Further Mathematics Rounding (Financial)

[Lear]

Hey all,
I will pose my question through an example.
Bob takes out a loan of 370,000 over a period of 23 years with an interest rate of 4.2% compounded monthly with monthly payments.
What would Bob's monthly payment be?

According to financial solver it is $2092.911905
Would this be rounded to 2092.92 or 2092.91? According to this commercial exam it is 2092.92 as if it was 2092.91 you would not completely pay it off after 23 years, which is logically correct. On the other hand though 2092.91 is the theoretical rounding.

As a counter example, 2012 VCAA EXAM 2 Question 3a in business related maths -
$40 000 is borrowed at an interest rate of 7.8% per annum.
Interest on the unpaid balance is charged to the loan account monthly.
Suppose the $40 000 loan is to be fully repaid in equal monthly instalments over five years.
a. Determine the monthly payment, correct to the nearest cent.

Financial solver gives us $807.2325
According to VCAA the answer is 807.23 yet if we simply only pay that much, we still owe around 18c by the end of 5 years.

What is the right answer?

Furthermore, say we have rounded our payment value to, say, 807.23 and further question asks us how much is paid off after 3 years, are we to use 807.23 as the payment or the one currently in financial solver (807.232512..) and then round it once we get the PV value.

Lastly, is there a difference between 'closest' and 'correct'?

[S_R_K]

Hey all,
I will pose my question through an example.
Bob takes out a loan of 370,000 over a period of 23 years with an interest rate of 4.2% compounded monthly with monthly payments.
What would Bob's monthly payment be?

According to financial solver it is $2092.911905
Would this be rounded to 2092.92 or 2092.91? According to this commercial exam it is 2092.92 as if it was 2092.91 you would not completely pay it off after 23 years, which is logically correct. On the other hand though 2092.91 is the theoretical rounding.

The best answer to the question would cite the exact value, since no rounding instruction is given. Unfortunately, it is unlikely that $2092.911905 is the exact value, so it is difficult to give a simple answer to this question.

If we're going to be more realistic, then what happens in this situation is that the bank is not restricted to deducting amounts from your loan account that are rounded to two decimal places (in units of $). The number that appears in the balance sheet is notional, and the bank can adjust it by increments of three or more decimal places. However, because the bank will report loan repayments to two decimal places on your balance sheet, you would notice that not all payments have the same size. This is not because the bank has decided to deduct an extra cent on one particular occasion; this is just because of the rounding involved in how the numbers are reported, and the accumulation of rounding errors in prior calculations. Any rounding that is still required at the end of the loan can be handled by adjusting the final payment.

As a counter example, 2012 VCAA EXAM 2 Question 3a in business related maths -
$40 000 is borrowed at an interest rate of 7.8% per annum.
Interest on the unpaid balance is charged to the loan account monthly.
Suppose the $40 000 loan is to be fully repaid in equal monthly instalments over five years.
a. Determine the monthly payment, correct to the nearest cent.

Financial solver gives us $807.2325
According to VCAA the answer is 807.23 yet if we simply only pay that much, we still owe around 18c by the end of 5 years.

What is the right answer?

I think the question is fine, and VCAA's suggested answer is correct. This is because VCAA states that the monthly repayments have equal size, and you are asked to round the value to the nearest cent. The important point here is that giving a rounded answer to this question does NOT imply that the rounded answer is the exact amount that the bank deducts from the loan balance each month.

Furthermore, say we have rounded our payment value to, say, 807.23 and further question asks us how much is paid off after 3 years, are we to use 807.23 as the payment or the one currently in financial solver (807.232512..) and then round it once we get the PV value.

The latter is the correct approach. Do not use rounded values in intermediate calculations (or at least don't use values that are rounded too much; it's usually fine to use values rounded to 4 or 5 decimal places for intermediate steps when the final answer should be 2 decimal places). If you are using the TI, it's good to know how to use the tvm.pmt etc. commands to recall exact values (or near enough).

Lastly, is there a difference between 'closest' and 'correct'?

"Write your answer correct to the nearest cent" means your answer should include at least 2 decimal places (if written in dollars). It may include more, as long as you write down all digits correctly (and the final digit is rounded correctly).
If the instruction specifically asks you to round to the nearest cent / dollar, then you must round. Writing extraneous decimal places shows that you have not followed the rounding instruction.

[Lear]

Cheers for the response

[studyingg]

Hey all,
I will pose my question through an example.
Bob takes out a loan of 370,000 over a period of 23 years with an interest rate of 4.2% compounded monthly with monthly payments.
What would Bob's monthly payment be?

According to financial solver it is $2092.911905
Would this be rounded to 2092.92 or 2092.91? According to this commercial exam it is 2092.92 as if it was 2092.91 you would not completely pay it off after 23 years, which is logically correct. On the other hand though 2092.91 is the theoretical rounding.

As a counter example, 2012 VCAA EXAM 2 Question 3a in business related maths -
$40 000 is borrowed at an interest rate of 7.8% per annum.
Interest on the unpaid balance is charged to the loan account monthly.
Suppose the $40 000 loan is to be fully repaid in equal monthly instalments over five years.
a. Determine the monthly payment, correct to the nearest cent.

Financial solver gives us $807.2325
According to VCAA the answer is 807.23 yet if we simply only pay that much, we still owe around 18c by the end of 5 years.

What is the right answer?

Furthermore, say we have rounded our payment value to, say, 807.23 and further question asks us how much is paid off after 3 years, are we to use 807.23 as the payment or the one currently in financial solver (807.232512..) and then round it once we get the PV value.

Lastly, is there a difference between 'closest' and 'correct'?

For the first question the correct answer would be 275 payments of $2092.92, and one final payment of $2089.17.  As paying $2092.92 for 23 years means that you paid extra, and 2092.91 would not meet the requirement of repaying the loan over 23 years.

[S_R_K]

For the first question the correct answer would be 275 payments of $2092.92, and one final payment of $2089.17.  As paying $2092.92 for 23 years means that you paid extra, and 2092.91 would not meet the requirement of repaying the loan over 23 years.

I would prefer not to get caught up in unnecessary to and fro here, but this is not correct. Your answer assumes that loan repayments are restricted to increments of one cent, which is not true. Financial institutions can (and do, as a matter of fact) make transactions where the dollar value exchanged involves fractions of cents.

Using your logic where the final payment is adjusted to handle any accumulated rounding errors, it is perfectly reasonable for Bob's monthly payment to be $2092.91, but his final payment is somewhat greater. (In fact, from the bank's point of view, this is preferable, since the accumulated interest yields a greater final payment to the bank!)

[studyingg]

For Lear's benefit, I think it’s important that you give them information that is within the scope of the study design, I recognise that your answer is completly true in a real-life situation, but I don't think that it's correct in terms of further maths. My teacher is an examiner, and he really emphasised the importance of this concept while he taught us financial, as it will likely be tested. I don't mean to start any 'to and fro' lol, but I just wanted to make the person who asked this question aware of this concept, just in case he already is not.

This is an example from the NHT 2018 exam where they test this idea.

[Lear]

Excellent example. Thanks @Studyingg

[S_R_K]

For Lear's benefit, I think it’s important that you give them information that is within the scope of the study design, I recognise that your answer is completly true in a real-life situation, but I don't think that it's correct in terms of further maths. My teacher is an examiner, and he really emphasised the importance of this concept while he taught us financial, as it will likely be tested. I don't mean to start any 'to and fro' lol, but I just wanted to make the person who asked this question aware of this concept, just in case he already is not.

This is an example from the NHT 2018 exam where they test this idea.

Yes, I don't want to confuse the issue, but notice in this case that VCAA explicitly give a rounding instruction, and also explicitly state that the first 11 repayments are equally sized and the final payment is adjusted to compensate for any prior rounding errors.

In other cases VCAA do not or may not give this additional instruction / information (although one would hope so). One should not assume that financial transactions automatically round to the nearest cent (they do not). Neither the study design, nor the general instructions on the exam, state that all financial transactions are rounded to the nearest cent. *** See edit below, this is misleading. ***

I don't think this is an entirely trivial point, because if we did assume that financial transactions should always be rounded to the nearest cent, then this would lead to errors when using values in intermediate steps - as highlighted above.

The general point is that if VCAA asks for a final answer to be rounded, then round as required. Otherwise, give exact value.

Edit: Apologies, but I did forget that VCAA do explicitly state on the exam instructions that final answers for the "financial mathematics" section should be rounded to the nearest cent. (I'm not convinced that this means that the answer to the first question in OP should be 2092.92, but that's a commercial exam, so it's probably irrelevant).