Learning Objectives for Section 3'3

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Learning Objectives for Section 3.3
Future Value of an Annuity Sinking Funds

• After this lesson you should be able to
• Compute the future value of an annuity.
• Solve problems involving sinking funds.
• Approximate interest rates of annuities.

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Definition of Annuity

• An annuity is any sequence of equal periodic
  payments.
• Ordinary annuity refers to payments being made at
  the end of each time interval.
• Future value is the sum of all of the payments
  plus interest.

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Future Value of an Ordinary Annuity

• FV future value (amount)
• PMT periodic payment
• i interest rate per period
• n total number of payments

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Example

• Example 1 Suppose a 1,000 payment is made at
  the end of each quarter and the money in the
  account is compounded quarterly at 6.5 interest
  for 15 years.
• How much is in the account after the 15-year
  period?
• How much interest was earned over the 15-year
  period?

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Example Solution

• Example 1 Suppose a 1,000 payment is made at
  the end of each quarter and the money in the
  account is compounded quarterly at 6.5 interest
  for 15 years.
• How much is in the account after the 15-year
  period?

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Example

• Example 1 Suppose a 1,000 payment is made at
  the end of each quarter and the money in the
  account is compounded quarterly at 6.5 interest
  for 15 years.
• How much interest was earned over the 15-year
  period?

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Example of Future Value

• Suppose a 1,000 payment is made at the end of
  each quarter and the money in the account is
  compounded quarterly at 6.5 interest for 15
  years.
• a) How much is in the account after the 15-year
  period?
• Solution

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Amount of Interest EarnedSolution

• b) How much interest was earned over the 15-year
  period?
• Solution
• Each periodic payment was 1,000. Over 15 years,
  15(4)60 payments were made for a total of
  60,000. Total amount in account after 15 years
  is 100,336.68. Therefore, amount of accrued
  interest is 100,336.68 - 60,000 40,336.68.

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Example 2
Example 2 Bob makes his first 1,000 deposit
into an IRA earning 6.4 compounded annually on
his 24th birthday (12 equal deposits in all.)
With no additional deposits, the money in the IRA
continues to earn 6.4 interest compounded
annually until Bob retires on his 65th birthday.
How much is in the IRA when Bob retires?
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Example 3
Example 3 Refer to example 2. John
procrastinates and does not make his first 1,000
deposit into an IRA until he is 36, but then he
continues to deposit 1,000 each year until he is
65 (30 deposits in all). If Johns IRA also
earns 6.4 compounded annually, how much is in
his IRA when he makes his last deposit on his
65th birthday?
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Sinking Fund

• Definition Any account that is established for
  accumulating funds to meet future obligations or
  debts is called a sinking fund.
• The sinking fund payment is defined to be the
  amount that must be deposited into an account
  periodically to have a given future amount.

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Sinking Fund Payment Formula

• Just a manipulation of the Future Value Formula

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Sinking Fund Payment Formula
Sinking Fund where FV future value
(amount) PMT periodic
payment i interest rate per
period (r/m) n total
of payments (periods)
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Sinking FundSample Problem

• Example 4 How much must Harry save each month
  in order to buy a new car for 12,000 in three
  years if the interest rate is 6 compounded
  monthly?

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Sinking FundSample Problem Solution

• How much must Harry save each month in order to
  buy a new car for 12,000 in three years if the
  interest rate is 6 compounded monthly?
• Solution

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Additional Example
Example 5 A new 2002 Toyota Sequoia Limited 4 ?
4 has an MSRP of 42,725. A buyer gets 7,500
for his trade-in and finances the balance of the
price at his local credit union with a 60-month
5.99 loan. Determine the amount of the car
payment.
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Multi-Dimensional Example
Not my age, by the way
Example 6 Forecasting Retirement Savings
Lets pretend that a 35-year-old college
professor contributes 200 to her CREF Stock
Account twice a month. (The CREF Stock Account
has earned an average of 10.39 annually since
1952.) She hopes to retire in 25 years with a
retirement account worth 800,000 or more. Her
current account balance is 15,000. a)
Assuming that she will be able to earn an annual
rate of 10.39 compounded semi-monthly, will she
be able to reach her retirement goal? b) Since
she ended up short of her investment goal, she
will need to increase her payment amount.
Determine the payment amount that will allow her
to reach her goal in 25 years.