Actuarial Outpost Increasing/Repeating Perpetuity
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#1
07-07-2012, 05:13 PM
 AppleSauce Member SOA Join Date: Sep 2011 Studying for FM Favorite beer: Island's Golden Ale Posts: 75
Increasing/Repeating Perpetuity

I'm confused why my solution to the following problem doesn't work.

"A loan is to be repaid by annual payments continuing forever, the first one due one year after the loan is made. Find the amount of the loan if the payments are 1,2,3,1,2,3... assuming an annual effective interest rate of 10%"

Here is my solution:

for either 1,2 or 3, they will have the same effective interest rate, j, for a period of 3 years.

1+j=(1.1)^3 => j=(1.1)^3-1

using j, we can find v where v=1/(1+j)

PV=(1/j)+(2/J)*[v^(1/3)]+(3/j)*[v^(2/3)]

PV=16.0045

But the answer is 19.37. What did I do wrong? I wrote the sum of the 3 perpetuities, one for 1, 2 and 3, having discounted the perpetuity for 2 by one third of a period and the perpetuity for 3 by two thirds of a period.
#2
07-07-2012, 05:23 PM
 StevieB Member Join Date: Sep 2011 Studying for 5/6/7/8/9 Posts: 191

your 1/j is a payment at time 1,4,7,etc... the pv 1/j is then the pv at time -2, since you are juming by 3 years with j

same problrm with the two others annuity
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#3
07-08-2012, 12:05 AM
 asrauf Member CAS SOA Join Date: Mar 2012 Location: Canada Studying for Studying for MFE Posts: 33

I guess you are making it more confusing. Best way to approach it, i think, is by finding PV of payments of 1,2 and 3 as 1/1.1 + 2/1.1^2 + 3/1.1^3 which is equal to 4.8159.
Now we will have a geometric series which is 4.8159[ 1 + 1/1.1^3 + 1/1.1^6 + 1/1.1^9 +...... and if we sum this series as 4.8159[1/(1 - 1/1.1^3)] we get 19.36544 which approximately equal to 19.37.
#4
07-09-2012, 02:32 PM
 AppleSauce Member SOA Join Date: Sep 2011 Studying for FM Favorite beer: Island's Golden Ale Posts: 75

StevieB, I see, going to a negative time is probably not a good idea.
asrauf, thats what the book did. I guess It's best to use the interest period as the standard for perpetuity problems.
#5
07-09-2012, 03:55 PM
 NoSpoon Member SOA Join Date: Jun 2012 Location: Coruscant Studying for MLC College: Senior, UMASS Amherst Favorite beer: BBC Steel Rail Posts: 36

You can actually treat the payment as a perpetuity with payment of the present value of the payments from 1,2, & 3.

However, in order to do this, you simply have to use the formula for a perpetuity due, not a perpetuity immediate:

PV = (1/i) * (1 + i)

In this case, i is (1.1^3)-1.
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