Actuarial Outpost ASM pg. 180 #1 help
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#1
03-14-2018, 11:02 PM
 ctbowne CAS Join Date: Jun 2017 College: UC Berkeley Posts: 2
ASM pg. 180 #1 help

ASM 12th ed. pg. 180 #1

A loan of 10,000 is to be amortized in 10 annual payments beginning 6 months after the date of the loan. The first payment, X, is half as large as the other payments. Interest is calculated at an annual effective rate of 5% for the first 4.5 years and 3% thereafter. Determine X.

I am having a difficult time understanding the book's answer, and I feel like I'll learn more if I can understand why my approach is incorrect. I'm sure I made some kind of newbie mistake here, and I am hoping for enlightenment!

If I understand the question correctly, first payment is six months out and then payments proceed one year apart. I just pulled everything back an additional 0.5 year, the first payment X back 0.5 year, the next 4 @ 5% by 1.5 years and the next 5 @ 3% by 5.5 years. All v is at 5%.

10,000 = Xv^.5 + 2Xv^1.5 a4|5% + 2Xv^5.5 a5|3%

Solving for X, I get 686.31. The correct answer is 656.

What am I doing wrong?
#2
03-14-2018, 11:16 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 30,854

Quote:
 Originally Posted by ctbowne ASM 12th ed. pg. 180 #1 A loan of 10,000 is to be amortized in 10 annual payments beginning 6 months after the date of the loan. The first payment, X, is half as large as the other payments. Interest is calculated at an annual effective rate of 5% for the first 4.5 years and 3% thereafter. Determine X. I am having a difficult time understanding the book's answer, and I feel like I'll learn more if I can understand why my approach is incorrect. I'm sure I made some kind of newbie mistake here, and I am hoping for enlightenment! If I understand the question correctly, first payment is six months out and then payments proceed one year apart. I just pulled everything back an additional 0.5 year, the first payment X back 0.5 year, the next 4 @ 5% by 1.5 years and the next 5 @ 3% by 5.5 years. All v is at 5%. 10,000 = Xv^.5 + 2Xv^1.5 a4|5% + 2Xv^5.5 a5|3% Solving for X, I get 686.31. The correct answer is 656. What am I doing wrong?
2Xv^1.5 a4|5% is that an immediate annuity or an annuity due?

2Xv^5.5 a5|3% again, immediate or due?
And is that v^5.5 at 5%? v^5.5 at 5% would not be right
#3
03-15-2018, 01:58 AM
 ctbowne CAS Join Date: Jun 2017 College: UC Berkeley Posts: 2

Hi Gandalf. I intended annuity immediate. But your question got me to realize my equation was unclear.

So I went back and tried to reformulate it from the start. Maybe this one uses better notation for annuities.

All annuity immediate and all v is @5%

I used the first payment time when X is paid as the comparison date.
PV immediately before first payment of X is 10,000(1.05)^ .5 = 10246.95.
PV immediately after payment of X is 10,246.95 - X
PV in terms of remaining payments at comparison date:
2X a4@5 + 2X v^4 a5@3

10,000(1.05)^.5 - X = 2X a4@5 + 2X v^4 a5@3.

Then, solving X = 655.70.

Now I understand the book's version, too.

 Tags asm 12th, interest rates, present value