = $6,099.98

Calculator* N I/Y PV PMT FV Solution 9 9 0 Compute –79,428.10 6,099.98


d. Tables: $7,500 × (FV of a Single Amount, n Periods, 9%) = $17,755.50 $17,755.50/$7,500 = 2.36740 Table Factor n = About 10 Periods

Formula: $7,500 × 1.09n = $17,755.50 1.09n = 2.36740 n = About 10 Periods

Calculator* N I/Y PV PMT FV Solution Compute 9 –7,500 0 17,755.50 10.00



part.

A3-12

Exercise A3–14

1. a. n = 8 i = 3% FV = ? $12,000 1/1/09 12/31/12

b. n = 8 i = 2% FV = ? $2,500 $2,500 $2,500 $2,500 3/31/09 6/30/09 9/30/09 12/31/09 …… 12/31/10 2. a. Tables: Future Value = $12,000 × (FV of a Single Amount, 8 Periods, 3%) = $12,000 × 1.26677 = $15,201.24





part.

A3-13


b. Tables: Future Value of Annuity = $2,500 × (FV of an Annuity, 8 Periods, 2%) = $2,500 × 8.58297 = $21,457.43

Formula: Future Value of Annuity = $2,500 × [(1.028 – 1)/0.02] = $21,457.42*

Calculator** N I/Y PV PMT FV Solution 8 2 0 –2,500 Compute 21,457.42*


Exercise A3–15

1. Tables: Future Value = $8,000 × (FV of a Single Amount, 5 Periods, 8%) = $8,000 × 1.46933 = $11,754.64

Formula: Future Value = $8,000 × 1.085 = $11,754.62*


*Differs from value calculated with the tables due to rounding.

**See footnote explanation for Cornerstone Exercise A3–1.


part.

A3-14


2. Tables: Future Value = $8,000 × (FV of a Single Amount, 7 Periods, 8%)

= $8,000 × 1.71382 = $13,710.56 = $13,710.56 – $8,000 = $5,710.56

Formula: Future Value = $8,000 × 1.087

= $13,710.59* = $13,710.59* – $8,000 = $5,710.59*



**See footnote explanation for Cornerstone Exercise A3–1. 3. Tables:

Future Value = $8,000 × [FV of a Single Amount, 20 (5 × 4) Periods, 2% (8%/4)] = $8,000 × 1.48595 = $11,887.60


= $11,887.58*





part.

A3-15

Exercise A3–16

1. n = 9

i = 1% FV = ? $70,000 6/30/09 3/31/10 2. Tables:

Future Value = $70,000 × (FV of a Single Amount, 9 Periods, 1%) = $70,000 × 1.09369 = $76,558.30


= $76,557.97*





part.

A3-16


3. Tables: Future Value = $70,000 × (FV of a Single Amount, 6 Periods, 1%)

= $70,000 × 1.06152 = $74,306.40 – $70,000 = $4,306.40


= $74,306.41* – $70,000 = $4,306.41


Interest = $74,306.41 – $70,000 = $4,306.41


Exercise A3–17

a. Tables: Present Value = $14,000 × (PV of a Single Amount, 7 Periods, 8%)

= $14,000 × 0.58349 = $8,168.86

Formula: Present Value = $14,000 × (1/1.087)

= $8,168.87*

Calculator** N I/Y PV PMT FV Solution 7 8 Compute 0 –14,000 8,168.87*




part.

A3-17

Exercise A3–17 (Continued)

b. Tables: $5,820 × (PV of a Single Amount, n Periods, 7%) = $10,000 $5,820 = ($10,000 × Table Factor) Table Factor = 0.58200 n = About 8 Periods

Formula: $5,820 × (1/1.07n) = $10,000

1/1.07n = 1.71821 1 = 1.71821 × 1.07n 1/1.71821 = 1.07n 0.582001036 = 1.07n

= 1.07 n = 8

Calculator* N I/Y PV PMT FV Solution Compute 7 –5,820 0 10,000 8


c. Tables:

Annual Cash Flow × (PV of an Annuity, 25 Periods, 9%) = $49,113 Annual Cash Flow × 9.82258 = $49,113 Annual Cash Flow = $5,000.01

Formula: Annual Cash Flow × {[1 – (1/1.0925)]/0.09} = $49,113

Annual Cash Flow × 9.82258 = $49,113 Annual Cash Flow = $5,000.01

Calculator* N I/Y PV PMT FV Solution 25 9 –49,113 Compute 0 5,000.01



part.

A3-18


d. Tables: $4,000 × (PV of a Single Amount, 4 Periods, i) = $2,542

i = $4,000 × Table Factor = $2,542 i = Table Factor = 0.63550 i = About 12%

Formula: $4,000 × [1/(1 + i)7] = $2,542

1/(1 + i)7 = $2,542/$4,000 1/(1 + i)7 = 0.63550 1/0.63550 = (1 + i)7

= 1 + i 1.12 – 1 = i i = 12%

Calculator* N I/Y PV PMT FV Solution 4 Compute –2,542 0 4,000 12


e. Tables:

$2,000 × (PV of an Annuity, 15 Periods, i) = $17,119 $2,000 × Table Factor = $17,119 Table Factor = 8.55950

i = About 8%

Formula: $2,000 × ({1 – [1/(1 + i)15]}/i) = $17,119 {1 – [1/(1 + i)15]}/i = $17,119/$2,000 {1 – [1/(1 + i)15]}/i = 8.55950 i = 8%

Calculator* N I/Y PV PMT FV Solution 15 Compute –17,119 2,000 0 8



part.

A3-19

Exercise A3–18

1. a. n = 2 i = 9% PV = ? $60,000 1/1/09 12/31/10

b. n = 10 i = 2% PV = ? $75,000 $75,000 $75,000 $75,000 1/1/09 3/31/09 6/30/09 9/30/09 … 6/30/11

2. a. Tables:

Present Value = $60,000 × (PV of a Single Amount, 2 Periods, 9%) = $60,000 × 0.84168 = $50,500.80


= $50,500.80

Calculator* N I/Y PV PMT FV Solution 2 9 Compute 0 –60,000 50,500.80



part.

A3-20

Exercise A3–18 (Continued)

Truck ...................................................................... 50,500.80 Note Payable .................................................... 50,500.80 b. Tables:

Present Value of Annuity = $75,000 × (PV of an Annuity, 10 Periods, 2%) = $75,000 × 8.98259 = $673,694.25 Journal entry if used tables: Property, Plant, and Equipment........................... 673,694.25 Note Payable .................................................... 673,694.25 Formula:

Present Value of Annuity = $75,000 × {[1 – (1/1.0210)]/0.02} = $673,693.88*

Calculator** N I/Y PV PMT FV Solution 10 2 Compute –75,000 0 673,693.88*


**See footnote explanation for Cornerstone Exercise A3–1. Journal entry if used formula or calculator: Building ................................................................. 673,693.88 Note Payable .................................................... 673,693.88 3. If used tables: Interest expense for the first quarter ended 3/31/09 is: 0.02 × $673,694.25 = $13,473.89 Interest Expense ................................................... 13,473.89 Note Payable* ........................................................ 61,526.11 Cash.................................................................. 75,000.00

*Interest – Cash Payment = Liability Change $75,000 – $13,473.89 = $61,526.11


part.

A3-21


Interest expense for the second quarter ended 6/30/09 is: 0.02 × New Liability Balance 0.02 × ($673,694.25 − $61,526.11) 0.02 × $612,168.14 = $12,243.36

Interest Expense ................................................... 12,243.36 Note Payable* ........................................................ 62,756.64 Cash.................................................................. 75,000.00

*$75,000 − $12,243.36 = $62,756.64 If used formula or calculator: Interest expense for the first quarter ended 3/31/09 is: 0.02 × $673,693.88 = $13,473.88 Interest Expense ................................................... 13,473.88 Note Payable* ........................................................ 61,526.12 Cash.................................................................. 75,000.00

*Interest – Cash Payment = Liability Change $75,000 – $13,473.88 = $61,526.12 Interest expense for the first quarter ended 6/30/09 is: 0.02 × New Liability Balance 0.02 × ($673,693.88 – $61,526.12) 0.02 × $612,167.76 = $12,243.36 Interest Expense ................................................... 12,243.36 Note Payable* ........................................................ 62,756.64 Cash.................................................................. 75,000.00

*$75,000 – $12,243.36 = $62,756.64 4. Interest expense for 2009 is: 0.09 × $50,500.80 = $4,545.07

Interest Expense ................................................... 4,545.07 Note Payable .................................................... 4,545.07


part.

A3-22

Exercise A3–19

1. Tables: Present Value = $200,000 × (PV of a Single Amount, 5 Periods, 6%)

= $200,000 × 0.74726 = $149,452.00


= $149,451.63*



**See footnote explanation for Cornerstone Exercise A3–1. 2. Tables:

Present Value = $200,000 × (PV of a Single Amount, 5 Periods, 10%) = $200,000 × 0.62092 = $124,184


= $124,184.26* Calculator** N I/Y PV PMT FV Solution 5 10 Compute 0 –200,000 124,184.26*




part.

A3-23


3. Tables: Present Value = $200,000 × (PV of a Single Amount, 10 Periods, 3%) = $200,000 × 0.74409 = $148,818


= $148,818.78*




Exercise A3–20

1. Tables: Future Value of Annuity = $4,200 × (FV of an Annuity, 7 Periods, 10%)

= $4,200 × 9.48717 = $39,846.11

Formula: Future Value of Annuity = $4,200 × [(1.107 – 1)/0.10]

= $39,846.12*

Calculator** N I/Y PV PMT FV Solution 7 10 0 –4,200 Compute 39,846.12*




part.

A3-24


2. The interest earned at any point in time equals the excess of the future value over the total amount of cash deposits. Thus, interest earned to the point in time just after the seventh deposit is:

Using tables: $39,846.11 – (7 × $4,200.00) = $39,846.11 – $29,400.00 = $10,446.11

Using formula or calculator: $39,846.12 – (7 × $4,200.00) = $39,846.12 – $29,400.00 = $10,446.12

Exercise A3–21

For a 10-week, $1-a-week plan: $1 × 10.0696 = $10.07

For a 30-week, $10-a-week plan: $10 × 30.6796 = $306.80 Number of Amount of Each Deposit Deposits $1 $5 $10 $50 10 $10.07 $ 50.35 $100.70 $ 503.48 20 20.30 101.48 202.95 1,014.77 30 30.68 153.40 306.80 1,533.98 40 41.23 206.13 412.25 2,061.25


part.

A3-25

Exercise A3–22

1. n = 30

i = 1% FV = ? $25,000 2. Tables:

Future Value = $25,000 × (FV of a Single Amount, 30 Periods, 1%) = $25,000 × 1.34785 = $33,696.25 Loss = $33,696.25 – $25,000 = $8,696.25


= $33,696.22*



**See footnote explanation for Cornerstone Exercise A3–1. Loss = $33,696.22 – $25,000 = $8,696.22 3. The loss is not recorded because it represents an opportunity missed. Ac-

countants call such losses opportunity losses. Although opportunity losses are not entered in the accounting records, it is important to know and under-stand such losses in managing a business.


part.

A3-26

Exercise A3–23

1. n = 25 i = 9% PV = ? $775,000 $775,000 $775,000 $775,000 1/1/09 12/31/09 12/31/10 12/31/11 … 12/31/33 2. Tables:

Present Value of Annuity = $775,000 × (PV of an Annuity, 25 Periods, 9%) = $775,000 × 9.82258 = $7,612,499.50

Formula: Present Value of Annuity = $775,000 × {[1 – (1/1.0925)]/0.09}

= $7,612,499.19*

Calculator** N I/Y PV PMT FV Solution 25 9 Compute –775,000 0 7,612,499.19*


**See footnote explanation for Cornerstone Exercise A3–1. 3. Journal entry using tables: Land ....................................................................... 7,612,499.50 Note Payable .................................................... 7,612,499.50 Journal entry using formula or calculator: Land ....................................................................... 7,612,499.19 Note Payable .................................................... 7,612,499.19


part.

A3-27

Exercise A3–24

1. n = 30 i = 1% PV = ? $50 $50 $50 $50 10/31/09 11/30/09 12/31/09 1/31/10 … 4/30/12 2. Tables:

Present Value of Annuity = $50 × (PV of an Annuity, 30 Periods, 1%) = $50 × 25.80771 = $1,290.39

Formula: Present Value of Annuity = 50 × {[1 – (1/1.0130)]/0.01}

= $1,290.39

Calculator* N I/Y PV PMT FV Solution 30 1 Compute 50 0 1,290.39


3. Notes Receivable .................................................. 1,290.39 Sales Revenue ................................................. 1,290.39 Cost of Goods Sold .............................................. 800.00 Inventory .......................................................... 800.00


part.

A3-28


4. Interest Period

Interest Revenue

Cash

Payment

Change in Receivable

Balance

Ending Receivable

Balance

10/31/09 — —– —– $1,290.39 11/30/09 $12.90 $50.00 $37.10 1,253.29 12/31/09 12.53 50.00 37.47 1,215.82

Total $25.43

5. The income effect in 2009 of this sale: Interest revenue .................................................... $ 25.43 Gross margin: Selling price ..................................................... $1,290.39 Less: Cost of goods sold................................ 800.00 490.39 2009 income effect .......................................... $515.82


part.

A3-29


part.

APPENDIX 3 TIME VALUE OF MONEY -...

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