$Year 12 Finance - Home loans$

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$Year 12 Finance - Home loans$

Mathew wishes to borrow from a bank for a home loan. The bank offers him an annual interest rate of 3% per annum, compounding monthly. He knows that he can afford to repay $2000 per month, and he wishes for the entire loan to be repaid in exactly 25 years.

What is the greatest amount that Mathew can borrow from the bank? Please include as much working as possible, question is worth 5 marks. (From year 12, Australia Adv maths)

Math Finance

Alice Verin

27

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25 years $\times$ 12 = 300 months
0.03 per annum / 12 = 0.0025 per month
The 2000 paid at the nth month will worth at the end of 25 years $$ 2000(1+0.0025)^{300-n}$$The sum of all these will amount to (we set $r=1.0025$) $$2000(r^{299}+\dots+r+1)=2000{r^{300}-1\over{r-1}}=2000{(1.0025)^{300}-1\over{0.0025}}= 892015.6 $$rounded to one decimal point. On the other hand, the amount $K$ borrowed now will worth $$Kr^{300}=K\times 2.115$$after 25 years. So we must have $$K\times 2.115=892015.6 \implies K=421,756.8$$
This is the maximum amount he can borrow.

Martin

795

Alice Verin

0

Hey, Would you not use the annuity formula for this question? And use a geometric progression? Since its a present value annuity problem?
Alice Verin

0

Is it not looking for the maximum amount he can borrow, for a home loan, surely thats a larger number?
- Martin
  
  0
  
  You are right, I will fix it
Martin

0

I fixed the solution, sorry for the mix-up. The above explanation is essentially the derivation of annuities formula.

$Year 12 Finance - Home loans$

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