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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