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#1
02-23-2019, 07:47 PM
 Mitsu96 Member Non-Actuary Join Date: Jan 2017 Location: Pennsylvania Studying for FM Posts: 55
SOA Sample Question #85 and #101

SOA Sample #85

The present value of a perpetuity paying 1 every two years with first payment due immediately is 7.21 at an annual effective rate of i.
Another perpetuity paying R every three years with the first payment due at the beginning of year two has the same present value at an annual effective rate of i + 0.01.
Calculate R.

The answer is 1.74.

This is how SOA does it:

I got i = 0.077511 and j = (1 + i)^3 - 1 = 0.2861788. I used the following equation in order to find R,
(v^2)(a-dot-infinity)(R) = 7.21

With v = 1/1+j = 0.7774969

and I got R = 2.65384, which is definitely incorrect. I can't understand where I am going wrong the SOA's solution is not helpful either.
Any help is appreciated! Thank you!

SOA Sample #101

A 30-year annuity is arranged to pay off a loan taken out today at a 5% annual effective interest rate. The first payment of the annuity is due in ten years in the amount of 1,000. The subsequent payments increase by 500 each year.
Calculate the amount of the loan.

The answer is 64,257.

This is how SOA does it:

My only concern is why are there 30 payments? Shouldn't there by only 20 payments considering the first payment is due in 10 years (i.e., deferred annuity)?

Thank you for your help!
#2
02-23-2019, 09:32 PM
 morbrenn Non-Actuary Join Date: Nov 2018 Studying for the final finals Posts: 10

30-year annuity means the annuity lasts for 30 years. In this case, from the 10th to the 40th year.
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#3
02-23-2019, 09:49 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,529

For the first one, the beginning of year 2 is time 1 and the v should be at the one year rate.
#4
02-24-2019, 05:41 PM
 Mitsu96 Member Non-Actuary Join Date: Jan 2017 Location: Pennsylvania Studying for FM Posts: 55

Quote:
 Originally Posted by morbrenn 30-year annuity means the annuity lasts for 30 years. In this case, from the 10th to the 40th year.
Oh, so it's not a deferred annuity? It's just an ordinary 30 - year annuity?
#5
02-24-2019, 05:42 PM
 Mitsu96 Member Non-Actuary Join Date: Jan 2017 Location: Pennsylvania Studying for FM Posts: 55

Quote:
 Originally Posted by Academic Actuary For the first one, the beginning of year 2 is time 1 and the v should be at the one year rate.
Got it! Thank you so much for your help!

 Tags annuity, financial mathematics, interest rate, perpetuity

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