Another annuity calculation - lump sum decreasing - formula needed

Andrew1930 · 2015-03-25 01:50:56

Hello All!

Following on from my last post (or rather, question and Bob's great solution) I have another annuity calculation / formula query - as follows :

"Andrew has $25,000 invested in a savings account, which is paying a nominal interest rate of 5% which is credited monthly.
Andrew wants to withdraw capital and interest monthly so that at the end of 8 years the account will have a zero balance.
How much can Andrew withdraw each month?"

As before, this is a "mock" exam question as preparation for an exam I have in April.

The answer given is $316, but I do not have a formula to learn and memorize.

Can anyone assist me?

Very grateful and thanks,

Andrew.

bobbym · 2015-03-25 09:51:29

Hi;

Bob · 2015-03-25 09:58:22

hi Andrew1930

5% credited monthly. ? This can mean one of two things.

(1) UK. You work out the monthly rate that is equivalent to 5% annually.

so r = 1.004074123

(2) But in some countries I have discovered that r = 1 + 0.05/12 = 1.004166666

Clearly, this will lead to very different answers. I think (1) is correct but you need to check.

I'll avoid the problem by just using r.

month 0: 25000

month 1: 25000xr - W where W is the amount withdrawn.

month 2:

month 3:

month n:

where a is the first term of the GP (written in reverse order) = 1

So you can put n = 8x12 and set this to zero to solve for W.

EDIT: Just tried with my r (as above) and got 315.18 rounded to the nearest cent.

and using (2) I get 316.50 which agrees with bobbym's answer.

Bob

FURTHER EDIT: In post 4 of your other thread, Phrontister gives a formula which has the term i/q. This is equivalent to (2)

You say you are sitting the exam in the UK. What syllabus? I'll look it up.

Andrew1930 · 2015-03-27 01:12:02

Hello Bob,

Very grateful for that - some queries :

1. We are using number (1) the UK.

2. In the final formula which you very kindly provide, you state "r" after the 25000. Is this the annual equivalent rate of 0.05116189788 or just 0.05?

3. Excuse my ignorance, what does the "." (the dot) between the "W" and the "a" mean?

4. And I am totally lost in terms of how to find W?

5. I can calculate the AER (UK Annual Equivalent Rate) OK, but what is the exact formula that I need to learn to input the values that I am given in the question? IE : $25,000, 5% credited monthly (convert to AER) and 8 years (or 96 monthly periods)?

Sorry!

The exam is the J10 Discretionary Investment Management exam of the Chartered Insurance Institute in London. I am very surprised that they have something of this level of mathematical complexity being examined.

Best regards,

Andrew.

Bob · 2015-03-27 09:24:53

Hello Andrew,

I'll try to cover all those questions in a mo. I'm a retired Uk maths teacher so I'm used to GCSE and A level syllabuses. These give much more detail than this:

http://www.cii.co.uk/media/5454608/j10_ … -15_v4.pdf

which I assume is what you are working towards. For instance, you'd get the required formulas and a (free) specimen exam paper plus lots of past papers also free. I'm also assuming you have one or more texts that give you some ideas about what to learn.

If you have an amount P and you want to add 5% interest then you'd do this:

But you'll get the same answer like this:

The number 1.05 is very useful when doing compound interest because you can keep applying the same multiplier again and again to the principle amount.

Also if you have to get the equivalent monthly rate you need to start with the yearly multiplier (1.05) and find the twelfth root:

That's the right multiplier to use throughout the question so that's what 'r' stands for everywhere. If I had different rates at different places in a formula I would use different symbols as needed.

I discovered that it is common elsewhere (ie. not the UK) to get a monthly rate by dividing the yearly rate by 12. Watch out for formulas you get off the internet in case they do this.

I hope that answers questions 1 and 2.

Q3. In algebra x is often used so it can get confusing if x is also used as a times sign. So you'll often find a dot used instead. So W.a just means W times a

Q4 I'll re-arrange my formula so W is the 'subject'. You want to know W so that after n= 96 (months) there is no money left.

The geometric progression (GP) is

This has first term a = 1, common ratio r = 1.004074123, and n = 8 times 12 = 96. I can leave the 'a' out as it is one:

with the above values:

So with a different annual rate (I), start by calculating the monthly rate:

and then for a starting amount P

Hopefully that deals with Q5.

I'm still a bit worried about this business of getting the monthly rate. Your question is in $ and the given answer is $316. That's what I got by doing 1.05 divided by 12 and also what bobbym got. They use this method (1.05/12) in the USA where bobbym hails from. Cii is a UK institute so why are they using a question with $ and a US method of calculating the monthly rate? It might be wise to query this with cii.

Hope that helps,

Bob

Math Is Fun Forum

#1 2015-03-25 01:50:56

Another annuity calculation - lump sum decreasing - formula needed

#2 2015-03-25 09:51:29

Re: Another annuity calculation - lump sum decreasing - formula needed

#3 2015-03-25 09:58:22

Re: Another annuity calculation - lump sum decreasing - formula needed

#4 2015-03-27 01:12:02

Re: Another annuity calculation - lump sum decreasing - formula needed

#5 2015-03-27 09:24:53

Re: Another annuity calculation - lump sum decreasing - formula needed

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