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 HOMEWORK QUESTIONS

Homework Questions: (Solutions below)

Note: Answers to # 1 – 4 should be to the nearest hundredth.

1.  Find the nominal (annual) interest rate, compounded annually, equivalent to 7.6%/a, compounded semi-annually.

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

3.  Find the nominal (annual) interest rate, compounded monthly, equivalent to 4.2%/a, compounded semi-annually.

4.  Find the nominal (annual) interest rate, compounded quarterly, equivalent to 6%/a, compounded monthly.

5.  Evaluate each of the following general annuities.    Include a complete time line diagram for each.

a)  \$500 per month for 60 months at 8.4%/a, compounded annually.

b)  \$700 per year for 12 years at 6.8%/a, compounded quarterly.

c)  \$750 every 3 months for 12 years at 8.4%/a, compounded semi-annually.

d)  \$50 per week for 6 years at 7%/a, compounded quarterly.

Solutions:

1.  Find the nominal (annual) interest rate, compounded annually, equivalent to 7.6%/a, compounded semi-annually.

Solution:

Ie. We must find the annual rate that is equivalent to 7.6%/a, compounded semi-annually.

Step 1:  Using the formula  A = P(1 + i)n, find the value of \$1 invested at 7.6%/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

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

Solution:

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

Step 1:  Using the formula  A = P(1 + i)n, find the value of \$1 invested at 6.6%/a, quarterly 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

3.  Find the nominal (annual) interest rate, compounded monthly, equivalent to 4.2%/a, compounded semi-annually.

Solution:

Ie. We must find the annual rate, compounded monthly that is equivalent to 4.2%/a, compounded semi-annually.

Step 1:  Using the formula  A = P(1 + i)n, find the value of \$1 invested at 4.2%/a, 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

4.  Find the nominal (annual) interest rate, compounded quarterly, equivalent to 6%/a, compounded monthly.

Solution:

Ie. We must find the annual rate, compounded quarterly 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, monthly 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

5.  Evaluate each of the following general annuities.    Include a complete time line diagram for each.

a)  \$500 per month for 60 months at 8.4%/a, compounded annually.

Solution:

Here the payment interval( monthly ) 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 monthly rate that is equivalent to 8.4%/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(1.084)1 = 1.084

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

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

Interest per.    0           1         2                             .  .  .                                                                         58         59         60   Accumulated value

Payment                    500       500                                                                                                          500       500       500

500

500(1.006744132)1

500(1.006744132)2

.

.

.

500(1.006744132)58

500(1.006744132)59

Hence the amount of the annuity is \$36 827.58.

b)  \$700 per year for 12 years at 6.8%/a, compounded quarterly.

Solution:

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

We must match the interest period to the payment interval.

Ie. We must find the yearly rate that is equivalent to 6.8%/a, compounded quarterly.

Step 1:  Using the formula  A = P(1 + i)n, find the value of \$1 invested at 6.8%/a, compounded quarterly after 1 year.

Step 2:  Let the equivalent yearly 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 period0           1         2                            .  .  .                                                                         10         11         12   Accumulated value

Payment                    700       700                                                                                                          700       700       700

700

700(1.069753736)1

700(1.069753736)2

.

.

.

700(1.069753736)10

700(1.069753736)11

Hence the amount of the annuity is \$12 503.78.

c)  \$750 every 3 months for 12 years at 8.4%/a, compounded semi-annually.

Solution:

Here the payment interval( ¼ year ) 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 quarterly 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 quarterly 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 with (1 + i) = 1.011230201.

Interest per.    0           1         2                             .  .  .                                                                          46        47        48   Accumulated value

Payment                    750       750                                                                                                          750       750       750

750

750(1.011230201)1

750(1.011230201)2

.

.

.

750(1.011230201)46

750(1.011230201)47

Hence the amount of the annuity is \$47 365.62.

d)  \$50 per week for 6 years at 7%/a, compounded quarterly.

Solution:

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

We must match the interest period to the payment interval.

Ie. We must find the weekly rate that is equivalent to 7%/a, compounded quarterly.

Step 1:  Using the formula  A = P(1 + i)n, find the value of \$1 invested at 7%/a, compounded quarterly after 1 year.

Step 2:  Let the equivalent weekly rate be i %.  (Note the equivalent yearly rate would be 52i %.)

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

A = 1(1 + i)52                           ** n = 52, 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 with (1 + i) = 1.001335401.

Interest period0           1         2                            .  .  .                                                                        310       311      312   Accumulated value

Payment                      50        50                                                                                                           50         50         50

50

50(1.001335401)1

50(1.001335401)2

.

.

.

50(1.001335401)310

50(1.001335401)311

Hence the amount of the annuity is \$19 336.61.

6.  Rahim has just purchased a new car for 34 500.  He borrows the money and will repay it in monthly instalments over 6 years.  The interest rate is 7.8%/a, compounded quarterly.  Find the monthly payment.

Solution:

Here the payment interval( monthly ) is different than the interest period ( quarterly).  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 7.8%/a, compounded quarterly.

Step 1:  Using the formula  A = P(1 + i)n, find the value of \$1 invested at 7.8%/a, compounded quarterly 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                                                                                                                                         70    71     72

Payment                          R         R        R                                                                                                                                          R      R      R

R(1.006458202)-1

R(1.006458202)-2

.

.

R(1.006458202)-70

R(1.006458202)-71

R(1.006458202)-72

This forms the following geometric series:

R(1.006458202)-72 + R(1.006458202)-71 + . . . + R(1.006458202)-2 + R(1.006458202)-1