     Home Simple & Compound Interest Present Value Ordinary Annuities Present Value Annuities General Annuities & Equivalent Rates Mortgages Review&Test 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 ( ¼ year ) for 10 years if interest is 8%/a, compounded annually.

Solution:

Here the payment interval( ¼ 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 ¼ year rate be i %.  (Note the equivalent yearly rate would be 4i %.)

Now find the value of \$1 invested at i % per ¼ 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 ( ½  year ) for 12 years if interest is 6%/a, compounded monthly.

Solution:

Here the payment interval( ½  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 ½  year rate be i %.  (Note the equivalent yearly rate would be 2i %.)

Now find the value of \$1 invested at i % per ½  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 ( ¼ 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 ¼ year rate be i %.  (Note the equivalent yearly rate would be 4i %.)

Now find the value of \$1 invested at i % per ¼  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 