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Simple & Compound Interest

Present Value

Ordinary Annuities

Present Value Annuities

General Annuities & Equivalent Rates

Mortgages

Review&Test

 

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 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.

Text Box: I  = interest due or earned
p = principal [amount borrowed                                
        or invested]
t  = time in years
r  = yearly interest rate as a decimal
A = amount repayable or accumulated
Text Box: Formulas:
 I = prt
A = p(1+ rt)
 

 

 


                                                                                                                       

 

 

 

 

 

 

 

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.

Text Box: P = principal [amount borrowed                                
       or invested]
n = number of interest periods
i  = interest rate per interest period as  
      as a decimal
A = accumulated amount (due or payable)
Text Box: Formula:
 A = P(1+ i)n
 

 

 


                                                                                                                       

 

 

 

 

 

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:

 

Text Box: Formulas:
 	
Text Box: PV = present value of a future amount                           
n = number of interest periods
i  = interest rate per interest period as  
      as a decimal
A = accumulated or future amount
 

 

 

 

 


                                                                                                                       

 

 

 

 

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.

Text Box: ·	The first payment is always at the end of the first interest period of the annuity.  This will be numbered 1 on your time line.
·	The accumulated sum of the future values of each payment at the end of the annuity’s term is called the amount or accumulated value of the annuity. This will be at the last number on your time line.
·	The future value of each payment is evaluated using the formula   A = P(1 + i)n
·	This accumulated amount forms a geometric series (see below).
·	A time line is very helpful in illustrating an annuity.
·	You can use either the geometric series formula OR the amount of an annuity formula (see below) to find the amount of an annuity.
 

 

 

 

 

 

 

 

 

 

Text Box: P = principal [amount borrowed                                
       or invested]
n = number of interest periods
i  = interest rate per interest period as  
      as a decimal
A = accumulated amount (due or payable)
Text Box: Formula:
 A = P(1+ i)n
 

 


                                                                                                                       

 

 

 

 

 

 

Text Box: a = the regular payment of the annuity 
n = number of payments or terms
Text Box: Geometric Series Formula:
 

 

 

 

 

 

 

 

 

Text Box: R = the regular payment of the annuity
n  = the number of payments or terms 
i   = interest rate per interest period
A = the accumulated amount of the annuity at the time of the last payment
Text Box: Amount of an Annuity Formula:
 

 

 

 

 

 

 

 

 

 


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.

Text Box: ·	The first payment is always at the end of the first interest period of the annuity.  This will be numbered 1 on your time line.
·	The accumulated sum of the present values of each payment at the end of the annuity’s term is called the present value of the annuity. This will be at the number 0 on your time line.
·	 
·	This accumulated sum forms a geometric series (see below).
·	A time line is very helpful in illustrating an annuity.
·	You can use either the geometric series formula OR the present value of an annuity formula (see below) to find the present value of an annuity.
 

 

 

 

 

 

 

 

 

 

 

 

 

 

Text Box: R = the regular payment of the annuity
n  = the number of payments or terms 
i   = interest rate per interest period
PV = the Present Value of the annuity at the time of the last payment
Text Box: Present Value of an Annuity Formula:
 

 

 

 

 

 

 

 

 


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.

 

           

 

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