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Simple & Compound Interest

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Ordinary Annuities

Present Value Annuities

General Annuities & Equivalent Rates

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 UNIT 11 : MATHEMATICS OF INVESTMENT

 LESSON 5: GENERAL ANNUITIES AND EQUIVALENT RATES

 

Definition: A general annuity is an annuity where the payment intervals are not the same as the interest intervals.

 

Example 1:

Monthly payments of $500 where interest is 6%/a, compounded monthly. Here the payment interval and the interest interval are the same 1 month.

This is an example of an ordinary annuity like those in previous lessons.

 

Suppose there are monthly payments of $500, but the interest is 6%/a, compounded semi-annually. Here the payment interval is 1 month, but the interest period is 6 months. They are not the same. This type of annuity is called a general annuity.

 

Example 2:

Find the amount of an annuity of $400 every 3 months ( year ) for 10 years if interest is 8%/a, compounded annually.

 

Solution:

Here the payment interval( year ) is different than the interest period (annual). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the quarterly rate that is equivalent to 8%/a, compounded annually.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 8%/a, compounded annually after 1 year.

A = 1(1.08)1 = 1.08

Step 2: Let the equivalent year rate be i %. (Note the equivalent yearly rate would be 4i %.)

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)4 ** n = 4, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Now find the amount of the annuity using the annuity formula.

Interest per. 0 1 2 . . . 118 119 120 Accumulated value

Payment 400 400 400 400 400

 


400

 

400(1.019426547)1

 

400(1.019426547)2

 

.

 

.

 

.

 

400(1.019426547)118

 

400(1.019426547)119

 

Hence the amount of the annuity is $186 603.56.

 

 

Example 3:

Find the amount of an annuity of $700 every 6 months ( year ) for 12 years if interest is 6%/a, compounded monthly.

 

Solution:

Here the payment interval( year ) is different than the interest period (monthly). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the semi-annual rate that is equivalent to 6%/a, compounded monthly.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 6%/a, compounded monthly after 1 year.

Step 2: Let the equivalent year rate be i %. (Note the equivalent yearly rate would be 2i %.)

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)2 ** n = 2, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Now find the amount of the annuity using the annuity formula.

 

Interest per. 0 1 2 . . . 22 23 24 Accumulated value

Payment 700 700 700 700 700

 


700

 

700(1.030377509)1

 

700(1.030377509)2

 

.

 

.

 

.

 

700(1.030377509)22

 

700(1.030377509)23

 

Hence the amount of the annuity is $24 212.83.

 

Example 4:

Find the amount of an annuity of $2000 every year for 15 years if interest is 8%/a, compounded quarterly.

 

Solution:

Here the payment interval( 1 year ) is different than the interest period ( year). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the annual rate that is equivalent to 8%/a, compounded quarterly.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 8%/a, compounded quarterly after 1 year.

Step 2: Let the equivalent annual rate be i %.

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)1 ** n = 1, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Now find the amount of the annuity using the annuity formula.

 

Interest per. 0 1 2 . . . 13 14 15 Accumulated value

Payment 2000 2000 2000 2000 2000

 


2000

 

2000(1.08243216)1

 

2000(1.08243216)2

 

.

 

.

 

.

 

2000(1.08243216)13

 

2000(1.08243216)14

 

 

 

Hence the amount of the annuity is $55 343.22 .

 

Example 5:

Find the equivalent nominal (annual) interest rate, compounded annually for 6.6%/a, compounded semi-annually.

Solution:

Ie. We must find the annual rate that is equivalent to 6.6%/a, compounded semi-annually.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 6.6%/a, compounded semi-annually after 1 year.

Step 2: Let the equivalent annual rate be i %.

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)1 ** n = 1, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Example 6:

Find the equivalent nominal (annual) interest rate, compounded quarterly equivalent to 6.6%/a, compounded semi-aanually.

Solution:

Ie. We must find the annual rate, compounded quarterly that is equivalent to 6.6%/a, compounded semi-annually.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 6.6%/a, compounded semi-annually after 1 year.

Step 2: Let the equivalent year rate be i %. (Note the equivalent yearly rate would be 4i %.)

 

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)4 ** n = 4, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Example 7:

Jeffrey has just purchased a new car for 24 400. He borrows the money and will repay it in monthly instalments over 5 years. The interest rate is 8.4%/a, compounded semi-annually. Find the monthly payment.

 

Solution:

Here the payment interval( monthly ) is different than the interest period ( semi-annual). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the monthly rate that is equivalent to 8.4%/a, compounded semi-annually.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 8.4%/a, compounded semi-annually after 1 year.

Step 2: Let the equivalent monthly rate be i %. (Note the equivalent yearly rate would be 12i %.)

Now find the value of $1 invested at i % per month after 1 year.

A = 1(1 + i)12 ** n = 12, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

The money in question is borrowed now at point 0 on the time line. Hence this is a PV general annuity question

 

Interest Period 0 1 2 3 58 59 60

Payment R R R R R R

R(1.006880554)-1

R(1.006880554)-2

.

.

R(1.006880554)-58

R(1.006880554)-59

 

R(1.006880554)-60

 

 

This forms the following geometric series:

R(1.006880554)-60 + R(1.006880554)-59 + . . . + R(1.006880554)-2 + R(1.006880554)-1

 

 

 

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