Chapter 5
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Chapter 5
Other Annuities Certain
• annuity due
an annuity due is one in which the payments are made at the beginning of the period.
Formula used to calculate the amount of annuities due .
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Or
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Formula used to calculate the present value of annuities due
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Or
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Example 1:
An investment of $200 is made at the beginning of each year for 10 years. If interest is 6% effective, how much will the investment be worth at the end of 10 years?
Solution:
Substitute R=200, n=10, and i = 6%
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Or
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Example 2:
A student wants to have $2500 for a trip after graduation 4 years from now. How much must she invest at the beginning of each year starting now if she gets 5% compounded annually on her saving?
Solution :
Substitute R=?, n=4, and i = 5%
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Or
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Example 3:
The premium on a life insurance policy is $60 a quarter, payable in advance. Find the cash equivalent of a year's premiums if the insurance company charges 6% converted quarterly for the privilege of paying a smaller amount every three months instead of all once for the year.
Solution :
Substitute R=60, n=4, and i = 0.06/4=0.015
Formula used to calculate the present value of annuities due
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Alternate solution:
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Example 4:
The beneficiary of a life insurance policy may take $10000 cash or 10 equal annual payments, the first to be made immediately. What is the annual payment if money is worth 6%?
Solution:
Substitute R=?, n=10, and i = 0.06
Formula used to calculate the present value of annuities due
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Alternate solution:
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Example 5:
A state lottery advertises a prize of $3000000. in fact, the winner will receive $150000 at the beginning of each year for 20 years. If money is worth 7.5%, find the true value of the prize.
Substitute R= 150000, n=20, and i = 0.075
Formula used to calculate the present value of annuities due
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Alternate solution:
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Exercise 5a :
1,3, 7, 9,11
Deferred annuity
A deferred annuity is one which the first payment is made not at the beginning or end of the first period, but at some later date.
Note1:
Differed annuity denoted by m periods
Number of payments denoted by n periods
Note 2: if the date of the first deferred payment is given
For example : if payments are made quarterly and the first payment is made in 4 years, means the interval of deferment is 15 periods.
Formulas used:
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Example 1:
Find the present value of a deferred annuity of $500 a year for 10 years that is deferred 5 years. Money is worth 6%
Solution :
Substituting n=10, m= 5, R= 500, i=6%
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Alternative solution:
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Example 2:
Find the present value of an annuity of $50 every 3 months for 5 years if the first payment is made in 3 years. Money is worth 5% converted quarterly.
Solution :
Substituting n=5*4=20, m= (3*4)-1=11, R= $50, i=(0.05/4)%
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Alternative solution:
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Example 3:
a women inherits $20000. instead of taking the cash, she invests money at 8% converted quarterly with the understanding that she will receive 20 equal quarterly payments with the first payment to be made in 5 years. Find the size of the payments.
Solution :
Substituting n=20, m= (5*4)-1=19, R= ?, i=(0.08/4)=0.02
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Alternative solution:
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EXERSISE 5B:
1, 3, 5, 7,9
• FORBORNE ANNUITY
A forborne annuity earns interest for two or more periods after the last payment.
The amount of n periods followed by p periods calculated by using the following formulas :
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Example 1:
Payments of $1000 are made semiannually for 10 years to an account paying 7% compounded semiannually. Find the value of this annuity 5 years after the last payment.
Solution:
Substituting n=20, p= 10, R= 1000, i=(0.07/2)=0.035
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Alternative solution:
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Example 2:
How much must be invested each year at 8% effective with the first payment in 1991 and the last in 2000 in order to be worth $100000 in the year 2010?
Solution:
Substituting n=10, p= 10, R= ?, i=0.08
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Alternative solution:
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