business mathematics jerome chapter 08
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McGraw-Hill Ryerson©
8 - 1Compound
Interest
Compound
Interest 88 88
McGraw-Hill Ryerson©
CompoundCompound
Chapter 8
McGraw-Hill Ryerson©
8 - 2Compound
Interest
Compound
Interest 88 88
Calculate the…
Learning ObjectivesLearning Objectives
After completing this chapter, you will be able to:
…Maturity Value of compound interest for Guaranteed Investment Certificates (GICs)
…Maturity Value(MV), Future Value (FV), and Present Value(PV) in
compound interest applications,
by both the algebraic method and the
pre-programmed financial calculator method
…Price of "strip" bonds
LO-1LO-1
McGraw-Hill Ryerson©
8 - 3Compound
Interest
Compound
Interest 88 88
Calculate the…
… … Redemption Value of a compound interest bearing Canada Savings Bond
…Payment on any date that is equivalent to one or more payments on other dates
…Economic Value of a payment stream
And be able to…
…Adapt the concepts and equations of compound interest to cases of compound growth
Learning ObjectivesLearning Objectives
LO-2LO-2
McGraw-Hill Ryerson©
8 - 4Compound
Interest
Compound
Interest 88 88
Compound
Interest
Compound
Interest 88 88
LO-1LO-1
McGraw-Hill Ryerson©
8 - 5Compound
Interest
Compound
Interest 88 88
To better understand how Compound Interest is calculated, let’s review how we calculate
Simple Interest!
Formula Formula I = Prt
The formula on which we base our calculation is…
Here we have an amount, the Principal, which is multiplied by the Interest Rate and the Time over
which the Interest is earned!
As we will now see, Compound Interest uses the Sum of P & I as a base on which to calculate
new Interest!
Compound
Interest
Compound
Interest 8 8 8 8
McGraw-Hill Ryerson©
8 - 6Compound
Interest
Compound
Interest 88 88
…the interest on the principal plus the
interest of prior periods
e.g. Principal + prior period interest = $1100.00
Interest for the next period is calculated on $1100.00.This method will continue over the life of the
loan or investment. (See later example)
$1000.00 $100.00
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 7Compound
Interest
Compound
Interest 88 88
…is the compounded amount and is the FINAL amount of the loan
or investment at the end of the last period!
Contrast this with…
...is the value of a loan or investment TODAY!
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 8Compound
Interest
Compound
Interest 88 88
…the calculation of interest over the life of the loan or investment
Example: Principal + prior period interest = $1100.00
Interest is now calculated on $1100.00
Let’s assume that the interest rate is 10% pa.
Principal(Compounded) * 0.10 = $110.00
New P $1210.00 to start next period
Graphically…
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 9Compound
Interest
Compound
Interest 88 88
100110
121
1000
1210
1331
1100100 100
110
Time(Years)0 1 2 3 4
Amount $1000Amount $1000
110
InterestInterestInterestInterest
100
InterestInterest
133.1
Compounding Period
Compounding Period
Compounding Period
Compounding Period
InterestInterest
121
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 10Compound
Interest
Compound
Interest 88 88
What happens if the interest rate changes during the life of
an investment?
Example…Example…
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 11Compound
Interest
Compound
Interest 88 88
You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively.
If you invested $1000 at the beginning of the term, how much
will you have at the end of the last year?
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 12Compound
Interest
Compound
Interest 88 88
$1000
Year 1 Year 2 Year 3 Year 4
$1040 $1123.20 $1010.88
$1000 * (1 + .04)
= $1040
$1040 * (1 + .08)
= $1123.20 = $1010.88 = $1101.86
$1123.20 *(1 - .10)
$1010.88 *(1 +.09)
…Alternative…Alternative
You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year?
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 13Compound
Interest
Compound
Interest 88 88
It is rare for interest to be compounded only once per year!
It is rare for interest to be compounded only once per year!
You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end
of the last year?1000(1.04)(1.08)(.90)(1.09) = $1101.86
1 -10%
Solving Alternative
Solving Alternative
Solve for all 4 years at
once!
Solve for all 4 years at
once!
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 14Compound
Interest
Compound
Interest 88 88
Compounding Frequencies and Periods
FrequencyFrequency No. per YearNo. per Year Period Period
Annually 1 1 year
Semiannually 2 6 months
Quarterly 4 3 months
Monthly 12 1 month
Daily 365 1 day
McGraw-Hill Ryerson©
8 - 15Compound
Interest
Compound
Interest 88 88 Formula Formula Development of a
n Total Number of PeriodsPeriods
Determining values for n and i
Nominal or Annual Rate ( j )
Periodic Rate per period ( i )Number of compoundings per year m
McGraw-Hill Ryerson©
8 - 16Compound
Interest
Compound
Interest 88 88 Formulae Formulae
To Determine nTo Determine n
To Determine iTo Determine i
# of Compounding Frequencies p.a.(m)Time(Years)
Annual Interest Rate(j)
# of Compounding Frequencies p.a. (m)
*
McGraw-Hill Ryerson©
8 - 17Compound
Interest
Compound
Interest 88 88
3 *3 *3 *
Annually Semiannually
Quarterly
= 3 = 6 = 12
1
2
4
nn
Determining values for n
If you compounded $100 for 3 years at 6%
annually, semiannually, or quarterly, what are the values for n and i ?
No.No.
# of Compounding Frequencies per year (m)Time(Years) *Formula
McGraw-Hill Ryerson©
8 - 18Compound
Interest
Compound
Interest 88 88
If you compounded $100 for 3 years at
6% annually, semiannually, or quarterly, what are the values for n and i ?
Determining values for i
Rate - iRate - i
6% /6% /6% /
1
2
4
= 6%
= 3%
= 1.5%
Annually Semiannually
Quarterly
Annual Interest Rate (j)# of Compounding Frequencies per
year(m)
Formula Formula
No.No.
McGraw-Hill Ryerson©
8 - 19Compound
Interest
Compound
Interest 88 88
Formula Formula Development of a for Future Value
PV= Present Value(Principal)
i = rate per period n = number of periods
FV = PV(1 + i)n
Where…
McGraw-Hill Ryerson©
8 - 20Compound
Interest
Compound
Interest 88 88
FV = PV(1 + i)n
Formula Formula
Steve Smith deposited $1,000 in a savings account for 4 years at a rate of 8%
compounded semiannually. What is Steve’s interest and compounded
amount?Extract necessary data...PV = n = i =
Solve…
Compound Interest- Future Value
4 X 2 = 8$1000
.08/2 = .04
McGraw-Hill Ryerson©
8 - 21Compound
Interest
Compound
Interest 88 88
FV = PV(1 + i)n
Formula Formula
Solve…
FV = $1000(1 + .04)8
= $1000(1.368569) = $1,368.57
Principal $1,000.00 + Interest 368.57Compounded $1,368.57
Using PV = $1000 n = 8 i= .04
Compound Interest- Future Value
McGraw-Hill Ryerson©
8 - 22Compound
Interest
Compound
Interest 88 88
BOTH ways will
be shown!
BOTH ways will
be shown!
Use a calculator and algebraic sequencing
Use the TI BAII Plus financial calculator!
There are two methods that can be used to
calculate compound interest:
McGraw-Hill Ryerson©
8 - 23Compound
Interest
Compound
Interest 88 88
Solve… $1000(1 + .04)8
.04
1
8
1000
$1,368.57$1,368.57
Use a calculator and algebraic sequencingUse a calculator and algebraic sequencing
1368.57
McGraw-Hill Ryerson©
8 - 24Compound
Interest
Compound
Interest 88 88
Find the following KEYS:Find the following KEYS:
The Power function Key. Used to calculate the
value of exponents.
Used to access symbols located “above”
another key, i.e. its acts like the
SHIFT key on a computer keyboard.
Use a calculator and algebraic sequencingUse a calculator and algebraic sequencing
Changes the sign of the data value of the number
being displayed.
McGraw-Hill Ryerson©
8 - 25Compound
Interest
Compound
Interest 88 88
Some calculators have the yx symbol above the calculator key.
(1.04)8 is…
The key stroke sequence to evaluate an EXPONENT that is…
1.04 8
1.368569
0.73069
PositivePositive
Find the following KEYS:Find the following KEYS:
Use a calculator and algebraic sequencingUse a calculator and algebraic sequencing
NegativeNegative (1.04)-8 is…1.04 8
McGraw-Hill Ryerson©
8 - 26Compound
Interest
Compound
Interest 88 88
This calculator can store up to 10 values.
Find the following KEYS:Find the following KEYS:
Use a calculator and algebraic sequencingUse a calculator and algebraic sequencing
Used to Store or save displayed values.
Used to Recall the saved values.
Let’s PractiseLet’s Practise
Therefore, the calculator must be informed as to
where the values are to be stored.
McGraw-Hill Ryerson©
8 - 27Compound
Interest
Compound
Interest 88 88 Use a calculator and algebraic sequencingUse a calculator and algebraic sequencing
Using the key
Using the key e.g. you want to store the value ’45’.
The key stroke sequence ‘to store’ is:45
..choose from 0 - 9
…’clear’ display
The key stroke sequence ‘to recall’ is:
…where you stored the value!
McGraw-Hill Ryerson©
8 - 28Compound
Interest
Compound
Interest 88 88
Some key Keys!Some key Keys!
McGraw-Hill Ryerson©
8 - 29Compound
Interest
Compound
Interest 88 88
The nominal interest rate (Interest/Year)
1. Number of compoundings (for lump payments)
2. Number of payments (for annuities)
Represents the Periodic Annuity Payment
(used in chapter 10)
Tells the calculator to compute (CPT)
Present Value or initial(first) lump sum
Find the following KEYS:Find the following KEYS:
Future Value or terminal(last) lump sum
McGraw-Hill Ryerson©
8 - 30Compound
Interest
Compound
Interest 88 88
However, we can now input the number of compoundings per year into the financial calculator. This can be performed by using
the symbol
Find the following KEYS:Find the following KEYS:
…it is rare for interest to be compounded only once per year!
…it is rare for interest to be compounded only once per year!
Previously, it was noted that
To access this symbol use:
…and you will see
McGraw-Hill Ryerson©
8 - 31Compound
Interest
Compound
Interest 88 88 The 12
is a default setting
The 12 is a
default setting
This display is referred to as “the worksheet”.
… represents the number of Payments per Year
… represents the number of Compoundings per Year
To access use:
Note: You can override these values by entering new ones!
…more…more
Appearsautomatically
Appearsautomatically
McGraw-Hill Ryerson©
8 - 32Compound
Interest
Compound
Interest 88 88
must be given the same value as
If the calculation
does not involve more than one payment
If the calculation
does not involve more than one payment
IllustrationIllustration
McGraw-Hill Ryerson©
8 - 33Compound
Interest
Compound
Interest 88 88
Setting a new value for P/Y will
automatically change the entry for C/Y to the same value
as the default, i.e. P/Y
Setting a new value for P/Y will
automatically change the entry for C/Y to the same value
as the default, i.e. P/Y
IllustrationIllustration
… represents the number of Compoundings per Year
In Compound Interest, P/Y must be
given the same value as C/Y.
In Compound Interest, P/Y must be
given the same value as C/Y.
…to scrollWe must key in this sequence to close any worksheet
you have opened.
We must key in this sequence to close any worksheet
you have opened.
McGraw-Hill Ryerson©
8 - 34Compound
Interest
Compound
Interest 88 88
There are two methods that can be used to
calculate compound interest:
Using the TI BAII Plus financial calculator!
McGraw-Hill Ryerson©
8 - 35Compound
Interest
Compound
Interest 88 88
Steve Smith deposited $1,000 in a savings account for 4 years at a rate of 8% compounded semiannually.
What is Steve’s interest and compounded amount?
Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator
Set the
frequency of
interest compounding
Set the
frequency of
interest compounding
Step 1Step 1
Input values into the
financial keys
Input values into the
financial keys
Step 2Step 2
Using
McGraw-Hill Ryerson©
8 - 36Compound
Interest
Compound
Interest 88 88
FV= 1368.57
8.0
2
1000
Set the frequency
of interest
compounding
Set the frequency
of interest
compounding
Step 1Step 1
4 * 2
0
Input values into the financial
keys
Input values into the financial
keys
Step 2Step 2
$1,368.57$1,368.57
Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator
Steve Smith deposited $1,000
in a savings account for
4 years at a rate of 8%
compounded semiannually.
What is Steve’s
interest and compounded
amount?
Steve Smith deposited $1,000
in a savings account for
4 years at a rate of 8%
compounded semiannually.
What is Steve’s
interest and compounded
amount?
McGraw-Hill Ryerson©
8 - 37Compound
Interest
Compound
Interest 88 88
…there is no need to keep inputting each time! 0
You only need to input the values that have changed!
The calculator remembers this step!
McGraw-Hill Ryerson©
8 - 38Compound
Interest
Compound
Interest 88 88
Cash FlowsCash Flows
… payments received e.g. receipts
Treated as:Treated as:
Positives +Positives + Negatives -Negatives -
..a term that refers to payments that can be either …
..a term that refers to payments that can be either …
… payments made e.g. cheques
McGraw-Hill Ryerson©
8 - 39Compound
Interest
Compound
Interest 88 88
What is the effect on the Future Value of
different Compounding Periods of
Interest?
McGraw-Hill Ryerson©
8 - 40Compound
Interest
Compound
Interest 88 88
If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly,
what are the final amounts that you would have at the end of the three (3) years ?
Compound Interest- Future Value
AnnualAnnual FVA = 100(1.06)3 $119.10$119.10
Semi-Semi- FVS = 100(1.03)6 $119.41$119.41
Semi = 6%/2
QuarterlyQuarterly FVQ = 100(1.015)12 $119.56$119.56
Quarterly = 6%/4
McGraw-Hill Ryerson©
8 - 41Compound
Interest
Compound
Interest 88 88
Fu
ture
Val
ue
S o
r F
VF
utu
re V
alu
e
S o
r F
V
1 2 3 4 5 6 7 8 9 10 110
50
100
150
200
250
FV=PV(1+i)n
Time(Years)
S=P(1+rt)
Original PrincipalOriginal Principal
Interest onOriginal Principal
Interest on
Interest
Compound Interest- Future Value
The Components of the Future Value of $100
McGraw-Hill Ryerson©
8 - 42Compound
Interest
Compound
Interest 88 88
ComparisonsComparisons
McGraw-Hill Ryerson©
8 - 43Compound
Interest
Compound
Interest 88 88
Al Jones deposited $1,000 in a savings account for 5 years at 10% p.a..
Al Jones deposited $1,000 in a savings account for 5 years at 10% p.a..
Annual Simple Interest Rate of 10%
Annual Simple Interest Rate of 10%
Annual Compound Rate of 10%
Annual Compound Rate of 10%
Simple Vs Compound Interest
What is Al’s
Simple Interest and
Maturity Value?
What is Al’s
Simple Interest and
Maturity Value?
What is Al’s
Interest and
Compounded Value?
What is Al’s
Interest and
Compounded Value?
McGraw-Hill Ryerson©
8 - 44Compound
Interest
Compound
Interest 88 88
Simple Vs Compound Interest
FV = PV(1 + i)nFormulae Formulae I = Prt
I = $1,000 * .10 * 5
= $500
FV = $1,000 + $500
= $1,500
FV = $1000(1.1)5
= $1,000 *1.6105
= $1,610.51
n = 5 * 1 = 5
I = FV – PV = $1610.51 - $1000
SimpleSimple CompoundCompound
Al Jones deposited $1,000 in a savings account for 5 years at 10%Al Jones deposited $1,000 in a savings account for 5 years at 10%
= $610.51
i = .10
Compare
Compare
McGraw-Hill Ryerson©
8 - 45Compound
Interest
Compound
Interest 88 88
0
200
400
600
800
1000
1200
1400
1600
1800
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Years to Maturity, n
Future Values of $100 at
Various Rates of Interest Compounded Annually
Future Values of $100 at
Various Rates of Interest Compounded Annually
100
6%
10%
8%
12%
Fu
ture
Val
ue
FV
Fu
ture
Val
ue
FV
Compound
Interest
Compound
Interest 88 88
McGraw-Hill Ryerson©
8 - 46Compound
Interest
Compound
Interest 88 88
Ending Balance
Ending Balance
Compounding Period
Compounding Period
$1,000$1,000
Nominal Rates of Interest Compared
$1,060.00
$1,060.90
$1,061.36
$1,061.83
Beginning Balance
Beginning Balance
Nominal Rate
Nominal Rate
+ 6%+ 6%
Annual
Semiannual
Quarterly
Daily
McGraw-Hill Ryerson©
8 - 47Compound
Interest
Compound
Interest 88 88
0
500
1000
1500
2000
2500
5 10 15 20 25
Fu
ture
Val
ue
FV
Fu
ture
Val
ue
FV
Time (years)
12% Compounded
monthly
Future Values of $100 at the same Nominal Rate but
Different Compounding Frequencies
12% Compounded
Annually
Compound
Interest
Compound
Interest 88 88
100
McGraw-Hill Ryerson©
8 - 48Compound
Interest
Compound
Interest 88 88
Calculate the Future Value of $2,000
compounded daily for 4 years
at 4.5%.
n = i =
= $2,000 * 1.1972 = $2,394.41= $2,000 * 1.1972 = $2,394.41FV = $2000(1+ .045/365)1460
FV = PV(1 + i)n
Formula Formula
Compounding Compounding DailyDaily Interest InterestCompounding Compounding DailyDaily Interest InterestCompound
Interest
Compound
Interest 88 88
4 * 365 = 1460 .045 /365 = 0.0001232
McGraw-Hill Ryerson©
8 - 49Compound
Interest
Compound
Interest 88 88
2394.41.045365
1
1460
2000
Solve FV = $2000(1+ .045/365)1460
= $2,394.41= $2,394.41
Compounding Compounding DailyDaily Interest InterestCompounding Compounding DailyDaily Interest Interest
McGraw-Hill Ryerson©
8 - 50Compound
Interest
Compound
Interest 88 88
FV= - 2394.414.5
2000
$2,394.41$2,394.41
Set the frequency
of interest
compounding
Set the frequency
of interest
compounding
Step 1Step 1
4 * 365
0
Input values into the
financial keys
Input values into the
financial keys
Step 2Step 2
Compounding Compounding DailyDaily Interest InterestCompounding Compounding DailyDaily Interest Interest
365
Calculate the Future Value of $2,000
compounded
daily for 4 years at 4.5%.
Calculate the Future Value of $2,000
compounded
daily for 4 years at 4.5%.
McGraw-Hill Ryerson©
8 - 51Compound
Interest
Compound
Interest 88 88
You invested $6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2%
compounded monthly. What amount will you have 41/2 years after the initial
investment?
Prepare a ‘time-line’ as part of the solution
McGraw-Hill Ryerson©
8 - 52Compound
Interest
Compound
Interest 88 88
You invested $6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2%
compounded monthly. What amount will you have 41/2 years after the
initial investment?
0 2 years 4.5 years
$6000
i = .045/4FV1 = PV2
FV1 = 6000(1+.045/4)8
= 6000(1.0936) = 6561.75
FV2
i = .052/12
FV2 = = 6561.75(1.1385)
= $7470.61
6561.75(1+.052/12)30
n = (2*4) = 8n = 2.5*12 = 30
McGraw-Hill Ryerson©
8 - 53Compound
Interest
Compound
Interest 88 88
You invested $6000 at 4.5% compounded
quarterly. After 2 years,
the rate changed to
5.2% compounded
monthly. What amount will you have
41/2 years after the initial
investment?
6000
Set the frequency
of interest
compounding
Set the frequency
of interest
compounding
Step 1Step 1
4 * 2 4.5
Input values into the financial
keys
Input values into the financial
keys
Step 2Step 2
$6,561.75$6,561.75
Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator
FV1 = PV2FV1 = PV2
FV2
4
6,571.75
McGraw-Hill Ryerson©
8 - 54Compound
Interest
Compound
Interest 88 88
You invested $6000 at 4.5% compounded
quarterly. After 2 years,
the rate changed to
5.2% compounded
monthly.
What amount will you have
41/2 years after the initial
investment?
7470.61
Set the frequency
of interest
compounding
Set the frequency
of interest
compounding
Step 1Step 1
2.5*12
5.2
Input values into the financial
keys
Input values into the financial
keys
Step 2Step 2
$7,470.61$7,470.61
Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator
FV2FV2
12
McGraw-Hill Ryerson©
8 - 55Compound
Interest
Compound
Interest 88 88
Prepare a ‘time-line’ as part of the solution
You borrowed $5000 at 7% compounded monthly. On the first and second anniversaries of the loan,
you made payments of $2500. What is the balance outstanding
immediately following the second payment?
McGraw-Hill Ryerson©
8 - 56Compound
Interest
Compound
Interest 88 88
0 1 year 2 years
$5000i = .07/12
FV1 - $2500 = PV2
FV1 = 5000(1+.07/12)12
= 5000(1.072290) = 5361.45PV2 = 5361.45 – 2500.00 = 2861.45
FV2
i = .07/12FV2 = = 2861.45(1.072290)
= $3068.30
2861.45 (1+.07/12)12
n = 12 n = 12
You borrowed $5000 at 7% compounded monthly. On the first and second anniversaries of the
loan, you made payments of $2500. What is the balance outstanding immediately following the second
payment?
New Balance New Balance
= $3068.30 – 2500.00
= $568.30
McGraw-Hill Ryerson©
8 - 57Compound
Interest
Compound
Interest 88 88
5000
Step 1Step 1
12
$2,861.45$2,861.45
Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator
FV1 – 2500 = PV2FV1 – 2500 = PV2
FV2
You borrowed $5000 at 7% compounded
monthly. On the 1st. and 2nd anniversaries of
the loan, you made payments of $2500.
What is the balance
outstanding immediately after the 2nd payment?
2500
7.0
12
FV= -5361.45-2861.45
McGraw-Hill Ryerson©
8 - 58Compound
Interest
Compound
Interest 88 88
-2861.45
$568.30$568.30
Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator
FV2FV2
You borrowed $5000 at 7% compounded
monthly. On the 1st. and 2nd anniversaries of
the loan, you made payments of $2500.
What is the balance
outstanding immediately after the 2nd payment?
2500
Step 2Step 2
- 2,861.45 3068.30 568.30
McGraw-Hill Ryerson©
8 - 59Compound
Interest
Compound
Interest 88 88
McGraw-Hill Ryerson©
8 - 60Compound
Interest
Compound
Interest 88 88 Formula for Present Value
PV = FV(1 + i)-nFormula Formula
Keys
i1
$PV$PV
This is the only change to the
usual sequence!
McGraw-Hill Ryerson©
8 - 61Compound
Interest
Compound
Interest 88 88
You expect to need $1,500 in 3 years. Your bank offers 4% interest compounded semiannually. How much money must
you put in the bank today (PV) to reach your goal in 3 years?
Calculating Present Value Calculating Present Value
Prepare the solution…(a) algebraically, and (b) by financial calculator
Prepare the solution…(a) algebraically, and (b) by financial calculator
McGraw-Hill Ryerson©
8 - 62Compound
Interest
Compound
Interest 88 88
PV = FV(1 + i)-nFormula Formula
i = .04/2 = .02
You expect to need $1,500 in 3 years. Your bank offers 4% interest compounded
semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years?
PV = $1500(1+.02)-6
n = 3 * 2 = 6
Calculating Present Value Calculating Present Value
1.02
6
1500
0.887971,331.96
= $1500 * .8880
= $1,331.96
(a)
McGraw-Hill Ryerson©
8 - 63Compound
Interest
Compound
Interest 88 88
3 * 2
4
2
1500
0PV= -1,331.96(b)
Calculating Present Value Calculating Present Value
You expect to need $1,500 in 3 years. Your bank offers 4% interest compounded
semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years?
$1331.96$1331.96
McGraw-Hill Ryerson©
8 - 64Compound
Interest
Compound
Interest 88 88
PV = FV(1 + i)-nFormula Formula
What amount must you invest now at 5% compounded daily to accumulate to $6000 after 1 year?
j =
m =
FV =
i =
n =
Calculating Present Value Calculating Present Value
PV = $6000(1+.05/365)-365
= $6000 * .9512
= $5,707.40
.05
365
6000
365
1
0.00011.0010.95125,707.405%
365
.05/365
1*365 = 365
$6000
McGraw-Hill Ryerson©
8 - 65Compound
Interest
Compound
Interest 88 88
What amount must you invest now at 5% compounded daily to accumulate to $6000 after 1 year?
1 * 365
5
365
PV= - 5,707.40
6000
0
Calculating Present Value Calculating Present Value
$5707.40$5707.40
McGraw-Hill Ryerson©
8 - 66Compound
Interest
Compound
Interest 88 88
Two payments of $2200 each must be made 1 and 4 years from now. If money can earn 5% compounded monthly,
what single payment 3 years from now would be
equivalent to the two scheduled payments?Draw a Time-lineDraw a Time-lineStep 1Step 1
Find the FV of the payment that is moved from Year 1 to Year 3
Find the FV of the payment that is moved from Year 1 to Year 3
Step 2Step 2
Find the PV of the payment that is moved from Year 4 to Year 3
Find the PV of the payment that is moved from Year 4 to Year 3
Step 3Step 3
Prepare the solution…(a) algebraically, and (b) by financial calculator
Prepare the solution…(a) algebraically, and (b) by financial calculator
Equivalent Payments Equivalent Payments
McGraw-Hill Ryerson©
8 - 67Compound
Interest
Compound
Interest 88 88
Two payments of $2200 each must be made 1 and 4 years from now. If money can earn 5% compounded
monthly, what single payment 3 years from now
would be equivalent to the two scheduled payments?
Draw a Time-lineDraw a Time-lineStep 1Step 1
$2200 $2200
0 1 year 2 years 3 years 4 years
i = .05/12 n = 2*12 = 24
Step 2Step 2Find the FV of the payment that is moved
from Year 1 to Year 3
Find the FV of the payment that is moved
from Year 1 to Year 3
FV1
= 2200(1+.05/12)24
= 2200(1.1049) = 2430.87
(a)
FV1PV1
PV2
FV2
Now 0
2430.87
Equivalent Payments Equivalent Payments
McGraw-Hill Ryerson©
8 - 68Compound
Interest
Compound
Interest 88 88 (b)
2*12
5
2200
0
Step 2Step 2Find the FV of the payment that is moved
from Year 1 to Year 3
Find the FV of the payment that is moved
from Year 1 to Year 3
$2200 $2200
0 1 year 2 years 3 years 4 years
i = .05/12 n = 2*12 = 24
FV1PV1
PV2
FV2
12
Now 0
2430.87
McGraw-Hill Ryerson©
8 - 69Compound
Interest
Compound
Interest 88 88
Find the PV of the payment that is moved
from Year 4 to Year 3
Find the PV of the payment that is moved
from Year 4 to Year 3
Step 3Step 3 i = .05/12 n =1*12=12
PV2
PV2 = 2200(1+.06/12)-12
= 2200(0.9513) = 2092.92
(a)
$2200 $2200
0 1 year 2 years 3 years 4 years
FV1PV1
$4523.79
Finally, this PV amount can be added to that put into memory…
0
2430.87
Equivalent Payments Equivalent Payments
McGraw-Hill Ryerson©
8 - 70Compound
Interest
Compound
Interest 88 88
n =1*12=12
$2200 $2200
0 1 year 2 years 3 years 4 years
PV2 FV1
PV1
(b)
1*12
2200 Some of the values have not
changed so there is no
need to enter them
again!
Some of the values have not
changed so there is no
need to enter them
again!
$4523.79
Finally, this PV amount can be added to that put into memory…
0
2430.87
2,092.924,523.79
McGraw-Hill Ryerson©
8 - 71Compound
Interest
Compound
Interest 88 88
What regular payment will an investor receive from a $10,000, 3 year, monthly payment GIC earning a nominal rate of 4.8%
compounded monthly?Interest rate per payment interval is:
i = j/m = .048/12
= 0.0040
…the monthly payment will be:
…the monthly payment will be:
PV * I = $10000 * 0.0040 = $40.00
Making a choice!…Making a choice!…
McGraw-Hill Ryerson©
8 - 72Compound
Interest
Compound
Interest 88 88
Suppose a bank quotes nominal annual interest rates of 6.6% compounded annually, 6.5% compounded semi-annually, and 6.4% compounded monthly on five-year GICs.
Making a choice!…Making a choice!…
Which rate should an investor choose for an investment of $1,000?
McGraw-Hill Ryerson©
8 - 73Compound
Interest
Compound
Interest 88 88
Suppose a bank quotes nominal annual interest
rates of
6.6% compounded
annually, 6.5% compounded
semi-annually, and 6.4% compounded monthly
on five-year GICs. Which
rate should an investor choose for
an investment of $1,000?
Suppose a bank quotes nominal annual interest
rates of
6.6% compounded
annually, 6.5% compounded
semi-annually, and 6.4% compounded monthly
on five-year GICs. Which
rate should an investor choose for
an investment of $1,000?
5*1
6.6
1000
0
1
1376.53
j = 6.6%
compoundedannually
5 * 2
6.5
2
1376.89
j = 6.5%
compoundedsemi-annually
j = 6.4%
compoundedmonthly5 * 12
6.4
12
1375.96ComparisonsComparisons
McGraw-Hill Ryerson©
8 - 74Compound
Interest
Compound
Interest 88 88 ComparisonsComparisons
the 6.5% compounded semi-annuallyprovides for the best
rate of return on investment!
the 6.5% compounded semi-annuallyprovides for the best
rate of return on investment!
ResultsResultsj = 6.6%
compounded
annuallyj = 6.5%
compoundedsemi-annuallyj = 6.4%
compounded
monthly
1376.531376.53
1376.891376.89
1375.961375.96
McGraw-Hill Ryerson©
8 - 75Compound
Interest
Compound
Interest 88 88
of Interest Rates
McGraw-Hill Ryerson©
8 - 76Compound
Interest
Compound
Interest 88 88
of Interest Rates
Fixed Rate
…the interest rate does not
change over the term of the GIC.
Fixed Rate
…the interest rate does not
change over the term of the GIC.
An investment in a GIC might have a…Step-up Rate
…the interest rate is increased every 6
months or every year according to a pre-
determined schedule.
Step-up Rate
…the interest rate is increased every 6
months or every year according to a pre-
determined schedule.
Variable Rate
... is adjusted every year or every 6
months to reflect market rates… may be a minimum
“floor” below which rates
cannot drop
Variable Rate
... is adjusted every year or every 6
months to reflect market rates… may be a minimum
“floor” below which rates
cannot drop
McGraw-Hill Ryerson©
8 - 77Compound
Interest
Compound
Interest 88 88
Regular Interest version
Regular Interest version
Compound Interest version
Compound Interest version
Interest is paid
to the investor every year or every 6
months
Interest is paid
to the investor every year or every 6
months
Interest is periodically
converted to principal and
paid at maturity
Interest is periodically
converted to principal and
paid at maturity
Payment of Interest
Payment of Interest
McGraw-Hill Ryerson©
8 - 78Compound
Interest
Compound
Interest 88 88
Canadian
Savings
Bonds
McGraw-Hill Ryerson©
8 - 79Compound
Interest
Compound
Interest 88 88
- Can be purchased from financial institutions but funds go to federal government to help finance its debt
- usual term is 10 or 12 years
- variable interest rates
- interest rate is changed on each anniversary, with minimum rates for subsequent 2 years
Canadian
Savings
Bonds
To view current rates of interest and redemption values
Go to http://www.cis-pec.gc.ca/
McGraw-Hill Ryerson©
8 - 80Compound
Interest
Compound
Interest 88 88
McGraw-Hill Ryerson©
8 - 81Compound
Interest
Compound
Interest 88 88
All CSBs issued up to 1988 (Series 1 to 43) have matured and are no longer earning interest.
The rates of interest for Series 45 to 70 for subsequent years to maturity will be announced at future dates.
All CSBs issued up to 1988 (Series 1 to 43) have matured and are no longer earning interest.
The rates of interest for Series 45 to 70 for subsequent years to maturity will be announced at future dates.
Canadian
Savings
Bonds
McGraw-Hill Ryerson©
8 - 82Compound
Interest
Compound
Interest 88 88
McGraw-Hill Ryerson©
8 - 83Compound
Interest
Compound
Interest 88 88
The fair market value of an investment is the
sum of the Present Values of the expected cash flows.
The discount rate used should be
the prevailing market determined rate of return
required on this type of
investment.
Concepts
McGraw-Hill Ryerson©
8 - 84Compound
Interest
Compound
Interest 88 88
McGraw-Hill Ryerson©
8 - 85Compound
Interest
Compound
Interest 88 88
… owner will receive a single payment (called the face value of the bond) on the
bond’s maturity date
… owner will receive a single payment (called the face value of the bond) on the
bond’s maturity date
… the maturity date could be as much as 30
years in the future. No interest will be received
in the interim!
… the maturity date could be as much as 30
years in the future. No interest will be received
in the interim!
McGraw-Hill Ryerson©
8 - 86Compound
Interest
Compound
Interest 88 88
Suppose a $10,000 face value strip bond matures 18 years from now.
The owner of this bond will receive a payment of $10,000 in 18 years.
What is the appropriate price to pay for the bond today if the prevailing rate of return is 5.75%,
compounded semi-annually?
FV = $10000
i = .0575/2 n = 18 * 2 = 36
PV = 10000(1+.0575/2)-36
= 10000(0.3605)
= $3604.50
McGraw-Hill Ryerson©
8 - 87Compound
Interest
Compound
Interest 88 88 Suppose a $10,000 face value strip bond
matures 18 years from now. The owner of this bond will receive a payment of $10,000 in 18 years.What is the appropriate price to pay for the bond today if the prevailing rate of return
is 5.75%, compounded semi-annually?
j = 5.75%
m = 2
FV = $10000
n = 18*2 = 36
18 * 2
5.75
2
PV = -3,604.50
10000
0
$3604.50$3604.50
McGraw-Hill Ryerson©
8 - 88Compound
Interest
Compound
Interest 88 88
This completes Chapter 8This completes Chapter 8