7.06 Investments and loans with financial solvers

Beyond using spreadsheets there are several options to explore compound interest investments and loans using technology. Most graphics and CAS calculators come with a built-in financial application which can be used to solve problems involving compound interest. These applications simply require you to enter in the known quantities (such as principal, interest, and number of compounding periods per year), and then the compound interest formula is applied or rearranged in the background to calculate the desired unknown quantity. These generic financial applications are known as Time-Value-Money solvers (TVM solvers) and there are many online versions which you can use to explore scenarios efficiently.

There are also many online calculators which offer to calculate values for specific problems such as loan repayment calculators or compound investment comparison calculators. 

The calculator applications and online TVM solvers typically use the following notation:

 

Remember that if you enter a positive value for $PV$PV then the solver will return a negative value for $FV$FV. This corresponds to borrowing: when you borrow you have a positive present value (the bank gives you money) but in the future you owe money to the bank, which is what the negative number represents. Conversely if you enter a negative number for $PV$PV then the solver returns a positive $FV$FV - this corresponds to investing.

• Borrowing money - the bank is giving you money - use positive value for $PV$PV
• Investing money - you are giving your money to the bank - use negative value for $PV$PV

Another difference is that the solvers are set up to deal with regular payments in addition to the accumulation of interest. If there is no payment then we set $PMT$PMT to $0$0.  We also set $P/Y=C/Y$P/Y=C/Y (the number of compounds per year) and then $N$N (number of payments of zero) is equal to the total number of compounding periods. 

You can use the following applet to explore the problems below:

Created with Geogebra

 

Worked example

Example 1

Suppose $\$10000$$10000 is invested in an account earning interest at $6%$6% p.a. compounded quarterly. How much is in the account after $3$3 years?

Note that since we are compounding quarterly the number of compounding periods per year is $4$4, and so we set both $C/Y$C/Y and $P/Y$P/Y equal to $4$4. Then since the amount is invested for $3$3 years we have $N$N equal to $4\times3=12.$4×3=12. When solving a problem using a financial solver you should always write down the values you are entering into the calculator and indicate which value you are solving for:

$N$N	$=$=	$4\times3=12$4×3=12	There are $4$4 quarters per year, and $3$3 years.
$I$I	$=$=	$6$6	Input as a percentage per annum.
$PV$PV	$=$=	$-10000$−10000	Negative because this is an investment  - we are giving the bank money.
$PMT$PMT	$=$=	$0$0	No payments are mentioned.
$FV$FV	$=$=	?	This is the value we are trying to find.
$P/Y$P/Y	$=$=	$4$4	Set equal to $C/Y$C/Y, so that $N$N is equal to the number of compounds.
$C/Y$C/Y	$=$=	$4$4	Compounding quarterly: there are $4$4 quarters per year.

 

Once you have entered all the known values you can tap or move the cursor (depending on your calculator) to the unknown and the calculated value will appear: $FV=11956.18$FV=11956.18

Finally, we should interpret the result: there will be $\$11956.18$$11956.18 in the account after $3$3 years.

 

The following demonstrate how the financial application appears in common calculator brands:

Casio Classpad

TI Nspire

 

Practice questions

Question 1

Question 2

Question 3

 

Reducing balance loans using a financial application

Often it's more convenient to analyse various situations for a reducible balance loan using an online financial solver or the financial facility of a CAS or graphics calculator. Remember that the present value will be the value of the loan and is entered as a positive value, as the lender is receiving money from the bank. The payments will be negative as these are paid to the bank.

 

Worked example

example 2

Audrey takes out a car loan for $\$24000$$24000. The finance company charge her $8.5%$8.5% interest compounded monthly.

How much should Audrey repay each month if she wants to repay this loan in $5$5 years?

Using the financial application on our calculator we enter the details as follows:

Compound Interest
N	$60$60	There are $12$12 months per year, and $5$5 years.
I%	$8.5$8.5	The annual interest rate
PV	$24000$24000	PV is positive as the bank is  giving Audrey money
PMT		This is the value we are trying to find.
FV	$0$0	The future balance will be zero.
P/Y	$12$12	$12$12 payments per year
C/Y	$12$12	Compounding monthly

Solving we get: PMT = $-492.40$−492.40

So Audrey should pay $\$492.40$$492.40 each month.

Note: PMT displays as a negative number since the payments will be made from Audrey to the financial institution.

 

Practice question

Question 4

 

What if number of payments $\ne$≠ number of compounding periods?  

In real life, banks usually calculate interest on loan accounts monthly but people can choose to make fortnightly or even weekly repayments.

When the number of payments is not equal to the number of compounding periods the financial solvers are great tool. 

The following demonstrate how the financial application appears in common calculator brands with an example involving a reduce balance loan with differing payment and compound periods, you can also use the following online TVM solver to explore these problems. 

Casio Classpad

TI Nspire

Practice question

Question 5