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Learning Objectives

1. Explain the concept of the time value of money. (LO1)

2. Calculate present values, future values, and annuities based on the number of time periods involved and the going interest rate. (LO2)

3. Calculate yield based on the time relationships between cash flows. (LO3)

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Future Value and Present Value

• Future Value (FV) is what money today will be worth at some point in the future.

• Present Value (PV) is what money at some point in the future is worth today.

0 1 2

PV FV FV

LO2

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Figure 9-1 Relationship of present value and future value

$1,000 present value

$

0 1 2 3 4

$1,464.10futurevalue

10% interest

Number of periods

LO2

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Future Value

FV = ?

N = 4PV = $1,000

464,1$000,1$ 4n (1.10)i)PV(1FVI = 10%

Alternatively,

FV = PV x FVIF

FVIF is the future value interest factor (Appendix A)FV = $1,000(1.464) = $1,464

LO2

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Future value of $1 (FVIF)

1 . . . .1.010 1.020 1.030 1.040 1.060 1.080 1.1002 . . . .1.020 1.040 1.061 1.082 1.124 1.166 1.2103 . . . .1.030 1.061 1.093 1.125 1.191 1.260 1.3314 . . . .1.041 1.082 1.126 1.170 1.262 1.360 1.4645 . . . .1.051 1.104 1.159 1.217 1.338 1.469 1.611

10 . . . .1.1051.219 1.344 1.480 1.791 2.159 2.59420 . . . .1.2201.486 1.806 2.191 3.207 4.661 6.727

An expanded table is presented in Appendix A

Percent

Period 1% 2% 3% 4% 6% 8% 10%

LO2

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Calculator (Texas Instruments BA-II Plus)

• The first step to use a business calculator is to clear the registers.

• Then set P/Y to 1.• Then PV = 1000• Then I/Y = 10• Then N = 4• Then PMT = 0• Finally CPT FV = -1464.10

LO2

Future Value

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Present Value

FV = $1,464.10

N = 4PV = ?

000,1$/10.464,1$ 4n (1.10)i)FV/(1PV

I = 10%

Alternatively,

PV = FV x PVIF

PVIF is the present value interest factor (Appendix B)PV = $1,464.10(0.683) = $1,000

LO2

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Present value of $1 (PVIF)

1 ..... 0.9900.980 0.971 0.962 0.943 0.926 0.9092 .....0.980 0.961 0.943 0.925 0.890 0.857 0.8263 .....0.971 0.942 0.915 0.889 0.840 0.794 0.7514 .....0.961 0.924 0.888 0.855 0.792 0.735 0.6835 .....0.951 0.906 0.863 0.822 0.747 0.681 0.62110 .....0.905 0.820 0.744 0.676 0.558 0.463 0.38620 .....0.820 0.673 0.554 0.456 0.312 0.215 0.149

An expanded table is presented in Appendix B

LO2

Percent

Period 1% 2% 3% 4% 6% 8% 10%

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Calculator (Texas Instruments BA-II Plus)

• Set FV = 1464.1• Then PMT = 0• Then I/Y = 10• Then N = 4

• Finally CPT PV = -1000.00

LO2

Present Value

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Nominal and Effective Annual Interest Rates

• Interest is often compounded quarterly, monthly, or semiannually.

• Effective interest rate takes into account compounding.

• Effective Annual Interest Rate = (1 + i/m)m – 1 where i = nominal annual interest rate m = number of compounding periods

per year

LO2

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Annuities

Annuity: a stream or series of equal payments to be received in the future.

• The payments are assumed to be received at the end of each period (unless stated otherwise).

• A good example of an annuity is a lease, where a fixed monthly charge is paid over a number of years.

LO2

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Figure 9-2Compounding process for annuity

Period 0 Period 1 Period 2 Period 3 Period 4

$1,000 for three periods @10%FV = $1,331

$1,000 for two periods @10%FV = $1,210

$1,000 for one period @10%FV = $1,100

$1,000 x 1.000 = $1,000

$4,641

LO2

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Future Value – Annuity

FV A(1 i) 1

iA

n

0 1 2 3 4

$1,000 $1,000 $1,000 $1,000

A = $1,000 FV = ?

641,4$

10%

110%)(11,000FV

4

A

FVA = A x FVIFA

FVIFA is the future value interest factor for an annuity (Appendix C)FVA = $1,000(4.641) = $4,641

LO2

Alternatively,

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Future value of an annuity of $1 (FVIFA)

1 . . . . 1.000 1.000 1.000 1.000 1.000 1.000 1.0002 . . . . 2.010 2.020 2.030 2.040 2.060 2.080 2.1003 . . . . 3.030 3.060 3.091 3.122 3.184 3.246 3.3104 . . . . 4.060 4.122 4.184 4.246 4.375 4.506 4.6415 . . . . 5.101 5.204 5.309 5.416 5.637 5.867 6.10510 . . . .10.46210.950 11.46412.00613.18114.487 15.93720 . . . .22.01924.297 26.87029.77836.78645.76257.27530 . . . .34.78540.588 47.57556.08579.058113.280164.490An expanded table is presented in Appendix C

LO2

Percent

Period 1% 2% 3% 4% 6% 8% 10%

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Calculator (Texas Instruments BA-II Plus)

• Set PMT = 1000

• Then PV = 0

• Then I/Y = 10

• Then N = 4

• Finally CPT FV = -4641.00

LO2

Future Value – Annuity

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Present Value – Annuity

i

i)(1

11

APVn

A

PV = ?

0 1 2 3 4

$1,000 $1,000 $1,000 $1,000

A = $1,000

PVA = A x PVIFAPVIFA is the present value interest factor for an annuity (Appendix D)PVA = $1,000(3.170) = $3,170

87.169,3$$

10%

10%)(1

11

1,000PV4

A

LO2

Alternatively,

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Present value of an annuity of $1 (PVIFA)

1 . . . . 0.990 0.980 0.971 0.962 0.943 0.9260.9092 . . . . 1.970 1.942 1.913 1.886 1.833 1.7831.7363 . . . . 2.941 2.884 2.829 2.775 2.673 2.5772.4874 . . . . 3.902 3.808 3.717 3.630 3.465 3.3123.1705 . . . . 4.853 4.713 4.580 4.452 4.212 3.9933.7918 . . . . 7.652 7.325 7.020 6.773 6.210 5.7475.33510 . . . .9.471 8.983 8.530 8.111 7.360 6.7106.14520 . . . .18.04616.35114.877 13.59011.4709.818 8.51430 . . . .25.80822.39619.600 17.29213.76511.258 9.427

An expanded table is presented in Appendix D

LO2

Percent

Period 1% 2% 3% 4% 6% 8% 10%

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Calculator (Texas Instruments BA-II Plus)

• Set PMT = 1000

• Then FV = 0

• Then I/Y = 10

• Then N = 4

• Finally CPT PV = -3169.87

LO2

Present Value – Annuity

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Determining The Annuity Value

• Formula Approach:

• Table Approach:

A = FVA/FVIFA (Appendix C)

A = PVA/PVIFA (Appendix D)• Calculator:FV (or PV), I/Y, N, CPT PMT

A FVi

(1 i) 1A n

A PVi

11

(1 i)

A

n

LO2

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Your uncle gives you $10,000 now. If you are able to earn 6% on these funds, how much can you withdraw at the end of each year for next 4 year?

PV = -$10,000 PMT = ?

0 1 2 3 4

A = PVA/PVIFA = $10,000/3.465 = $2,886

Alternatively, N = 4I/Y = 6PV = -10,000FV = 0CPT PMT = $2,885.91

LO2

Determining The Annuity Value

PMT = ? PMT = ? PMT = ?

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Table 9-1Relationship of present value to annuity

1 . . . .$10,000.00$600.00 $2,886.00$7,714.00

2 . . . . 7,714.00 462.84 2,886.00 5,290.84

3 . . . . 5,290.84 317.45 2,886.00 2,722.29

4 . . . . 2,722.29 163.71 2,886.00 0

YearBeginningBalance

Annual Interest (6%)+ - Annual

Withdrawal= Ending

Balance

LO2

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Table 9-2Payoff table for loan (amortization table)

1 . . . .$40,000 $4,074 $3,200 $ 874$39,126

2 . . . . 39,126 4,074 3,130 944 38,182

3 . . . . 38,182 4,074 3,055 1,019 37,163

PeriodBeginningBalance

AnnualPayment

AnnualInterest

(8%)

Repaymenton

Principal

EndingBalance

LO2

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Formula Summary Formula

AppendixFuture value—–single amount . . (9-1) FV = PV(1 + i)n A

Present value—–single amount . (9-3) B

Future value—–annuity . . . . . . . (9-4a) C

Future value—–annuity in advance . . . . . . . . . . . . . . . . . (9-4b) –

Present value—annuity . . . . . . . (9-5a) D

PV FV1

(1 i)n

FV A(1 i) 1

iA

n

FV A(1 i) (1 i)

iA BGN

n 1

PV A1

1(1 i)

iA

n

LO2

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Formula SummaryFormula

AppendixPresent value—annuity in

advance . . . . . . . . . . . . . . . . (9-5b) –

Annuity equalling a future

value . . . . . . . . . . . . . . . . . . (9-6a) C

Annuity in advance equalling a future value . . . . . . . . . . . . (9-6b) –

Annuity equalling a presentvalue . . . . . . . . . . . . . . . . . . (9-7a) D

Annuity in advance equalling a – present value . . . . . . . . . . . (9-7b)

PV A(1 i)

1(1 i)iA BGN

n 1

A FVi

(1 i) 1A n

A FVi

(1 i) (1 i)BGN A n 1

A PVi

11

(1 i)

A

n

A PVi

(1 i)1

(1 i)

BGN A

n 1

LO2

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Two Questions to Ask in Time Value of Money Problems

First, Future Value or Present Value?Future Value: Present (Now) FuturePresent Value: Future Present (Now)

Second, Single amount or Annuity?Single amount: one-time (or lump) sumAnnuity: same amount per year for a number of

years

LO2

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Special Considerations in Time Value of Money – Deferred Annuity

A1 A2 A3 A4 A5

$1,000$1,000$1,000$1,000$1,000PV = ?

0 1 2 3 4 5 6 7 8

LO2

e.g.: Find the PV of an Annuity of $1,000 being paid for 5 years beginning 4 years in the future

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Deferred Annuity

A1 A2 A3 A4 A5

$1,000 $1,000 $1,000 $1,000 $1,000PV = ?

0 1 2 3 4 5 6 7 8

First: Find the PV of an annuity of $1,000 to be received at the end of the year for 5 years with I= 8% N = 5, I/Y = 8%, PMT = $1,000, FV = 0 CPT PV = $3,993

End of third period—Beginning of fourth period

$3,993

LO2

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Second: Find the PV of $3,993 to be received at the end of year 3 discounted back to the present (with I = 8%). N = 3, I/Y = 8%, PMT = 0, FV = $3,993, CPT PV = $3,170

Deferred Annuity (final part)

$3,170 $3,993 A1 A2 A3 A4 A5

Present (single amount) $1,000 $1,000 $1,000 $1,000 $1,000value

0 1 2 3 4 5 6 7 8

End of third period—Beginning of fourth period

LO2

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Perpetuities

• Equal Payments

• Growing Payments

• Growing annuity (with End Date)

i

APV

gi

APV

1

n

n i

g

giAPV

1

11

11

LO2

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• The second step is to calculate the monthly payment.

0 1 240

PV = $80,000

PMT = ? FV = 0

Canadian MortgagesLO2

Using a calculatorN = 240PV = -80,000I/Y = 0.65582FV = 0CPT PMT = $662.69

N = 240(20 yrs. x 12)

I/Y = .65582