Chapter 5 - Time Value of Money

INTRODUCTION TO

CORPORATE CORPORA TE FINANCE FINANCE Laurence Booth • W. Sean Cleary Chapter 5 – Time Value of Money

Prepared by Ken Hartviksen and Robert Ironside

CHAPTER 5 Time Value of Money

CHAPTER 5 Time Value of Money

Lecture Agenda • • • • • • • • •

Learning Objectives Important Terms Types of Calculations Compounding Discounting Annuities and Loans Perpetuities Effective Rates of Return Summary and Conclusions  – Concept Review Questions CHAPTER 5 – Time Value of Money

5-3

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 Know how to construct an amortization table

CHAPTER 5 – Time Value of Money

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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 Time value of money

CHAPTER 5 – Time Value of Money

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Types of Calculations Time Value of Money

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 Time Value of Money

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.  – Conversely, if you are promised a cash flow in the future, it‟s present value today is less than what is promised! CHAPTER 5 – Time Value of Money

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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 Time Value of Money

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 I1

2 I2

3 I3

CHAPTER 5 – Time Value of Money

n

… In+P

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Simple Interest Example PROBLEM: Invest $1,000 today for a fiveyear term and receive 8 percent annual simple interest.

Year 1 2 3 4 5

Beginning Amount $1,000 1,080 1,160 1,240 1,320

Ending Amount $1,080 1,160 1,240 1,320 $1,400

How much will you accumulate by the end of five years? Value (time n)  P  (n  P  k) Value5

 $1,000  (5  $1,000  .08)  $1,000  (5  $80)  $1,000  $400  $1,400

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Simple Interest General Formula

[ 5-1]

Value (time n)  P  (n  P  k)

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 Time Value of Money

Compound Interest Compounding (Computing Future Values)

Compound interest is interest that is earned on the principal amount invested and on any accrued interest.

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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: The solution in one simple step : Year

Beginning Amount

1

$1,000.00

2

1,080.00

1,166.40

FV 2  P(1  .08)(1  .08)  P(1  .08)  $1,166.40

3

1,166.40 

1,259.71

FV 3  P(1  .08)(1  .08)(1  .08)  P(1.08)  $1,259.71

4

1,259.71

1,360.49

FV 4  P(1.08)(1.08)(1.08)(1.08)  P (1.08)  $1,360.49

5

1,360.49

1,469.33

FV 5  P(1  .08)  $1,469.33

FV n

FV5



Ending Amount

n Value  P  ( 1  k) Future

PV 0( 1  k)  $1,080.00

FV 1

n

P(1  .08)  $1,080 1

2

$1,000(1.08)

5

$1,469.33 3

4

5

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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 1 2 3 4 5

Beginning Amount $1,000.00 1,080.00 1,166.40 1,259.71 1,360.49

Ending Amount $1,080.00 1,166.40 1,259.71 1,360.49 1,469.33

CHAPTER 5 – Time Value of Money

Interest earned in the year $80.00 $86.40 $93.31 $100.78 $108.84

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Compound Interest General Formula

[ 5-2]

FV n



n

PV 0( 1  k)

Where: FV= future value P = principal invested n = number of years k = interest rate CHAPTER 5 – Time Value of Money

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Compound Interest General Formula

[ 5-2]

FV n



n

PV 0( 1  k)

( 1  k)n is known as the compoundinterest factor  CVIF

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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

(See Figure 5-1 to compare simple and compound interest effects over time)

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Compound Interest Simple versus Compound Interest 5-1 FIGURE 8,000 7,000 6,000    S 5,000    R    A    L    L 4,000    O    D 3,000

2,000 1,000 0

1

2

3

4

5

6

Simple

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Compound

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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

Table 5-1 Ending Wea lth of $1,000 Invested From 1938 to 2005 in Various Asset Classes

Government of Canada treasury bills Government of Canada bonds Canadian stocks U.S. stocks

Annual Arithmetic Average (%)

Annual Geometric Mean (%)

5.20 6.62 11.79 13.15

5.11 6.24 10.60 11.76

Yeark-End Value, 2005 ($) $29,711 61,404 946,009 1,923,692

Source: Data are fro m the Canadian Institut e of A ctuaries

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Compound Interest Solution Using a Financial Calculator (TI BA II Plus) Input the following variables: 0

→

PMT

; -1,000 →

Press

CPT

(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

; 10 →

I/Y

; and

5→

N

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

5 FV n  $1,000(1.10)  $1,610.51

PV is entered with a negative sign to reflect investors must pay money now to get money in the future. Answer = $1,610.51 CHAPTER 5 – Time Value of Money

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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. 2. 3. 4. 5.

Place cursor in cell on spreadsheet Using the pull-down menu, choose, INSERT, FUNCTION Choose financial functions Choose FV 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. ) CHAPTER 5 – Time Value of Money

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Using Excel to Solve for FV Built-in Formula Function Arguments and Solution

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Compound Interest Underlying Assumptions

Notice the compound interest assumptions that are embodied in the basic formula: FV2 = $1,000 × (1+k1) × (1+k2) FVn= PV0 × (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 the end of some time period (t= 1, 2, 3, 4,…) CHAPTER 5 – Time Value of Money

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Compound Interest Underlying Assumptions – Timing of Cash Flows

Time = 0

Time = 1

Time = 2

Time of Investment

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Compound Interest Formula (For a single cash flow)

FVn=PV0 (1+k)n Where: FVn= 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 n = the time or number of periods the investment is allowed to grow

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CVIFk,n

(For a single cash flow)

Tables of Compound Value Interest Factors can be created:

Period 1 2 3 4 5 6 7 8 9 10

1% 1.0100 1.0201 1.0303 1.0406 1.0510 1.0615 1.0721 1.0829 1.0937 1.1046

2% 1.0200 1.0404 1.0612 1.0824 1.1041 1.1262 1.1487 1.1717 1.1951 1.2190

3% 1.0300 1.0609 1.0927 1.1255 1.1593 1.1941 1.2299 1.2668 1.3048 1.3439

CVIF k 5%,n10 years  (1  .05)10

 1.6289 4% 1.0400 1.0816 1.1249 1.1699 1.2167 1.2653 1.3159 1.3686 1.4233 1.4802

5% 1.0500 1.1025 1.1576 1.2155 1.2763 1.3401 1.4071 1.4775 1.5513 1.6289

CHAPTER 5 – Time Value of Money

6% 1.0600 1.1236 1.1910 1.2625 1.3382 1.4185 1.5036 1.5938 1.6895 1.7908

7% 1.0700 1.1449 1.2250 1.3108 1.4026 1.5007 1.6058 1.7182 1.8385 1.9672 5 - 34

CVIFk,n

(For a single cash flow) 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 8 9 10

1% 1.0100 1.0201 1.0303 1.0406 1.0510 1.0615 1.0721 1.0829 1.0937 1.1046

2% 1.0200 1.0404 1.0612 1.0824 1.1041 1.1262 1.1487 1.1717 1.1951 1.2190

3% 1.0300 1.0609 1.0927 1.1255 1.1593 1.1941 1.2299 1.2668 1.3048 1.3439

4% 1.0400 1.0816 1.1249 1.1699 1.2167 1.2653 1.3159 1.3686 1.4233 1.4802

5% 1.0500 1.1025 1.1576 1.2155 1.2763 1.3401 1.4071 1.4775 1.5513 1.6289

CHAPTER 5 – Time Value of Money

6% 1.0600 1.1236 1.1910 1.2625 1.3382 1.4185 1.5036 1.5938 1.6895 1.7908

7% 1.0700 1.1449 1.2250 1.3108 1.4026 1.5007 1.6058 1.7182 1.8385 1.9672

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CVIFk,n

(For a single cash flow) 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 11 12 13 14 15 16

1% 1.0100 1.0201 1.0303 1.0406 1.0510 1.0615 1.0721 1.0829 1.0937 1.1046 1.1157 1.1268 1.1381 1.1495 1.1610 1.1726

2% 1.0200 1.0404 1.0612 1.0824 1.1041 1.1262 1.1487 1.1717 1.1951 1.2190 1.2434 1.2682 1.2936 1.3195 1.3459 1.3728

3% 1.0300 1.0609 1.0927 1.1255 1.1593 1.1941 1.2299 1.2668 1.3048 1.3439 1.3842 1.4258 1.4685 1.5126 1.5580 1.6047

4% 1.0400 1.0816 1.1249 1.1699 1.2167 1.2653 1.3159 1.3686 1.4233 1.4802 1.5395 1.6010 1.6651 1.7317 1.8009 1.8730

5% 1.0500 1.1025 1.1576 1.2155 1.2763 1.3401 1.4071 1.4775 1.5513 1.6289 1.7103 1.7959 1.8856 1.9799 2.0789 2.1829

6% 1.0600 1.1236 1.1910 1.2625 1.3382 1.4185 1.5036 1.5938 1.6895 1.7908 1.8983 2.0122 2.1329 2.2609 2.3966 2.5404

CHAPTER 5 – Time Value of Money

7% 1.0700 1.1449 1.2250 1.3108 1.4026 1.5007 1.6058 1.7182 1.8385 1.9672 2.1049 2.2522 2.4098 2.5785 2.7590 2.9522

8% 1.0800 1.1664 1.2597 1.3605 1.4693 1.5869 1.7138 1.8509 1.9990 2.1589 2.3316 2.5182 2.7196 2.9372 3.1722 3.4259

9% 1.0900 1.1881 1.2950 1.4116 1.5386 1.6771 1.8280 1.9926 2.1719 2.3674 2.5804 2.8127 3.0658 3.3417 3.6425 3.9703

10% 1.1000 1.2100 1.3310 1.4641 1.6105 1.7716 1.9487 2.1436 2.3579 2.5937 2.8531 3.1384 3.4523 3.7975 4.1772 4.5950

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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!

Number of  years to double 

72 Annual compoundinterest rate

If  you expect toearn a 4.5% rate on your money it will double in :



72  16 years 4.5

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CVIFk,n

(For a single cash flow) 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

1.8000

1.6000

1.4000

1.2000    0    0  . 1.0000    1     $    f    o    V    F

0.8000

0.6000

0.4000

0.2000

0.0000 1

2

3

4

5

6

7

8

9

10

Year

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CVIFk,n

(For a single cash flow) 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. Future Value

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

8.0000

7.0000

6.0000

5.0000    0    0  .    1    $    f 4.0000   o    V    F

3.0000

2.0000

1.0000

0.0000 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Time

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. CHAPTER 5 – Time Value of Money

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Types of Problems in Compounding Time Value of Money Skills

Types of Compounding Problems •

There are really only four different things you can be asked to find using this basic equation: FVn=PV0 (1+k)n  –  –  –  –

Find the initial amount of money to invest (PV0) Find the Future value (FV n) Find the rate (k) Find the time (n)

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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.

FVt=PV0 (1+k)n $20,000= $10,000 (1+r)5 2=(1+r)5 21/5=1+r 1.14869=1+r r = 14.869% CHAPTER 5 – Time Value of Money

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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

FVt=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

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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.

FVt=PV0 (1+k)n $300,000= $150,000 (1+.08)n 2=(1.08)n log 2 =log 1.08 × n 0.301029995 = 0.033423755 × n t = 9.00 years

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Types of Compounding Problems Solving for the Future Value (FVn) •

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.

FVt=PV0 (1+k)n FV5= $650,000 (1+.08)5 FV5 = $650,000 × 1.469328077 FV5 = $955,063.25 CHAPTER 5 – Time Value of Money

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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

FVn=PV0 (1+k)n $40,000= PV0 (1.1)4 PV0 = $40,000/1.4641=$27,320.53

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Compound Interest Discounting (Computing Present Values)

[ 5-3]

PV 0 

FV n

(1  k )

n

 FV n 

CHAPTER 5 – Time Value of Money

1 ( 1  k)

n

5 - 47

Annuities Time Value of Money Concepts

Annuity

• An annuity is a finite series of equal and periodic

cash flows.

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Annuities and Perpetuities Ordinary Annuity Formula

[ 5-5]

1   1  (1  k ) n  PV 0  PMT   k     

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

Time = 0

Time of Investment

Time = 1

n=0

PMT1

Time = 2

Time = 3

Time = n

PMT2

PMT3

PMTn

An annuity is a finite series of equal and periodic cash flows where PMT1=PMT2=PMT3=…=PMTn CHAPTER 5 – Time Value of Money

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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 over a period of time to accumulate a future sum.

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Annuities and Perpetuities Ordinary Annuities

Compound Value Annuity Formula (CVAF)

( 1  k)  1 FV n  PMT   PMT(CVAF) k  n

[ 5-4]

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Future Value of An Annuity n ( 1  k)  1 FV n  PMT  k 

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

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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

Sum =

= $3,310

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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?

If these assumptions

$1,0001 × (1.1) × (1.1) = $1,210

don’t hold…you can’t

+ $1,0002 × (1.1)

= $1,100

+ $1,0003

= $1,000

Sum =

= $3,310

use the formula.

The CVAF assumes that time zero (t=0) (today) you decide to invest, but you don’t make the first investment until one year from today. The Future

Value you forecast is the value of the entire fund (a series of investments together with the accumulated interest) at the end of some year n = 1 or n = 2 …in this case n = 3. NOTE: the rate of interest is assumed to remain

unchanged throughout the forecast period. CHAPTER 5 – Time Value of Money

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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

year later!!!

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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%. This is known as an annuity due rather than a regular annuity.

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Annuity Due First cash flow occurs at n=0

Time = 0

PMT1

Time = 1

Time = 2

PMT2

PMT3

Time = 3

PMTn

Time = n

No PMT

An annuity due is a finite series of equal and periodic cash flows where PMT1=PMT2=PMT3=…=PMTn but the first payment occurs at time=0. CHAPTER 5 – Time Value of Money

5 - 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 in three years if your money is assumed to earn 10%.

$1,0001 × (1.1) × (1.1) × (1.1) + $1,0002 × (1.1) × (1.1) + $1,0003 × (1.1) Sum =

= = = =

$1,331 $1,210 $1,100 $3,641

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!

FV3 (Annuity due)= $1,000 × {[(1.1)3 - 1].1}× (1+ k) =$1,000 × 3.31 × 1.1 = $3,310 × 1.1 = $3,641 CHAPTER 5 – Time Value of Money

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Annuities and Perpetuities Future Value of an Annuity Due Formula

[ 5-6]

 ( 1  k)n  1 FV n  PMT  (1  k )  k   

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Annuities and Perpetuities Present Value of an Annuity Due

[ 5-7]

1   1  (1  k ) n  PV 0  PMT   (1  k) k     

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Discounting Cash Flows Time Value of Money …

What is Discounting? • Discounting is the inverse of compounding.

PVIF k ,n 

1 CVIF k ,n

CHAPTER 5 – Time Value of Money



1 (1  k )

n

5 - 64

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 × PVIFk,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 CHAPTER 5 – Time Value of Money

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PVIFk,n

(For a single cash flow)

Tables of present value interest factors can be created:

PVIF k ,n  Period 1 2 3 4 5 6 7 8 9 10

1% 0.9901 0.9803 0.9706 0.9610 0.9515 0.9420 0.9327 0.9235 0.9143 0.9053

2% 0.9804 0.9612 0.9423 0.9238 0.9057 0.8880 0.8706 0.8535 0.8368 0.8203

3% 0.9709 0.9426 0.9151 0.8885 0.8626 0.8375 0.8131 0.7894 0.7664 0.7441

1 (1  k )

4% 0.9615 0.9246 0.8890 0.8548 0.8219 0.7903 0.7599 0.7307 0.7026 0.6756

5% 0.9524 0.9070 0.8638 0.8227 0.7835 0.7462 0.7107 0.6768 0.6446 0.6139

CHAPTER 5 – Time Value of Money

n

6% 0.9434 0.8900 0.8396 0.7921 0.7473 0.7050 0.6651 0.6274 0.5919 0.5584

7% 0.9346 0.8734 0.8163 0.7629 0.7130 0.6663 0.6227 0.5820 0.5439 0.5083

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PVIFk,n

(For a single cash flow) 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. PVIF k 7%,n 10 Period 1 2 3 4 5 6 7 8 9 10

1% 0.9901 0.9803 0.9706 0.9610 0.9515 0.9420 0.9327 0.9235 0.9143 0.9053

2% 0.9804 0.9612 0.9423 0.9238 0.9057 0.8880 0.8706 0.8535 0.8368 0.8203

3% 0.9709 0.9426 0.9151 0.8885 0.8626 0.8375 0.8131 0.7894 0.7664 0.7441

4% 0.9615 0.9246 0.8890 0.8548 0.8219 0.7903 0.7599 0.7307 0.7026 0.6756

5% 0.9524 0.9070 0.8638 0.8227 0.7835 0.7462 0.7107 0.6768 0.6446 0.6139

CHAPTER 5 – Time Value of Money

6% 0.9434 0.8900 0.8396 0.7921 0.7473 0.7050 0.6651 0.6274 0.5919 0.5584



1 10

(1  .07 )



0.5083

7% 0.9346 0.8734 0.8163 0.7629 0.7130 0.6663 0.6227 0.5820 0.5439 0.5083

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PVIFk,n

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

PVIF k ,n  Period 60 70 80 90 100 110

5% 0.0535 0.0329 0.0202 0.0124 0.0076 0.0047

10% 0.0033 0.0013 0.0005 0.0002 0.0001 0.0000

15% 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000

20% 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

25% 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

30% 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1 (1  k )

n

35% 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

The present value of $10 million promised 100 years from today at a 10% discount rate is = $10,000,000 * 0.0001 = $1,000!!!!

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The Reinvestment Rate Time Value of Money Concepts

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

investment can be decomposed as follows...

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FV4 of $10,000 @ 8%

Rate of Interest =

Time 1 2 3 4

Principal at Beginning of the Year $10,000.00 $10,800.00 $11,664.00 $12,597.12

8.00%

Interest $800.00 $864.00 $933.12 $1,007.77

End of Period Value of the Fund (Principal plus Interest) $10,800.00 $11,664.00 $12,597.12 $13,604.89

Of course we can find the answer using the formula: FV4 =$10,000(1+.08)4

CHAPTER 5 – Time Value of Money

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

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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  – the interest rate is unchanging over the period of the analysis.

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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

$2,000

2 $2,000

3 $2,000

4 $2,000

5 $2,000

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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

$2,000

2 $2,000

3 $2,000

4 $2,000

5 $2,000

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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….

0

1

$2,000

2 $2,000

3 $2,000

4 $2,000

5

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

$2,000 $2,000 invested at 8% for 1 year

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

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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 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

CHAPTER 5 – Time Value of Money

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.

5 - 76

FV of an Annuity Demonstrated Annuity Assumptions:

A demonstration 

- focal point is time zero - the first cash flow occurs at time one 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 0 1 $2,000 1.3605 2 $2,000 1.2597 3 $2,000 1.1664 4 $2,000 1.0800 5 $2,000 1.0000 Future Value of Annuity = FV(5)

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

Future Value

$2,720.98 $2,519.42 $2,332.80 $2,160.00 $2,000.00 $11,733.20

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

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 CHAPTER 5 – Time Value of Money

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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  – the interest received is reinvested at that same rate of interest for the remaining time until maturity.

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PV of an Annuity Demonstrated Annuity Assumptions:

A demonstration 

- focal point is time zero - the first cash flow occurs at time one 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%:

PVIF for 1 year 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

CHAPTER 5 – Time Value of Money

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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.

CHAPTER 5 – Time Value of Money

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Perpetuities Time Value of Money Concepts

Perpetuities 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)  – Preferred stock  – Consol bonds (bonds with no maturity date) CHAPTER 5 – Time Value of Money

5 - 82

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

Time = 0

Time of Investment

Time = 1

n=0

PMT1

Time = 2

Time = 3

Time = α

PMT2

PMT3

PMTα

A perpetuity is an infinite series of equal and periodic cash flows where PMT1=PMT2=PMT3=…=PMTα CHAPTER 5 – Time Value of Money

5 - 83

Perpetuities Perpetuity Formula

P V 0 

[ 5-8]

P M T  k 

Where: PV0 = Present value of the perpetuity PMT = the periodic cash flow k = the discount rate CHAPTER 5 – Time Value of Money

5 - 84

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 Napoleonic Napoleonic War. If the the 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?

CHAPTER 5 – Time Value of Money

5 - 85

Perpetuity: Solution

PV 0 



PMT  k 

$1, 000  0.03 0.05

$30  0.05  $600

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Nominal Versus Effective Rates Time Value of Money Concepts

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

CHAPTER 5 – Time Value of Money

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Nominal versus Effective Rates General Formula for Effective Annual Rate

[ 5-9]

k   (1 

QR m

) 1 m

CHAPTER 5 – Time Value of Money

5 - 89

Calculating the Effective Rate m

k  Effective

 QR   1  1  m  

Where: k Effective = Effective annual interest rate QR = the quoted interest rate M = the number of compounding periods per year

CHAPTER 5 – Time Value of Money

5 - 90

Example: Effective Rate Calculation • A bank is offering loans at 6%, compounded monthly. What is

the effective annual interest rate on their loans?

 

k  Effective   1 

QR 

m

 1 m  12

 .06   1  1   12   6.17%

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Nominal versus Effective Rates Continuous Compounding Formula

[ 5-10]

k   eQR  1

CHAPTER 5 – Time Value of Money

5 - 92

Continuous Compounding • When compounding occurs continuously, we

calculate the effective annual rate using e, the base of the natural logarithms (approximately 2.7183)

k Effective  eQR  1

CHAPTER 5 – Time Value of Money

5 - 93

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%

Continuous

10.5171%

CHAPTER 5 – Time Value of Money

5 - 94

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   m QR  1  k Eff    1 m  

CHAPTER 5 – Time Value of Money

5 - 95

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 m

 QR   f  1 k Per    1   m  Period   Where: f = the payment frequency per year CHAPTER 5 – Time Value of Money

5 - 96

Nominal versus Effective Rates Formula for Effective Rates for “Any” Period

[ 5-11]

k   ( 1 

QR m

m

) f  -1

CHAPTER 5 – Time Value of Money

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Loans and Loan Amortization Tables Time Value of Money Concepts

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.

CHAPTER 5 – Time Value of Money

5 - 99

Blended Interest and Principal Loan Payments - formula

Principal  PMT(PVAFk, n ) 1  1  (1  k) n Principal  PMT  k   

    

Where: Pmt = the fixed periodic payment t= the amortization period of the loan r = the rate of interest on the loan CHAPTER 5 – Time Value of Money

5 - 100

Blended Interest and Principal Loan Payments - example

1   1  (1  k ) n  Principal  PMT   r     1   1   (1.08) 20  $10,000  Pmt  .08    

Calculator Approach: 10,000 PV 0 FV 20 N 8 I/Y CPT PMT $1,018.52

$10,000 Pmt   $1,018.52 9.818147

t= 20 years

Where:

CHAPTER 5 – Time Value of Money

Pmt = unknown

r = 8% 5 - 101

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

tax-deductible expense.

CHAPTER 5 – Time Value of Money

5 - 102

Loan Amortization Tables Using an Excel Spreadsheet

Principal = Rate = Term = PVAF = Payment = Year 1 2 3 4 5

$100,000 8.0% 5 3.99271 $25,045.65 Principal Interes t 100,000.00 8,000.00 82,954.35 6,636.35 64,545.06 5,163.60 44,663.02 3,573.04 23,190.41 1,855.23

Paym ent 25,045.65 25,045.65 25,045.65 25,045.65 25,045.65

CHAPTER 5 – Time Value of Money

Retired Principal 17,045.65 18,409.30 19,882.04 21,472.60 23,190.41

Ending Balance 82,954.35 64,545.06 44,663.02 23,190.41 0.00

5 - 103

Loan or Mortgage Arrangements Effective Rate for Any Period Formula

[ 5-11]

k  Eff   ( 1 

QR m

CHAPTER 5 – Time Value of Money

m

) f  -1

5 - 104

Loan Amortization Example with Solution

• First calculate the annual payments

PV Annuity

 1  1  k   n   PMT      k   

PMT  



PV  Annuity

 1  1  k   n      k    5,000

Calculator Approach: 5,000 PV 0 FV 3 N 10 I/Y CPT PMT $2,010.57

 1  1.10  3      0.10  

 $2,010.57 CHAPTER 5 – Time Value of Money

5 - 105

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

2

3,489.43

2,010.57

348.94

1,661.63

1,827.80

3

1,827.80

2,010.57

182.78

1,827.78

0

CHAPTER 5 – Time Value of Money

5 - 106

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

CHAPTER 5 – Time Value of Money

5 - 107

Calculating the Balance O/S after the 1 st Year

1  1  k  n  PVt 1  PMT    k    1  1.10 2   2,010.57   .10    $3,489.42

CHAPTER 5 – Time Value of Money

5 - 108

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

CHAPTER 5 – Time Value of Money

5 - 109

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: m

 QR   f  k Per    1  1  m  Period   2 12

 .06   1  1  2    .004938622 or  0.4938622% CHAPTER 5 – Time Value of Money

5 - 110

Calculating the Monthly Payment • Now, calculate the monthly payment on the mortgage PVt  0

1  1  k  n   PMT    k   

PMT  



PV t  0

1  1  k  n    k    100,000

Calculator Approach: 100,000 PV 0 FV 300 N .4938622 I/Y CPT PMT $639.81

1  1.004938622 300    .004938622  

 $639.81 CHAPTER 5 – Time Value of Money

5 - 111

Monthly Mortgage Loan Amortization Table Principal = Quoted rate = Effective annual Rate = Effective monthly Rate = Term = Term in months = PVAF = Payment = Month 1 2 3 4 5

$100,000 6.0% 6.090% (Assum ing semi-annual compounding) 0.49386% 25 years 300 156.297225 $639.81 Retired Ending Principal Interest Payment Principal Balance 100,000.00 493.86 639.81 145.94 99,854.06 99,854.06 493.14 639.81 146.67 99,707.39 99,707.39 492.42 639.81 147.39 99,560.00 99,560.00 491.69 639.81 148.12 99,411.88 99,411.88 490.96 639.81 148.85 99,263.03

CHAPTER 5 – Time Value of Money

5 - 112

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.

CHAPTER 5 – Time Value of Money

5 - 113

Concept Review Questions Time Value of Money

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 quote APR (Annual Percentage Rates) that are stated (nominal) the published rate is often much lower than the actual rate charged depending on the frequency of compounding. This is why reading the fine print is so important!

CHAPTER 5 – Time Value of Money

5 - 115

Internet Links

• •

Planning tools and online courses through TD Canada Trust Online tools and calculators through RBC Royal Bank

CHAPTER 5 – Time Value of Money

5 - 116