Chapter 5 - Time Value of Money

(1)

INTRODUCTION TO

CORPORA

TE F

INANCE

Laurence Booth

•

• W. Sean Cleary

W. Sean Cleary

Chapter 5

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CHAPTER 5

CHAPTER 5 Time Value of Money

(3)

CHAPTER 5

CHAPTER 5 Time Value of Money

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Lecture Agenda

•

• Learning Objectives

Learning Objectives

•

• Important Terms

Important Terms

•

• Types of Calculations

Types of Calculations

•

• Compounding

Compounding

•

• Discounting

Discounting

•

• Annuities and Loans

Annuities and Loans

•

• Perpetuities

Perpetuities

•

• Effective Rates of Return

Effective Rates of Return

•

• Summary and Conclusions

Summary and Conclusions

–

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Learning Objectives

• Understand the importance of the time value of money

• Understand the difference between simple interest and compound interest

• Know how to solve for present value, future value, time or rate

• Understand annuities and perpetuities

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Important Chapter Terms

• Amortize • Annuity • Annuity due • Basis point • Cash flows • Compound interest

• Compound interest factor (CVIF) • Discount rate • Discounting • Effective rate • Lessee • Medium of exchange • Mortgage • Ordinary annuities • Perpetuities

• Present value interest factor (PVIF)

• Reinvested

• Required rate of return

• Simple interest

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Types of Calculations

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Before We Get Started

Types of Calculations

Ex Ante:

– Calculations done „before-the-fact‟ – It is a forecast of what might happen

– All forecasts require assumptions

• It is important to understand the assumptions underlying any

formula used to ensure that those assumptions are consistent with the problem being solved.

– As a forecast, while you may be able to calculate the answer to a

high degree of accuracy…it is probably best to round off the

answer so that users of your calculations are not misled.

Ex Post:

– Calculation done „after -the-fact‟

– It is an analysis of what has happened

– It is usually possible, and perhaps wise to express the result as accurately as possible.

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The Basic Concept

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The Time Value of Money Concept

• Cannot directly compare $1 today with $1 to be

received at some future date

– Money received today can be invested to earn a rate of return

– Thus $1 today is worth more than $1 to be received at some future date

• The interest rate or discount rate is the variable that

equates a present value today with a future value at

some later date

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Opportunity Cost

Opportunity cost = Alternative use

–

The opportunity cost of money is the interest rate that

would be earned by investing it.

–

It is the underlying reason for the time value of money

–

Any person with money today knows they can invest

those funds to be some greater amount in the future.

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Choosing from Investment Alternatives

Required Rate of Return or Discount Rate

• You have three choices:

1. $20,000 received today

2. $31,000 received in 5 years 3. $3,000 per year indefinitely

• To make a decision, you need to know what

interest rate to use.

–

This interest rate is known as your

required rate of

return

or

discount rate.

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Simple Interest

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Simple Interest

Simple interest is interest paid or received on only the

initial investment (or principal).

At the end of the investment period, the principal plus

interest is received.

0 1 2 3 …

n

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Simple Interest

Example

PROBLEM:

Invest $1,000 today for a five-year term and receive 8

percent annual simple interest.

How much will you accumulate by the end of five years?

Year Beginning Amount Ending Amount

1 $1,000 $1,080 2 1,080 1,160 3 1,160 1,240 4 1,240 1,320 5 1,320 $1,400 ) 08 . 000 , 1 $ 5 ( 000 , 1 $ 5         Value k) P (n P e n) Value (tim

(16)

Simple Interest

General Formula

k)

P

(n

P

e n)

Value (tim







[ 5-1] Where: P = principal invested n = number of years k = interest rate

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Simple Interest

Simple interest problems are rare.

In finance we are most interested in COMPOUND

INTEREST.

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Compound Interest

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Compound Interest

Compounding (Computing Future Values)

Compound interest is interest that is earned on the

principal amount invested and on any accrued

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Compound Interest

Example

PROBLEM:

Invest $1,000 today for a five-year term and receive 8 percent

annual

compound interest

. How much will the accumulated

value be at time 5.

SOLUTION:

Year Beginning Amount Ending Amount 1 $1,000.00 $1,080.00 2 1,080.00 1,166.40 3 1,166.40 1,259.71 4 1,259.71 1,360.49 5 1,360.49 1,469.33 (1 .08) $1,469.33 49 . 360 , 1 $ ) 08 . 1 ( ) 08 . 1 )( 08 . 1 )( 08 . 1 )( 08 . 1 ( 71 . 259 , 1 $ ) 08 . 1 ( ) 08 . 1 )( 08 . 1 )( 08 . 1 ( 40 . 166 , 1 $ ) 08 . 1 ( ) 08 . 1 )( 08 . 1 ( 080 , 1 $ ) 08 . 1 ( 1 5 5 4 4 3 3 2 2 1 1                         P FV P P FV P P FV P P FV P FV k) ( P Value Future n

$1,469.33

8)

$1,000(1.0

FV

1 :

step

simple

one

in

solution

The

5

0

    n n

PV

(

k)

FV

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Compound Interest

Example of Interest Earned on Interest

PROBLEM:

Invest $1,000 today for a five-year term and receive 8 percent annual compound interest.

The Interest-earned-on-Interest Effect:

Interest (year 1) = $1,000 × .08 = $80

Interest (year 2 ) =($1,000 + $80)×.08 = $86.40

Interest (year 3) = ($1,000+$80+$86.40) × .08 = $93.31

Year Beginning Amount Ending Amount

Interest earned in the year

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Compound Interest

General Formula

Where: FV= future value P = principal invested n = number of years k = interest rate [ 5-2]

1

0 n n

PV

(

k)

FV

 

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Compound Interest

General Formula

[ 5-2]

1

0 n n

PV

(

k)

FV

 

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Compound Interest

Simple versus Compound Interest

Compounding of interest magnifies the returns on an

investment.

Returns are magnified:

• The longer they are compounded

• The higher the rate they are compounded

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Compound Interest

Simple versus Compound Interest

5-1 FIGURE D O L L A R S 8,000 7,000 6,000 5,000 4,000 3,000 2,000

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Compound Interest

Compound Interest at Varying Rates

Compounding of interest magnifies the returns on an

investment.

Returns are magnified:

• The longer they are compounded

• The higher the rate they are compounded

(See Table 5-1 that demonstrates the cumulative effect of higher rates of return earned over time.)

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Compound Interest

Compounded Returns over Time for Various Asset Classes

Annual Arithmetic Average (%) Annual Geometric Mean (%) Yeark-End Value, 2005 ($)

Government of Canada treasury bills 5.20 5.11 $29,711

Government of Canada bonds 6.62 6.24 61,404

Canadian stocks 11.79 10.60 946,009

U.S. stocks 13.15 11.76 1,923,692

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Compound Interest

Solution Using a Financial Calculator (TI BA II Plus)

PMT PV I/Y N

Input the following variables:

0 → ; -1,000 → ; 10 → ; and 5 →

CPT FV

Press (Compute) and then

PMT refers to regular payments FV is the future value

I/Y is the period interest rate N is the number of periods

PV is entered with a negative sign to reflect investors must pay money now to get money in the future.

Answer = $1,610.51 $1,610.51 0) $1,000(1.1 1 . % 10 000 , 1 $ 5 0     n n n FV k) ( PV FV years five for at invested of value Future

(29)

Compound Interest

Solution Using a Excel Spreadsheet

• Electronic spreadsheets have built-in formulae that

can assist in the solution of problems

• Electronic spreadsheets can also be created to

solve complex problems using both built-in

functions, defined mathematical algorithms and

relationships.

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Compound Interest

Solution Using a Excel Spreadsheet Built-in Formula

Determining the Future Value of $1,000 invested

for forty years at 10%:

1. Place cursor in cell on spreadsheet

2. Using the pull-down menu, choose, INSERT, FUNCTION 3. Choose financial functions

4. Choose FV

5. Fill in the appropriate function arguments as follows:

=FV (rate, nper, pmt, pv, type)

=FV (0.10, 40, 0, 1000,0) which yields → -45,259.26

(The answer is expressed as a negative because we entered the investment as a positive number. )

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Using Excel to Solve for FV

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Compound Interest

Underlying Assumptions

Notice the compound interest assumptions that are

embodied in the basic formula:

FV

₂

= $1,000

×

(1+k

₁

)

×

(1+k

₂

)

FV

_n

= PV

₀

×

(1+k)

n

Assumptions:

• The rate of interest does not change over the periods of compound interest

• Interest is earned and reinvested at the end of each period

• The principal remains invested over the life of the investment

• The investment is started at time 0 (now) and we are

determining the compound value of the whole investment at

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Compound Interest

Underlying Assumptions – Timing of Cash Flows

Time = 0 Time = 1 Time = 2

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Compound Interest Formula

(For a single cash flow)

FV

_n

=PV

₀

(1+k)

n

Where:

FV_n= the future value (sum of both interest and principal) of the investment at some time in the future

PV₀= the original principal invested

k= the rate of return earned on the investment

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CVIF

_k,n

Tables of Compound Value

Interest Factors can be

created:

Period 1% 2% 3% 4% 5% 6% 7% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026

6289

.

1 )

05 .

1 (

10 10 %, 5







  _n _years k

CVIF

(36)

CVIF

_k,n

The table shows that the longer you invest…the greater the amount of

money you will accumulate.

It also shows that you are better off investing at higher rates of return.

Period 1% 2% 3% 4% 5% 6% 7% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672

(37)

CVIF

_k,n

How long does it take to double or triple your investment? At 5%...at 10%? Period 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.3310 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.4641 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.6105 6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.7716 7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.9487 8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 1.8509 1.9926 2.1436 9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.3579 10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.5937

(38)

The Rule of 72

• If you don‟t have access to time value of money tables or a financial

calculator but want to know how long it takes for your money to

double…use the rule of 72!

years 16 4.5 72 : in double it will money your on rate 4.5% a earn expect to you If rate interest compound Annual 72 double to years of Number   

(39)

CVIF

_k,n

Let us predict what happens with an investment if it is invested at 5%

…show the accumulated value after t=1, t=2, t=3, etc.

Period 1% 2% 3% 4% 5% 1 1.0100 1.0200 1.0300 1.0400 1.0500 2 1.0201 1.0404 1.0609 1.0816 1.1025 3 1.0303 1.0612 1.0927 1.1249 1.1576 4 1.0406 1.0824 1.1255 1.1699 1.2155 5 1.0510 1.1041 1.1593 1.2167 1.2763 6 1.0615 1.1262 1.1941 1.2653 1.3401 7 1.0721 1.1487 1.2299 1.3159 1.4071 8 1.0829 1.1717 1.2668 1.3686 1.4775 9 1.0937 1.1951 1.3048 1.4233 1.5513 10 1.1046 1.2190 1.3439 1.4802 1.6289 FV 1.2000 1.4000 1.6000 1.8000

(40)

CVIF

_k,n

Let us predict what happens with an investment if it is invested at 5% and 10%

…show the accumulated value after t=1, t=2, t=3, etc.

Period 5% 10% 1 1.0500 1.1000 2 1.1025 1.2100 3 1.1576 1.3310 4 1.2155 1.4641 5 1.2763 1.6105 6 1.3401 1.7716 7 1.4071 1.9487 8 1.4775 2.1436 9 1.5513 2.3579 10 1.6289 2.5937 Future Value 0.0000 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 6 17 18 19 20 Time F V o f $ 1 . 0 0

Notice: compound interest creates an exponential curve and there will be a substantial difference over the long term when you can earn higher rates of return.

(41)

Types of Problems in Compounding

(42)

Types of Compounding Problems

•

There are really only four different things you can be asked

to find using this basic equation:

FV

_n

=PV

₀

(1+k)

n

– Find the initial amount of money to invest (PV₀)

– Find the Future value (FV_n)

– Find the rate (k)

(43)

Types of Compounding Problems

Solving for the Rate (k)

• Your have asked your father for a loan of $10,000 to get you started in a business. You promise to repay him $20,000 in five years time.

• What compound rate of return are you offering to pay?

• This is an ex ante calculation.

FV_t=PV₀ (1+k)n

$20,000= $10,000 (1+r)5

(44)

Types of Compounding Problems

Solving for Time (n) or Holding Periods

• You have $150,000 in your RRSP (Registered Retirement Savings Plan). Assuming a rate of 8%, how long will it take to have the plan grow to a value of $300,000?

– This is an ex ante calculation

FV_t=PV₀(1+k)n $300,000= $150,000 (1+.08)n 2=(1.08)n ln 2 =ln 1.08 × n 0.69314 = .07696 × n t = 0.69314 / .076961041 = 9.00 years

(45)

Types of Compounding Problems

Solving for Time (n) – using logarithms

• You have $150,000 in your RRSP (Registered Retirement Savings Plan). Assuming a rate of 8%, how long will it take to have the plan grow to a value of $300,000?

– This is an ex ante calculation.

FV_t=PV₀ (1+k)n

$300,000= $150,000 (1+.08)n

2=(1.08)n

(46)

Types of Compounding Problems

Solving for the Future Value (FV_n)

• You have $650,000 in your pension plan today. Because you have retired, you and your employer will not make any further

contributions to the plan. However, you don‟t plan to take any

pension payments for five more years so the principal will continue to grow.

• Assuming a rate of 8%, forecast the value of your pension plan in 5 years.

– This is an ex ante calculation.

FV_t=PV₀ (1+k)n

FV₅= $650,000 (1+.08)5

FV₅ = $650,000 × 1.469328077 FV₅ = $955,063.25

(47)

Types of Compounding Problems

Finding the amount of money to invest (PV₀)

•

You hope to save for a down payment on a home. You hope

to have $40,000 in four years time; determine the amount

you need to invest now at 6%

– This is a process known as discounting

– This is an ex ante calculation

FV

_n

=PV

₀

(1+k)

n

(48)

Compound Interest

Discounting (Computing Present Values)

1

1 )

1 (

0 _n n _n n

k)

(

FV

k

FV

PV











[ 5-3]

(49)

Annuities

(50)

Annuity

• An annuity is a finite series of equal and periodic

cash flows.

(51)

Annuities and Perpetuities

Ordinary Annuity Formula

)

1 (

1

0



















k

PMT

PV

n [ 5-5]

(52)

Ordinary Annuity

Involve end-of-period payments – First cash flow occurs at n=1

An annuity is a finite series of equal and periodic cash flows where PMT₁=PMT₂=PMT₃=…=PMT_n Time = n PMT_n Time = 0 Time of Investment _n=0 Time = 1 PMT₁ Time = 2 PMT₂ Time = 3 PMT₃

(53)

Future Value of An Ordinary Annuity

• An example of a compound annuity would be where

you save an equal sum of money in each period

(54)

Annuities and Perpetuities

Ordinary Annuities

Compound Value Annuity Formula (CVAF)

1

1 PMT(CVAF)

k

k)

(

PMT

FV

n n









[ 5-4]

(55)

Future Value of An Annuity

Example:

How much will you have at the end of three years if you save $1,000 each year for three years at a rate of 10%?

1

k k) ( PMT FV n n







(56)

Future Value of An Annuity

Example:

How much will you have at the end of three years if you save $1,000 each year for three years at a rate of 10%?

FV₃ = $1,000 × {[(1.1)3 _{- 1] / .1} =$1,000} _× _{3.31 = $3,310}

What does the formula assume?

$1,000₁ × (1.1) × (1.1) = $1,210 + $1,000₂ × (1.1) = $1,100

+ $1,000₃ = $1,000

(57)

Future Value of An Annuity

Assumptions

FVA₃ = $1,000 × {[(1.1)3 _{- 1].1} =$1,000} _× _{3.31 = $3,310}

What does the formula assume? $1,000₁ × (1.1) × (1.1) = $1,210 + $1,000₂ × (1.1) = $1,100

+ $1,000₃ = $1,000

Sum = = $3,310

The CVAF assumes that time zero (t=0) (today) you decide to invest, but If these assumptions don’t hold…you can’t use the formula.

(58)

Adjusting your solution to the

circumstances of the problem

•

The time value of money formula can be applied to any

situation…what you need to do is to understand the assumptions underlying the formula…then adjust your

approach to match the problem you are trying to solve.

• In the foregoing problem…ít isn‟t too logical to start a savings program…and then not make the first investment until one

(59)

Example of Adjustment

(An Annuity Due)

You plan to invest $1,000 today, $1,000 one year

from today and $1,000 two years from today.

What sum of money will you accumulate at time 3 if

your money is assumed to earn 10%.

(60)

Annuity Due

First cash flow occurs at n=0

An annuity due is a finite series of equal and periodic cash flows where PMT₁=PMT₂=PMT₃=…=PMT_n but the first payment occurs at time=0. Time = n PMT_n Time = 0 Time = 1 PMT₁ Time = 2 PMT₂ Time = 3 PMT₃ No PMT

(61)

Example of Adjustment

An Annuity Due

You plan to invest $1,000 today, $1,000 one year from today and $1,000 two years from today.

What sum of money will you accumulate in three years if your money is assumed to earn 10%.

You should know that there is a simple way of adjusting a normal annuity

to become an annuity due…just multiply the normal annuity result by (1+k)

and you will convert to an annuity due!

$1,000₁ × (1.1) × (1.1) × (1.1) = $1,331 + $1,000₂ × (1.1) × (1.1) = $1,210

+ $1,000₃ × (1.1) = $1,100

(62)

Annuities and Perpetuities

Future Value of an Annuity Due Formula

)

1

1 k

(

k

k)

(

PMT

FV

n n















_

_



[ 5-6]

(63)

Annuities and Perpetuities

Present Value of an Annuity Due

k)

(1

)

1 (

1

0





















k

PMT

PV

n [ 5-7]

(64)

Discounting Cash Flows

Time Value of Money …

(65)

What is Discounting?

•

Discounting is the inverse of compounding.

n

k

n

k

CVIF

PVIF

)

1 (

1

1 ,

,



_

(66)

Example of Discounting

You will receive $10,000 one year from today. If you had the money today, you could earn 8% on it.

What is the present value of $10,000 received one year from now at 8%?

PV₀=FV₁ × PVIF_k,n = $10,000 × (1/ 1.081₎

PV₀ = $10,000 × 0.9259 = $9,259.26

NOTICE: A present value is always less than the absolute value of the cash flow unless there is no time value of money. If there is no rate of interest then PV = FV

(67)

PVIF

_k,n

Tables of present value interest factors can be created:

Period 1% 2% 3% 4% 5% 6% 7% 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 n n k

k

PVIF

)

1 (

1

,



_

(68)

PVIF

_k,n

Notice – the farther away the receipt of the cash flow from today…the lower the present value…

Notice – the higher the rate of interest…the lower the present value.

Period 1% 2% 3% 4% 5% 6% 7% 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 5083 . 0 ) 07 . 1 ( 1 10 10 %, 7      n k PVIF

(69)

PVIF

_k,n

If someone offers to pay you a sum 50 or 60 years hence…that promise is „pretty-much‟ worthless!!!

n n k

k

PVIF

)

1 (

1

,



_

Period 5% 10% 15% 20% 25% 30% 35% 60 0.0535 0.0033 0.0002 0.0000 0.0000 0.0000 0.0000 70 0.0329 0.0013 0.0001 0.0000 0.0000 0.0000 0.0000 80 0.0202 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 90 0.0124 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 100 0.0076 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 110 0.0047 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

(70)

The Reinvestment Rate

(71)

The Nature of Compound Interest

• When we assume compound interest, we are implicitly assuming that any credited interest is reinvested in the next period, hence, the growth of the fund is a function of interest on the principal, and a

growing interest upon interest stream….

• This principal is demonstrated when we invest $10,000 at 8% per

annum over a period of say 4 years…the future value of this

(72)

FV

₄

of $10,000 @ 8%

Rate of Interest = 8.00%

Time

Principal at Beginning

of the Year Interest

End of Period Value of the Fund (Principal plus Interest) 1 $10,000.00 $800.00 $10,800.00 2 $10,800.00 $864.00 $11,664.00 3 $11,664.00 $933.12 $12,597.12 4 $12,597.12 $1,007.77 $13,604.89

Of course we can find the

answer using the

formula:

FV₄ =$10,000(1+.08)4

=$10,000(1.36048896) =$13,604.89

(73)

Annuity Assumptions

• When using the unadjusted formula or table values

for annuities (whether future value or present value)

we always assume:

– the focal point is time 0

– the first cash flow occurs at time 1

– intermediate cash flows are reinvested at the rate of interest for the remaining time period

(74)

FV of an Annuity Demonstrated

When determining the Future Value of an Annuity…we

assume we are standing at time zero, the first cash flow

will occur at the end of the year and we are trying to

determine the accumulated future value of a series of five

equal and periodic payments as demonstrated in the

following time line...

0 1

2

3

4

5

(75)

FV of an Annuity Demonstrated

We could be trying find out how much we would

accumulate in a savings fund…if we saved $2,000 per year

for five years at 8%…but we won’t make the first deposit

in the fund for one year...

0 1

2

3

4

5

(76)

FV of an Annuity Demonstrated

The time value of money formula assumes that each

payment will be invested at the going rate of interest for the

remaining time to maturity….

This final $2,000 is contributed to the fund, but is assumed not to earn any interest.

$2,000 invested at 8% for 4 years $2,000 invested at 8% for 3 years

$2,000 invested at 8% for 2 years

$2,000 invested at 8% for 1 year

0 1

2

3

4

5

(77)

FV of an Annuity Demonstrated

Annuity Assumptions: A demonstration

- focal point is time zero

- the first cash flow occurs at time one

Future value of a $2,000 annuity at the end of five years at 8%:

Time Cashflow CVIF Future Value

0

1 $2,000 1.3605 $2,720.98 2 $2,000 1.2597 $2,519.42

CVIF for 4 years at 8% (4 years is the remaining time to maturity.)

Notice that the final cashflow is just

(78)

FV of an Annuity Demonstrated

Annuity Assumptions: A demonstration - focal point is time zero

You can always discount or compound the individual cash flows…however it may be quicker to use an annuity formula.

Future value of a $2,000 annuity at the end of five years at 8%:

Time Cashflow CVIF Future Value

0 1 $2,000 1.3605 $2,720.98 2 $2,000 1.2597 $2,519.42 3 $2,000 1.1664 $2,332.80 4 $2,000 1.0800 $2,160.00 5 $2,000 1.0000 $2,000.00

Future Value of Annuity = FV(5) $11,733.20

Using the formula: FV(5) = PMT(CVAF t=5, r=8%) = $2,000 [(((1 + r)t)-1) / r] = $2,000(5.8666) = $11,733.20

CVIF for 4 years at 8% (4 years is the remaining time to maturity.)

Notice that the final cashflow is just received, it doesn't receive any interest.

(79)

FV of an Annuity Demonstrated

In summary the assumptions are:

–

focal point is time zero

–

we assume the cash flows occur at the end of every

year

–

we assume the interest rate does not change during

the forecast period

(80)

PV of an Annuity Demonstrated

Annuity Assumptions: A demonstration

- focal point is time zero

You can always discount or compound the individual cash flows…however it may be quicker to use an annuity formula.

Present value of a five year $2,000 annual annuity at 8%:

Time Cashflow PVIF Present Value

0 1 $2,000 0.9259 $1,851.85 2 $2,000 0.8573 $1,714.68 3 $2,000 0.7938 $1,587.66 4 $2,000 0.7350 $1,470.06 5 $2,000 0.6806 $1,361.17

Present Value of Annuity = $7,985.42

Using the formula: PV = PMT(PVIFA n=5, k=85) = $2,000 [1- 1/(1 + k)n] / k = $2,000(3.9927) = $7,985.40

(81)

The Reinvestment Rate Assumption

• It is crucial to understand the reinvestment rate assumption that is built-in to the time value of money.

• Obviously, when we forecast, we must make

assumptions…however, if that assumption not realistic…it is

important that we take it into account.

• This reinvestment rate assumption in particular, is important in the

yield-to-maturity calculations in bonds…and in the Internal Rate of Return (IRR) calculation in capital budgeting.

(82)

Perpetuities

(83)

Perpetuities

• A perpetuity is an infinite annuity

• An infinite series of payments where each payment

is equal and periodic.

• Examples of perpetuities in financial markets

includes:

–

Common stock (with a no growth in dividend

assumption)

(84)

Perpetuity

Involve end-of-period payments – First cash flow occurs at n=1

A perpetuity is an infinite series of equal and periodic cash flows where PMT₁=PMT₂=PMT₃=…=PMT_α Time = α PMT_α Time = 0 Time of Investment _n=0 Time = 1 PMT₁ Time = 2 PMT₂ Time = 3 PMT₃

(85)

Perpetuities

Perpetuity Formula Perpetuity Formula 0 0

k

P

PM

MT

T

P

PV

V



[ 5-8] [ 5-8]

Where:

PV

(86)

Perpetuity:

An

Example

While acting as executor for a distant relative, you

discover a $1,000 Consol Bond issued by Great Britain

in 1814,

issued to

help fund

the Na

poleonic War.

If th

e

bond pays annual interest of 3.0% and

other long U.K.

Government bonds are currently paying 5%,

what would

each $1,000 Consol Bond sell for in the market?

(87)

Perpetuity: Solution

 



0 0

$

$1

1,, 0

00

00 0 0

0..0

03

3

0.05

0.05 $30

$30

0.05

0.05 PMT

PMT

PV

k



(88)

Nominal Versus Effective Rates

(89)

Nominal Versus Effective Interest Rates

• So far, we have assumed annual compounding

• When rates are compounded annually, the quoted

rate and the effective rate are equal

• As the number of compounding periods per year

increases, the effective rate will become larger than

the quoted rate

(90)

Nominal versus Effective Rates

General Formula for Effective Annual Rate

1 )

1 (







m

QR

k

[ 5-9]

(91)

Calculating the Effective Rate

1

m Effective

QR

k

m





 

_

_







Where:

k _Effective = Effective annual interest rate

(92)

Example: Effective Rate Calculation

•

A bank is offering loans at 6%, compounded monthly. What is

the effective annual interest rate on their loans?

12

1

1 .06

1

12 6.17%

m Effective

QR

k

m





 

_

_











 

_

_









(93)

Nominal versus Effective Rates

Continuous Compounding Formula

1 



QR

e

k

[ 5-10]

(94)

Continuous Compounding

• When compounding occurs continuously, we

calculate the effective annual rate using e, the base

of the natural logarithms (approximately 2.7183)

1

QR Effective

(95)

10% Compounded At Various Frequencies

Compounding

Frequency

Effective Annual

Interest Rate

2 10.25%

4 10.3813%

12 10.4713%

52 10.5065%

365 10.5156%

(96)

Calculating the Quoted Rate

•

If we know the effective annual interest rate, (k

_Eff

) and we

know the number of compounding periods, (m) we can

solve for the Quoted Rate, as follows:



1 

m1

1

Eff

QR





_



k





_

m

(97)

When Payment & Compounding Periods Differ

• When the number of payments per year is different

from the number of compounding periods per year,

you must calculate the interest rate per payment

period, using the following formula

1

m f Per Period

QR

k

m







_



_







(98)

Nominal versus Effective Rates

Formula for Effective Rates for “Any” Period

1

1 )

-m

QR

(

k

f m





[ 5-11]

(99)

Loans and Loan Amortization Tables

(100)

Loan Amortization

–

A blended payment loan is repaid in equal periodic

payments

–

However, the amount of principal and interest varies

each period

–

Assume that we want to calculate an amortization

table showing the amount of principal and interest

paid each period for a $5,000 loan at 10% repaid in

three equal annual instalments.

(101)

Blended Interest and Principal Loan

Payments - formula



















k

k)

(1

1

1 PMT

Principal

)

PMT(PVAF

Principal

n n k, Where:

(102)

Blended Interest and Principal Loan

Payments - example

52 .

018 ,

1 $

818147

.

9

000 ,

10 $

Pmt

.08

)

08 .

1 (

1

1 Pmt

000 ,

10 $

r

)

1 (

1

1 PMT

Principal

20





































k

n Where: Pmt = unknown t= 20 years r = 8%

Calculator Approach:

10,000

PV

0 FV

20 N

8 I/Y

CPT PMT

$1,018.52

(103)

How are Loan Amortization Tables Used?

• To separate the loan repayments into their constituent components.

– Each level payment is made of interest plus a repayment of some portion of the principal outstanding on the loan.

– This is important to do when the loan has been taken out for the

purposes of earning taxable income…as a result, the interest is a

(104)

Loan Amortization Tables

Using an Excel Spreadsheet

Principal = $100,000 Rate = 8.0% Term = 5 PVAF = 3.99271 Payment = $25,045.65 Retired Ending Year Principal Interes t Paym ent Principal Balance

1 100,000.00 8,000.00 25,045.65 17,045.65 82,954.35 2 82,954.35 6,636.35 25,045.65 18,409.30 64,545.06 3 64,545.06 5,163.60 25,045.65 19,882.04 44,663.02 4 44,663.02 3,573.04 25,045.65 21,472.60 23,190.41 5 23,190.41 1,855.23 25,045.65 23,190.41 0.00

(105)

Loan or Mortgage Arrangements

Effective Rate for Any Period Formula

1

1 )

-m

QR

(

k

f m Eff





[ 5-11]

(106)

Loan Amortization

Example with Solution

•

First calculate the annual payments













3 1 1 1 1 5,000 1 1.10 0.10 $2,010.57 n Annuity Annuity n k PV PMT k PV PMT k k     _{ }           _{ }          _        

Calculator Approach:

5,000

PV

0 FV

3 N

10 I/Y

CPT PMT

$2,010.57

(107)

Amortization Table

Period Principal: Start of

Period

Payment Interest Principal Principal: End of Period

1

5,000.00 2,010.57 500.00 1,510.57 3,489.43

(108)

Calculating the Balance O/S

• At any point in time, the balance outstanding on the

loan (the principal not yet repaid) is the PV of the

loan payments not yet made.

• For example, using the previous example, we can

calculate the balance outstanding at the end of the

first year, as shown on the next page

(109)

Calculating the Balance O/S after the 1

st

_Year









1 2

1

1.10 2,010.57

.10

$3,489.42

n t k PV PMT k   



_{ }



















_



















(110)

Canadian Residential Mortgages

• A Canadian residential mortgage is a loan with one

special feature

–

By law, banks in Canada can only compound the

interest twice per year on a conventional mortgage,

but payments are typically made at least monthly

• To solve for the payment, you must first calculate

the correct periodic interest rate

(111)

Canadian Residential Mortgages

•

For example, suppose we want to calculate the monthly

payment on a $100,000 mortgage amortized over 25 years

with a 6% annual interest rate.

•

First, calculate the monthly interest rate:

2

1

1 .06

m f Per Period QR k m





 

_

_











(112)

Calculating the Monthly Payment

•

Now, calculate the monthly payment on the mortgage













0 0 300 1 1 1 1 100,000 1 1.004938622 .004938622 $639.81 n t t n k PV PMT k PV PMT k k     



_{ }





















_{ }



















_

















Calculator Approach:

100,000

PV

0 FV

300 N

.4938622

I/Y

CPT PMT

$639.81

(113)

Monthly Mortgage Loan Amortization

Table

Principal = $100,000 Quoted rate = 6.0%

Effective annual Rate = 6.090% (Assum ing semi-annual compounding) Effective monthly Rate = 0.49386%

Term = 25 years Term in months = 300

PVAF = 156.297225 Payment = $639.81

Retired Ending Month Principal Interest Payment Principal Balance

(114)

Summary and Conclusions

In this chapter you have learned:

– To compare cash flows that occur at different points in time

– To determine economically equivalent future values from values that occur in previous periods through compounding.

– To determine economically equivalent present values from cash flows that occur in the future through discounting

– To find present value and future values of annuities, and

– To determine effective annual rates of return from quoted interest rates.

(115)

Concept Review Questions

Time Value of Money

(116)

Concept Review Question 1

Quoted versus Effective Rates

Why can effective rates often be very different from

quoted rates?

The more frequently interest is compounded the higher the effective

rate of return.

Because financial institutions are legally only required to

Because financial institutions are legally only required to quote APRquote APR

(Annual Percentage Rates) that are stated (nominal) the published

rate is often much lower

rate is often much lower than the actual rate charged depending onthan the actual rate charged depending on

the frequency of compounding.

This is why reading the fine print is so important!

(117)

Internet Links

•

• Planning tools and online courses throughPlanning tools and online courses through TD Canada TrustTD Canada Trust

•