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Some sample solutions of annuity problems
Typical MAT 112 problems:
What is the value of an ordinary annuity at the end of 10 years if $300 is deposited each quarter into an account earning 7 % compounded quarterly? Also, of this total value, how much did you contribute and how much is from the interest? For 40 deposits of $300 each with [pic], we find the accumulated value as [pic]
The total interest earned is the difference between the amount in the account and what you actually deposited: 17170.24 – (300)(40) = $5170.24.
An individual deposits $600 per month into an account paying 7.2 % compounded monthly. How much money will be in the account in 5 years? These 60 deposits have an accumulated value of [pic]
A person wishes to have $300,000 in an account 16 years from now. How much should be deposited each quarter in an account paying 8 % compounded quarterly in order to achieve this goal. For 64 deposits of $X each, we are given the accumulated value [pic]. Thus, solving for X we find [pic]
Changing the deposit size:
What if the size of the deposits changes at some point? For example, suppose 12 monthly deposits of $100 each during the first year are followed by 24 monthly deposits of $150 each over the next two years. If the nominal interest is [pic], find the accumulated value at the end of the 3 years.
Consider the following two ways of thinking about these deposits.
Approach #1
We may consider and sum the first 12 deposits separately from the final 24 deposits.
The first 12 deposits yield [pic] but taking this forward (with interest) to time t = 36 gives a value of [pic]
The next 24 deposits yield [pic] as of the final deposit at time t = 36.
Thus taken together, the accumulated value at time t = 36 is [pic]
Approach #2
We may consider the extra $50 a month during the final 24 deposits as a separate annuity and sum these $50 deposits separately.
Typical MAT 112 problems:
What is the value of an ordinary annuity at the end of 10 years if $300 is deposited each quarter into an account earning 7 % compounded quarterly? Also, of this total value, how much did you contribute and how much is from the interest? For 40 deposits of $300 each with [pic], we find the accumulated value as [pic]
The total interest earned is the difference between the amount in the account and what you actually deposited: 17170.24 – (300)(40) = $5170.24.
An individual deposits $600 per month into an account paying 7.2 % compounded monthly. How much money will be in the account in 5 years? These 60 deposits have an accumulated value of [pic]
A person wishes to have $300,000 in an account 16 years from now. How much should be deposited each quarter in an account paying 8 % compounded quarterly in order to achieve this goal. For 64 deposits of $X each, we are given the accumulated value [pic]. Thus, solving for X we find [pic]
Changing the deposit size:
What if the size of the deposits changes at some point? For example, suppose 12 monthly deposits of $100 each during the first year are followed by 24 monthly deposits of $150 each over the next two years. If the nominal interest is [pic], find the accumulated value at the end of the 3 years.
Consider the following two ways of thinking about these deposits.
Approach #1
We may consider and sum the first 12 deposits separately from the final 24 deposits.
The first 12 deposits yield [pic] but taking this forward (with interest) to time t = 36 gives a value of [pic]
The next 24 deposits yield [pic] as of the final deposit at time t = 36.
Thus taken together, the accumulated value at time t = 36 is [pic]
Approach #2
We may consider the extra $50 a month during the final 24 deposits as a separate annuity and sum these $50 deposits separately.