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#1




Loan question
A borrower took out a loan of 100,000 and promised to repay it with a payment at the end of each year for 30 years.
The amount of each of the first ten payments equals the amount of interest due. The amount of each of the next ten payments equals 150% of the amount of interest due. The amount of each of the last ten payments is X. The lender charges interest at an annual effective rate of 10%. Calculate X. (A) 3,204 (B) 5,675 (C) 7,073 (D) 9,744 (E) 11,746 So I'm kind of lost. But here is my intuition: Obviously I got the wrong answer but any guidance would be greatly appreciated. __________________________________________________ _____________________________________________ Edit: So Mimetex is really poorly optimized here. That strange group of letters after 15000a is supposed to be overline. I literally copy pasted the annuity code for each coefficient: Code:
100000=10000a_{\overline{10}10}+v^{10}15000a_{\overline{10}10}+v^{20}Xa_{\overline{10}10} Last edited by Futon; 11102017 at 06:04 PM.. 
#3




[quote=Gandalf;9162286]
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doesn't yield me the right answer either. Edit: again, it supposed to be a overline 10 at the second coefficient. 
#4




In a given year if you pay (1+k) x i x Loan Balance at the end of the year, k x i x Loan Balance will reduce principal so the remain proportion is (1 ki). If you continued to do this the Balance would decline by that proportion each year.

#5




Sorry it's not clicking. I finally get that 150% decreases the principal (sorry didn't read Gandalf's comment clearly enough). But I have a hard time finding the pattern and making an arithmetic annuity for it.
1000000(.1)(1.5)1000000(.1)= 5000 95000(.1)(1.5)95000(.1)=4750 90250(.1)(1.5)90250(.1)=4512.5 Thus no pattern. Am I not supposed to use annuities for this question? Edit: Nevermind I did find a pattern. 5000/4750 = 4750/4512.5 = 1.05263 I got X=31389.31128 which is very wrong. Last edited by Futon; 11102017 at 08:01 PM.. 
#6




Don't discount the amounts of principal repayment. It's simpler to look at the pattern for the outstanding loan balance than the pattern for the principal repayments.

#7




The problem with you latest attempt is that it says the total paid in year 11 is 5000. No, 5000 is the amount that reduces principle.
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Then write the equation, looking forward, at the end of year 20. The present value of a level annuity of X must equal that balance. ETA: ninja'd by AA's response, which he posted as I was composing mine. 
#8




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Favorite Quote(s): Spoiler: Last edited by NattyMo; 11112017 at 12:32 AM.. 
#9




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(Fixed in line [2] by putting a space between X and v^10.)
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#10




@Gandalf @Academic Actuary
I took a day break from this problem hoping to be able to solve this with a fresh mind. Year 11: If I pay (1.5)x(.1)x100000=15000, (.5)x(.1)x(100000)=5000 will reduce principle so the remaining proportion is 100000(1(.5)(.1))=95000 Year 12: 1000000(1(.5)(.1))(1(.5)(.1))= 90250 Year 13: 1000000(1(.5)(.1))(1(.5)(.1))(1(.5)(.1))=85737.5 Year 20: 1000000(1(.5)(.1))^10=59873.69392 59873.69392= R x (a angle 10) R= 9744 ooooooooooooooooooooooooooooooooooooooh :o Thanks a lot guys. Much appreciated. @NattyMo Oh you're right! I wonder why. 
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