Home Simple & Compound Interest Present Value Ordinary Annuities Present Value Annuities General Annuities & Equivalent Rates Mortgages Review&Test

UNIT 11  :  MATHEMATICS OF INVESTMENT

LESSON 7: UNIT SUMMARY AND TEST

Simple Interest:

When you borrow or invest money, interest is paid or earned.  If the interest is calculated only on the money originally invested, it is called simple interest.

Compound Interest:

If interest is calculated at the end of each year (or interest period) and added on at this point, then this is called compound interest.

Time Value of Money:

The investment above can be illustrated on a time line.  Time = 0 means today.  The arrow moving to the right shows the investment increasing over the 3 year period.

0                              1                                  2                                      3

\$5000                                                                                                              \$5000(1.05)3

Note:  We can evaluate a give sum of money at any point using a time line simply using the compound interest formula, the correct interest rate and the correct number of time periods.

Example : Using different compounding periods.

a)  Shelby invested \$8 000 in a 5-year term deposit which pays interest at a rate of 5 %/a [per annum], compounded semi-annually.  What will the investment be worth at the end of the 5 year period?

Solution:

Because the interest is compounded every 6 months, we must adjust both the number of interest periods n and the interest rate i.

If interest is paid twice a year, then the number of interest periods [compounding periods] will be 5 x 2 = 10.  Hence n = 10.

The appropriate time line is shown below.  Since there are 10 interest periods, we will put a mark every 6 months over the 5-year period.

0                1                2                       .   .   .  .   .                                        4              5

\$8000                                                                                                                                        \$8000(1.025)10

Present Value:

Example:

Mr. And Mrs. Trinh would like to have \$500 000 available when they retire in 20 years.  How much should they invest now if interest is 6%/a, compounded semi-annually?

Solution:

The appropriate time line is shown below.  Since there are 40 interest periods, we will put a mark every 6 months over the 20-year period.

0       1       2       3       4                       .     .    .                                                        38   39  40

500000(1.03)-40                                                                                                                                                                                                      500000

They should invest \$153 278.42 today to achieve their goal.

Ordinary Annuities:

Definition: A sequence of payments made at regular intervals is called an annuity.

Interest Period   0       1       2       3                                                                                                      18   19  20

Payment                       200       200       200                                                                                                                                      200   200  200

An ordinary annuity has the following properties.

Example:

For the past 5 years Amane has been depositing \$100 every month into an investment account .  If the interest rate is 5.4%/a, compounded monthly, how much has she accumulated at the time of her last deposit?  Include a time line diagram in your solution.

Solution:

\$100 at the end of every month for 5 years with interest at 4.25%/a, compounded monthly.

Interest period0       1       2                     .  .  .                                                    238    239   240   Accumulated value

Payment                    100       100                                                                                                          100       100       100

100

100(1.0045)1

100(1.0045)2

.

.

.

100(1.0045)59

100(1.0045)59

Example:

Find the annual payment for an annuity of 10 years duration at a rate of 5.6%/a, compounded annually, that will amount to \$10 000 at the time of the last payment.

Solution:

Let the yearly payment be \$R, with the first payment at the end of the first year; i = 0.056 and 1 + i = 1.056

Interest period0       1       2                     .  .  .                                                     8        9       10               Accumulated value

Payment                       R         R                                                                                                            R           R          R

R

R(1.056)1

R(1.056)2

.

.

.

R(1.056)8

R(1.056)9

Present Value of an  Annuity:

Definition 1: A sequence of payments made at regular intervals is called an annuity.

Definition 2: When we calculate the present values of the sequence of payments made at regular intervals this is called the Present Value of the annuity.

A Present Value annuity has the following properties.

Example 1: Finding the Present Value of an annuity.

Heywood recently won \$5 000 000 in the lottery.  He plans to purchase an annuity that will pay him \$50 000 every year for 25 years and spend the rest.  How much of his winnings would he need to pay today for that annuity  if  interest is 7.6%/a, compounded annually?  Include the series and a time line diagram in the solution.

Solution:

We calculate the present values of the 25 future payments of \$50 000 each.  Notice the arrows go to the left for a present value annuity.

Interest Period   0          1         2          3                                                                                                                                         23   24    25

Payment( 1000’s)           50       50                                                                                                                                                   50    50    50

50000(1.076)-1

50000(1.076)-2

.

.

50000(1.076)-23

50000(1.076)-24

50000(1.076)-25

This forms the following geometric series:  Note – write the last term first.

50 000(1.076)-25 + 50000(1.076)-24 +  . . . + 50000(1.076)-2 + 50000(1.076)-1

Hence Michael needs \$552 492.55 of his winnings to purchase the annuity. He would have \$5 000 000 - \$ 552 492.55 = \$4 447 507.45 left to spend.

Alternate Solution:

Notice this present value amount is much less than the total amount paid out over 25 years which would be 25 x \$50 000=\$1 250 000.

General Annuities:

Definition:   An annuity where the payment intervals are not the same as the interest intervals.

Example:

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.

Mortgages:

Example 1:

a)  Camille has just purchased a new house near Brantford.   She needs a mortgage of \$150 000 after her down payment..  She will repay it in monthly instalments over 25 years.  The interest rate is 6.6%/a, compounded semi-annually.  Find the monthly payment.

b)  Determine the total interest paid over the 25 year period.

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 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 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(1.005425865)-1

R(1.005425865)-2

.

.

R(1.005425865)-298

R(1.005425865)-299

R(1.005425865)-300

This forms the following geometric series:

R(1.005425865)-300 + R(1.005425865)-59 + . . . + R(1.005425865)-2 + R(1.005425865)-1

b)  Determine the total interest paid over the 25 year period.

Total amount repaid = 1013.85 x 300 = \$304 095.00

Mortgage amount                               = \$150 000

Interest paid = \$304 095 - !50 000    =\$154 095

Hence The total interest paid over 25 years is \$154 095.