Equation of Value questions

[Devanshi Maheshwari]

A salaried person aged exactly 35 now wishes to make 15 annual payments starting today into a pension plan.
His aim is to provide for the expenses to be incurred, towards his daughter’s education, when he is aged 55.
The expenses to be incurred are an initial lump sum of Rs.50,000 payable at age 55, and an annuity certain of Rs.20,000
per annum payable half-yearly in arrear during the next 6 years.
In calculating how much annual payment to invest each year, the person has assumed that
• an effective rate of interest of 7% per annum will be achieved during the 20-year period, such that,
• the annuity certain can be purchased at a price that will yield a nominal rate of interest of 6% per annum
convertible half-yearly.
(a) Assuming that the amount of each payment during any period of 5 years is half of the amount of each payment in
the subsequent 5 years, calculate the amount of the first annual payment.
(b) Under the pension plan,
• the rate of interest actually earned over the 20-year period is 6% per annum effective, and
• the annuity certain is purchased at age 55 at a nominal rate of 5% per annum convertible half-yearly.
Assuming that the person makes the payments as in (a) above, calculate the revised amount of the initial lump sum that
will be available.

in part (a) I tried equating 50,000v^20+20,000a(12)@3%*v^26 to P1*a due (10) @ 7% + 2P1*adue (5)@7%v^5+4P1*adue (5)@7%v^10, but the answer is not matching, is my equation incorrect?

 

[Greg Ardan]

Hi Devanshi
Happy New Year! I can take a look at this next week....but before I do, would you mind just letting me know which question this is taken from?

I couldn't immediately find it in either the course notes or the X Assignments. For an efficient answer, it would really help if you just let me know where it's from, so I can ensure what I say is fully consistent with the solution etc.

Many thanks

 

[Devanshi Maheshwari]

Hello, Happy New Year!
This a question I found in one of my friend's notes and is not available in course notes or X assignments.

 

[Greg Ardan]

Hi Devanshi

Just on the materials point...we're of course conscious that students may well wish to study together (in fact, I'd encourage this, it can be very helpful). However, can I just politely point you to the copyright information, which is printed on all our materials: "Unless prior authority is granted by ActEd, you may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of the study material. You must take care of your study material to ensure that it is not used or copied by anybody else. Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the Institute and Faculty of Actuaries or through your employer."

As I'm sure you'll appreciate, ActEd are a commercial organisation, so it is important that people are not using our materials without paying for them. I'll assume you do have your own material which you have purchased, and that you were just studying with a friend, but it's extremely important that you do take the copyright conditions of our materials seriously.

Putting that aside, I've taken a quick look at your query, though I don't have the specific question or solution to hand. I think the method you're tried to use is basically correct, essentially equating present value of income with present value of outgo, but there are a few errors along the way.

Left hand side of equation (present value of income)

You've used 50,000v^20+20,000a(12)@3%*v^26

If you're doing everything in present value terms, I think the v^26 should be v^20? (the annuity starts in 20 years time, and ends in 26 years' time)

For the annuity, I'm not sure it's quite right - and also just be careful with how you're writing it down. I initially interpreted 20,000a(12)@3% to mean an annuity of 20,000 per annum payable monthly in arrears, at an interest rate of 3% p.a, which wouldn't be correct, but may not be what you meant. What you need to do is to either value an annuity of 20,000 per annum, for 6 years, payable twice per year in arrears, which would be 20,000a(2)<6>@i (you'd need to calculate i based on i(2) being 6%) or to work instead in periods of 6 months. In that latter case you'd value an annuity of 10,000 per 6 monthly period, payable for 12 6-monthly periods at the 6-monthly interest rate of 3%. I think that's perhaps what you've tried to do, but used 20,000 rather than 10,000. But you need to make clear what you've done, and that it's for 6 years (or 12 6-monthly periods). Write something like 10,000a<12>@3% per half year. And as per above, I think that needs to be multiplied by v^20, not v^26.

Right hand side of equation (present value of outgo)

You've used P1*a due (10) @ 7% + 2P1*adue (5)@7%v^5+4P1*adue (5)@7%v^10.

I think the first adue (10) should be adue(5)? So that it's 3 lots of "5 year" payment streams, 15 years of payments in total.

I've not fully checked there's nothing else awry, and as I say I'm not familiar with this question (the reference to R makes me wonder if it's not even a UK question?). But hopefully this is helpful, and you can get it to work based on this.

 

[Oliviagreen8824]

Hi everyone, I’ve been working through this pension valuation problem and could use a little help verifying my equation setup for part (a).


Here’s what I’ve tried:


To match the present value of the outgo (expenses at age 55), I set up the following:


Outgo:
50,000 × v²⁰ + 20,000 × a₍₁₂₎<6>@3% × v²⁰
(Where v = 1/(1 + i) and i = 0.07, and the annuity is paid half-yearly in arrears, so I used i(2) = 6%, making i(half-yearly) = 3%.)


Income:
P₁ × a_due<5>@7% + 2P₁ × a_due<5>@7% × v⁵ + 4P₁ × a_due<5>@7% × v¹⁰
(Assuming payment pattern increases every 5 years, with 15 total payments.)


But the values don’t seem to balance, and I’m wondering if I’ve applied the timing or annuity formula incorrectly. Should the annuity be valued as 10,000 every 6 months instead of 20,000 annually? And should the discounting be up to v²⁰ rather than v²⁶?


Would really appreciate any feedback from those who’ve tackled similar problems—or even tips from those juggling actuarial studies with ILM assignment help, like me!


Thanks in advance!

 

[James Nunn]

Hi Olivia

Please can I check where the pension valuation problem question comes from? More specifically, is it from an ActEd product or an IFoA past exam paper please?

Thanks