Actuarial Outpost Prospective and Retrospective Methods Question
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 Financial Mathematics Old FM Forum

#1
03-17-2018, 01:09 PM
 Nagsinde2002 SOA Join Date: May 2017 College: University of Wisconsin - Madison (1st year) Posts: 13
Prospective and Retrospective Methods Question

Hello, I was looking at Example 37.2 from "A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2" by Marcel B. Finan and I am confused why they set it up like this:

The problem reads: A loan is being repaid with 16 quarterly payments, where the first 8 payments are each \$200 and the last 8 payments are each \$400. If the nominal rate of interest convertible quarterly is 10%, use both the prospective method and the retrospective method to find the outstanding loan balance immediately after the first six payments are made.

Using the prospective method, it was set up like this:
B(p,6) = 200(v + v^2) + 400(v + v^2 + ....+ v^8)

If the question asks for the loan balance after 6th payment shouldn't it be:
200 * (annuity-immediate of n = 10) and not include the 400.

If someone can clarify this, that would be helpful.
#2
03-17-2018, 07:50 PM
 xplicitator Member SOA Join Date: Sep 2014 College: SFU Posts: 41

Since on the time diagram, we are at time 6. If you wrote 200 * (annuity immediate with n=10), then you are saying that starting at time 8, only payments of 200 are made. But in reality, at time 6-8, it's still 200, but starting with time 9, payments are 400 each rather than just 200. So 200(v + v^2) accounts for time 6 to 8 and then 400(v + v^2 + ....+ v^8) starting at time 9. Hope that helps!
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#3
03-17-2018, 08:11 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,056

Quote:
 Originally Posted by xplicitator Since on the time diagram, we are at time 6. If you wrote 200 * (annuity immediate with n=10), then you are saying that starting at time 8, only payments of 200 are made. But in reality, at time 6-8, it's still 200, but starting with time 9, payments are 400 each rather than just 200. So 200(v + v^2) accounts for time 6 to 8 and then 400(v + v^2 + ....+ v^8) starting at time 9. Hope that helps!
You need a v^2 multiplying the second series.
#4
03-17-2018, 09:05 PM
 xplicitator Member SOA Join Date: Sep 2014 College: SFU Posts: 41

Yes that I forgot! That v^2 in front of the 400 series is to discount for the 200 time payments.
__________________
P, FM, MFE

VEE: Statistics, Economics, Corporate Finance