LEARNING OBJECTIVES

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ANNUITIES
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LEARNING OBJECTIVES

• Calculate the Payment Size, Number of Payments,
  Present and Future Value for
• Ordinary Annuities
• Annuities Due
• Calculate the Present Value of Deferred Annuities
  and Perpetuities
• Calculate Mortgage Payments for the initial loan
  and its renewals
• Calculate Mortgage loan balances and amortization
  periods to reflect prepayments of principal
• Calculate Payments for Sinking Funds

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An Annuity is a series of equal payments at
regular intervals (monthly, quarterly,
semi-annually, annually etc.). PMT Amount of
Each Payment in an Annuity n Number of
Payments in the Annuity PV Present Value of an
Annuity FV Future Value of an Annuity i
Periodic Interest Rate Nominal Interest
Rate ----------------------------------
- number of compoundings
per year
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STEPS TO SOLVING ANNUITY PROBLEMS
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Step 1 Identify type of Annuity SIMPLE OR
COMPLEX

• Step 3 Identify the subtype of annuity -
• ORDINARY, DUE, OR DEFERRED

Step 4 Does the annuity involve - AMORTIZATION
(FV 0) or ACCUMULATION (PV 0)
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GENERAL OR COMPLEX ANNUITIES
When payment period is not equal to the
compounded period. Convert the given interest
rate to the equivalent rate compounded as often
as the payment period. (1 i1)n1 (1 i2)n2
Examples Payments are quarterly and interest
rate is 10 compounded semi-annually.   (1
.1/2)2 (1 i2)4 i 2.4695 quarterly   Payment
s are monthly and interest rate is 10 compounded
semi-annually. (1 .1/2)2 (1 i2)12 i
0.8165 monthly   Payments are monthly and
interest rate is 10 compounded quarterly.   (1
.1/4)4 (1 i2)12 i 0.82648 monthly
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ACCUMULATION (PV has no meaning) A series of
periodic payments, each of which is a saving to
accumulate a given amount at the end of the
term.  Periodic Savings in an Accumulation
Periodic Payment  Savings at any time
Outstanding Balance
AMORTIZATION (FV has no meaning) A series of
periodic payments, each of which pays interest on
the unpaid balance and repays part of the
principal.  Periodic Expense in an Amortization
Periodic Payment  Debt at any time Outstanding
Balance
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ORDINARY ANNUITIES
Mr. Smith plans to deposit 1000 at the end of
each year into his RRSP for the next ten years.
If his investments earn 9 per year compounded
annually, what will be the value of his RRSP
after the last payment?
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ORDINARY ANNUITIES
Mr. Jones wants to save enough money to send his
two children to college. They are three years
apart in age, so he wants to have a sum of money
that will provide 3000 a year for six years.
Find the single sum required one year before the
first withdrawal if interest is 7 compounded
annually.
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ORDINARY ANNUITIES
A contract calls for payments of 100 at the end
of every 3 months for 12 years and an additional
payment of 2000 at the end of 12 years. What is
the present value of the contract if money is
worth 8 compounded semi-annually.
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ORDINARY ANNUITIES
A 10,000 loan is amortized at 10 compounded
annually, over five years. Find the size of the
payments.
Find how much of the second payment has gone to
the principal amount and how much has gone to the
interest amount. What is the balance on the loan
after the second payment?
For payments three to five, how much has gone to
the principal and how much has gone to interest?
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CALCULATOR FUNCTIONS
Finding how a payment is distributed between
interest payment and the principal payment, and
the remaining balance.
For the second payment.
2 AMRT (display PRIN 1,801.77)  
AMRT (display BAL -6,560.25)  
AMRT (display INT 836.20)
Finding how much the total payments that went
toward reducing the outstanding balance and how
much went to paying interest
For payments 3 to 5.
3 P1/P2 5 ACC (display ?PRN 6,560.25) ACC
(display ?INT 1,353.67)
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AMORTIZATION SCHEDULE
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ORDINARY ANNUITIES
Calculate the amount of interest included in the
accumulated value of 600.00 deposits made at the
end of each month for 5 years. The interest rate
is 13.5 compounded monthly.
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ORDINARY ANNUITIES
Construct an Amortization schedule to repay a
loan of 5,600 with 8 monthly payments using 10
compounded monthly.
Find the total principal repaid and the total
interest paid over the amortization using the
amortization schedule.
Find the totals (interest and principal) for
payments 4 to 7.
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PV 5600 i 10/12 n 8 PMT 726.51 (round
up from 726.504)

2822.97 principal and 83.07 interest.
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ANNUITIES DUE
Mary deposits 100 at the beginning of each month
for 3 years in an account paying 6 interest
compounded monthly. How much is in her account
at the end of 3 years?
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ANNUITIES DUE
The monthly rent for a townhouse is 315 payable
at the beginning of each month. What is the
equivalent yearly rental payable in advance if
money is worth 9 compounded monthly?
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ANNUITIES DUE
A debt of 10,000 with interest at 11 compounded
quarterly is to be paid off by 8 equal quarterly
payments, the first is due today. Find the
quarterly payment.
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DEFERRED ANNUITIES
What sum of money should be set aside on a
childs birth to provide 8 semi-annual payments
of 1500 to cover the expenses for university
education, if the first payment is to be made on
the childs 19th birthday? The fund will earn
interest at 9 compounded semi-annually.   Solve
as Ordinary and as Due.
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DEFERRED ANNUITIES
A woman wins 100,000 in a 6/49 draw. She takes
only 20,000 in cash and invests the balance at
8 compounded monthly, with the understanding
that she will receive 180 equal monthly payments
with the first one to be made in 4 years. Find
the size of the payments   Solve as Ordinary and
as Due.
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DEFERRED ANNUITIES
How much money does a person need now in order to
receive 300 monthly for 5 years, the first
payment to be made 4 years and 7 months from now,
if the interest rate is 5 compounded
semi-annually?
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PERPETUITIES
A company is expected to pay 3.50 every six
months on a share of its preferred stock. If
money is worth 9 compounded semi-annually, what
should a share of the stock be selling
for? Solve as ordinary and due.
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AMORTIZATION AND SINKING FUNDS
These are two methods of paying off long term
loans. Amortization A series of periodic
payments, each of which pays interest on the
unpaid balance and repays part of the
principal.   Periodic Expense in an Amortization
Periodic Payment Debt at any time Outstanding
Balance Sinking Funds Pay the interest on the
loan at the end of each interest period and
create a sinking fund to accumulate the principal
at the end of the term of the loan.   Periodic
Expense in a Sinking Fund Interest Payment
Sinking Fund Deposit Debt at Any Time Original
Principal The Amount in the Sinking Fund
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AMORTIZATION
A debt of 10,000 is to be amortized with
semi-annual payments over 3 years with interest
at 10 compounded semi-annually. What is the
periodic expense of the debt.
If you wanted to payoff the debt in 1.5 years,
what would the payment be?
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SINKING FUND
A debt of 10,000 is to be repaid over 3 years by
semi-annual payments into a sinking fund.
Interest is paid at 10 semi-annually. The
sinking fund earns interest at 6 compounded
semi-annually. What is the periodic expense of
the debt.
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Payments For Fund n 6 i 6/2 FV
10,000 PMT 1545.98 Accumulated Interest For
Loan (FV-PV) n 6 i 10/2 PV 10,000 FV
13,400.96 Accumulated Interest
3,400.96 (13400.96-10000) Payments For
Accumulated Interest on Loan n 6 I 10/2 FV
3,400.96 PMT 500.00 Periodic Expense Fund
Deposit Interest Payments on Loan 1545.98
500 2045.98
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