Need help with Differing Payment Period and Geometric Frequency

[eragonngo]

Problem:

"Melanie wants to buy an annuity of 15 years providing 30 semi-annual payments, the first payment being six months after the purchase of the annuity. During a year, the semi-annual payments are the same amounts. After the first year, at the beginning of each year, the payment level are decreased by 4%. If the semi-annual interest rate is i(2) = 10%, capitalized two times a year, how much should cost the annuity ?"

My attempt solving this problem:

Year 1: R*(a(2]i)*v^0
Year 2: (96/100)R*(a(2]i)*v^2
Year 3: (92/100)R*(a(2]i)*v^4
...
Year 15: (28/100)R*(a(2]i)*v^28

=> PV = R*(a(2]i)*v^0 + (96/100)R*(a(2]i)*v^2 + ... + (28/100)R (a(2]i)*v^28

=> PV = R*(a(2]i)* Sigma (from k=0 to k=14) of {(1 -4%*k)*[v^(2k)]}


--------------------------------------------------------------


I got stuck it the geometric series ...
Is my understanding about this question correct T____T

Sorry for taking your time reading this question (i)

And thanks for your concerning (ii) ( not sure if (i) and (ii) are similar ?)

Sincerely,

[liujeffqi]

i not quite understand what u did but it quite long and you won't have that much time in exam condition so you need something short here is what i do

do you know the geometric formula for varying annuity?

if not better download a forumla sheet from here where people post their formula sheets:
http://www.actuarialoutpost.com/actu...ad.php?t=92252
the one i will using is this one with a P in front

1st you need put first two payment into 1 because the annuity increase every two payment
so P*S_(2)_0.05
then annual effective rate: 1.05^2-1=0.1025
then plug in equation P*S_(2)_0.05*{[1-(0.96/1.1025)^15]/(0.1025+0.04)}= P*2.05*6.1374=12.5816P

well since it didn't give P, i think this is the answer because we pretty much have no clue how much money is involved. unless answer is something else let me know.

[eragonngo]

Thanks for your concerning ( it is likely that you are the only one here helping me to solve my questions in detail T___T ) But I think this question is not about the geometric frequency at all, I think it's about arithmetic metric progression. Because the payment R will decrease 4% every 2 times
let's X = P*S_2_0.05 = 2.1025P
For the end of 1st year ( 2 times ) : X= 40%X + 15*4%X
i=(1+i(2)/2)^2 -1 = 0.1025
For the end of 2dn year (4 times): 96%*X=40%X + 14*4%X
...
For the end of 15th year (30 times):44%*X== 40%X +4%X

a_(15)_0.1025=a_n_i=7.49876
(Da)_n_i = (n - a_n_i)/i=
PV = 40%X*a_n_i + 4%X*(Da)_n_i
= X*2.9995 + 7.297X
= 21.6484P

Oh ! It's 4% decrease in payment level each yar, I got it wrong. Thought it is decrease 4% R each year >___< Sorry ! Your solution is right.

About the amount of P is a little bit ambiguous for me too ><. Because in problem 2, I have to face with a ambiguous payment too.

Problem 2:

About Perpetuity. The following questions are related.
(a) Consider an annuity with payments of 1/m made each month. The nominal interest rate compounded each month is i(12) = 4%. Use the geometric series to express the present value of this annuity in terms of a_n_i and i.
(b) Using the previous assumptions, consider a perpetuity of 1/m, that is an infinite
series of payments of 1/m. Express the present value of the perpetuity of 1/m in terms of i(12).
(c) Consider the perpetuity in (b). Express the present value of the perpetuity in terms of i (12).


I have already emailed her, asking about that question
( My attempt is like : a) PV = 1/m * 1/(1+i(12)/12) * [(1-v^n)]/([i(12)/12]/{1+[i(12)/12]})
note: all i(12)/12 is replaced by (1+i)^-12 -1

but she says this will not consist 1/m (which means I have to change 1/m to an equation respects to a_n_i and i ) but I just don't know how ?

[liujeffqi]

Originally Posted by eragonngo

Thanks for your concerning ( it is likely that you are the only one here helping me to solve my questions in detail T___T ) But I think this question is not about the geometric frequency at all, I think it's about arithmetic metric progression. Because the payment R will decrease 4% every 2 times
let's X = P*S_2_0.05 = 2.1025P
For the end of 1st year ( 2 times ) : X= 40%X + 15*4%X
i=(1+i(2)/2)^2 -1 = 0.1025
For the end of 2dn year (4 times): 96%*X=40%X + 14*4%X
...
For the end of 15th year (30 times):44%*X== 40%X +4%X

a_(15)_0.1025=a_n_i=7.49876
(Da)_n_i = (n - a_n_i)/i=
PV = 40%X*a_n_i + 4%X*(Da)_n_i
= X*2.9995 + 7.297X
= 21.6484P

Oh ! It's 4% decrease in payment level each yar, I got it wrong. Thought it is decrease 4% R each year >___< Sorry ! Your solution is right.

About the amount of P is a little bit ambiguous for me too ><. Because in problem 2, I have to face with a ambiguous payment too.

Is it R and P are both level payment? some text use R some use P i am kind use both. unless it is P_t then it is means principle payment. otherwise i use both as level payment. so what you thought should be right also just didn't solve the right way.

if u have trouble see it draw a time diagram it will help you out should be like

payment__R____R__0.96R___0.96R____0.92R ....
----------|-----|-----|-------|---------|
time_____0.5___1___1.5_____2_______2.5 ....

if u want use R*a_2_i instead of R*s_2_i is fine too just the 2nd part you will use the geometric annuity due instead of immediate
it will be like R* a_2_i * a(dot)_15_[(0.1025+0.04)/0.96]=12.5816R

Originally Posted by eragonngo

Problem 2:

About Perpetuity. The following questions are related.
(a) Consider an annuity with payments of 1/m made each month. The nominal interest rate compounded each month is i(12) = 4%. Use the geometric series to express the present value of this annuity in terms of a_n_i and i.
(b) Using the previous assumptions, consider a perpetuity of 1/m, that is an infinite
series of payments of 1=m. Express the present value of the perpetuity of 1/m in terms of i(12).
(c) Consider the perpetuity in (b). Express the present value of the perpetuity in terms of i (12).


I have already emailed her, asking about that question
( My attempt is like : a) PV = 1/m * 1/(1+i(12)/12) * [(1-v^n)]/([i(12)/12]/{1+[i(12)/12]})
note: all i(12)/12 is replaced by (1+i)^-12 -1

but she says this will not consist 1/m (which means I have to change 1/m to an equation respects to a_n_i and i ) but I just don't know how ?

hmm i am not quite sure because question didn't say 1/m is paid at end or beginning of month because one is immediate, one is due. anyway i think the question is ask you to setup like annuities payable monthly but work towards perpetuity, it is too much work and i don't think i will go though the hassle to do all the proof, it will kill too much brain cell of me. good luck but at end solution should be same as perpetuity (1/m)a_inf =(1/m)(1/i) if it is immediate or (1/m)a(dot)_inf=(1/m)(1+a_inf)=(1/m)(1/d) for due. it is a perpetuity after all

part b i am not quite get what is 1=m thing? so 1/m become 1/1? then it will be 1/(0.04/12)=300

so is c not quite get what is it asking....otherwise it is just same answer as b.

if she ever give u the answers please let me know i am kind curious

[eragonngo]

Hmmm, I still don't get the term: " at the beginning of each year, the payment level are decreased by 4%" . So it's 100*(1-4%) in year 2(payment 3). But at the beginning of year 3 (payment 5): Will it be 100*((1-4%)^2) of Geometric frequency or 100*(1-2*4%) of Arithmetic frequency T___T ?
Now she adds more data in question 2) where m = 12 => PV = (1/12)*a_(n/12)_(i(12)/12) . And I will represent it in term of a_n_i and i ... Hope it will be good ..
But at the case b) and c) while she states they are the same. Still can't get it. Can you enlighten little me this time, master T____T

The solution will be post on Friday 28/09/2012 (UTC -5.00) ... hope I will survive

[liujeffqi]

oh my mistake ya it is 100*((1-4%)^2) of Geometric frequency forget to change it sorry about that.

well it your sure it is PV = (1/12)*a_(n/12)_(i(12)/12) not PV = (1/12)*a_(12n)_(i(12)/12)? but anyway since it is Perpetuity n should be inf. so it should still ends up like what i have (1/m)a_inf =(1/m)(1/i) just become (1/12)a_inf =(1/12)(1/i) and i still don't get what is 1=m thing so if it is just asking to plug in m=1 then answer will 300.. and c i don't see anything new same function same plug in so answer is 300 also if what i guessing is right. but i don't know when the soultion is posted do you care to post it out so i can get what she is really asking? if u don't want post it out message is fine too

Thanks

[eragonngo]

Aw my mistake, it's 1/m in question b). So it's like a_inf = (1/m)*(1/i). But question a) isn't about Perpetuity, isn't it ?

[eragonngo]

Got question a) like: (1/12)*a_n_i* (i / ((1+i)^(-12) -1)) ... Damm this week 's assigment is pure Math .. not even actuarial science and finance in this case =)) I will post the solution and assignment tomorrow :P