ordinary annuities ordinary annuities1010 mcgraw-hill ryerson© 10-1 chapter 10 o rdinary a nnuities...
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Ordinary
Annuities
Ordinary
Annuities
10101010
10-1
McGraw-Hill Ryerson©
Chapter 10
Ordinary AnnuitiesOO AA
McGraw-Hill Ryerson©
Ordinary
Annuities
Ordinary
Annuities
10101010
10-2
McGraw-Hill Ryerson©
Calculate the…
Define and distinguish between…
Learning ObjectivesLearning ObjectivesAfter completing this chapter, you will be able to:
… Future Value and Present Value of ordinary simple annuities
… ordinary simple annuities and ordinary general annuities
… fair market value of a cash flow stream that includes an annuity
LO-1LO-1
LO-2LO-2
LO-3LO-3
Ordinary
Annuities
Ordinary
Annuities
10101010
10-3
McGraw-Hill Ryerson©
Calculate the…
Learning ObjectivesLearning Objectives
… Present Value of and period of deferral of a deferred annuity
… principal balance owed on a loan immediately after any payment
… Future Value and Present Value of ordinary general annuities
LO-4LO-4
LO-5LO-5
LO-6LO-6
Ordinary
Annuities
Ordinary
Annuities
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10-4
McGraw-Hill Ryerson©
TerminologyTerminology
- A series of equal payments at regular intervals
Term of the Annuity
- the time from the beginning of the first payment period to the end of the last payment period
Future ValuePresent Valuethe future dollar amount of a series of payments plus interest
the amount of money needed to invest today in order to receive a series of payments for a given number of years
in the future
AnnuityLO-1LO-1
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
TerminologyTerminology
… is the amount of each payment in an annuityPMTPMT
… is the number of payments in the annuitynpayment interval
ordinary annuities
… is the time between successive payments in an annuity
… are ones in which payments
are made
at the end of each payment
interval
Ordinary
Annuities
Ordinary
Annuities
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10-6
McGraw-Hill Ryerson©
TerminologyTerminology
Suppose you obtain a personal loan
to be repaid by
payment interval
Term
ordinary annuities 48 equal monthly payments
48 months or 4years.
1 month
first payment will be due 1 month after you receive the loan,
i.e. at the end of the first payment interval
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Annuities
Ordinary
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TerminologyTerminology
PMT
0 1 2 3 nn-1 Intervalnumber
Term of the annuity
Payment interval
… for an n-payment Ordinary Annuity
PMT PMT PMTPMT
Ordinary
Annuities
Ordinary
Annuities
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Ordinary Annuity
Ordinary
Simple Annuities
Ordinary
Simple AnnuitiesOrdinary
General Annuities
Ordinary
General Annuities
Monthly payments,
and interest is
compounded monthly
Monthly payments,
and interest is
compounded monthly
Monthly payments,
but interest is
compounded semi-annually
Monthly payments,
but interest is
compounded semi-annually
The payment interval
=
the compounding interval
The payment interval
=
the compounding interval
The payment interval
differs from
the compounding interval
The payment interval
differs from
the compounding interval
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
$1000
$1000 (1.04)1n = 1
Sum = FV of annuity
0 1 2 3 4 Intervalnumber
$1000 $1000 $1000
$1000 (1.04)2n = 2
$1000 (1.04)3n = 3
…the sum of the future values of all the payments
Assume that there are four(4) annual $1000 payments with interest at 4%
Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
LO-2LO-2
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
= $4246.46
= $1000 +FV of annuity
$1000$1000 (1.04)1n = 1
Sum = FV of annuity
0 1 2 3 4 Intervalnumber
$1000 $1000 $1000
$1000 (1.04)2n = 2
$1000 (1.04)3n = 3
Assume that there are four(4) annual $1000 payments with interest at 4%
$1000(1.04) + $1000(1.04)2 + $1000(1.04)3
= $1000 +$1040+ $1081.60 +$1124.86
Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
Ordinary
Annuities
Ordinary
Annuities
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ResultResult
$500
$500(1+.03/12)
Sum = FV of annuity
0 1 2 3 4 Month
$500 $500 $500
$500(1+.03/12)3
Suppose that you vow to save $500 a month for the next four months, with your first deposit one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now.
$500(1+.03/12)2
$ 500.00501.25502.50503.76
$2,007.51
Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
Now imagine that you save $500 every month for the next three years. Although the same logic applies, I
certainly don’t want to do it this way!
Since your ‘account’ was empty when you began… PV = 0
n = 3 yrs * 12 payments per year = 36 payments
Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
Using the …
Ordinary
Annuities
Ordinary
Annuities
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36
You save $500 every month for the next three years. Assume your savings can earn 3% converted monthly.
Determine the total in your account three years from now.
3
500
Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
012
Using the formulaUsing the formula
NoteNote
Keys direction
P/Y= 120FV = 18810.28
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
…the sum of the future values of all the payments
Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
FV = PMT (1+ i)n - 1[ i ]Formula Formula
Ordinary
Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
You save $500 every month for the next three years. Assume your savings can earn 3% converted monthly.
Determine the total in your account three years from now.
Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
0.0025[FV = PMT (1+ i)n - 1i ] 1.0025 1.09410.094137.620618810.28
12.03
500
361
1
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
You vow to save $500/month for the next four months, with your first deposit one month from today.
If your savings can earn 3% converted monthly, determine the total in your account four months from now.
Since your ‘account’ was empty when you began… PV = 0
n = 4 paymentsPMT = -500
Solving earlier Question using Annuities
Solving earlier Question using Annuities
Ordinary
Annuities
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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 madee.g. cheques
Therefore…Therefore…
Ordinary
Annuities
Ordinary
Annuities
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Therefore…Therefore…
…when you are making payments, or even making deposits to
savings,
Really payments to
the bank!
Really payments to
the bank!these are cash outflows,
and therefore the values must be negative!
Cash Flow Sign Convention
Using the …
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
You vow to save $500/month for the next four months,
with your first deposit one month from today. If your savings can earn
3% converted monthly, determine
the total in your account four
months from now.
You vow to save $500/month for the next four months,
with your first deposit one month from today. If your savings can earn
3% converted monthly, determine
the total in your account four
months from now.
PV = 0 n = 4 payments PMT -500
Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
4
3
500
012 FV = 2007.51
We already have these from before, so
we don’t have to enter them again!
We already have these from before, so
we don’t have to enter them again!
Formula solutionFormula solution
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
12.03
500
41
1
You vow to save $500/month for the next four months, with your first deposit
one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now.
You vow to save $500/month for the next four months, with your first deposit
one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now.
Formula Formula [FV = PMT (1+ i)n - 1i ]
PMT = $500
n = 4
i = .03/12 = 0.0025
0.00251.0025 1.01000.01004.01502007.51
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
Not seeing the total picture!Not seeing the total picture!
When you use formula or a calculator’s financial functions to
calculate an annuity’s Future Value,
the amount each payment
contributes to the future value is
NOT apparent!
Ordinary
Annuities
Ordinary
Annuities
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10% Compounded Annually10% Compounded Annually
$10.00$10.00
YearsYears0 1 2 3 4 5
14.64
13.31
12.10
11.00
10.00
Contribution$
$61.05$61.05
FV ContributionsFV Contributions
$10.00$10.00
$10.00$10.00
$10.00$10.00
$10.00$10.00
FVFV
Ordinary
Annuities
Ordinary
Annuities
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Future Value of an
Ordinary Simple Annuity
Future Value of an
Ordinary Simple Annuity
You decide to save $75/month for the next four years. If you invest all of these savings in an account
which will pay you 7% compounded monthly, determine:
a) the total in the account after 4 years b) the amount you deposited c) the amount of interest earned
Extract necessary data...
PMT = = 7 n = 4 * 12 = 48 - $75
PV = 0 FV = ?
Solve…
Total Deposits = $75* 48 = $3,600
= 12
Ordinary
Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
You decide to save $75/month for the
next four years. If you invest all of these savings
in an account which will pay you 7%
compounded monthly, determine:
a) the total in the account after 4
years b) the amount you deposited
c) the amount of interest
earned
You decide to save $75/month for the
next four years. If you invest all of these savings
in an account which will pay you 7%
compounded monthly, determine:
a) the total in the account after 4
years b) the amount you deposited
c) the amount of interest
earned
487
75012
Formula solutionFormula solution
FV……….. $4,140.69
Interest Earned = $ 540.69Deposits…... 3,600.00
P/Y = 12FV = 4140.69
Ordinary
Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
FV $4,140.69= Interest Earned $540.69
- Deposits 3,600.00
Formula Formula [FV = PMT (1+ i)n - 1i ]
0.005833 1.0058331.322050.32205
12.07
75
481
1
55.209244140.6927You decide to save $75/month for the
next four years. If you invest all of these savings
in an account which will pay you 7%
compounded monthly, determine:
a) the total in the account after 4
years b) the amount you deposited
c) the amount of interest
earned
You decide to save $75/month for the
next four years. If you invest all of these savings
in an account which will pay you 7%
compounded monthly, determine:
a) the total in the account after 4
years b) the amount you deposited
c) the amount of interest
earned
Ordinary
Annuities
Ordinary
Annuities
10101010
10-26
McGraw-Hill Ryerson©
…the sum of the present values of all the payments
PV = PMT 1-(1+ i)-n[ i ]
PresentValue of an
Ordinary Simple Annuity
PresentValue of an
Ordinary Simple Annuity
Formula Formula
Ordinary
Annuities
Ordinary
Annuities
10101010
10-27
McGraw-Hill Ryerson©
$1000
Sum = PV of annuity
$1000 $1000 $1000
…the sum of the present values of all the payments
Assume that there are four(4) annual $1000 payments with interest at 4%
Present Value of an
Ordinary Simple Annuity
Present Value of an
Ordinary Simple Annuity
$1000 (1.04)-1 n = 1
$1000 (1.04)-2n = 2
$1000 (1.04)-3n = 3
$1000 (1.04)-4n = 4
0 1 2 3 4 Interval Number
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
= $3629.90
PV of annuity
= $1000(1.04)-1 + $1000(1.04)-2 + $1000(1.04)-3 += $961.54 + $924.56 + $889.00 + $854.80
$1000 $1000 $1000 $1000
Assume that there are four(4) annual $1000 payments with interest at 4%
Present Value of an
Ordinary Simple Annuity
Present Value of an
Ordinary Simple Annuity
$1000 (1.04)-1 n = 1
$1000 (1.04)-2 n = 2
$1000 (1.04)-3 n = 3
$1000 (1.04)-4 n = 4
0 1 2 3 4 Interval Number
$1000 (1.04)-4
Sum = PV of annuity
Ordinary
Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
Present Value of an
Ordinary Simple Annuity
Present Value of an
Ordinary Simple Annuity
You overhear your friend saying the he is repaying a loan at $450 every month for the next nine months.
The interest rate he has been charged is 12% compounded monthly. Calculate the amount of the
loan, and the amount of interest involved.
… Interest - use 12, not .12 when using financial calculator
… Interest - use 12, not .12 when using financial calculator … At the end of the loan, you don’t owe any money, so FV = 0
… n = 9 payments
…Since you are making payments, not receiving them, PMT = -450…Since you are making payments, not receiving them, PMT = -450
Solve…
… Repaid 9 payments at $450 = $4,050
Ordinary
Annuities
Ordinary
Annuities
10101010
10-30
McGraw-Hill Ryerson©Formula solutionFormula solution
You overhear your friend saying the he is repaying a
loan at $450 every month for the next nine months. The interest rate he has been charged is
8% compounded monthly. Calculate the amount of the
loan, and the amount of interest
involved.
You overhear your friend saying the he is repaying a
loan at $450 every month for the next nine months. The interest rate he has been charged is
8% compounded monthly. Calculate the amount of the
loan, and the amount of interest
involved.
98
450
012
PV = 3,918.24
Amount Borrowed (PV) $ 3,918.24
Interest Paid =
Repaid.…………………. 4,050.00
$ 131.76
Ordinary
Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
Formula Formula i
PV = PMT 1-(1+ i)-n[ ]
- Borrowed $3,918.24
= Interest Charged $131.76
Repaid $4,050.00
12.08
450
91
1
0.0066671.0066670.94195-0.05804793,918.24 You overhear your friend saying the he is repaying a
loan at $450 every month for the next nine months. The interest rate he has been charged is
8% compounded monthly. Calculate the amount of the
loan, and the amount of interest
involved.
You overhear your friend saying the he is repaying a
loan at $450 every month for the next nine months. The interest rate he has been charged is
8% compounded monthly. Calculate the amount of the
loan, and the amount of interest
involved.
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
Contribution of Each Payment to an Annuity’s
Present Value
Ordinary
Annuities
Ordinary
Annuities
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$10.00$10.00
YearsYears0 1 2 3 4 5
Contribution$
9.09
8.20
7.51
6.83
6.21
$37.91$37.91
PV ContributionsPV Contributions
$10.00$10.00
$10.00$10.00
$10.00$10.00
$10.00$10.00
$10.00$10.00
PVPV
Ordinary
Annuities
Ordinary
Annuities
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10-34
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…of a cash flow stream that includes an annuity
Ordinary
Annuities
Ordinary
Annuities
10101010
LO-3LO-3
Ordinary
Annuities
Ordinary
Annuities
10101010
10-35
McGraw-Hill Ryerson©
You have received two offers on a building lot that you want to sell.
Ms. Armstrong’s offer is $25,000 down plus a $100,000 lump sum payment
five years from now.
Mr. Belcher has offered $20,000 down plus $5000 every quarter for
five years.
Compare the economic values of the two offers if money can earn 5% compounded annually.
LO-3LO-3
Ordinary
Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
The economic value of a payment stream on a particular date (focal date) refers to a single amount that is an economic substitute for the
payment stream
On what information should we
focus?
On what information should we
focus?
WE need to choose a focal date, and determine the values of the two offers at that focal date.
(Obvious choices would be today, the date of the offers, or the end of the term i.e. 5 years from now.)
ocu
Back to Offer Comparison Back to Offer Comparison
Ordinary
Annuities
Ordinary
Annuities
10101010
10-37
McGraw-Hill Ryerson©Preparing Time Lines
Mr. BelcherMs. Armstrong
$20,000 down
plus $5000 every quarter for five years
$25,000 down
plus a $100,000 lump sum payment five
years from nowFocal Date: TodayFocal Date: Today
You have received two offers on a building lot that you want to sell. Ms. Armstrong’s offer is $25,000
down plus a $100,000 lump sum payment five years from now. Mr. Belcher has offered $20,000
down plus $5000 every quarter for five years. Compare the economic values of the two offers if
money can earn 5% compounded annually.
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
$20,000$20,000$20,000$20,000
$20,000
Years0 1 2 3 4
Time Lines
$20,000 down plus $5,000 every quarter for five years
$25,000 down plus a $100,000 lump sum payment five years from nowAA
BB
$25,000
$20,000
Ms. Armstrong
Mr.Belcher$5000 every quarter
5
$100,000
Ordinary
Annuities
Ordinary
Annuities
10101010
10-39
McGraw-Hill Ryerson©
You have received two offers on a
building lot that you want to sell. Ms.
Armstrong’s offer is $25,000 down plus a $100,000 lump sum payment five years
from now. Mr. Belcher has offered $20,000
down plus $5000 every quarter for five years.
Compare the economic values of the two offers if money can earn 5% compounded annually.
Step 1–Determine today’s value of Ms. Armstrong’s offer
today’s value of
lump sum
today’s value of
lump sum
5
100,000
1 5
25,000
PV= 78352.692 103,352.62 today’s value of Ms. A’s
total offer
today’s value of Ms. A’s
total offer
Step 2…Step 2…
0
Ordinary
Annuities
Ordinary
Annuities
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McGraw-Hill Ryerson©
Step 2 – Determine today’s value of Mr. Belcher’s offer.
4
1
5 0
4500
20
P/Y = 4 C/Y = 1 0PV = 79,376.93
20000
99,376.93 today’s value of
lump sum
today’s value of
lump sum
today’s value of Mr. B’s
total offer
today’s value of Mr. B’s
total offer
You have received two offers on a
building lot that you want to sell. Ms.
Armstrong’s offer is $25,000 down plus a $100,000 lump sum payment five years
from now. Mr. Belcher has offered $20,000
down plus $5000 every quarter for five years.
Compare the economic values of the two offers if money can earn 5% compounded annually.
Ordinary
Annuities
Ordinary
Annuities
10101010
10-41
McGraw-Hill Ryerson©
$103,352.62
99,376.93
$ 3,975.69
Better off accepting Ms. Armstrong’s offer!
Ms. Armstrong
Mr.Belcher
Total Value
of each offer
Total Value
of each offer
Difference in Offers
Ordinary
Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
The required monthly payment on a five-year loan, bearing 8% interest,
compounded monthly, is $249.10.
Since you are “borrowing” money, you are looking for PV… and FV = 0 once you have repaid the loan!
n = 5 yrs * 12 payments per year = 60 payments
Since you are “borrowing” money, you are looking for PV… and FV = 0 once you have repaid the loan!
n = 5 yrs * 12 payments per year = 60 payments
a) What was the original principal amount of the loan?b) What is the balance owed just after the twentieth payment?
a) What was the original principal amount of the loan?b) What is the balance owed just after the twentieth payment?
Calculating the Original Loan
and a Subsequent Balance
Calculating the Original Loan
and a Subsequent Balance
LO-4LO-4
Ordinary
Annuities
Ordinary
Annuities
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Original Principal = PV of all 60 payments
PMT = FV = n = i = c =249.10 0 5*12 = 60 .08/12 1
12
0 8
60
0 PV = 12,285.22 Original loan value Original loan value
249.10
The required
monthly payment on a five-year loan,
bearing 8% interest,
compounded monthly, is $249.10.
a) What was the original principal
amount of the loan?
b) What is the balance owed just after the twentieth
payment?
The required
monthly payment on a five-year loan,
bearing 8% interest,
compounded monthly, is $249.10.
a) What was the original principal
amount of the loan?
b) What is the balance owed just after the twentieth
payment?
Ordinary
Annuities
Ordinary
Annuities
10101010
10-44
McGraw-Hill Ryerson©
Balance after 20 payments = PV
of 40 payments leftPMT = FV = n = i =249.10 0 60 - 20 = 40 .08
40
PV = 8,720.75 New loan balance
New loan balance
We will leave it to you to do the algebraic solution…!
We will leave it to you to do the algebraic solution…!
The required
monthly payment on a five-year loan,
bearing 8% interest,
compounded monthly, is $249.10.
a) What was the original principal
amount of the loan?
b) What is the balance owed just after the twentieth
payment?
The required
monthly payment on a five-year loan,
bearing 8% interest,
compounded monthly, is $249.10.
a) What was the original principal
amount of the loan?
b) What is the balance owed just after the twentieth
payment?
Ordinary
Annuities
Ordinary
Annuities
10101010
10-45
McGraw-Hill Ryerson©
A Deferred Annuity may be viewed as an ordinary annuity that does not begin until a time interval (named the period of deferral)
has passed
LO-5LO-5
Ordinary
Annuities
Ordinary
Annuities
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Deferred AnnuitiesDeferred Annuities
A Deferred Annuity may be viewed as
an ordinary annuity that does not begin
until a time interval (named the period
of deferral) has passed
A Deferred Annuity may be viewed as
an ordinary annuity that does not begin
until a time interval (named the period
of deferral) has passed
d = Number of payment intervals in the period of deferral
Two-step procedure to find PV:Two-step procedure to find PV:
Calculate the present value, PV1,
of the payments at the end of the period of deferral — this is just the
PV of an ordinary annuity Calculate the present value,
PV2, of the STEP 1 amount
at the beginning of the period of deferral
Ordinary
Annuities
Ordinary
Annuities
10101010
10-47
McGraw-Hill Ryerson©
… your friend saying the he is repaying a loan at $450 every month for four months. The interest rate he has been charged is 8% compounded monthly. Calculate the amount of the loan, and
the amount of interest involved.
… your friend saying the he is repaying a loan at $450 every month for four months. The interest rate he has been charged is 8% compounded monthly. Calculate the amount of the loan, and
the amount of interest involved.
…this same friend doesn’t begin to repay his loan for another 11 months, at a rate $500 every month for four months. The interest
rate is still 8% compounded monthly. Determine the size of the loan.
…this same friend doesn’t begin to repay his loan for another 11 months, at a rate $500 every month for four months. The interest
rate is still 8% compounded monthly. Determine the size of the loan.
Solve…
Ordinary
Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
$500 $500 $500 $500
…of the Annuity
Present Value of a
Deferred Annuity
Present Value of a
Deferred Annuity
10 11 12 13 14 Months0
PVPV
Step 1 – Determine PV of Annuity 10 months from now
Hint: (Use Compound Discount) Step 2 - Discount for 10 months to get today’s Loan ValueStep 2 - Discount for 10 months to get today’s Loan Value
Ordinary
Annuities
Ordinary
Annuities
10101010
10-49
McGraw-Hill Ryerson©
…this same friend doesn’t begin to
repay his loan for
another 11 months, at a rate $500
every month for four
months. The interest rate is still
8% compounded
monthly. Determine the size
of the loan.
…this same friend doesn’t begin to
repay his loan for
another 11 months, at a rate $500
every month for four
months. The interest rate is still
8% compounded
monthly. Determine the size
of the loan.
12
0
0 8
4
10
PV = 1967.11 FV = - 1967.11PV = 1840.65 value 10 months from now
value 10 months from now
loan value today
loan value today
500
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10101010
10-50
McGraw-Hill Ryerson©
The payment interval
differs from
the compounding interval
The payment interval
differs from
the compounding interval
e.g. A typical Canadian mortgage has Monthly payments, but the interest is
compounded semi-annually
Using calculators…Using calculators…
LO-6LO-6
Ordinary
Annuities
Ordinary
Annuities
10101010
10-51
McGraw-Hill Ryerson©
For those who are using this type of calculator,
the C/Y
worksheet will now be used
For those who are using this type of calculator,
the C/Y
worksheet will now be used
See following REVIEW
For those who are using a non-financial calculator,
new formulae will be added to find
the solution
For those who are using a non-financial calculator,
new formulae will be added to find
the solution
See following
Ordinary
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10101010
10-52
McGraw-Hill Ryerson©
We can input the number of compoundings per year into the
financial calculator. This can be performed by using
the symbolTo access this symbol use:
…and you will see
Ordinary
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Ordinary
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10101010
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McGraw-Hill Ryerson©
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!
…Example…Example
Appearsautomatically
Appearsautomatically
Ordinary
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Ordinary
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10101010
10-54
McGraw-Hill Ryerson©
12
2
P/Y = 12.00C/Y = 12.00
UsingC/Y = 2.00
Adding New Formulae
Typical Canadian mortgageInterest is
compounded semi-annually
and payments are each month.
Typical Canadian mortgageInterest is
compounded semi-annually
and payments are each month.
Ordinary
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10101010
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McGraw-Hill Ryerson©
to calculate the equivalent periodic rate that matches the payment interval
C =
number of interest compoundings per yearnumber of payments per year
Use c to determine i2 Step 2Step 2
Use i2 = (1+i)c - 1
Use this equivalent periodic rate as the value for “i”
in the appropriate simple annuity formula
Step 3
Step 3
…Example…Example
Step 1Step 1 Determine the number of Interest periods per compounding interval
Adding New FormulaeAdding New Formulae
Ordinary
Annuities
Ordinary
Annuities
10101010
10-56
McGraw-Hill Ryerson©
Typical Canadian mortgage
6% Interest is compounded
semi-annually and
payments are each month.
Find C and i2.
Typical Canadian mortgage
6% Interest is compounded
semi-annually and
payments are each month.
Find C and i2.
C =
number of interest compoundings per yearnumber of payments per year
2 12
0.166666
Step 1Step 1 To determine the number of Interest periods per compounding interval
= C
Use c to determine i2 Step 2Step 2
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10101010
10-57
McGraw-Hill Ryerson©
Use c to determine i2 Step 2Step 2
i2 = (1+i)c - 1
i2 = (1+ .06/2).16666 -1Typical
Canadian mortgage
6% Interest is compounded
semi-annually
and payments are each month.
Find C and i2.
Typical Canadian mortgage
6% Interest is compounded
semi-annually
and payments are each month.
Find C and i2.
1.03
1
0.166666 = i2 1.0049 0.0049
…another example…another example
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Annuities
Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
5% interest is
compounded monthly
and payments
are each week
5% interest is
compounded monthly
and payments
are each week
MortgageMortgage
Step 1Step 1 To determine the number of compoundings
C =
number of interest compoundings per yearnumber of payments per year
12 52
0.23076 = C
Use c to determine i2 Step 2Step 2
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10101010
10-59
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1
Use c to determine i2 Step 2Step 2
i2 = (1+i)c - 1
i2 = (1+ .05/12).2308 -1
1
= i2
0.05 12
0.0041667 1.0041667
5% interest is
compounded monthly
and payments
are each week
5% interest is
compounded monthly
and payments
are each week
MortgageMortgage
0.230769 1.00096 0.00096
…another example…another example
Ordinary
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Ordinary
Annuities
10101010
10-60
McGraw-Hill Ryerson©
You decide to save $50/month for the next three years. If you invest all of these savings in an account
which will pay you 7% compounded semi-annually, determine the total in the account after 3 years.
Is the following a
General Annuity?
The payment interval differs from
the compounding interval
The payment interval differs from
the compounding interval
CriteriaCriteria
As the Criteria have been met, therefore,
we need to determine CAs the Criteria have been met, therefore,
we need to determine C
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Annuities
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10101010
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Find i2 Step 2Step 2 i2 = (1+i)c - 1
i2 =
1.035
1
0.1666
(1+ .07/2).1666-1
0.00575
You decide to save $50/month for the next three years.
If you invest all of
these savings in an account
which will pay you 7%
compounded semi-annually, determine the
total in the account after
3 years.
i2 =
Step 1Step 1 Find c
Use i2Step 3
Step 3
1.00575 0.00575
2 12
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10101010
10-62
McGraw-Hill Ryerson©
Formula Formula [FV = PMT (1+ i)n - 1i ]
You decide to save $50/month for the next
three years. If you invest all of
these savings in an account
which will pay you 7%
compounded semi-annually, determine the
total in the account after
3 years.
PMT = PV = n = i = c = i2 =
50 0 3*12 = 36.07/2 2/12 = .16666 0.00575
1
50
36
1
0.00575 1.00575 1.229255
Use i2 in the appropriate formulaStep 3
Step 3
0.229255 39.8702 1993.51
Solve…
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Annuities
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10101010
10-63
McGraw-Hill Ryerson©
P/Y = 12 C/Y = 12C/Y = 2
You decide to save $50/month for the next
three years. If you
invest all of these savings in an
account which will pay you 7%
compounded semi-annually, determine the
total in the account after
3 years.
12
2
50
0
36
7
0FV = 1993.51
Ordinary
Annuities
Ordinary
Annuities
10101010
10-64
McGraw-Hill Ryerson©
…your calculator retains at least two more digits than you see displayed!
Improving the
Accuracy of
Calculated Results
C =
number of interest compoundings per yearnumber of payments per year
the value for c can be a repeating decimal
SAVE c in memory…
when you need the exponent for
Simply the c value from memory!
The value for i2 should be saved in
memory as soon you calculate it! it later!
Ordinary
Annuities
Ordinary
Annuities
10101010
10-65
McGraw-Hill Ryerson©
Reid David made annual deposits of $1,000 to Fleet Bank, which pays 6% interest compounded annually. After 4 years, Reid makes no more
deposits.
What will be the balance in the account 10 years after the last deposit?
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Ordinary
Annuities
10101010
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McGraw-Hill Ryerson©
…of the Annuity
1 2 3 14 40
FV1FV1
Step 2 – Determine FV using compound interest
FV2FV2
Step 1 – Determine FV1 of Annuity 10 years from now
Years
$1000 $1000$1000 $1000
Reid David made annual deposits of $1,000 to Fleet Bank, which pays 6% interest compounded
annually. After 4 years, Reid makes no more deposits. What will be the balance in the account
10 years after the last deposit?
Ordinary
Annuities
Ordinary
Annuities
10101010
10-67
McGraw-Hill Ryerson©
Step 1 – Determine FV1 of Annuity 10 years from now
1
1
6 0
4
P/Y = 1.00C/Y = 1.00 value at end of 4 years
value at end of 4 years
Step 2…Step 2…
0
1000
FV = 4374.62
Reid David made annual
deposits of $1,000 to Fleet Bank, that pays 6%
interest
compounded annually. After 4 years, Reid
makes no more deposits.
What will be the balance in the
account 10 years after the
last deposit?
Reid David made annual
deposits of $1,000 to Fleet Bank, that pays 6%
interest
compounded annually. After 4 years, Reid
makes no more deposits.
What will be the balance in the
account 10 years after the
last deposit?
Ordinary
Annuities
Ordinary
Annuities
10101010
10-68
McGraw-Hill Ryerson©
0
10
Formula solutionFormula solution
Step 2 – Determine FV2 using compound interest
FV = 4374.62 FV = 7834.27 value 14 years from now
value 14 years from now
Reid David made annual
deposits of $1,000 to Fleet Bank, that pays 6%
interest
compounded annually. After 4 years, Reid
makes no more deposits.
What will be the balance in the
account 10 years after the
last deposit?
Reid David made annual
deposits of $1,000 to Fleet Bank, that pays 6%
interest
compounded annually. After 4 years, Reid
makes no more deposits.
What will be the balance in the
account 10 years after the
last deposit?
Ordinary
Annuities
Ordinary
Annuities
10101010
10-69
McGraw-Hill Ryerson©
Formula Formula [FV = PMT (1+ i)n - 1i ]
n = i = c =1000 0.06
1.06
1000
4
1 0.06
PMT =
1.262477 0.262477 4374.62
4 1
Step 1 – Determine FV of Annuity 4 years from now
value at end of 4 years
value at end of 4 years
Step 2…Step 2…
Reid David made annual
deposits of $1,000 to Fleet Bank, that pays 6%
interest
compounded annually. After 4 years, Reid
makes no more deposits.
What will be the balance in the
account 10 years after the
last deposit?
Reid David made annual
deposits of $1,000 to Fleet Bank, that pays 6%
interest
compounded annually. After 4 years, Reid
makes no more deposits.
What will be the balance in the
account 10 years after the
last deposit?
Ordinary
Annuities
Ordinary
Annuities
10101010
10-70
McGraw-Hill Ryerson©
1.06 10
Step 2 – Determine FV using compound interest
Reid David made annual deposits of
$1,000 to Fleet Bank, which pays
6% interest
compounded
annually. After 4 years, Reid
makes no more deposits.
What will be the balance in the account 10 years after the last deposit?
n = i =4374.62 0.06PV = 10
1.262477 0.262477 4374.62 value 14 years from now
value 14 years from now 1.1708477 7834.27
FV = PV(1 + i)nFormula Formula
Ordinary
Annuities
Ordinary
Annuities
10101010
10-71
McGraw-Hill Ryerson©
How much more interest will Reid David accumulate over the 14 years if his
account earns 6%
compounded daily?
1
365
1000
0
4
6
P/Y = 10 value at end of 4 years
value at end of 4 yearsC/Y = 1 C/Y = 365
Step 1 – Determine FV of Annuity 4 years from now
0FV = 4386.52
Ordinary
Annuities
Ordinary
Annuities
10101010
10-72
McGraw-Hill Ryerson©
0 3650365
FV = 4386.52 How much more interest will Reid David accumulate over the 14 years if his
account earns 6%
compounded daily?
value 14 years from now
value 14 years from nowP/Y = 1P/Y = 3650FV = 7992.37
Step 2 – Determine FV in 10 years using compound interest
Ordinary
Annuities
Ordinary
Annuities
10101010
10-73
McGraw-Hill Ryerson©
InterestInterest
$7,992.37$7,992.37 $7,834.27$7,834.27
Ordinary
Annuities
Ordinary
Annuities
10101010
10-74
McGraw-Hill Ryerson©
This completes Chapter 10This completes Chapter 10