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name: Investigation
filter: "search: Investigation 2"

Context §

The Task §

You are required to investigate the process of purchasing a car for personal use, and selling it after 5 years for an appropriate amount that you decide. You are approved for a loan of $30,000, and you should assume you have an appropriate income for a school/university student.

To complete this investigation, you must: 1. “Purchase” a car by taking a loan of $30,000, showing research and information on all initial complementary expenses (insurance, tax etc) 2. Use your understanding of the ATMAA Financial maths topic in your report. All mathematics must be shown, using screenshots of your classpad or showing working out. Online calculators will not be sufficient. 3. Present your findings in a written report and include all steps of the Mathematical Investigation Process. This also includes discussion and justification of the decisions and assumptions you made.

Instructions §

Instructions:

• This investigation consists of a Take Home section only. There will be no validation test.
• You will be required to undertake a statistical investigation using the mathematical investigation process and present your findings in a report.
• You may discuss your investigation with others, but your data, analysis and report must be your own work.
• You may use the Internet as an additional resource.
• Ask your teacher for guidance on the Mathematical Investigation Process if necessary.
• Your report is due on Wednesday 23 August 2023 and must be submitted via Turnitin on Seqta.

Working §

Introduction §

In this investigation, I will investigate the process of purchasing a car from the context of the average School/University Student, using it for 5 years, then selling it. Context includes;

• Approval for a loan up to $30,000.
• Income slightly above Minimum wage. ($24.27 per hour)
• Operates under WA Laws & Regulations.
• Am to-be enrolled in a Full-time University exactly 1 year into the loan.
  • Thus, reduction in Income as there is less time to work. I will investigate how a Uni student can keep up with the loan, taking into account all complementary expenses (Insurance, tax, Fuel Costs, Living Costs etc.), Comparing Mutliple different loans in the context of different vehicles in an attempt to get the best deal out of the options available to me. `

Income §

I will calculate income weekly, this allows me to include any benefits that re-occur on a weekly basis, that primarily being Penalty Payments, for hours worked on Saturday, I get 150% of my base Income, and on Sundays I earn 200% of my base income. I will assume that I am taking a gap year for the first year, following that hours will be restricted to the weekend, and 4 hours on fridays.

Year 1 - Hours §

Year 2, 3, 4, & 5 Hours §

Modelling Income Mathematically §

I can model my income with the following equation: Where; = Hourly Salary = Hours worked at that rate = Pay-rate (e.g. 150% > 1.5) W = Weekly Income I will also be taking into account the national Wage Growth, according to the Australian Bureau of Statistics, wage growth is currently averaging 3.6% Per Year , thus we can expect, that on average, my wage will increase by 3.6% year on year.

Since after the first year I am drastically decreasing my hours as I would move into University, I will calculate this 3.6% based on my current hourly wage. After the first year, I can model it as a geometric sequence using the weekly Wage, simply assuming hours remain constant from there on.

We now calculate the Hourly wage, after growth, after a year; 25.14}=25.143 = 24.27 \times 1.036653.64}=(25.14 \times 4) + (25.14 \times 1.5 \times 4)+(25.14 \times 2 \times 8)W_{y+1}=W_y \times 1.036, \quad W_2=653.64$$ Where . I provided because you can not use this sequence from since that year has another factor influencing it. In the classpad I simply used , where , and so on.

Classpad Working out

Representing Wage Growth Explicitly

Here is the same thing representated as an Explicit Geomtric Sequence: where , and .

Yearly Income After Tax §

I now define a equation to model my Post-Tax Income; Then;

I won’t utilise this equation until the end, throughout the rest of the report I will keep a number of weekly expenses, that I will then add on at the end.

Cost of Living §

Now we must adjust these values for the cost of living, I utilised the government provided living cost estimator, excluding the accomodation option as the average Uni student is living with their parents, or has accomodation provided for them by their parents. Thus;

Weekly Expenses: 220

I also excluded Transport as that is a factor dependant on fuel economy of the vehicle, something that will be discussed in detail later.

Archive

Data Collection - Cars & Banks §

Plan §

The Plan is to collect all the relevant data into seperate notes, this allows me to include bank specific information alongside each one. I will include each relevant offer in a sperate Entry, e.g.

• Beyond Bank (Low Rate Loan)
• Beyond Bank (Special Offer)

This allows me to seperate out the requirements for each loan, for example the special offer is only relevant for cars less than two years old, where as the Low Rate Loan, wihle a higher rate, can have any car sold within the last 8 years as a security, or no security at all.

One advantage of this approach is that once I have decided on my car, I can run calculations for all loans, determien which ones are theoretically the best, then eliminate the ones that choice of car or I, am inelligble for, With a wide variety of Cars & Loans I can determine the absolute best choices for both.

Data §

Data in these tables are links to other notes, these links do not work. However some relevant metadata from them is shown in the table

Collected Banks Data §

let pages = dv.pages('"Maths/Unit 3/Finance/Banks"')
dv.table(
["Bank", "Fixed Compar.", "Max Age"],
pages.map(b => [
b.file.link,
`**${b.FixedComparison}**`,
b.MaxFixedAge
])
)

Note: All Banks compound daily.

NAB is different

The NAB loan listed here is Variable, since the dataview query above only includes Fixed. Here it is;

let pages = dv.pages(‘“Maths/Unit 3/Finance/Banks/NAB”’) dv.table( [“Bank”, “Variable Compar.”, “Max Age”], pages.map(b => [ b.file.link, **${b.VariableComparison}**, b.MaxVariableAge ]) )

Collected Cars Data §

function calculateTotalAmount(principal, interestRate, years, compoundsperyear) {
// Convert interest rate to decimal and get the daily interest rate
let decimalInterestRate = interestRate / 100;
let dailyInterestRate = decimalInterestRate / 365;
// Calculate the number of compounds in total over the entire loan
let compounds = years * compoundsperyear;
// Calculate the compound factor
let compoundFactor = Math.pow(1 + dailyInterestRate, compounds);5
// Calculate the total amount to be paid
let totalAmount = principal * compoundFactor;
return totalAmount.toFixed(2); // Round to two decimal places
}

function notgonnawork(principal, interestRate, years, compoundsperyear, Payment, PaymentsPerYear) {
let i = 0
let Price = principal
let ActualPayment = Payment / 12
let ActualRate = 1 + (interestRate / 100) / compoundsperyear
while (i < (compoundsperyear*years) ) {
	let Value = Price * ActualRate
	Price = Value
	i++
}
console.log(Price)
return Price.toFixed(2);
}


let pages = dv.pages('"Maths/Unit 3/Finance/Cars"')
dv.table( 
[
"Car", "Price", "Age"
],
pages.map(c =>
[
c.file.link, c.Price, c.Age
])
)

Comparing Loans §

Loans have two defining characteristics, their Interest Rate, and their Vehicle Age Limit, usually we’d define a better loan as one which has a lower interest rate as that means we pay less Interest, however the Age Limit presents a complexity. As a result of the diminishing depreciation on cars, Older cars tend to be disproportionality cheaper than newer cars. However, loans with lower interest rates tend to set a lower Age Limit, case-in-point, the Beyond Bank Special Offer offers a 6.36% Fixed Interest Rate, but is only applicable to cars that were manufactured less than 2 years ago. Since these cars have seen the least of their depreciation, they are much more expensive, theoretically meaning you pay more interest overall, while cheaper cars are subject to much higher interest rates, also theoretically meaning you pay more interest overall. To objectively compare loans, I will convert the nominal Interest Rates provided into Effective Interest Rates, this allows me to represent a loan’s daily compound interest rate as a single simple interest rate allowing for more representative single year comparisons. This rate can’t actually be used for calculations beyond a single year timeframe though it allows for a better representation of loans that I can then use in the next section.

Once I have compared loans, I will select 1 bank based on it’s Effective Interest Rate, and another based on both that Effective Interest as well as it’s Age Limit, this one should allow the cheapest Car I’ve found, the Suzuki Swift (8yr old), I can there compare the total interest Accrued across the loans. The outcome could be any of the following;

• lower accumulation on a older, but cheaper car outweighs the higher Interest Rate of the loan, thus the proportion of interest paid to car value is the best out of the other options.
• The lowest Interest Rate on a more age-limited loan outweighs the increased Accumulation of the expensive price on a brand new car, thus the proportion of interest paid to car value is the best out of the other options.
• The lower interest rate on a more age limited loan provides a more optimimal Interest Paid to Car Value ratio when paired with a moderatly priced and “middle-aged” car.

In order to satisfy any of these scenarios, I will calculate Total Interest based on the Principal (starting amount) of three samples from my overall collection of cars. This sample should be representative of the three scenarios described above;

• A Expensive, but brand new car (2023 Kia Cerato)
• A Moderately priced, Middle-aged car (2019 Holden Trax)
• A Old but cheap car (2015 Suzuki Swift GL)

Calculations §

Comparison Rates Disclaimer

I will utilise the Fixed Comparison Rate, they are meant to represent a $30,000 loan taken over 5 years, this is very similar to our maximum loan, however for the lower end ones this could over-estimate the cost, I will keep this in mind during the comparison phase.

Where; Nominal Interest Rate Effective Interest Rate

Beyond Bank (LRL) §

Nominal Fixed Compensation Rate: 8.27%

Beyond Bank (SO) §

Nominal Fixed Compensation Rate: 6.36% Max Age: 2 Years

Commbank §

Nominal Fixed Compensation Rate: 9.04%

Heritage §

Nominal Fixed Compensation Rate: 7.04%

Westpac §

Nominal Fixed Compensation Rate: 10.67%

Bankwest §

Nominal Fixed Compensation Rate: 8.34%

When looking at these interest rates, we can see a trend emerge where the higher the Nominal Interest Rate, the more the Effective Interest Rate rises by; For example the lowest, , the , where as the highest, , it’s . This is inherent, as a higher interest rate means that it accumulates faster, the more that it accumulates, the more it grows, and so on.

Given the information above I have chosen;

• Beyond Bank (SO) - 6.36%
  • Kia Cerato
• Heritage Bank - 7.04%
  • 2019 Holden Trax
• Beyond Bank (LRL) - 8.27%
  • 2015 Suzuki Swift GL

Comparing Total Accumulation §

BB-SO (6.36%) Kia Cerato
= 4648.82\text{Repayments}=$125.15 \text{ Per Week}4648.82+27890\boxed{$32738.82}4648.82 \div 27890 = 0.1666841 = \boxed{16.67%}$ of Car's value paid in Interest

Classpad - Working out Payments Classpad - Finding of Interest

Heritage (7.04%) 2019 Holden Trax
= 2394.74\text{Repayments}=$58.45 \text{ Per Week}2394.74+12800\boxed{$15194.74}2394.74 \div 12800 = 0.187089 = \boxed{18.71%}$ of Car's value paid in Interest

Classpad - Working out Payments Classpad - Finding of Interest

BB-LRL (8.27%) 2015 Suzuki Swift GL
= 2417.38\text{Repayments}=$51.57 \text{ Per Week}2417.38+10990\boxed{$13407.38}2417.38 \div 10990 = 0.21996 = \boxed{22.00%}$ of Car's value is paid in Interest.

Classpad - Working out Payments Classpad - Finding of Interest

Given the results above, there is a clear positive relationship between the value/!age (youth) of the car, and the proportion of it’s value that you will have to pay back extra as Interest. The best Interest-wise deal appears to be the most expensive car, with the lowest I.R you pay back the least of it’s actual value as Interest, however it’s repayments are more than double that of the mid-range car at $125.15, and alongside that the Interest Paid is double that off the mid-range. The difference in the Interest Rate between it & the mid-range is only , and the Car to Value Ratio difference, which takes into account the car’s value in relation to the interest paid for these cars specifically is between them. The Mid-range Car (Holden Trax) with Heritage seems to be the most appealing/realistic, it is only 58.45$ Per Week in repayments, while being only 4 Years old in contrast to 8 Years like the Suzuki.

Ontop of that, comapring it to the cheapest car, the Suzuki, it accumulates less total interest despite having a higher Principal. Comparatively, you pay back less of the cars value of as interest, “saving” around 418.56$ in Interest. That’s enough to cover the entire Licensing Duty & License Transfer Fee.

Finally, the Low-range car, This one offers the cheapest repayments, at only 51.5722%3.29%5.33%\boxed{$1810}=(12800-10990)$181030,000** loan over 5 Years, the estimated loan is likely to become slightly more inaccurate the further you stray away from $30,000 Though this shouldn’t have too much of an impact.

Variable Interest Loans §

NAB offers a Variable Interest Loan who’s comparison rate ranges from 7.91% to 20.83%, we make the generous assumption that a University Student is able to get a loan in the average between these two values ), and that Interest Rates decline from their current peak of that they achieved, and maintained since June, causing this variable interest rate loan to hit it’s best potential rate of 7.91% within a years time. I estimate a year since the RBA was able to get the Cash Rate up to it’s current value in a little over a year, thus, it’d likely take a similar amount of time to reduce them back to if they were to decrease them now.

I’ll use the Holden Trax as my example car, as the Loan has a Vehicle Age Limit of 7 Years, thus the Suzuki is not Eligible.

Once running the;

Classpad Calculations

We first calculate the payments, as if this where a regular loan. Then find the of Interest payed that year, as well as the balance at that point.

We then take the and use it as the new value to calculate new Repayments And then use Amortisation;

Thus,

We get 3303.19}$ as the total Interest accrued, this is;

1. Still more Interest than the Holden Trax with Heritage Bank, which works out to 2451.34$ in a year.
2. Not Guarranteed, this only works out if the RBA drastically dropped interest rates harder than it has ever done before, and that it would do so starting right now.

Thus, the Variable Rate loan is not going to be considered as even if unknwon variables outside of our control go perfectly in our favour in a historically unprecedentated fashion, we still work out to a loan that has a higher minimum rate than others we can get with lower rates that are guarrenteed to stay that way. By choosing a fixed loan instead, even in today’s high interest rate environment, we can state the price of the money we’re borrowing, and plan around that without the possibility of change. (Interest Rates represent the price of borrowing money)

Reducing Balance Depreciation §

Since Car Depreciation is atleast partly specualitive, meaning that if it is not already documented, we must look into a broad array of past pricing statistics to find a reliable depreciation rate for the particular model, I’ve opted to use this website, It offers a generalisation that should hold true for most cars on average. It states that cars depreciate their value at a rate of 25% within the first year, after that point they lose 14.5% of their value per-year on average. With this in-mind, we can utilise a geometric sequence to calculate the per-year reducing balance depreciation, often referred to as Diminishing Value Depreciation. If It is more than one year old;

If it is less than one year old (brand new)

Where; = Purchase Price of the Car =

Total Remaining Value §

Kia Cerato 7994.29}\color{red} -19,895.71\text{Retained } {7994.29 \div 27890 =}\boxed{28.66%} \text{ of Value}$)

Holden Trax 4891.93}\color{red} -7908.07\text{Retained } {4891.93 \div 12800 =}\boxed{38.2182%} \text{ of Value}$)

Suzuki Swift 4200.18}\color{red} -6789.82\text{Retained } 4200.18 \div 10990 = \boxed{38.218198%} \text{ of Value}$)

Classpad Working Out

Kia Cerato §

Holden Trax §

Suzuki Swift §

Unsurprisingly, The Kia Cerato fairs the worst at retaining it’s value, since it has to endure the higher Initial purcahse depreciation. (often referred to as the value lost after driving off the lot) Thus second-hand / older cars who have already been through this fair a lot better in terms of retaining value after the 5 year period is up. This is important as it puts the Second-hand cars in a much better position for being sold. After we finish paying off the loan, we can get a more significant portion of the original loan payment back by selling the vehicle;

Excess §

After Depreciation, despite whichever one you sell, your Net Expenses, that is, the money paid for the overall loan & it’s Interest, is offset by the sale of the Car at it’s estimated 5 Year Book Value, such that you actually pay less than the Original Value of the Car over that 5 Years. This means that there is an excess, not only is the Interest offset by the sale of the car, some of the original, already paid-off loan is also offset, sort of like a bit of your loan payment you get back after selling the car.

I could get the Holden Trax with Heritage, paying 58.45(12800-10302.81)=\boxed{2497.19}$ excess at the end of it. We Achieve this value by taking the original value of the vehicle from the Total Loan Cost After Car Sold (T.L.C After Car Sold), which includes all the accumulated interest. if this value is not negative, that means that the T.L.C After Car sold is lower than the value of the car itself, meaning the sale of the car produces an overall excess.

instead of the Holden Trax I could get the Suzuki Swift GL, going for 51.5710990-9207.2=\boxed{$1782.8}\boxed{$714.39}\boxed{$6.88 \text{ Per Week}}=($58.45-$51.57)\boxed{$1788.8}=(6.88 \times 52)$10990$2789027890 - 24744.53 =\boxed{$3145.47}10990-9207.2=\boxed{$1782.8}125.15 - 51.57 = \boxed{$73.58 \text{ Per Week}}3145.47-1782.8=\boxed{$1362.67}$73.58 \times 260 = \boxed{\color{green}$19130.80}$

Fuel Economy §

The Fuel economy will have a significant impact on my living costs. Choosing a more efficient car will reduce my overall petrol payments significantly over the 5 year period, possibly enough to justify a more expensive car over the 5 year period.

Western Australians usually drive 12,100Km every year. Fuel prices are also on average, 1.93/L. This equation tells us the Yearly fuel cost with each car, dependant on it’s fuel economy that we found above. Thus; §