INTRODUCTION TO
INTRODUCTION TO
CORPORA
CORPORA
TE F
TE F
INANCE
INANCE
Laurence Booth
Laurence Booth
•
•
W. Sean Cleary
W. Sean Cleary
Chapter 5
CHAPTER 5
CHAPTER 5
Time Value of Money
CHAPTER 5
CHAPTER 5
Time Value of Money
Lecture Agenda
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
–
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
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
Types of Calculations
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.
The Basic Concept
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
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.
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.
Simple Interest
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
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
Simple Interest
General Formula
k)
P
(n
P
e n)
Value (tim
[ 5-1] Where: P = principal invested n = number of years k = interest rateSimple Interest
Simple interest problems are rare.
In finance we are most interested in COMPOUND
INTEREST.
Compound Interest
Compound Interest
Compounding (Computing Future Values)
Compound interest is interest that is earned on the
principal amount invested and on any accrued
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
5
0
n nPV
(
k)
FV
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 = $93.31
Year Beginning Amount Ending Amount
Interest earned in the year
Compound Interest
General Formula
Where: FV= future value P = principal invested n = number of years k = interest rate [ 5-2]1
0 n nPV
(
k)
FV
Compound Interest
General Formula
[ 5-2]1
0 n nPV
(
k)
FV
Compound Interest
Simple versus Compound Interest
Compounding of interest magnifies the returns on an
investment.
Returns are magnified:
• The longer they are compounded
• The higher the rate they are compounded
Compound Interest
Simple versus Compound Interest
5-1 FIGURE D O L L A R S 8,000 7,000 6,000 5,000 4,000 3,000 2,000
Compound Interest
Compound Interest at Varying Rates
Compounding of interest magnifies the returns on an
investment.
Returns are magnified:
• The longer they are compounded
• The higher the rate they are compounded
(See Table 5-1 that demonstrates the cumulative effect of higher rates of return earned over time.)
Compound Interest
Compounded Returns over Time for Various Asset Classes
Annual Arithmetic Average (%) Annual Geometric Mean (%) Yeark-End Value, 2005 ($)
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
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
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.
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. )
Using Excel to Solve for FV
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)
nAssumptions:
• 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
Compound Interest
Underlying Assumptions – Timing of Cash Flows
Time = 0 Time = 1 Time = 2
Compound Interest Formula
(For a single cash flow)
FV
n=PV
0(1+k)
nWhere:
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
CVIF
k,n
(For a single cash flow)
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 1.40266289
.
1
)
05
.
1
(
10 10 %, 5
n years kCVIF
CVIF
k,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% 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
CVIF
k,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% 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
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
CVIF
k,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 1.2000 1.4000 1.6000 1.8000
CVIF
k,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.
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.
Types of Problems in Compounding
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 (FVn)
– Find the rate (k)
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
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
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
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
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)
nCompound Interest
Discounting (Computing Present Values)
1
1
)
1
(
0 n n n nk)
(
FV
k
FV
PV
[ 5-3]Annuities
Annuity
•
An annuity is a finite series of equal and periodic
cash flows.
Annuities and Perpetuities
Ordinary Annuity Formula
)
1
(
1
1
0
k
k
PMT
PV
n [ 5-5]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=…=PMTn Time = n PMTn Time = 0 Time of Investment n=0 Time = 1 PMT1 Time = 2 PMT2 Time = 3 PMT3
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
Annuities and Perpetuities
Ordinary Annuities
Compound Value Annuity Formula (CVAF)
1
1
PMT(CVAF)
k
k)
(
PMT
FV
n n
[ 5-4]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
1
k k) ( PMT FV n n
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
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.
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
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%.
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=…=PMTn but the first payment occurs at time=0. Time = n PMTn Time = 0 Time = 1 PMT1 Time = 2 PMT2 Time = 3 PMT3 No PMT
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
Annuities and Perpetuities
Future Value of an Annuity Due Formula
)
1
1
1
k
(
k
k)
(
PMT
FV
n n
[ 5-6]Annuities and Perpetuities
Present Value of an Annuity Due
k)
(1
)
1
(
1
1
0
k
k
PMT
PV
n [ 5-7]Discounting Cash Flows
Time Value of Money …
What is Discounting?
•Discounting is the inverse of compounding.
n
n
k
n
k
k
CVIF
PVIF
)
1
(
1
1
,
,
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
PVIF
k,n
(For a single cash flow)
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
,
PVIF
k,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.
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
PVIF
k,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!!!
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 0.0000The Reinvestment Rate
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
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
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
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
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
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
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
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 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.
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
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%:
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
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.
Perpetuities
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)
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=…=PMTα Time = α PMTα Time = 0 Time of Investment n=0 Time = 1 PMT1 Time = 2 PMT2 Time = 3 PMT3
Perpetuities
Perpetuities
Perpetuity Formula Perpetuity Formula 0 0k
k
P
PM
MT
T
P
PV
V
[ 5-8] [ 5-8]Where:
Where:
PVPerpetuity:
Perpetuity:
An
An
Example
Example
While acting as executor for a distant relative, you
While acting as executor for a distant relative, you
discover a $1,000 Consol Bond issued by Great Britain
discover a $1,000 Consol Bond issued by Great Britain
in 1814,
in 1814,
issued to
issued to
help fund
help fund
the Na
the Na
poleonic War.
poleonic War.
If th
If th
e
e
bond pays annual interest of 3.0% and
bond pays annual interest of 3.0% and
other long U.K.
other long U.K.
Government bonds are currently paying 5%,
Government bonds are currently paying 5%,
what would
what would
each $1,000 Consol Bond sell for in the market?
Perpetuity: Solution
Perpetuity: Solution
0 0$
$1
1,, 0
00
00
0 0
0..0
03
3
0.05
0.05
$30
$30
0.05
0.05
PMT
PMT
PV
PV
k
k
Nominal Versus Effective Rates
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
Nominal versus Effective Rates
General Formula for Effective Annual Rate
1
)
1
(
mm
QR
k
[ 5-9]Calculating the Effective Rate
1
1
m EffectiveQR
k
m
Where:
k Effective = Effective annual interest rate
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
1
12
6.17%
m EffectiveQR
k
m
Nominal versus Effective Rates
Continuous Compounding Formula
1
QRe
k
[ 5-10]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
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%
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
m11
Eff
QR
k
m
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
1
m f Per PeriodQR
k
m
Nominal versus Effective Rates
Formula for Effective Rates for “Any” Period
1
1
)
-m
QR
(
k
f m
[ 5-11]Loans and Loan Amortization Tables
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.
Blended Interest and Principal Loan
Payments - formula
k
k)
(1
1
1
PMT
Principal
)
PMT(PVAF
Principal
n n k, Where:Blended Interest and Principal Loan
Payments - example52
.
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
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
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
Loan or Mortgage Arrangements
Effective Rate for Any Period Formula
1
1
)
-m
QR
(
k
f m Eff
[ 5-11]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
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 3,489.43Calculating 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
Calculating the Balance O/S after the 1
stYear
1 21
1
1
1.10
2,010.57
.10
$3,489.42
n t k PV PMT k
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
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
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
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
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.
Concept Review Questions
Concept Review Questions
Time Value of Money
Concept Review Question 1
Concept Review Question 1
Quoted versus Effective Rates
Quoted versus Effective Rates
Why can effective rates often be very different from
Why can effective rates often be very different from
quoted rates?
quoted rates?
The more frequently interest is compounded the higher the effective
The more frequently interest is compounded the higher the effective
rate of return.
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
(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.
the frequency of compounding.
This is why reading the fine print is so important!
Internet Links
Internet Links
•
• Planning tools and online courses throughPlanning tools and online courses through TD Canada TrustTD Canada Trust
•