TVM Applications Chapter

Chapter 6

Time Value of Money

UPS, Walgreens, Costco, American Air, Dreamworks Intel (note 10 page 28)

1

TVM Applications

Accounting issue Chapter

Notes receivable (long-term receivables) 7

Long-term assets 10

Long-term intangibles (patents, copyrights) 12 Notes payable (long-term liabilities) 14

Investments 17

Installment contracts 18

Pensions and other postretirement benefits (OPEB) 20

Leases 21

TVM calculations used in many fair value calculations Use of fair value increasing, TVM more important

2

Ch 6: TVM Calculator

§  TI-83 Plus, TI-84 Plus (Int. Algebra)

§  TI BA II Plus

§  HP 10B, HP 17B, HP 12C

§  Casio FC-200V, Sharp EL733A

§  Rite-Aid

FV (Future Value) and PV (Present Value) 3

Chapter Overview

§  Explain time value of money concept

§  Differentiate simple, compound interest

§  Solve future and present value of $1

problems

§  Solve future and present value of

annuities (ordinary, annuity due)

§  Solve deferred annuities, bonds, and

expected cash flows problems

4

Learning Objectives

§  Identify topics using time value of money (TVM) §  Distinguish between simple and compound interest §  Use appropriate compound interest tables. §  Identify variables needed to solve TVM problems §  Solve future and present value of $1 problems §  Solve future value of ordinary, annuity due problems §  Solve present value ordinary, annuity due problems §  Solve deferred annuities and bonds problems §  Solve expected cash flows problems

5

Learning Objectives

§  Identify topics using time value of

money (TVM)

§  Distinguish between simple and

compound interest

§  Identify variables needed to solve TVM

problems

Time Value Of Money

§  Required by GAAP

§  Many applications in accounting

§  Value assets and liabilities

§  Any amount more due > 1 year

§  Long-term budgeting

§  Basis for all finance

In general, time value of money calculations must be made whenever a dollar amount will change hands

more than one year from today. 7

TVM Applications

§  In life

§  Saving for retirement: 401(k), IRA, 529

§  Mortgage payments

See WebAccess sample problems and website

8

Time Value Of Money

§  All money earns interest over time

§  $1 today

≠ $1 tomorrow

§ 

Every dollar in the future is part principal

and part interest

9

Time Value Of Money

Amount of cash is small

Interest = Principal × Rate × Time

Interest = $10 × 3% × 30/360

Interest = $0.025

10

Time Value Of Money

Amount of cash is large

Interest = Principal × Rate × Time

Interest = $10,000,000,000 × 3% × 30/360

Interest = $25,000,000

Interest = $10,000,000,000 × 3% × 1

Interest = $300,000,000

See Intel Annual Report, note 10, page 28 ₁₁

Interest

§  Payment to rent money

§  Compensation to lender for use of $

§  Compensation for risk

§  Inflation §  Risk of default

§  Compensation for profit

Interest

§  Difference between

§  Beginning balance

= $100

§  Ending balance

= $106

§  To lender, interest revenue

§  To borrower, interest expense

13

Simple Interest

§  Interest rate per period

§  Time is number of periods

§  Rate and time must be same periods

§  Year, semi-annual, quarter, month

Interest = Principal × Rate × Time

Interest rate usually stated as rate per year, must convert to rate per period

14

Interest Rate Per Period

Compounding Annual Rate Periods per Year Rate per Period

Annual 12% ÷ 1 = 12% Semiannual 12% ÷ 2 = 6% Quarterly 12% ÷ 4 = 3% Monthly 12% ÷ 12 = 1% 15

Number of Periods

Compounding Years Periods per Year Periods

Annual 20 × 1 = 20 Semiannual 20 × 2 = 40 Quarterly 20 × 4 = 80 Monthly 20 × 12 = 240

16

Compounding Annual Rate Periods per Year Rate per Period

Annual 12% ÷ 1 = 12% Semiannual 12% ÷ 2 = 6%

Quarterly 12% ÷ 4 = 3% Monthly 12% ÷ 12 = 1%

Compounding Years Periods per Year Periods

Annual 20 × 1 = 20 Semiannual 20 × 2 = 40 Quarterly 20 × 4 = 80 Monthly 20 × 12 = 240

Interest rate per period = Interest rate per year / periods per year

Number of periods = Number of years ✕ periods per year

17

Simple Interest: Yearly

§  Borrow $100,000

§  For 20 years

§  Annual interest rate of 12%

Interest = $100,000 × 12% × 20

Interest = $240,000

Interest calculated on original principal only

Interest = Principal × Rate × Time

Simple Interest: Monthly

§  Borrow $100,000

§  For 20 years

§  Annual interest rate of 12%

Interest = Principal × Rate × Time

Interest = $100,000 × 1% × 240

Interest = $240,000

Same total interest 19

Simple Interest

Present value (PV) $1,000

Interest rate per year 10%

Number of years 3

Compounding periods per year None

Period Principal Interest Ending Balance

1 1,000 100 1,100

2 1,000 100 1,200

3 1,000 100 1,300

Interest calculated on original principal only 20

Compound Interest

§  Earn interest on

§  Initial investment

§  Interest accumulated in previous periods

21

Present value (PV) $1,000

Interest rate per year 10%

Number of years 3

Compounding periods per year 1

Interest rate period 10%

Number of periods 3

Period Beg Balance Interest Ending Balance

1 1,000 100 1,100

2 1,100 110 1,210

3 1,210 121 1,331

One compounding interval per year

22

Present value (PV) $1,000

Interest rate per year 10%

Number of years 3

Compounding periods per year 2

Interest rate period 5%

Number of periods 6

Period Beg Balance Interest Ending Balance

1 1,000 50 1,050 2 1,050 53 1,103 3 1,103 55 1,158 4 1,158 58 1,215 5 1,215 61 1,276 6 1,276 64 1,340

Two compounding intervals per year

23

Simple and Compound

Present value (PV) $1,000

Interest rate per year 10%

Number of years 3

Interest Calculation Amount

Simple $1,300

Compounded annually $1,331

Compounded semiannually $1,340

Simple Interest ($100,000, 12% per period, 3 periods) Period Beg Bal Rate Interest End Bal

1 100,000 12% 12,000 112,000 2 100,000 12% 12,000 124,000 3 100,000 12% 12,000 136,000

Total interest 36,000 Compound Interest

Period Beg Bal Rate Interest End Bal 1 100,000 12% 12,000 112,000 2 112,000 12% 13,440 125,440 3 125,440 12% 15,053 140,493

Total interest 40,493 Simple interest compared to compound interest

Principal $100,000

Rate 12% per period

Time 3 periods

25

Simple and Compound

Principal Rate Time Interest Simple $100,000 × 12% × 20 $240,000 Simple $100,000 × 1% × 240 $240,000

Compounding Interval Interest

Annually $864,629

Semiannually $928,572

Quarterly $964,089

Monthly $989,255

Interest same regardless of time periods

Interest different for each compounding interval 26

Five Tables in Textbook

1.  Future Value of $1

2.  Present Value of $1

3.  Future Value: Ordinary Annuity of $1

4.  Present Value: Ordinary Annuity of $1

5.  Present Value: Annuity Due of $1

27

Time Value Of Money

§  Money variables

§  Present value

§  Future value

§  Annuity

§  Other variables

§  Interest rate (per period)

§  Time (number of periods)

§  Annuity timing (beginning or end of period)

28

Problem Solving

§  What are you given?

§  What do you need to compute?

§  Draw a timeline

§  Carefully count periods

§  Write down formulas (FV=PV×FV$1)

§  Solve for unknowns

§  Double check what you need to calc

§  Ask: Does answer make sense?

29

Memorize These Formulas

Future value × PV$1 factor = Present value

Present value × FV$1 factor = Future value Annuity × FVAnnuity$1 factor = Future value Annuity × PVAnnuity$1 factor = Present value

Learning Objectives

§  Compute future value of single amount

31

Time Value Of Money

Present

Value

Future

Value

32

Time Value Of Money

Present value

Principal

Original investment

Future value

Principal + interest

Maturity value

33

Given PV Calculate FV

Single amount

Present value

Single amount

Future value

Accumulating interest

Principal Interest

34

Given PV Calculate FV

If we make an investment today, how

much will it grow to in the future?

Today Present

Value FutureValue

Interest compounding periods

35

Given PV Calculate FV

§  Invest $10,000 today and earn 20%

compounded quarterly for three years

§  Calculate future value

$10,000 Present Value Future Value Unknown Unknown Interest 36

Given PV Calculate FV

§  Invest $10,000 today and earn 20%

compounded quarterly for three years

§  Calculate future value

Data Given

Present value $10,000 Interest rate per year 20% Number of years 3 Compounding periods per year 4 Interest rate per period 5% Number of periods 12 37

Data Given

Present value $10,000 Interest rate per year 20% Number of years 3 Compounding periods per year 4 Interest rate per period 5% Number of periods 12 Future Value of $1 Periods 4% 5% 6% 11 1.539 1.710 1.898 12 1.601 1.796 2.012 13 1.665 1.886 2.133

See TVM tables on WebAccess

38

Calculation of Future Value

PV × FV$1 = FV $10,000 × 1.796 = FV $17,960 = FV Future Value of $1 Periods 4% 5% 6% 11 1.539 1.710 1.898 12 1.601 1.796 2.012 13 1.665 1.886 2.133

Present value × FV$1 factor = Future Value

39

Interest

Future value

− Present value

Interest

40

Given PV Calculate FV

§  Invest $10,000 today and earn 20%

compounded quarterly for three years

§  Calculate future value

$10,000 Present Value Future Value $17,960 $7,960 Interest 41

How Does it Work?

Given present value calculate future value

Given: Present value (PV) $4,000 Interest rate per year (R) 10% Years of investment (Y) 3 Compounding periods per year (c) 2 Calculate: Interest rate per period (i = R / c) 5% Number of periods (n = Y × c) 6 Future value of $1 factor 1.340

Future value $5,360

Period Beginning Balance Interest Balance Ending 1 4,000 200 4,200 2 4,200 210 4,410 3 4,410 221 4,631 4 4,631 232 4,863 5 4,863 243 5,106 6 5,106 254 5,360

Given present value calculate future value

Given: Present value (PV) $4,000 Interest rate per year (R) 10% Years of investment (Y) 3 Compounding periods per year (c) 2 Calculate: Interest rate per period (i = R / c) 5% Number of periods (n = Y × c) 6 Future value of $1 factor 1.340

Future value $5,360

43

Given PV Calculate FV

Present value × FV$1 factor = Future value

FV$1 factor Present value = Future value

FV$1 factor Present value Future value = 44

Learning Objectives

§  Compute present value of single

amount

45

Given FV Calculate PV

How much of future amount is

original investment (principal)?

Today Present

Value FutureValue

Interest compounding periods

Discounting interest

Principal Interest

46

Given FV Calculate PV

§  Need $90,000 at end of five years, earn

12% compounded semi-annually

§  Calculate present value

Unknown Present Value Future Value $90,000 Unknown Interest 47

Given FV Calculate PV

§  Need $90,000 at end of five years, earn

12% compounded semi-annually

§  Calculate present value

Data Given

Future value $90,000 Interest rate per year 12% Number of years 5 Compounding periods per year 2 Interest rate per period 6% Number of periods 10 48

Present Value of $1 Periods 5% 6% 7% 9 0.645 0.592 0.544 10 0.614 0.558 0.508 11 0.585 0.527 0.475 Data Given Future value $90,000 Interest rate per year 12% Number of years 5 Compounding periods per year 2 Interest rate per period 6% Number of periods 10

49

Calculation of Present Value

FV × PV$1 = PV $90,000 × 0.558 = PV

$50,220 = PV

Future value × PV$1 factor = Present value

Present Value of $1 Periods 5% 6% 7% 9 0.645 0.592 0.544 10 0.614 0.558 0.508 11 0.585 0.527 0.475 50

Interest

Future value

− Present value

Interest

51

Given FV Calculate PV

§  Need $90,000 at end of five years, earn

12% compounded semi-annually

§  Calculate present value

$50,220 Present Value Future Value $90,000 $39,780 Interest 52

How Does it Work?

Given future value calculate present value

Given: Future value (FV) $100,000 Interest rate per year (R) 14% Years of investment (Y) 6 Compounding periods per year (c) 1 Calculate: Interest rate per period (i = R / c) 14% Number of periods (n = Y × c) 6 Present value of $1 factor 0.456

Present value $45,600 53 Period Beginning Balance Interest Ending Balance 1 45,600 6,384 51,984 2 51,984 7,278 59,262 3 59,262 8,297 67,559 4 67,559 9,458 77,017 5 77,017 10,782 87,799 6 87,799 12,201 100,000

Given future value calculate present value

Given: Future value (FV) $100,000 Interest rate per year (R) 14% Years of investment (Y) 6 Compounding periods per year (c) 1 Calculate: Interest rate per period (i = R / c) 14% Number of periods (n = Y × c) 6 Present value of $1 factor 0.456

Present value $45,600

Given FV Calculate PV

PV$1 factor Future value = Present Value

PV$1 factor

Future value Present Value =

Future Value × PV$1 factor = Present value

55

Learning Objectives

§  Given present value and future value,

solve for interest rate or number of

periods

56

FV = PV (1 +

i

)

n

Future Value Present Value Interest Rate Number of Compounding Periods

Solving for Other Values

§  Four variables in time value of money

§  Given three calculate fourth

57

Manipulating Equation

Present value × FV$1 factor = Future value

FV$1 factor Present value = Future value

FV$1 factor

Present value Future value =

58

Calculate Rate: FV$1 Table

§  Borrow $1,000 today and repay $1,082

at end of two periods

§  Calculate interest rate per period

$1,000 Present Value Future Value $1,082 $82 Interest 59

Calculate Rate: FV$1 Table

§  Borrow $1,000 today and repay $1,082

at end of two periods

§  Calculate interest rate per period

Calculation of FV$1 Factor

PV × FV$1 = FV $1,000 × FV $1 = $1,082 FV$1 = $1,082 ⁄ $1,000

FV$1 = 1.082

Future Value of $1

Periods 3% 4% 5%

1 1.030 1.040 1.050 2 1.061 1.082 1.103 3 1.093 1.125 1.158

Future Value of $1 Table

PV × FV$1 = FV $1,000 × FV $1 = $1,082 FV$1 = $1,082 ⁄ $1,000

FV$1 = 1.082 See row 2 of FV$1 table

Solve this question using TVM calculator, not TVM table 61

Manipulating Equation

PV$1 factor Future value = Present Value

PV$1 factor

Future value Present Value =

Future Value × PV$1 factor = Present value

62

Calculate Rate: PV$1 Table

§  Borrow $1,000 today and repay $1,082

at end of two periods

§  Calculate interest rate per period

Present Value of $1 Table

FV × PV$1 = PV $1,082 × PV $1 = $1,000 PV$1 = $1,000 ⁄ $1,082

PV$1 = 0.924

See row 2 of PV$1 table 63

Present Value of $1 Table

FV × PV$1 = PV $1,092 × PV $1 = $1,000 PV$1 = $1,000 ⁄ $1,082

PV$1 = 0.924 See row 2 of PV$1 table

Present Value of $1

Periods 3% 4% 5%

1 0.971 0.962 0.952 2 0.943 0.925 0.907 3 0.915 0.889 0.864 Solve this question using TVM calculator, not TVM table ₆₄

Calculate Periods: FV$1 Table

§  Deposit $47,811 today and accumulate

$70,000 at 10% compounded annually

§  Calculate number of periods

$47,811 Present Value Future Value $70,000 $22,189 Interest 65

Calculate Periods: FV$1 Table

§  Deposit $47,811 today and accumulate

$70,000 at 10% compounded annually

§  Calculate number of periods using FV

Future Value of $1 Table

PV × FV$1 = FV $47,811 × FV $1 = $70,000 FV$1 = $70,000 ⁄ $47,811

FV$1 = 1.464

Future Value of $1 Table

PV × FV$1 = FV $47,811 × FV $1 = $70,000 FV$1 = $70,000 ⁄ $47,811

FV$1 = 1.464 See 10% column of FV$1 table

Future Value of $1 Periods 9% 10% 11% 1 1.090 1.100 1.110 2 1.188 1.210 1.232 3 1.295 1.331 1.368 4 1.412 1.464 1.518 5 1.539 1.611 1.685 Solve this question using TVM calculator, not TVM table 67

Calculate Periods: PV$1 Table

§  Deposit $47,811 today and accumulate

$70,000 at 10% compounded annually

§  Calculate number of periods using PV

Present Value of $1 Table

FV × PV$1 = PV $70,000 × PV $1 = $47,811 PV$1 = $47,811 ⁄ $70,000

PV$1 = 0.683

See 10% column of PV$1 table 68

Present Value of $1 Table

FV × PV$1 = PV $70,000 × PV $1 = $47,811 PV$1 = $47,811 ⁄ $70,000

PV$1 = 0.683 See 10% column of PV$1 table

Present Value of $1 Periods 9% 10% 11% 1 0.917 0.909 0.901 2 0.842 0.826 0.812 3 0.772 0.751 0.731 4 0.708 0.683 0.659 5 0.650 0.621 0.593 Solve this question using TVM calculator, not TVM table 69

Learning Objectives

§  Explain the difference between an

ordinary annuity and an annuity due

§  Compute the future value of both an

ordinary annuity and an annuity due

70

Annuities

§  Series of equal periodic payments

§  Equal amounts

§  Equal time periods

§  Defined period of time

Financial calculators: PMT key, specify END or BEG Pay $2,500 at the end of each quarter for five years

Payments are called “Rents”

71

Ordinary Annuity

§  Payments made at end of period

Present Value

Future Value Interest compounding periods

Payment 1 Payment 2 Payment 3

+

+

No payment

Annuity Due (in Advance)

§  Payments made at beginning of period

Present

Value FutureValue

Interest compounding periods

Payment 1 Payment 2 Payment 3

+

+

No

payment

Only use annuity due when specifically stated 73

§  Annuity amount:

$10,000

§  Interest rate per period:

4%

Period Beginning Balance Interest Payment Balance Ending 1 0 0 10,000 10,000 2 10,000 400 10,000 20,400 3 20,400 816 10,000 31,216

Period Payment Beginning Balance Interest Balance Ending 1 10,000 10,000 400 10,400 2 10,000 20,400 816 21,216 3 10,000 31,216 1,249 32,465 Ordinary Annuity Annuity Due 74

Future Value Ordinary Annuity

§  Make regular principal investments

§  Calculate future value

Interest Principal

75

Future Value Ordinary Annuity

§  Equal payments made each period

§  Payments, interest accumulate

Today Present

Value

Future Value Interest compounding periods

Payment 1 Payment 2 Payment 3

+

+

76

Calculation of FV of Ordinary Annuity

Annuity × FVAnnuity$1 factor = FV $1,000 × 4.993 = FV

$4,993 = FV

Annuity × FVAnnuity$1 factor = Future value

Future Value of Ordinary Annuity of $1

Periods 14% 15% 16% 3 3.440 3.473 3.506 4 4.921 4.993 5.066 5 6.610 6.742 6.877

77

Given Annuity Calculate FV

§  Invest $5,000 at end of each quarter, at

16% compounded quarterly, for 5 years

§  Calculate future value

Data Given

Annuity $5,000 Interest rate per year 16% Number of years 5 Compounding periods per year 4 Interest rate per period 4% Number of periods 20 78

Data Given

Annuity $5,000 Interest rate per year 16% Number of years 5 Compounding periods per year 4 Interest rate per period 4% Number of periods 20

Future Value of Ordinary Annuity of $1

Periods 3% 4% 5% 18 23.414 25.645 28.132 19 25.117 27.671 30.539 20 26.870 29.778 33.066

79

Calculation of FV of Ordinary Annuity

Annuity × FVAnnuity$1 factor = FV $5,000 × 29.778 = FV

$148,890 = FV

Annuity × FVAnnuity$1 factor = Future value

Future Value of Ordinary Annuity of $1

Periods 3% 4% 5% 18 23.414 25.645 28.132 19 25.117 27.671 30.539 20 26.870 29.778 33.066

80

Future Value Ordinary Annuity

Future value

− Annuity (Amount × number)

Interest

81

How Does it Work?

Given ordinary annuity calculate future value Given: Annuity [also called PMT] $10,000

Interest rate per year (R) 8%

Years of investment (Y) 3

Payments / compounding periods per year (c) 2 Calculate: Interest rate per period (i = R / c) 4% Number of periods (n = Y × c) 6 Future value of ordinary annuity of $1 factor 6.633 Future value of ordinary annuity $66,330

82

Period

Beginning

Balance Interest Payment

Ending Balance 1 0 0 10,000 10,000 2 10,000 400 10,000 20,400 3 20,400 816 10,000 31,216 4 31,216 1,249 10,000 42,465 5 42,465 1,699 10,000 54,164 6 54,164 2,166 10,000 66,330

Given ordinary annuity calculate future value

Given: Annuity [also called PMT] $10,000 Interest rate per year (R) 8% Years of investment (Y) 3 Payments / compounding periods per year (c) 2 Calculate: Interest rate per period (i = R / c) 4% Number of periods (n = Y × c) 6 Future value of ordinary annuity of $1 factor 6.633 Future value of ordinary annuity $66,330

83

Future Value Ordinary Annuity

FVAnnuity$1 factor Annuity = Future value

FVAnnuity$1 factor

Annuity Future value =

Annuity × FVAnnuity$1 factor = Future value

Future Value Annuity Due

§  Similar calculations

§  Use FV annuity due table

§  Use FV ordinary ann table × (1 + rate)

85

Calculation of FV of Ordinary Annuity

Annuity × FVAnnuity$1 factor = FV $5,000 × 29.778 = FV

$148,890 = FV

Future Value of Ordinary Annuity of $1

Periods 3% 4% 5% 18 23.414 25.645 28.132 19 25.117 27.671 30.539 20 26.870 29.778 33.066

Calculation of FV of Annuity Due

FV Ordinary Annuity × (1 + rate) = FV Annuity Due $148,890 × (1 + 0.04) = FV Annuity Due

$154,846 = FV Annuity Due 86

§  Annuity amount:

$10,000

§  Interest rate per period:

4%

Period

Beginning

Balance Interest Payment

Ending Balance 1 0 0 10,000 10,000 2 10,000 400 10,000 20,400 3 20,400 816 10,000 31,216 Period Payment Beginning Balance Interest Ending Balance 1 10,000 10,000 400 10,400 2 10,000 20,400 816 21,216 3 10,000 31,216 1,249 32,465 Ordinary Annuity Annuity Due 87

Learning Objectives

§  Compute the present value of an

ordinary annuity and an annuity due

88

PV Ordinary Annuity

What amount today is equivalent to a

series of payments in the future?

Today Present

Value

Payment 1 Payment 2 Payment 3

Principal Interest 89

PV Ordinary Annuity

§  Withdraw $10,000 at end of each year

§  For 4 years

§  Earn 10% compounded annually

§  How much do you need to invest today?

Given Annuity Calculate PV

§  Pay $7,000 at end of each six months,

10% compounded semi-annually,7years

§  Calculate present value

Data Given

Annuity $7,000 Interest rate per year 10% Number of years 7 Compounding periods per year 2 Interest rate per period 5% Number of periods 14 91

Data Given

Annuity $7,000 Interest rate per year 10% Number of years 7 Compounding periods per year 2 Interest rate per period 5% Number of periods 14

Present Value of Ordinary Annuity of $1

Periods 4% 5% 6% 13 9.986 9.394 8.853 14 10.563 9.899 9.295 15 11.118 10.380 9.712

92

Calculation of PV of Ordinary Annuity

Annuity × PVAnnuity$1 factor = PV $7,000 × 9.899 = PV

$69,293 = PV

Annuity × PVAnnuity$1 factor = Present value

Present Value of Ordinary Annuity of $1

Periods 4% 5% 6% 13 9.986 9.394 8.853 14 10.563 9.899 9.295 15 11.118 10.380 9.712 93

PV Ordinary Annuity

Annuity (Amount × number)

− Present value

Interest

94

How Does it Work?

Given ordinary annuity calculate present value

Given: Annuity [also called PMT] $2,500 Interest rate per year (R) 7% Years of investment (Y) 6 Payments / compounding periods per year (c) 1 Calculate: Interest rate per period (i = R / c) 7% Number of periods (n = Y × c) 6 Present value of ordinary annuity of $1 factor 4.767 Present value of ordinary annuity $11,918

95

Period Beginning

Balance Interest Payment

Balance

Reduction Balance Ending

1 11,918 834 2,500 1,666 10,252 2 10,252 718 2,500 1,782 8,470 3 8,470 593 2,500 1,907 6,563 4 6,563 459 2,500 2,041 4,522 5 4,522 317 2,500 2,183 2,339 6 2,339 161 2,500 2,339 0

Given ordinary annuity calculate present value

Given: Annuity [also called PMT] $2,500 Interest rate per year (R) 7% Years of investment (Y) 6 Payments / compounding periods per year (c) 1 Calculate: Interest rate per period (i = R / c) 7% Number of periods (n = Y × c) 6 Present value of ordinary annuity of $1 factor 4.767 Present value of ordinary annuity $11,918

PV Ordinary Annuity

PVAnnuity$1 factor Annuity = Present value

PVAnnuity$1 factor

Annuity Present value =

Annuity × PVAnnuity$1 factor = Present value

97

Present Value Annuity Due

§  Similar calculations

§  Use PV annuity due table

§  Use PV ordinary ann table × (1 + rate)

98

Learning Objectives

§  Annuity problems: Solving for annuity

amount, interest rate, number of periods

Present value of ordinary annuity used as example

99

Manipulating Equation

PVAnnuity$1 factor Annuity = Present value

PVAnnuity$1 factor

Annuity Present value =

Annuity × PVAnnuity$1 factor = Present value

100

Calculate Annuity

§  Borrow $39,550 for 5 years at 24%

interest, compounded semi-annually

§  Calculate semi-annual annuity amount

Present Value of Annuity of $1

Periods 11% 12% 13% 9 5.537 5.328 5.132 10 5.889 5.650 5.426 11 6.207 5.938 5.687

101

Present Value of Annuity of $1

Periods 11% 12% 13% 9 5.537 5.328 5.132 10 5.889 5.650 5.426 11 6.207 5.938 5.687

Present Value of Annuity of $1

Annuity × PVAnnuity$1 factor = PV Annuity × 5.650 = $39,550 Annuity = $39,550 ⁄ 5.650

Annuity = $7,000

Calculate Rate

§  Borrow $20,442 today and pay $3,000

at end of each period for 12 periods

§  Calculate interest rate per period

Present Value of Annuity of $1

Annuity × PVAnnuity$1 factor = PV $3,000 × PVAnnuity$1 = $20,442 PVAnnuity$1 = $20,442 ⁄ $3,000

PVAnnuity$1 = 6.814

See row 12 of PVAnnuity$1 table 103

Present Value of Annuity of $1

Periods 9% 10% 11%

11 6.805 6.495 6.207 12 7.161 6.814 6.492 13 7.487 7.103 6.750

Present Value of Annuity of $1

Annuity × PVAnnuity$1 factor = PV $3,000 × PVAnnuity$1 = $20,442 PVAnnuity$1 = $20,442 ⁄ $3,000

PVAnnuity$1 = 6.814 See row 12 of PVAnnuity$1 table

Solve this question using TVM calculator, not TVM table 104

Calculate Periods

§  Borrow $17,118 today and pay $2,000

at end of each period at 8% per period

§  Calculate number of periods

Present Value of Annuity of $1

Annuity × PVAnnuity$1 factor = PV $2,000 × PVAnnuity$1 = $17,118 PVAnnuity$1 = $17,118 ⁄ $2,000

PVAnnuity$1 = 8.559

See 8% column of PVAnnuity$1 table 105

Present Value of Annuity of $1

Periods 7% 8% 9% 14 8.745 8.244 7.786 15 9.108 8.559 8.061 16 9.447 8.851 8.313

Present Value of Annuity of $1

Annuity × PVAnnuity$1 factor = PV $2,000 × PVAnnuity$1 = $17,118 PVAnnuity$1 = $17,118 ⁄ $2,000

PVAnnuity$1 = 8.559 See 8% column of PVAnnuity$1 table

Solve this question using TVM calculator, not TVM table ₁₀₆

Learning Objectives

§  Compute the present value of a

deferred annuity

107

PV of Deferred Annuity

§  First cash flow of annuity occurs more

than one period in future

1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10 Present Value? $12,500 $12,500 1 2 3 4

PV of Deferred Annuity

§  Today: January 1, 2010

§  Beginning: December 31, 2012

§  Annuity will pay $12,500 a year

§  At end of each year for 2 years

§  Rate of return, 12%

§  Calculate PV

1/1/10 12/31/10 12/31/11 12/31/12 12/31/13 Present Value $12,500 $12,500 109 1/1/10 12/31/10 12/31/11 12/31/12 12/31/13 Present Value? $12,500 $12,500

PV of Deferred Annuity: #1

Two Step Process

1.  Calculate PV of annuity as of beginning of annuity period 2.  Discount single value to its present value at time zero

110 1/1/10 12/31/10 12/31/11 12/31/12 12/31/13 Present Value? $12,500 $12,500

PV of Deferred Annuity: #1

PV ordinary annuity n = 2, i = 12% Annuity = $12,500 PV factor = 1.690 PV = $21,126 PV single amount n = 2, i = 12% FV = $21,126 PV factor = 0.797 PV = $16,841 111

PV of Deferred Annuity: #2

1/1/10 12/31/10 12/31/11 12/31/12 12/31/13 Present Value $12,500 $12,500

PV annuity for entire period = 3.03735 PV annuity for period with no payments = 1.69005

$12,500 × (3.03735 − 1.69005) = $16,841

112

Learning Objectives

§  Application of time value of money

§  Notes receivable / Notes payable

§  Bonds

§  Effective interest amortization

§  Expected cash flow

113

Monetary Assets, Liabilities

§  Monetary assets

§  Cash and claims to receive cash

§  Amount fixed or determinable

§  Monetary liabilities

§  Obligations to pay cash

§  Amount fixed or determinable

Monetary Assets, Liabilities

§  Time frame important

§  Cash exchanged one year or less

§  Value at face value

§  Use simple interest (if interest rate stated)

§  Cost of using PV > benefit

•  Receive utility bill, $500; pay in 30 days •  Made sale on account, $1,000; collect in 60 days •  Loan $15,000 to vendor, 8% interest, due in 90 days

115

Monetary Assets, Liabilities

§  Time frame important

§  Cash exchanged more than one year

§  Use compound interest

§  Value at present value of future cash flows

116

Note With Interest Rate

§  Purchase equipment

§  Sign note, face value, $1,000

§  Interest rate, 4.5%

§  Market rate, 4.5%

§  Due in two years

§  Pay $1,092 in two years (FV)

Date Description Debit Credit

Jan 1 Equipment 1,000

2011 Note payable 1,000 117

Note With Interest Rate

Date Description Debit Credit

Dec 31 Interest expense (1,000 × 4.5%) 45

2011 Interest payable 45

Date Description Debit Credit

Dec 31 Interest expense (1,045 × 4.5%) 47

2012 Interest payable 47

Date Description Debit Credit

Dec 31 Note payable 1,000

2012 Interest payable 92

Cash 1,092 118

Unreasonable Stated Rate

§  Exchanging cash for non-cash asset

§  Time frame greater than one year

§  Discount future amount at market rate

Stated rate on note 15%, market rate for borrower 6%

119

Differing Rates 1

§  Purchase inventory on January 2, 2011

§  FMV inventory unknown

§  Seller accepts note

§  Face value, $100,000

§  Stated interest rate, 2%

§  Term, 4 years (due 12/31/2014)

§  Buyer’s interest rate from bank, 10%

1/2/11 12/31/11 12/31/12 12/31/13 12/31/14 Present Value

Differing Rates 1

Future Value

Calculation of Future Value i = 2% (stated rate), n=4 PV × FV$1 = FV $100,000 × 1.08243 = FV $108,243 = FV 121 1/2/11 12/31/11 12/31/12 12/31/13 12/31/14 Present Value

Differing Rates 1

Future Value

Calculation of Present Value i = 10% (market rate), n=4

FV × PV$1 = PV $108,243 × 0.68301 = PV

$73,931 = PV 122

Differing Rates 1

Date Description Debit Credit

1/2/11 Inventory 73,931

Discount on note payable 34,312

Note payable 108,243

123

Differing Rates 1

§  Recognize interest expense for period

Date Description Debit Credit

Dec 31 Interest expense (73,931 × 10%) 7,393

2011 Discount on note payable 7,393

Date Description Debit Credit

1/2/11 Inventory 73,931

Discount on note payable 34,312

Note payable 108,243

Date Description Debit Credit

Dec 31 Int exp ((73,931 + 7,393) × 10%) 8,132

2012 Discount on note payable 8,132 124

Differing Rates 2

§  Purchase inventory on January 2, 2011

§  FMV inventory unknown

§  Seller accepts note

§  Face value, $100,000

§  Stated interest rate, 8%

§  Term, 4 years (due 12/31/2014)

§  Buyer’s interest rate from bank, 5%

125 1/2/11 12/31/11 12/31/12 12/31/13 12/31/14 Present Value

Differing Rates 2

Future Value

Calculation of Future Value i = 8% (stated rate), n=4

PV × FV$1 = FV $100,000 × 1.36049 = FV

1/2/11 12/31/11 12/31/12 12/31/13 12/31/14 Present Value

Differing Rates 2

Future Value

Calculation of Present Value i = 5% (market rate), n=4

FV × PV$1 = PV $136,049 × 0.82270 = PV

$111,928 = PV 127

Differing Rates 2

Date Description Debit Credit

1/2/11 Inventory 111,928

Discount on note payable 24,121

Note payable 136,049

Date Description Debit Credit

Dec 31 Interest expense (111,928 × 5%) 5,596

2011 Discount on note payable 5,596

Date Description Debit Credit

Dec 31 Int exp ((111,928 + 5,596) × 5%) 5,876

2012 Discount on note payable 5,876 128

Learning Objectives

§  Bonds issued at discounts, premiums

§  Effective interest amortization

129

Need Two Billion Dollars

§  Intel needs cash to build new factory

§  Large debt broken into small pieces

$2,000,000,000

Sell 2,000,000 $1,000 bonds

130

Bonds

§  Receive cash when issued

§  Promise to pay

§  Face value on maturity date (future value)

§  Interest semiannually (ordinary annuity)

§  Issue price of bond is

§  PV of future value + PV of ordinary annuity

131

Issued At Par

General Electric issued bonds

§  Face value of $50 million

§  Mature in five years

§  Coupon interest rate of 9%

§  Issued par, market rate = coupon rate

Date Description Debit Credit

Cash 50,000,000

Bonds payable 50,000,000

Calculate Annuity

Coupon rate used to compute annuity

(periodic interest payments)

Interest payment = Face value × Coupon rate × Time

133

Interest Payments

Semi-annual interest payments

I = P × R × T

I = $50,000,000 × 0.09 × 6/12

I = $2,250,000

Interest only loan

Date Description Debit Credit

Interest expense 2,250,000

Cash 2,250,000

134

Payment At Maturity

Make last interest payment

Date Description Debit Credit

Interest expense 2,250,000

Cash 2,250,000

Pay principal (face value) in full

Date Description Debit Credit

Bonds payable 50,000,000

Cash 50,000,000 135

Two Interest Rates

Rate printed on bond called

§  Coupon rate

§  Stated rate

§  Contract rate

Market interest rate called

§  Effective-rate

§  Yield-to-maturity

136

Two Interest Rates

Coupon interest rate

§  Determines semi-annual payment

Market interest rate

§  Determines bond market price (PV)

§  Effective interest expense

Market rate > coupon rate, bond issued at discount Coupon rate > market rate, bond issued at premium

137

Issued At Discount

A $1,000 bond issued at a discount

§  Market rate > coupon rate

§  Bond sells for less than face value

§  For example

§  Quoted at 88 3/8

§  Sells for 88 3/8% of face value

§  Bought or sold for $ 883.75

Bond issued at discount

Face value

$5,000

Term

3 years

Coupon interest rate

7%

Market interest rate

10%

Issue price

$4,618

Discount

$382

Compounded semi-annually

Market interest rate per period

5%

Number of periods

6

Interest payment = Face value × Coupon rate × Time $175 = $5,000 × 0.07 × 1/2

139

Present Value Bond

Present

Value Payment 1 Payment 2 Payment 3 Principal Interest

Single amt 140

Present Value Bond

Present

Value Payment 1 Payment 2 Payment 3 Single amt $5,000 $175 $175 $175 $3,730 $888 $4,618

Discount at market rate, 10%, semi-annually

141

Issue Price Of Bond

Present Value of the Face Value (a single amount) + Present Value of the Interest Payments (an annuity) = Present value of Bond (Issue Price of the Bond)

Use market rate of interest

to calculate present value

142

Issue Price Of Bond

3,730

$ Present Value of the Face Value

+ 888 Present Value of the Annuity

= $ 4,618 Present Value of the Bonds

Also called issue price of bonds,

or market value of bonds ₁₄₃

(A) Beginning Balance A*MR*1/2= (B) Effective Interest (C) Annuity Payment B−C= (D) Discount Amortized F(up)−D= (F) Discount Remaining A+D= (G) Ending Balance 0 382 4,618 1 4,618 231 175 56 326 4,674 2 4,674 234 175 59 267 4,733 3 4,733 237 175 62 205 4,795 4 4,795 240 175 65 140 4,860 5 4,860 243 175 68 72 4,928 6 4,928 246 175 72 0 5,000

Effective interest expense Beg Bal × Market Rate × Time

$4,618 × 10% × 1/2 = $231

Interest payment Face value × Coupon rate × time

$5,000 × 7% × 1/2 = $175 Bond discount effective-interest amortization schedule

Zero Coupon Bond

Present

Value Payment 1 Payment 2 Payment 3 Principal Interest

Single amt 145

Zero Coupon Bond

Present Value of the Face Value (a single payment) + Present Value of the Interest Payments (an annuity) = Issue Price of the Bond

146

Issued At Premium

A $1,000 bond issued at a premium

§  Market rate < coupon rate

§  Bond sells for more than face value

§  For example

§  Quoted at 110 ¼

§  Sells for 110.25% of face value

§  Bought or sold for $1,102.50

147

Bond issued at premium

Face value

$6,000

Term

3 years

Coupon interest rate

12%

Market interest rate

8%

Issue price

$6,627

Discount

$627

Compounded semi-annually

Market interest rate per period

6%

Number of periods

6

Interest payment = Face value × Coupon rate × Time $360 = $6,000 × 0.12 × 1/2

148

Present Value Of Bond

4,740

$ Present Value of the Face Value

+ 1,887 Present Value of the Annuity

= $ 6,627 Present Value of the Bonds

$6,627 is greater than face amount of $6,000, bonds are issued at premium of $627.

149 (A) Beginning Balance A*MR*1/2= (B) Effective Interest (C) Annuity Payment C−B= (D) Premium Amortized F(up)−D= (F) Premium Remaining A−D= (G) Ending Balance 0 627 6,627 1 6,627 265 360 95 532 6,532 2 6,532 261 360 99 433 6,433 3 6,433 257 360 103 330 6,330 4 6,330 253 360 107 223 6,223 5 6,223 249 360 111 112 6,112 6 6,112 244 360 112 0 6,000

Effective interest expense Beg Bal × Market Rate × Time

$6,627 × 8% × 1/2 = $265

Interest payment Face value × Coupon rate × time

$6,000 × 12% × 1/2 = $360 Bond premium effective-interest amortization schedule

Learning Objectives

§  Expected cash flow

151

Cash Flow Issues

§  Amount

§  Timing

§  Uncertainty

152

Expected Cash Flows

§  Concepts Statement No. 7 requires

expected cash flow approach that uses

a range of cash flows and incorporates

the probabilities of those cash flows

§  FASB states a company should

discount expected cash flows by the

risk-free rate of return

153

Expected Cash Flows

§  Pure Rate

(2% to 4%)

§  No possibility of default

§  No expectation of inflation

§  Expected Inflation Rate

(0% or more)

§  Credit Risk Rate

( 0% or more)

Risk-Free Rate of Return

Pure rate + Expected inflation rate = Risk Free Rate

154

Expected Cash Flows

§  Future cash flow uncertain

§  Estimate amount using expected value

§  Discount to PV using risk-free rate

Amount Probability Expected Value

$100,000 10% $10,000 $200,000 60% $120,000 $300,000 30% $90,000

Expected value $220,000 ₁₅₅

Expected Cash Flows

§  Expected value paid at end of 5 years

§  Assume risk free rate of 5%

§  Calculate present value

Amount Probability Expected Value

$100,000 10% $10,000 $200,000 60% $120,000 $300,000 30% $90,000

Calculation of Present Value

FV × PV$1 = PV $220,000 × 0.784 = PV

$172,480 = PV

Future value × PV$1 factor = Present value

Present Value of $1 Periods 4% 5% 6% 4 0.855 0.823 0.792 5 0.822 0.784 0.747 6 0.790 0.746 0.705 157

Learning Objectives

§  Leases

§  Pension obligations

158

Present Value of Annuities

§  Financial instruments typically specify

equal periodic payments

§  Pension obligations

§  Long-term leases

159

Long-Term Leases

§  Certain long-term leases require

recording of an asset and liability at

present value of future lease payments

§  Make periodic payments (annuity)

160

Pension Obligations

§  Pension plans create obligations that

must be paid during retirement periods

§  To calculate amounts which must be

paid today to pension plan use present

value of estimate of future amount paid

during retirement

161

Memorize These Formulas

Future Value × PV$1 factor = Present value

Present value × FV$1 factor = Future value Annuity × FVAnnuity$1 factor = Future value Annuity × PVAnnuity$1 factor = Present value

End of Chapter