Finance - calculator free
In most cases, questions involving finance require the use of a basic calculator or a CAS calculator with sequence or finance apps. However, there are some situations where we may be asked to demonstrate an understanding of concepts used for finance questions, without the assistance of any calculator. Examples of questions that do not require a calculator can be seen below.
Note that the Year 12 Applications 2019 formula sheet contains the following formulae that may be useful.
Worked examples
Example 1 - effective interest rates
Ivan was keen to compare interest rates offered by different banks, so he decided to construct a table showing the effective annual rates of interest (%). Part of his table is shown below.
| Compounding Period | 4% | 5% | 6% | 7% |
|---|---|---|---|---|
| Quarterly | $4.060$4.060 |
$5.095$5.095 |
$6.136$6.136 |
$7.186$7.186 |
| Monthly | $4.074$4.074 |
$5.116$5.116 |
$6.168$6.168 |
$7.229$7.229 |
| Daily | $4.081$4.081 |
$5.127$5.127 |
$6.183$6.183 |
$7.250$7.250 |
Ivan wants to borrow $$8000$8000 to purchase a new TV. A bank offers to lend him the money at the rate of $5$5% p.a. for one year. He plans to pay off the entire loan (including the interest) at the end of the year.
(a) Which compounding period should he sign up for?
Think: Looking at the numbers in the $5$5% p.a. column, the highest effective rate is that of daily.
Do: Write 'He should sign up for daily compounding'.
(b) What is the reason for your choice in part (a)?
Think: Daily compounding means every day the interest is added to the balance.
Do: Write 'If the interest is added daily the compounding is more effective and the amount will grow at a faster rate'.
(c) Determine the interest Ivan would earn by investing $$1000$1000 for a year, earning $4$4% p.a. with interest compounded quarterly.
Think: Look in the table to find the effective rate is $4.060$4.060% p.a.
Do: $1000\times0.04060=\$40.60$1000×0.04060=$40.60
(d) Ivan's sister has $$4000$4000 to invest for a year. She has been offered a rate of $5$5% p.a., with interest compounded quarterly. Determine the value of her investment at the end of the year.
Think: Use the table to find the effective interest rate is $5.095$5.095%
Do: Find $5.095$5.095% of $4000$4000. Note that without a calculator it's easier to find $5.095$5.095% of $1000$1000 and then multiply by $4$4.
| $\text{Value }$Value | $=$= | $4000\times0.05095$4000×0.05095 |
| $=$= | $=4\times1000\times0.05095$=4×1000×0.05095 | |
| $=$= | $=\$203.80$=$203.80 |
Example 2 - Interpreting Graphs
Will and Grace both invest $500. Will’s investment compounds annually at 12.5% p.a. and Grace’s investment pays simple interest at the rate of 20% p.a. The graph below shows the balance, $A$A at the start of each year, $n$n.
(a) Which graph (blue or red) represents Will's investment?
Think: Simple interest is a straight line graph where the constant gradient represents the increase each year. Will's investment is compound interest which means exponential growth and a curved shape.
Do: Write 'Will's investment is the blue graph'.
(b) Use the graph to determine the year in which Will's investment balance exceeds Grace's balance.
Think: Look at the graph and determine where the two graphs intersect.
Do: Write 'Will's investment first exceeds Grace's investment in $2028$2028.
Example 3 - Interpret a sequence
Heidi borrows money for a holiday to Croatia. The monthly balance ($A$A) for the loan after $n+1$n+1 months, can be modelled by the recursive rule
$A_{n+1}=1.0045\times A_n-150$An+1=1.0045×An−150 ; $A_0=5400$A0=5400
(a) How much did Heidi borrow?
Think: The amount borrowed is the starting amount. In this case $A_0$A0
Do: Write 'She borrowed $$5400$5400'.
(b) What is the annual interest rate?
Think: Each month the amount is being multiplied by $1.0045$1.0045. This is an increase of $0.0045$0.0045 or $0.45$0.45% per month. Multiply this by twelve to find the annual rate.
Do: $12\times0.45=5.4$12×0.45=5.4% p.a.
(c) What monthly payment is Heidi making to pay off the loan?
Think: Each month there is $-150$−150 in the rule.
Do: Write 'She is paying off $$150$150 per month'.
Example 4 - Express as sequence
To save money for a car, Jess started a savings account with her bank. She made an initial deposit of $$200$200, and makes a regular deposit, at the start of each month for the next month.
The table below shows the progress of her savings for the first few months.
| Month(n) | Balance at start of month($) | Interest for month($) | Balance at end of month ($) |
|---|---|---|---|
| 1 | $200$200 | $5.00$5.00 | $205$205 |
| 2 | $305$305 | $7.63$7.63 | $312.63$312.63 |
| 3 | $412.63$412.63 | $10.32$10.32 | $422.95$422.95 |
(a) Calculate the annual interest rate.
Think: Calculate the monthly interest rate using the simplest numbers in the table which are found in row 1. Multiply the monthly rate by twelve to find the annual rate.
Do: $\frac{5}{200}\times100=2.5$5200×100=2.5% per month
which is $2.5\times12=30$2.5×12=30 % p.a.
(b) How much is Jess depositing each month?
Think: The difference between the balance at the end of one month and the start of the next month is $$100$100.
Do: Write ' She is depositing $$100$100'.
(c) Write a recursive rule to determine the balance at the start of each month.
Think: The pattern for each month is increase by $2.5$2.5% and then add $$100$100. Increase by $2.5$2.5% is the same as multiplying by $1.025$1.025. Remember to include a starting value too.
Do: Write $B_{n+1}=1.025\times B_n+100$Bn+1=1.025×Bn+100 ; $B_0=200$B0=200 or $B_1=305$B1=305
Alternatively: $B_n=1.025\times B_{n-1}+100$Bn=1.025×Bn−1+100 ; $B_0=200$B0=200 or $B_1=305$B1=305
example 5 - interpreting graphs
Match the graph with the best description of the investment.
Graphs:
| Graph letter | Graph | Graph letter | Graph |
|---|---|---|---|
| A |  |
B |  |
| C |  |
D |  |
| E |  |
F |  |
Descriptions:
| Number | Type of financial situation | Graph letter |
|---|---|---|
| 1 | Savings account earning $36%$36% p.a. interest compounded monthly | ? |
| 2 | Savings account earning $24%$24% p.a. interest compounded monthly | ? |
| 3 | Annuity earning $5%$5% p.a. compounded monthly and with a monthly payment of $\$4000$$4000 | ? |
| 4 | Investment account earning $4.5%$4.5% p.a simple interest | ? |
| 5 | Perpetuity earning $4.2%$4.2% p.a. compounded weekly | ? |
| 6 | Investment account earning $5.5%$5.5% p.a simple interest | ? |
Think:
- A savings account will grow so the graph will be exponentially increasing and $36%$36% p.a. compounded monthly will grow faster than $24%$24% p.a compounded annually so will be steeper.
- An annuity with a payment has a decreasing balance so it is the graph with a downwards slope.
- A perpetuity has a constant balance so it will be the flat line graph.
- An investment with simple interest increases by the same amount each time so it will be a straight line graph. $5.5%$5.5% p.a is higher than $4.5%$4.5% p.a so the gradient will be steeper on the $5.5%$5.5% graph.
Do: Write 'The answers are:'
| Number | Type of financial situation | Graph letter |
|---|---|---|
| 1 | Savings account earning $36%$36% p.a. interest compounded monthly | B |
| 2 | Savings account earning $24%$24% p.a. interest compounded monthly | A |
| 3 | Annuity earning $5%$5% p.a. compounded monthly and with a monthly payment of $\$4000$$4000 | F |
| 4 | Investment account earning $4.5%$4.5% p.a simple interest | C |
| 5 | Perpetuity earning $4.2%$4.2% p.a. compounded weekly | D |
| 6 | Investment account earning $5.5%$5.5% p.a simple interest | E |
Finance - calculator assumed
Most finance questions require the use of a calculator. Your CAS calculator has the sequence facility, or the financial application that can be used to answer these types of questions. Try the questions below to practice using both of these tools.
Practice questions
Question 1
Paul has $15$15 years until retirement and wishes to have a savings of $\$165000$$165000 in that time by making regular monthly payments into an account paying $4.5%$4.5% compounded daily. Complete the table showing the settings required for a financial solver and hence determine the monthly payment for Paul to reach the required amount if he currently has $\$2500$$2500 in the account.
Assume there are $365$365 days in a year.
Round the monthly payment to the nearest cent.
Variable Value $N$N $\editable{}$ $I\left(%\right)$I(%) $\editable{}$$%$% $PV$PV $\editable{}$ $Pmt$Pmt - $FV$FV $\editable{}$ $P/Y$P/Y $\editable{}$ $C/Y$C/Y $\editable{}$ Monthly payment$=\$$=$$\editable{}$
Question 2
Frank wins the lottery and decides to deposit the winnings in a high interest savings account. He has the following two choices.
Option $A$A: $5.25%$5.25% p.a. compounded monthly
Option $B$B: $5.3%$5.3% p.a. compounded weekly
Determine the effective annual interest rate for option $A$A as a percentage.
Round your answer to three decimal places.
Determine the effective annual interest rate for option $B$B as a percentage.
Round your answer to three decimal places.
Assume there are $52$52 weeks in a year.
Another bank offers option $C$C with an interest rate of $5.35%$5.35% p.a.
If the effective rate is also $5.35%$5.35%, how many times a year must the interest be compounded?
Weekly
AMonthly
BQuarterlyCDaily
DYearly
EWhich option should Frank choose?
Option $A$A
AOption $B$B
BOption $C$C
C
Question 3
Elizabeth is given $\$3500$$3500 as a $21$21st birthday present and decides to invest the money in an account where interest is compounded quarterly. She decides to also make a $\$75$$75 per quarter deposit into the account.
The table shows the balance of Elizabeth’s account over the first $5$5 quarters.
| Quarter | Balance at start of quarter ($\$$$) | Interest ($\$$$) | Deposit ($\$$$) | Balance at end of quarter ($\$$$) |
|---|---|---|---|---|
| $1$1 | $3500.00$3500.00 | $119.00$119.00 | $75$75 | $3694.00$3694.00 |
| $2$2 | $3694.00$3694.00 | $125.60$125.60 | $75$75 | $3894.60$3894.60 |
| $3$3 | $3894.60$3894.60 | $132.42$132.42 | $75$75 | $4102.02$4102.02 |
| $4$4 | $4102.02$4102.02 | $139.47$139.47 | $75$75 | $4316.49$4316.49 |
| $5$5 | $4316.49$4316.49 | $146.76$146.76 | $75$75 | $4538.25$4538.25 |
Calculate the annual interest rate for Elizabeth’s account as a percentage.
Round your answer to one decimal place.
Complete the recursive rule, where $P_n$Pn describes Elizabeth's balance after the $n$nth quarter.
$P_n=$Pn=$\editable{}$$P_{n-1}+$Pn−1+$\editable{}$, $P_1=$P1=$\editable{}$
After four quarters, Elizabeth realises she is not going to make her savings goal which is to have $\$8000$$8000 at then end of three years.
By only depositing $\$75$$75 per quarter, by how much is Elizabeth short?
Round your answer to the nearest cent.
What deposit does Elizabeth need to make each quarter to reach her goal of $\$8000$$8000 at the end of three years?
Round your answer to the nearest cent.