Maths/finance games: how much money is required

davis

This is a discussion that came up in the office, and I got fixated on the maths: our local young idealist reckoned that if he had enough money to earn e.g. 1000 GBP per month (inflation-adjusting) in interest, he'd give up work forever, and just tit about. Now, if we assume that 1000 GBP per month is sufficient (which it might be assuming you've got no dependants or mortgage etc, but let's just assume it is), how much does he need to have in order to earn "enough"?

Using trial and error, interest of 6% and inflation of 5%, I got to around 900000 quid being required (think it was something like 880000). The bit that annoyed me is that I couldn't resolve that into an equation. I got to the "Time Value of Money" but that doesn't adjust for inflation. Anyone got a clue how I'd write that? Obviously we're assuming ridiculous static values for both inflation and interest earned...

The outcome of the discussion was that if you're the kind of person willing to work to accumulate that much money, you're unlikely to want to give up and exist on 1000/month.

petemadoc

This could get very complicated if you calculate interest monthly on a compound basis but to just make things nice and easy.

If you were lucky enough to get an interest rate of 6%

£1000 per month makes £12000 per year.

£200000 + 6% interest = 212000

Not sure what inflation has got to do with anything other than each year you'd need to increase the amount you earn by 5% to compensate. You'd also be taxed on the interest you earn so the above figure is based on a gross income. You wouldn't pay much tax on a 12K income.

petemadoc

davis wrote:

The outcome of the discussion was that if you're the kind of person willing to work to accumulate that much money, you're unlikely to want to give up and exist on 1000/month.

Most probably true

rolf_f

PeteMadoc wrote:

Not sure what inflation has got to do with anything other than each year you'd need to increase the amount you earn by 5% to compensate.

That's the point. The presumption is £1000 a month without having to do anything so you have to account for inflation. Your calc is the easy one but obviously the capital devalues.

davis

Yeah, I was assuming payment and interest compounded monthly. It's that increasing payment that's the bugger; if you need to increase the monthly payments, then the monthly interest earnings need to go up by at least that amount to avoid eventually being depleted...

dhope

Yep, assume 12k per year to make life simple.

12,000 / 6% = 200,000

But really interest is less than that atm

12,000 / 6% = 200,000
12,000 / 5% = 240,000
12,000 / 4% = 300,000
12,000 / 3% = 400,000
12,000 / 2% = 600,000
12,000 / 1% = 1,200,000

Pick your rate of interest

clarkey cat

P(1 + i) + P(1 + i)i = P(1 + i)(1 + i)

davis

clarkey cat wrote:

P(1 + i) + P(1 + i)i = P(1 + i)(1 + i)

Umm... huh? I can't work out your notation. Also, assuming "i" is interest, then I can't see a term for inflation (or vice-versa).

clarkey cat

i = simple interest on the relevant time period over which it is earned - monthly.

in terms of inflation I'd just deduct the monthly RPI % from the sum of the simple interest + deposit as it stands at the end of that month.

I'm not an accountant though!

sketchley

Do you want to the £1000 per month + inflation to come from interest alone or from a combination of interest and the capital investment. If the later over what time period should the capital investment be spent? If the former do you want to be left with the same capital investment at the end or not or do you want the capital investment to grow with inflation as well.

I could do the maths if you want but my brain will hurt.

scrumpydave

My time has come at last! This can be solved using actuarial functions, in particular the annuity function. If you want the present value of a series of payments for n years at a rate of interest i then the formula you need is:

PV = (1-v^n)/i

where v = 1/(1+i)

If we ignore the fact that the payments are monthly (which we could allow for if we wanted to be more precise), assume that n tends to infinity (i.e. assume that you will need this income forever) and include inflation in the interest rate, your answer is given by:

12000 * 1/(r+i)

Where r is your assumed rate of return and i is inflation.

For example if r+i is 5% you would need 12000/0.05 = 240000

You could of course just ask an annuity provider for a quote, and depending on how long you can expect to live you may get it cheaper as the capital will effectively be gradually returned to you as well.

That's the last time I expect my actuarial qualification to be useful on a cycling forum...

scrumpydave

And by the way clarkey cat, it's an actuary you want, not an accountant...

davis

Sketchley wrote:

Do you want to the £1000 per month + inflation to come from interest alone or from a combination of interest and the capital investment. If the later over what time period should the capital investment be spent? If the former do you want to be left with the same capital investment at the end or not or do you want the capital investment to grow with inflation as well.

I could do the maths if you want but my brain will hurt.

The former, with the remaining capital growing by the amount required to maintain an indefinite supply of inflation-linked income in perpetuity. I thought I'd be able to work it out, but after skipping through geometric progressions to the binomial theorem (where my brain melted), I gave in and just stuffed it into a spreadsheet for trial-and-error games.

scrumpydave

This stupid forum automatically changes "could_of" to "could have".

davis

scrumpydave wrote:

If we ignore the fact that the payments are monthly (which we could allow for if we wanted to be more precise), assume that n tends to infinity (i.e. assume that you will need this income forever) and include inflation in the interest rate, your answer is given by:

12000 * 1/(r+i)

Where r is your assumed rate of return and i is inflation.

For example if r+i is 5% you would need 12000/0.05 = 240000

That can't be right. You've got inflation and interest on the same side of the division, whereas in reality if interest was higher, you'd need less capital. The inverse is true for inflation, so they'd have to be against each other.

London_Falcon

Because someone beat me to the maths I'll only comment on the fact that £1000 per month may sound like a lot to your young idealist but try raising a family on that and maintaining the same lifestyle.

I reckon £2m would set me up for life pretty handsomely though. Certainly be able to afford the N+1 bike.....

scrumpydave

davis wrote:

scrumpydave wrote:

If we ignore the fact that the payments are monthly (which we could allow for if we wanted to be more precise), assume that n tends to infinity (i.e. assume that you will need this income forever) and include inflation in the interest rate, your answer is given by:

12000 * 1/(r+i)

Where r is your assumed rate of return and i is inflation.

For example if r+i is 5% you would need 12000/0.05 = 240000

That can't be right. You've got inflation and interest on the same side of the division, whereas in reality if interest was higher, you'd need less capital. The inverse is true for inflation, so they'd have to be against each other.

Sorry davis - you're absolutely right. It should be 12000 * 1/(i-r).

London_Falcon

davis wrote:

scrumpydave wrote:

If we ignore the fact that the payments are monthly (which we could allow for if we wanted to be more precise), assume that n tends to infinity (i.e. assume that you will need this income forever) and include inflation in the interest rate, your answer is given by:

12000 * 1/(r+i)

Where r is your assumed rate of return and i is inflation.

For example if r+i is 5% you would need 12000/0.05 = 240000

That can't be right. You've got inflation and interest on the same side of the division, whereas in reality if interest was higher, you'd need less capital. The inverse is true for inflation, so they'd have to be against each other.

Unless you express inflation as negative interest. If r=5% and i=-4% your real rate of interest is 5%-4%=1%

sketchley

davis wrote:

Sketchley wrote:

Do you want to the £1000 per month + inflation to come from interest alone or from a combination of interest and the capital investment. If the later over what time period should the capital investment be spent? If the former do you want to be left with the same capital investment at the end or not or do you want the capital investment to grow with inflation as well.

I could do the maths if you want but my brain will hurt.

The former, with the remaining capital growing by the amount required to maintain an indefinite supply of inflation-linked income in perpetuity. I thought I'd be able to work it out, but after skipping through geometric progressions to the binomial theorem (where my brain melted), I gave in and just stuffed it into a spreadsheet for trial-and-error games.

In which case it's easy I think. 6% growth per year or which 5% need to be used to increase the fund to counter the effect of inflation in the next year. That means your £12k needs to come from the remaining 1% of interest. £12k is 1% of £1.2m which is your answer. In this example assuming that interest stays at 6% and inflation at 5% so the investment grows by 5% each year and you are just taking out 1% out income. The income grows by 5% each year because the investment grows by 5% each year. I maybe wrong but I do think it's that simple.

davis

scrumpydave wrote:

davis wrote:

scrumpydave wrote:

If we ignore the fact that the payments are monthly (which we could allow for if we wanted to be more precise), assume that n tends to infinity (i.e. assume that you will need this income forever) and include inflation in the interest rate, your answer is given by:

12000 * 1/(r+i)

Where r is your assumed rate of return and i is inflation.

For example if r+i is 5% you would need 12000/0.05 = 240000

That can't be right. You've got inflation and interest on the same side of the division, whereas in reality if interest was higher, you'd need less capital. The inverse is true for inflation, so they'd have to be against each other.

Sorry davis - you're absolutely right. It should be 12000 * 1/(i-r).

Interesting -- if I try to make it monthly (for accuracy, innit), would this be correct?:

1000*1/((e^(ln(1.06)/12))-(e^(ln(1.05)/12)))


(Obviously assuming interest of 6%, and inflation of 5%. That gives me a figure of ~1.26million. Have you got a link for this stuff? Or is it basically the time value of money equations with n tending to infinity?

davis

Wow. There we go. I was overcomplicating it.

Thanks to Sketchley (best explanation ever), scrumpydave, and london_falcon.

scrumpydave

There is a simple way which can be done by adjusting the assumed annual rate of interest but I can't remember the conversion formula. Alternatively you can convert the interest rate and payments to months [i.e. i(monthly) = (1+i)^(1/12) - 1]. Your method using logs would only work if 6% was a continually earned annualised rate (called the force of interest). My formula using "i" assumes a single annual interest payment for compounding purposes.

For all the other approximations inherent in all of this I think this is more bother than it's worth to be honest. Why not just assume you can outperform inflation long term by 1% and that gives you a factor of 100 to apply to whatever you want to get back each year.

scrumpydave

davis - no link I'm afraid, but try searching for "annuity functions" online. The forumlae are very simple and the proofs aren't too hard either.

davis

Yeah, I did wonder about the effect of the logarithms, but they've been close enough for me before to not worry. I like the factor of 100 rule though...

Also, "online", you say? Hmmm. Interesting. If only there was some type of engine for searching through all of this...

Cheers

kurako

If you want 1000 per month then that is actually fairly easy to calculate:

http://www.investopedia.com/articles/03 ... z1gi8OpCBm

If you assume you're not going to live forever then you can calculate the value of a bond paying the same ampount. This is the annuity less another one which starts on the day you plan to pop you clogs:

http://www.investopedia.com/university/ ... z1gi8OpCBm

If you want to preserve 1000 in line with inflation then you will need an index-linked gilt of some sort or other.

http://www.fixedincomeinvestor.co.uk/x/ ... tml?id=206

I'll leave deriving the equation as an exercise. Ha ha ;-)

DrLex

scrumpydave wrote:

My time has come at last! [...]

That's the last time I expect my actuarial qualification to be useful on a cycling forum...

I remember doing Duxbury calculations in my former life when negotiating clean break settlements. Always made for some eye-watering numbers.

scrumpydave

DrLex wrote:

scrumpydave wrote:

My time has come at last! [...]

That's the last time I expect my actuarial qualification to be useful on a cycling forum...

I remember doing Duxbury calculations in my former life when negotiating clean break settlements. Always made for some eye-watering numbers.

I'm sure David Cameron would agree with you there!