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99The Time Value of Money

The Time Value of Money

©2009 McGraw-Hill Ryerson Limited

Prepared by:

Michel PaquetSAIT Polytechnic

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Chapter 9 - Outline

• Basic Idea of Time Value of Money• Application to Capital Budgeting Decision

and Cost of Capital• Future Value and Present Value• Nominal and Effective Annual Interest Rates• Annuities• Summary and Conclusions

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

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Time Value of

0 1 2 3

$1 $1 $1 $1> > >

Why?because interest can be earned on the money

The connecting piece or link between present (today) and future is the interest or discount rate

Basic Idea: Money has different values at different points in time.

LO1

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Capital Budgeting Decision and Cost of Capital

• Capital budgeting decision is about

comparing current outlays with future benefits:

• Cost of capital is the discount rate used in this comparison process.

-$1,000 $60 $100 $200

0 1 2 3

LO1

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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 1.100

2 . . . . 1.020 1.040 1.061 1.082 1.124 1.166 1.210

3 . . . . 1.030 1.061 1.093 1.125 1.191 1.260 1.331

4 . . . . 1.041 1.082 1.126 1.170 1.262 1.360 1.464

5 . . . . 1.051 1.104 1.159 1.217 1.338 1.469 1.611

10 . . . . 1.105 1.219 1.344 1.480 1.791 2.159 2.594

20 . . . . 1.220 1.486 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.• 1000 PV• 10 I/Y• 4 N• 0 PMT• 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 . . . . . 0.990 0.980 0.971 0.962 0.943 0.926 0.909

2 . . . . . 0.980 0.961 0.943 0.925 0.890 0.857 0.826

3 . . . . . 0.971 0.942 0.915 0.889 0.840 0.794 0.751

4 . . . . . 0.961 0.924 0.888 0.855 0.792 0.735 0.683

5 . . . . . 0.951 0.906 0.863 0.822 0.747 0.681 0.621

10 . . . . . 0.905 0.820 0.744 0.676 0.558 0.463 0.386

20 . . . . . 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)

• 1464.1 FV

• 0 PMT

• 10 I/Y

• 4 N

• 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 in the real world.

• A nominal annual interest rate assumes annual compounding.

• It must be adjusted to reflect the more frequent than annual 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

Alternatively, FVA = A x FVIFA

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

LO2

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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 1.000

2 . . . . 2.010 2.020 2.030 2.040 2.060 2.080 2.100

3 . . . . 3.030 3.060 3.091 3.122 3.184 3.246 3.310

4 . . . . 4.060 4.122 4.184 4.246 4.375 4.506 4.641

5 . . . . 5.101 5.204 5.309 5.416 5.637 5.867 6.105

10 . . . . 10.462 10.950 11.464 12.006 13.181 14.487 15.937

20 . . . . 22.019 24.297 26.870 29.778 36.786 45.762 57.275

30 . . . . 34.785 40.588 47.575 56.085 79.058 113.280 164.490

An 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)

• 1000 PMT

• 0 PV

• 10 I/Y

• 4 N

• 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

Alternatively, PVA = A x PVIFA

PVIFA 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

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Present value of an annuity of $1 (PVIFA)

1 . . . . 0.990 0.980 0.971 0.962 0.943 0.926 0.909

2 . . . . 1.970 1.942 1.913 1.886 1.833 1.783 1.736

3 . . . . 2.941 2.884 2.829 2.775 2.673 2.577 2.487

4 . . . . 3.902 3.808 3.717 3.630 3.465 3.312 3.170

5 . . . . 4.853 4.713 4.580 4.452 4.212 3.993 3.791

8 . . . . 7.652 7.325 7.020 6.773 6.210 5.747 5.335

10 . . . . 9.471 8.983 8.530 8.111 7.360 6.710 6.145

20 . . . . 18.046 16.351 14.877 13.590 11.470 9.818 8.514

30 . . . . 25.808 22.396 19.600 17.292 13.765 11.258 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)

• 1000 PMT

• 0 FV

• 10 I/Y

• 4 N

• 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 A = ? A = ? A = ? A = ?

0 1 2 3 4

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

Alternatively, 4 N6 I/Y10000 PV0 FVCPT PMT = -2885.91

LO2

Determining The Annuity Value

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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 9-2

Payoff 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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Determining Yield on an Investment

• A $10,000 investment will generate $1,490 a year for the next 10 years, what is the yield on the investment?

$1,490 $1,490PV = $10,000

0 1 10

Using a calculator10 N-10000 PV1490 PMT0 FVCPT I/Y = 7.996(%)

i = ?

LO3

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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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Finding Present Value (first part)

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

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Finding Present Value (second part)

A1 A2 A3 A4 A5

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

0 1 2 3 4 5 6 7 8

Each number represents the end of the period; that is, 4 represents the end of the fourth period.

End of third period—Beginning of fourth period

$3,993

LO2

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Finding Present Value (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

• Perpetuity

• Growing perpetuity

• Growing annuity (with end date)

i

APV

gi

APV

1

n

n i

g

giAPV

1

11

11

LO2

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Canadian Mortgages: An Integrated Application

• A 20-year, $80,000 mortgage carries an annual interest rate of 8%. How much is the monthly payment?

• The first step is to calculate the effectively monthly interest rate from the semi-annual compounded annual interest rate.

0 1 2 3 4 5 6

FV = 1.04

PV = -1

Using a calculator, 6 N-1 PV0 PMT1.04 FVCPT I/Y = 0.65582(%)

LO3

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• The second step is to calculate the monthly payment.

0 1 240

PV = $80,000

A = ? A = ?

Using a calculator,240 N80000 PV0.65582 I/Y0 FVCPT PMT = -662.69

Canadian Mortgages: An Integrated Application

LO2

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Summary and Conclusions

• The financial manager uses the time value of money approach to value cash flows that occur at different points in time.

• A dollar invested today at compound interest will grow to a larger value in future. That future value, discounted at compound interest, is equated to a present value today.

• Cash values may be single amounts, or a series of equal amounts (annuity).

• Five variables are involved in a time value of money problem. The fifth variable can be determined if given the other four.