Actuarial Outpost Perpetuity question
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 Financial Mathematics Old FM Forum

#1
11-20-2017, 04:37 PM
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128
Perpetuity question

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.
(A) 1.23
(B) 1.56
(C) 1.60
(D) 1.74
(E) 1.94

My approach:

$\frac{1}{j}(1+j)=7.21$

$j=.161$

$i=.0775$

$1.0875=(1+k)^{\frac{1}{3}}$

$k=.286178755$

Since the first payment is at t=2, I want the perpetuity formula to be at t=0 or t=3. I chose t=3.

$\frac{R}{k}(1.0875)=7.21$

$R=1.897$

What did I do wrong?

Last edited by Futon; 11-20-2017 at 05:34 PM..
#2
11-20-2017, 05:24 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,118

When you post questions, post what the author says the answer choice is.
#3
11-20-2017, 05:34 PM
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128

#4
11-20-2017, 05:48 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,118

Your calculations look OK to me. What's the source of the question?
#5
11-20-2017, 05:51 PM
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128

Quote:
 Originally Posted by Gandalf Your calculations look OK to me. What's the source of the question?
This is one of the questions from the sample SOA exams. I think should've included the solution as well, sorry:

This solution does nothing for me. I do not understand why they are discounting the second perpetuity back by a year making the perpetuity at time = 1.

Also that mysterious "+1".

It's a very unintuitive solution imo.
#6
11-20-2017, 07:54 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,118

Their solution is correct. Yours has the payment in the 3-year annuity at the end of year 2, not the beginning. The payment is at t=1, not t=2.
#7
11-20-2017, 08:55 PM
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128

Quote:
 Originally Posted by Gandalf Their solution is correct. Yours has the payment in the 3-year annuity at the end of year 2, not the beginning. The payment is at t=1, not t=2.
Ah right. "beginning of year two" blinded me. Why the +1 though? I would've multiplied by (1+i)^3 in addition to (1+i)^-1 since it is a perpetuity-due. Is there a reason why he didn't do that?
#8
11-20-2017, 10:54 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,118

Either is fine. Thinking of the every year case, you can do either:

perpetuity-due = perpetuity-immediate + 1
or
perpetuity-due = (perpetuity-immediate)(1+i).

Neither formula is more right than the other.
#9
11-20-2017, 11:09 PM
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128

Quote:
 Originally Posted by Gandalf Either is fine. Thinking of the every year case, you can do either: perpetuity-due = perpetuity-immediate + 1 or perpetuity-due = (perpetuity-immediate)(1+i). Neither formula is more right than the other.
Thank you