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The Investor Aid Help

Using the Investor Aid Calculator

Compounding

Annuities

Mortgages and Loans


Compounding

Value at Maturity

Years

Interest


Annuities

Value at Maturity

Annuity Required

Years

Present Value


Mortgages and Loans

Size of Payments

Size of Mortgage/Loan

Number of Years

Ammortization Table


Value at Maturity

Formula:	V = P * (1 + i) ^ n

Where:

V = value at maturity

P = principal

i  = interest

n = total number of conversion periods

Example:

You invest $1000 on Jan 01, 1994 in a Government Savings Bond (GSB).  The current rate of interest is 6% compounded semiannually.  What will your GSB be worth on Jan 01, 1996.

Solution:

First determine the value of each term in the expression.

P=1000

i=0.06/2	{interest compounded semiannually}

n=2x2=4	{interest compounded semiannually provides (4) periods}

V = 1000 * (1 + 0.06 / 2) ^ 4

V = 1125.51

On Jan 01, 1996 your GSB will be worth $1125.51.


Years

Formula:

n = ln(V / P)/ln(1 + i)

Where:

n = total number of conversion periods

V = value at maturity

P = principal

i  = interest

Example:

How long will it take for $1,500 to accumulate to $2,146 if the interest rate is 12% compounded monthly.

Solution:

First determine the value of each term in the expression.

V=$2,146

P=$1,500

i=0.12/12 = 0.01	interest is calculated monthly

Substituting into our formula we have

n = ln(2,146 / 1,500) / ln(1.01)

n = 35.99 or 36

Since interest rate is compounded monthly we perform the final calculation to convert the above result to years.

Years = 36/12

Years	= 3

It will take 3 years for $1,500 to accumulate to $2,146 if the interest rate is 12% compounded monthly.


Interest

Formula:

i = (V / P) ^ (1/n) - 1

Where:

i  = interest

V =  value a maturity

P = principal

n = total number of conversion periods

Example:

At what interest rate, compounded monthly for 6.5 years, will $1,500 accumulate to an amount of $2,500 ?

Solution:

First determine the value of each term in the expression.

V=$2,500

P=$1,500

n=6.5x12=78

Substituting into our formula we have:

i = (2,500 / 1,500) ^ (1/78) -1

i = (1.006571)-1

i = 0.006571

The interest rate per month is 0.006571.  To find the rate per year we multiply this monthly rate by 12, which yields 0.078852.  Rounding this result gives 7.86%.


Value at Maturity {Annuity}

Formula:

V = [A(1+f)^n - 1] / f

Where:

V =  value a maturity

A = annuity amount

f  = effective rate of interest

n = total number of conversion periods

Formula for effective rate of interest is:

f=(1+i)^c - 1

Where:

i = rate of interest per conversion period

c=(number of interest conversion periods per year)

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(number of payment periods per year)

Example:

Find the value at maturity of an annuity that requires payments of $1,000 at the end of each month for six years, it the interest rate is 18% compounded semiannually.

Solution:

First determine the value of each term in the expression.

A=$1,000

i  = 0.18/2 = 0.09

n = 12x6 = 72

c = 2/12

The effective rate of interest is

f = (1 + 0.09) ^ (2/12) - 1

f = 0.0144665

Substituting into our annuity formula we have.

V = 1,000*[(1 + 0.0144665) ^ 72 - 1] / 0.0144665

V = 125,300.05

The value at maturity of an annuity that requires payments of $1,000 at the end of each month for six years, if the interest rate is 18% compounded semiannually is $125,300.05


Annuity Required

Formula:

AR = VM / [(1+f)^n - 1] / f

Where:

AR = annuity required

VM = value at maturity

f  = effective rate of interest

n = total number of conversion periods

Formula for effective rate of interest is:

f=(1+i)^c - 1

Where:

i = rate of interest per conversion period

c=(number of interest conversions periods per year)

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(number of payment periods per year)

Example:

You have just decided to invest in a fund that will provide $20,000 for your childs education.  You expect to contribute to the fund for 15 years.  What will be the required payment at the end of every 2-month period if the interest rate paid by the fund is 10% compounded annually ?

Solution:

First determine the value of each term in the expression.

VM = $20,000

i  = 0.10

n = 15x6 = 90

c = 1/6

The effective rate of interest is

f = (1 + 0.10)^(1/6) - 1

f = 0.0160118

Substituting this result into our formula we have

AR = 20,000 / [(1+ 0.0160118)^90 - 1] / 0.0160118

AR = 100.79

If payments of $100.79 are made at the end of every 2 months for the next 15 years at an interest rate of 10% compounded annually, there will be $20,000 in the fund.


Years

Formula:

n = ln(1 + VM*f/A) / ln(1 + f)

Where:

n = total number of conversion periods

VM = value at maturity

A = annuity amount

f  = effective rate of interest

The formula for effective rate of interest is:

f=(1+i)^c - 1

Where:

i = rate of interest per conversion period

c=(number of interest conversions periods per year)

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(number of payment periods per year)

Example:

Youve started depositing money into a retirement fund.  Once you have $100,000 accumulated you plan to retire.  You will deposit $500 at the end of every 3 months to a RRSP that pays interest at 8% compounded annually.  When can you retire ?

Solution:

First determine the value of each term in the expression.

VM = $100,000

A = $500

i  = 0.08

c = 1/4	(Deposit every 3 months results to 4 times a year)

The effective rate of interest is:

f = (1 + 0.08)^(1/4) -1

f = 0.01942655

Substituting this result into our formula we have:

n = ln(1 + 100,000*0.01942655/500) / ln(1 + 0.01942655)

n = 82.4434

Since you are contributing 4 times a year then we divide the number of payments by 4, yielding 20.61 years.

Therefore if you  deposit $500 at the end of every 3 months to a RRSP that pays interest at 8% compounded annually until youve accumulated $100,000 you will retire in 20.61 years.


Present Value

Formula:

PV = A* [1-(1+f)^-n] / f

Where:

PV = present value of an annuity

A = annuity amount

f  = effective rate of interest

n = total number of conversion periods

The formula for effective rate of interest is:

f=(1+i)^c - 1

Where:

i = rate of interest per conversion period

c=(number of interest conversions periods per year)

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(number of payment periods per year)

Example:

A retirement annuity is offered to you which will provide a monthly income of $1,000 for the next (10) years.  The bank informs you that the current interest rate is 6% compounded semiannually, and that for your retirement fund they will  maintain this rate for the next (10) years.  How much should you pay for this annuity.

Solution:

First determine the value of each term in the expression.

A=$1,000

i  = 0.06/2 = 0.03	{interest is compounded semiannually}

n = 10x12 = 120	{monthly income for 10 years gives 120 periods}

c = 2/12 = 1/6

The effective rate of interest is

f = (1 + 0.03)^(1/6) - 1

f = 0.0049386

Substituting the effective interest rate into the PRESENT VALUE formula we have:

PV = 1,000 [1 - (1+0.0049386)^-120] / 0.0049386

PV = 90,374.25

You should pay no more than $90,374.25 for an annuity of $1,000 per month for ten years if the current interest rate is 6% compounded semiannually, and will remain so for the next (10) years.


Size of Payments

Formula:

P = VM / [1 - (1+f)^-n] / f

Where:

P = payment required

VM = value of mortgage or loan

f  = effective rate of interest

n = total number of conversion periods

Formula for effective rate of interest is:

f=(1+i)^c - 1

Where:

i = rate of interest per conversion period

c=(number of interest conversions periods per year)

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(number of payment periods per year)

Example:

You have just bought a home at a cost of  $100,000.  Your payments will be made monthly over 25 years at an interest rate of 11% compounded semiannually.  What will be the amount of your monthly payments ?

Solution:

First determine the value of each term in the expression.

VM = $100,000

i  = 0.11 / 2 = 0.055

n = 25x12 = 300

c = 2/12 = 1/6

The effective rate of interest is

f = (1 + 0.055) ^ (1/6) - 1

f = 0.0089633

We can now calculate the monthly payments.

P = 100,000 / [1- (1+0.0089633) ^  - 300] / 0.0089633

P =  962.53

For a $100,000 home and monthly payments over a 25 year period, at an interest rate of 11% compounded semiannually, the monthly mortgage payments will be $962.53.


Size of Mortgage/Loan

Formula:

M = P / [1 - (1+f) ^ -n] / f

Where:

M = size of mortgage

P = amount of payments

f  = effective rate of interest

n = total number of conversion periods

Formula for effective rate of interest is:

f=(1+i)^c - 1

Where:

i = rate of interest per conversion period

c=(number of interest conversions periods per year)

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(number of payment periods per year)

Example:

After analyzing your financial situation you conclude that the maximum monthly mortage payment you can afford is $750.00 for a 25 year mortgage.  If current interest rates are 8.75% compounded semiannually, what size of mortgage or loan can you afford.

Solution:

First determine the value of each term in the expression.

P = 750.00

i  = 0.0875 / 2 = 0.04375

n = 25x12 = 300

c = 2/12 = 1/6

The effective rate of interest is

f = (1 + 0.04375) ^ (1/6) - 1

f = 0.0071621

Substituting the effective rate of interest into our formula:

M = 750[1 - (1 + 0.0071621) ^ -300] / 0.0071621

M = 92,409.30

With a monthly payment of $750.00 of a 25 year mortgage, at an interest rate of 8.75% compounded semiannually, the size of your mortgage is $92,409.30.


Number of Years

Formula:

n = -ln[1 - M*f / P] / ln[1 + f]

Where:

n = number of payments

M = size of mortgage

P = amount of payments

f  = effective rate of interest

Formula for effective rate of interest is:

f = (1+i)^c - 1

Where:

i = rate of interest per conversion period

c=(number of interest conversions periods per year)

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(number of payment periods per year)

Example:

At a meeting with your banker you are advised that the monthly payment on a $100,000 mortgage is $962.53, based on payments over 25 years and an interest rate of 11% compounded semiannually.  Instead, you decide to make monthly payments of $1,050.  If you continue to make $1,050 payments over the life of the mortgage, how long will it take you to repay it ?

Solution:

First determine the value of each term in the expression.

M = $100,000

P = $1,050

i  = 0.11 / 2 = 0.055

n = 25x12 = 300

c = 2/12 = 1/6

The effective rate of interest is

f = (1 + 0.055)^(1/6) - 1

f = 0.0089633

Substituting the effective rate of interest into our formula we have:

n = -ln[1 - 100,000*0.0089633/1,050] / ln[1 + 0.0089633]

n = 215.37

Since there are 12 payments per year:

Number of Years 	= 215.37 / 12

= 17.95 years

Therefore if you make monthly payments of $1,050 instead of $962.35, you will pay off your mortgage in 17.95 years instead of 25 years.


Ammortization Table

This command displays a table showing the regular payments one must make on a specific loan or mortgage.  Along with the payments, the portions of the payment consisting of principal and interest are also displayed.  As you go down the ammortization table youll notice that the principal amount increases and the interest amount decreases.

The procedure to calculate an ammortization table is as follows.

Given the details of a mortgage settlement, calculate the regular Mortgage Payments of the loan.

Then for each row in the Amortization Table calculate the principal and interest amounts of the payment.

The formulas are as follows:

Principle Portion:

Principal = Mortgage Payment / (1+UnitInterest) ^ (TotPeriods-I+1)

Where:

UnitInterest is the interest per unit of time.  For a mortgage this is usually on a monthly basis.

TotPeriods is the total number of periods of the mortgage.  For a 25 year mortgage with monthly payments the TotPeriods is (25x12)=300.

I is the current unit of time, which is an integer value and varies from (1 to TotPeriods)

Interest Portion:

Interest = Mortgage Payment - Principal


Using the Investor Aid Calculator

Using the Compound, Annuity, and Mortgage Data Grids

Manoeuvering in the Amortization Table

Copying Data to Other Applications


Using the Compound, Annuity, and Mortgage Data Grids

The following keys are used in manipulating data in the data grid.

<Numeric keys 0-9>

Enters numeric data into the selected cell.  The actual editing is performed in the edit box located above the data grid.

<Enter key>

If the focus is on the edit box, then the contents of the edit box are transferred to the selected grid cell, and the focus is transferred to the adjacent right cell.

If the grid has the focus, the contents of the cell above the selected cell are copied to the selected cell.  The focus is then transferred to the adjacent right cell.

<F2 key>

If the selected cell is numeric then its contents are copied to the edit box, and the focus is transferred to the edit box.  Editing can then continue.

<Delete key>

Deletes the contents of the currently selected cell.

<Right, left, down, up arrow keys>

Selects an adjacent cell in accordance to the arrow key pressed.


Manoeuvering in the Amortization Table

<Page Down>

Scrolls the grid (1) page forward.

<Page Up>

Scrolls the grid (1) page back.

<Ctrl Home>

Scrolls to the first page in the grid.

<Ctrl End>

Scrolls to the last page in the grid.

<Home>

Puts the focus on the left most cell in the current row.

<End>

Puts the focus on the right most cell in the current row.


Copying Data to Other Applications

The Investor Aid Calculator can copy data to other Windows applications like Microsoft Excel and Word for Windows.  The mechanism for doing so is the Windows clipboard.  Once data is inserted into the clipboard it can be pasted by the receiving application.  To copy grid data to the clipboard following the procedure described below.

Using the mouse, click the top left corner of the selection area and drag the mouse to the bottom right corner of the selection area.

Note: To copy the entire grid, click the cell where row 0 and col 0 intersect.

Open the Grid Menu and Click the Copy to Clipboard command.

Shortcut: Instead of opening the Grid Menu press <CTRL C> on the keyboard.

To paste clipboard data to the receiving application follow the procedure described in the receiving applications manual.