Actuarial Outpost Annuity Question - Unknown Interest rate
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 Financial Mathematics Old FM Forum

#1
07-06-2017, 11:24 PM
 gpecci Member CAS SOA Join Date: Jun 2017 Studying for FM College: The Ohio State University - Junior Posts: 30
Annuity Question - Unknown Interest rate

Question:
Using an annual effective interest rate j>0, you are given:
(i) The present value of 2 at the end of each year for 2n years, plus an additional 1 at the end of each of the first n years, is 36
(ii) The present value of an n-year deferred annuity-immediate paying 2 per year for n years is 6.
Calculate j
a) .03 b) .04 c) .05 d) .06 e) .07
I understand the solution listed, but why is the way I solved the question giving me a different answer?

#2
07-07-2017, 01:25 AM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 4,226

Your second line should be 3 angle n + 2 *v^n * angle n.
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Last edited by Breadmaker; 07-07-2017 at 11:20 AM.. Reason: Forgot the 2.
#3
07-07-2017, 08:35 AM
 NattyMo Member Non-Actuary Join Date: Nov 2016 Posts: 606

Draw a timeline. It helps visualize how the payments are being made.

$
36 = 3a_{\overline{n|}} - a_{\overline{n|}}
$

... does not follow from ...

$
36 = 2a_{\overline{2n|}} + 1a_{\overline{n|}}
$

There are several ways to approach this, aside from the answer you provided.

After drawing a timeline, here's how I approached it:

\begin{align}
\textrm{From (i):}
36 &= 2a_{\overline{2n|}} + a_{\overline{n|}} &\textrm{[1]}
36 &= 2a_{\overline{n|}} + 2a_{\overline{n|}}v^n + a_{\overline{n|}} &\textrm{[2]}
36 &= 3a_{\overline{n|}} + 2a_{\overline{n|}}v^n &\textrm{[3]}
\textrm{From (ii):}
6 &= 2a_{\overline{n|}}v^n &\textrm{[4]}
\textrm{Substitute [4] into [3].}
36 &= 3a_{\overline{n|}} + 6 &\textrm{[5]}
30 &= 3a_{\overline{n|}} &\textrm{[6]}
10 &= a_{\overline{n|}} &\textrm{[7]}
10 &= \frac{1 - v^n}{j} &\textrm{[8]}
\textrm{Substitute [7] into [4].}
6 &= 2(10)v^n &\textrm{[9]}
0.3 &= v^n &\textrm{[10]}
\textrm{Substitute [10] into [8].}
10 &= \frac{1 - 0.3}{j} &\textrm{[11]}
j &= \frac{0.7}{10} &\textrm{[12]}
j &= 0.07 &\textrm{[13]}
\end{align}
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#4
07-08-2017, 11:30 AM
 textgameno100 SOA Join Date: May 2017 Location: Vietnam-HCMC Studying for P College: University of science Posts: 8

Correct:
line 2: PV = 2(a_2n) + (a_n)
line 3: PV = 3(a_2n) - (a_n)*v^n