Ordinary Annuities… ordinary simple annuities and ordinary general annuities
… fair market value of a cash flow stream that includes an annuity
Calculate the…
Learning 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
Calculate the…
Term of the Annuity
- the time from the beginning of the first payment period to the end of the last payment period
Future Value
Present Value
the 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
Terminology
Annuity
LO-1
PMT
n
… is the time between successive payments in an annuity
… are ones in which payments are made at the end of each payment interval
Terminology
Suppose you obtain a personal loan to be repaid by
payment interval
ordinary annuities
1 month
first payment will be due 1 month after you receive the loan, i.e. at the end of the first payment interval
Terminology
Term
PMT
PMT
PMT
PMT
Terminology
The payment interval
…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
n = 1
$1000 (1.04)2
$1000 (1.04)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
n = 1
$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
Result
0
1
2
3
4
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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
Future Value of an Ordinary Simple Annuity
Using the …
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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
0
12
Note
…the sum of the future values of all the payments
Future Value of an Ordinary Simple Annuity
(1+ i)n
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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
0.0025
1.0025
1.0941
0.0941
37.6206
18810.28
12
.03
500
36
1
1
FV
= PMT
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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
..a term that refers to payments that can be either …
… payments made
e.g. cheques
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…when you are making payments, or even making deposits to savings,
these are cash outflows, and therefore the values must be negative!
Therefore…
Cash Flow Sign Convention
4
FV = 2007.51
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
12
3
500
0
We already have these from before, so we don’t have to enter them again!
Formula solution
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0.0025
1.0025
1.0100
0.0100
4.0150
2007.51
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.
FV
= PMT
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!
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Contribution
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
PMT =
= 7
n =
= 12
Deposits…... 3,600.00
P/Y = 12
FV = 4140.69
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
12
Formula
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
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…the sum of the present values of all the payments
PresentValue of an Ordinary Simple Annuity
Formula
PV
= PMT
…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
$1000 (1.04)-1
$1000 (1.04)-2
$1000 (1.04)-3
$1000 (1.04)-4
Interval Number
n = 1
n = 2
n = 3
n = 4
= $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
$1000 (1.04)-1
$1000 (1.04)-2
$1000 (1.04)-3
$1000 (1.04)-4
Interval Number
$1000 (1.04)-4
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
… 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
… Repaid 9 payments at $450 = $4,050
Solve…
Interest Paid =
Repaid.…………………. 4,050.00
Formula 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.
12
$ 131.76
[
]
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.
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Contribution of Each Payment to an Annuity’s Present Value
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Contribution
Ordinary
Annuities
10
LO-3
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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-3
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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?
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
$25,000 down
plus a $100,000 lump sum payment five years from now
Focal 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.
Preparing 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 now
A
B
$25,000
$20,000
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Step 1–Determine today’s value of Ms. Armstrong’s offer
5
100,000
5
25,000
0
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.
today’s value of lump sum
1
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Step 2 – Determine today’s value of Mr. Belcher’s offer.
4
1
5
0
4500
20
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.
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Ms. Armstrong
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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
What was the original principal amount of the loan?
What is the balance owed just after the twentieth payment?
Calculating the Original Loan and a Subsequent Balance
LO-4
0
8
60
0
Original loan value
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?
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PMT =
FV =
n =
i =
249.10
0
PV = 8,720.75
We will leave it to you to do the algebraic solution…!
New loan balance
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?
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10-*
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-5
Deferred 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
d = Number of payment intervals in the period of deferral
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
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… 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.
Solve…
10
11
12
13
14
Months
0
PV
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 Value
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10-*
…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.
0
0
8
4
10
loan value today
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e.g. A typical Canadian mortgage has Monthly payments, but the interest is compounded semi-annually
Using calculators…
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For those who are using this type of calculator, the C/Y worksheet will now be used
For those who are using a non-financial calculator, new formulae will be added to find the solution
See following REVIEW
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We can input the number of compoundings per year into the financial calculator. This can be performed by using the symbol
To access this symbol use:
…and you will see
… represents the number of Payments per Year
… represents the number of Compoundings per Year
Note: You can override these values by entering new ones!
…Example
To access use:
Using
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to calculate the equivalent periodic rate that matches the payment interval
C =
number of payments per year
Use c to determine i2
Step 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
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Typical Canadian mortgage
6% Interest is compounded semi-annually and payments are each month. Find C and i2.
2
12
= C
C =
number of payments per year
0.166666
To determine the number of Interest periods per compounding interval
Step 1
Step 2
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(1+ .06/2).16666 -1
Typical Canadian mortgage
6% Interest is compounded semi-annually and payments are each month. Find C and i2.
1.03
1
= i2
1.0049
0.0049
Step 2
5% interest is compounded monthly and payments are each week
Mortgage
Step 1
number of payments per year
0.23076
Step 2
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Step 2
5% interest is compounded monthly and payments are each week
Mortgage
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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
Criteria
As the Criteria have been met, therefore, we need to determine C
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0.1666
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.
Step 1
Find c
Use i2
Step 3
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Formula
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.
Use i2 in the appropriate formula
Step 3
FV = 1993.51
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.
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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!
…your calculator retains at least two more digits than you see displayed!
Improving the Accuracy of Calculated Results
C =
number of payments per year
SAVE c in memory…
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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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FV2
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?
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Step 1 – Determine FV1 of Annuity 10 years from now
value at end of 4 years
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?
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FV = 4374.62
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?
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Step 1 – Determine FV of Annuity 4 years from now
value at end of 4 years
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?
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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
0.06
PV =
10
1.262477
0.262477
4374.62
FV = PV(1 + i)n
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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
Step 1 – Determine FV of Annuity 4 years from now
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FV = 7992.37
How much more interest will Reid David accumulate over the 14 years if his account earns 6% compounded daily?
value 14 years from now
Step 2 – Determine FV in 10 years using compound interest
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