# Simple maths for investors

Maths is not my strong suit, but it doesn’t half1 come in handy for keeping a grip on your investments.

In this post we’ll look at a few simple investing maths techniques to help you stay on top of your financial life.

Apologies in advance if you find it all too simple, or too hard, or too clumsy. When it comes to maths, I’m like a foreigner with an amusing accent.

### Percentages

To do percentages on your phone calculator, first divide the percentage by 100 and forget the % sign. So:

10% / 100 = 0.1
1% / 100 = 0.01
0.1% / 100 = 0.001

Now you have the percentage as a decimal figure you can multiply with other numbers in your life to find percentage values.

For example, let’s say you hold £10,000 in a global index tracker that sports an Ongoing Charge Figure (OCF) of 0.1% and you want to know the cost in £££s.

Take the percentage figure, 0.1%, and divide by 100:

0.1% / 100 = 0.001 (the percentage as a decimal figure)

Multiply the decimal by your holding:

£10,000 x 0.001 = £10 (the OCF’s cost per year)

From there, you can decide if a 0.09% global tracker is worth switching to. First, find the percentage saving between the two trackers as a decimal figure:

0.1% – 0.09% = 0.01%

0.01% / 100 = 0.0001

How much does that save us on our £10,000 holding?

£10,000 x 0.0001 = £1 (saved per year)

Quick, call Martin Lewis!

To convert fractions to decimals: divide the top by the bottom. That is, divide the numerator by the denominator.

¼ = 0.25

⅔ = 0.667

Multiply the decimal by 100 to get the percentage:

0.25 x 100 = 25%

0.667 x 100 = 66.7%

Also handy:

x% of y = y% of x

So:

8% of 50 = 50% of 8

50% of 8 = 4

8% of 50 = 4

### Gains

How much is a 20% gain worth to you?

So:

(20 + 100) / 100 = 1.2

Multiply this number by your holding:

£100 x 1.2 = £120 (your holding after a 20% gain on £100)

### Losses

Losing money is not so much fun. Maybe the market has plunged 20%. Or the taxman is after you.

How much are you out of pocket?

Divide your loss by 100 to get the decimal: 20% / 100 = 0.2

£100 x 0.2 = £20 (loss)

Or you might wonder: What have I got left after the taxman takes his 20%?

Well, after a 20% loss you’ve got 80% left:

£100 x 0.8 = £80 (remaining)

### Regaining losses

What gain do you need to recover from a 20% loss?

20% / 80% x 100 = 25% (gain needed to make good the loss)

Where 20 is the percentage lost, 80 is the percentage left after the loss, and 25% is the gain needed to breakeven.

Here’s the proof:

£80 x 1.25 = £100 (the amount we had before the 20% loss)

You can also use these calculations to work out how much tax relief you’re due. For example, a 20% tax-payer should get 25% of their net contribution returned to their SIPP in tax relief.

Play around with the numbers and you’ll see very big losses can be hard to recover from.

Picture a 75% loss:

75% / 25% x 100 = 300%

You need a 300% cumulative gain to climb back out of the hole. *shudder*

### Compounding gains and losses

We rarely make a single gain or loss and that’s the end of it. Our portfolios are advancing and retreating every year, every day – every nanosecond if you check too often.

You can string a sequence of returns together like this:

£100 x 1.05 x 0.9 x 0.85 x 1.11 x 1.32 = £118

…where the returns are 5%, -10%, -15%, 11%, 32%.

Negative compounding can set you back quickly. Consider how your salary is corroded by annual inflation unmitigated by an offsetting pay rise:

£100 x 0.96 x 0.98 x 0.97 x 0.95 x 0.97 = £84

Now your salary buys 16% less than it did five years ago. Or it’s fallen to 84% of its former value.

You can also use this method to see by how much a tracker is lagging behind its index every year.

Returning to positive territory, those magic compound interest calculators work the same way:

£100 x 1.1 x 1.1 x 1.1 x 1.1 x 1.1 = £161

Where a £100 lump sum accumulates 10% interest once a year for five years.

You can save on typing by multiplying your interest rate by the power of the number of years you’ll compound.

So 1.1 is multiplied by the power of 10 if you’ll earn the 10% interest for 10 years. (This is assuming annual interest payments – if interest is compounded daily or monthly then you’ll need to dive deeper into the formula.)

You can also quickly use this formula – or a compound interest calculator – to check the damage done by expensive funds:

Expensive active fund

£10,000 lump sum invested
£6,000 invested every year
40 years compounding
5% expected return minus 0.75% OCF = 4.25% return p.a.
Future value = £683,497

Cheap tracker fund

£10,000 lump sum invested
£6,000 invested every year
40 years compounding
5% expected return minus 0.1% OCF = 4.9% return p.a.
Future value = £809,781

£809,781 / £683,497 = 1.185
1.185 x 100 = 118.5
118.5% – 100% = 18.5%

You earn 18.5% more by avoiding high fees.

### Weighted average

You might want to know how much each asset class contributes to your overall portfolio expected return, or to the overall cost of your portfolio.

Enter weighted averages!

Let’s say your portfolio breaks down like this:

 Index fund Allocation (%) OCF (%) Global index tracker 50 0.2 UK index tracker 10 0.1 UK bond index tracker 40 0.15

The weighted average will tell you the overall OCF of your portfolio, after taking into account how your money is allocated across the differently priced funds.

First, multiply each fund’s allocation by its OCF.

For instance, the global index tracker’s weighted OCF = 0.5 x 0.2 = 0.1%

Then add up each weighted OCF to find your portfolio’s total OCF:

 Index fund Allocation (%) OCF (%) Weighted OCF (%) Global index tracker 50 0.2 0.5 x 0.2 = 0.1 UK index tracker 10 0.1 0.1 x 0.1 = 0.01 UK bond index tracker 40 0.15 0.15 x 0.15 = 0.06 Total portfolio OCF 100 0.17%

Go through the same process to estimate your portfolio’s expected return, but replace the OCFs with each asset class’s expected return figure.

How about if you have two different accounts on two different platforms and want to know your overall return?

Weighted average swings into action again:

Account one is worth £100,000 and returned 7%.

Account two is worth £250,000 and returned 5%.

Overall return = [(£100,000 x 7%) + (£250,000 x 5%)] / (£100,000 + £250,000) = 5.57%

Here’s the slow-motion action replay:

£100,000 x 0.07 = £7,000 (the return from account one)

£250,000 x 0.05 = £12,500 (the return from account two)

£7,000 + £12,500 = £19,500 (overall return)

£19,500 / £350,000 x 100 = 5.57% (the weighted average return of your accounts)

### Pound cost averaging

Pound cost averaging can come off as some kind of witchcraft – shares go into free-fall and somehow you come up smelling of roses. But it’s the average price that counts.

En guarde!

You buy one share at £100
The market tumbles 50%
You buy two more shares at £50 each
You’ve paid £200 paid for three shares

£200 / 3 = £66.67 (average price paid)

When the market share price bounces back above your average share price then you’re in profit:

3 x £66.68 = £200.04 (back in the black)

### Profit and loss

How do you calculate a profit or loss as a percentage?

Like this:

(Price sold – Price bought) / (Price bought) x 100 = profit or loss %

For example, the price bought is the amount you originally paid for the investment, say £100, and then we sold at £110:

£110 – £100 = £10 (profit)

£10 / £100 x 100 = 10% (gain)

You probably paid some trading fees along the way and maybe got a dividend, too:

((Price sold – Price bought) + Income gain – Costs) / Price bought x 100

£110 – £100 = £10 (profit as before)

£10 + £4 – £12 = £2 (after adding a £4 dividend and subtracting £12 in trading fees)

£2 / £100 x 100 = 2% gain.

(Welcome to The Investor’s life!)

### Financial independence: How much do I need?

The big kahuna – what will it cost you to stick it the man?

First calculate how much annual income you need for financial independence (FI).

Let’s say you can live on £25,000 a year.

Now choose a sustainable withdrawal rate (SWR) that you’ll use to drawdown your wealth when you reach FI.

Let’s say 3%, which is 0.03 in decimal.

£25,000 / 0.03 = £833,333 (the stash you need to accumulate in today’s money to declare FI at a 3% SWR)

The proof:

£833,333 x 0.03 = £25,000
£25,000 / £833,333 x 100 = 3%

Side note: 3% SWR is a more conservative version of the ‘4% rule’ commonly bandied about. Because it’s more conservative, it’s safer. This is not a party political broadcast.

‘Multiply by 25’ is another way of expressing the 4% rule: 1 / 0.04 = 25.

Multiply your annual income target by 25 and you get the same result as dividing by 0.04. Multiplying by 33.333 is the same deal but for a 3% SWR.

The rule of 400 is a quick way of estimating how much capital you’ll need to support a monthly expense in your FI golden age.

Say you spend £40 a month on the gym:

£40 x 400 = £16,000 (the capital to support that spending during FI using a 3% SWR)

Now you can decide whether that gym is really ‘giving you joy’ or not.

You can work out the yield of an annuity using SWR maths, too. If the annuity company wants £833,333 in exchange for an income of £25,000 then they’re offering you a yield of 3%:

£25,000 / £833,333 x 100 = 3% yield.

You can compare that yield to your other investment options.2

### Probability: Will you need all that money?

You and your significant other are planning a long and happy retirement. The longer you expect to last, the more conservative your SWR should be.

But you can be too conservative, if you’re not realistic about your survival chances in old age.

It’s easy to be squeamish about this but what are the chances of either of you surviving to, say, 100?

Jill has a 15.4% probability of living to 100 according to the Office For National Statistics (ONS) life expectancy calculator.

Jack has an 11% chance.

What’s the probability they’ll still need an income for two at 100?

0.154 x 0.11 x 100 = 1.7%.

Gosh. I don’t know who the hell Jack and Jill are but I’ve got a lump in my throat.

Quick, let’s get Vulcan on this!

The probability formula that determines our couple’s joint chances is:

p(A and B) = p(A) x p(B)

We need a different formula to figure the chance of Jack OR Jill surviving to 100… [I can’t look! – The Investor]

First, add their individual probabilities together:

0.154 + 0.11 = 0.264

Now multiply their individual probabilities together:

0.154 x 0.11 = 0.01694

Subtract the second number from the first:

0.264 – 0.01694 x 100 = 24.7% the chance of Jack OR Jill surviving to 100.

The or probability formula is:

p(A or B) = p(A) + p(B) – p(A and B)

This formula is used because Jack or Jill’s survival are not mutually exclusive events. They can both survive and that’s OK. They love each other.

When the events are mutually exclusive you’d use:

p(A or B) = p(A) + p(B)

There can be only one!

Chances of Jack and Jill not making it to 100? Just for completeness:

0.846 x 0.89 x 100 = 75.3%

If you’re dying to try this for yourself then the ONS publishes extensive life expectancy data for the UK.

Another side note: Here’s the Monevator take on life expectancy and your retirement plan.

Back in the present, Jack and Jill are planning a 40-year retirement from age 60 with a 3% SWR. They’ve baked in a 10% failure rate.

The failure rate means that in 10% of their scenarios, they’d need to lower their income at some point because of a poor sequence of returns. But for the failure rate to matter, Jack and Jill have to be alive to worry about it at the time.

The probability of that is:

0.1 x 0.247 x 100 = 2.47% (failure rate)

Where there’s a 10% chance of failure within 40 years and – from our sums above – a 24.7% chance of either of our two protagonists living to 100.

The two events are independent of each other3 so we multiply their probabilities together and end up with a 97.5% chance of success!

Of course success could mean that everybody is dead, but let’s set that aside for the moment.

The 10% failure rate only matters 25% of the time, which turns out to be a very small 2.5% probability of both events occurring simultaneously.

Earlier we calculated an FI target of £833,000 – but inflation is going to weaken the potency of your stash like an intestinal worm. This means every year your target will be less capable of delivering the purchasing power you require.

To maintain purchasing power, you must upweight your target by the inflation rate.

If inflation is 3% this year then multiply your target by 1.03 after 12-months:

£833,333 x 1.03 = £858,333 (adjusted for 3% inflation)

Next year, multiply £858,333 by year two’s inflation number.

Perhaps it’s 2.5%:

£858,333 x 1.025 = £879,791 (FI target after two years of inflation adjustment)

Adjust last year’s target by this year’s inflation rate every year.

If you stick to £833,333 then your portfolio will be like a puffed-up bodybuilder – it will still look good but isn’t really that strong because your returns are nominal. We live in today’s money, and inflation adjustments keep our returns real.

The same upweighting technique applies to your investment contributions and your target income.

The UK’s inflation figures are published by the ONS. Personally, I adjust for Retail Price Index (RPI) inflation because it contains housing costs and Consumer Price index (CPI) inflation does not.

Also RPI is always higher, and I’m a glutton for punishment.

RPI has come in for much official criticism, however, and so now there’s CPIH – the Consumer Prices Index including owner occupiers’ housing costs.

So you could use CPIH if RPI looks depressing and you’re not into government conspiracy theories about how they suppress the true inflation figures to keep us peons in our place.

[Editor’s note: Please don’t comment about inflation conspiracies below, this paragraph was permitted for leavening purposes only.]

### Tax costs

Your effective tax rate is the total amount you pay in taxes, divided by your gross income.

Perhaps you earn £75,000 and pay £23,000 in income tax and National Insurance (NI).

£23,000 / £75,000 x 100 = 30.7% (your effective tax rate)

After income tax and NI, each £1 of spending costs you:

£1 / 69.3 x 100 = £1.44

Your marginal rate of tax is the amount you pay on each additional pound of income you earn.

How much will I need to cover tax?

If you need £25,000 income to live on then:

£25,000 – £12,500 tax-free personal allowance = £12,500 (taxed at 20%)

£12,500 / 0.8 = £15,625 (the gross income you need)

The proof:

£15,625 x 0.8 = £12,500 after 20% tax.

I haven’t factored in ISAs and personal savings allowance and so on, but you get the idea.

### Platforms: choosing the cheapest

To compare fixed fee Platform A with percentage fee-based Platform B, make the following calculation:

Tot up all the charges you’d incur with the most competitive of the fixed fee platforms you can find on our broker table.

Make sure you count any annual fees, platform fees, dealing charges, and other relevant costs.

Take your fixed fee cost and compare that against the most competitive percentage fee platform.

Total annual fixed fee platform costs divided by percentage fee platform cost = your breakeven point.

Fixed fees at Platform A = £80
Percentage fee at Platform B = 0.25%
£80 / 0.0025 = £32,000 breakeven point.

The breakeven point – £32,000 in this example – refers to your portfolio’s size.

In the example above, we’re better off with platform A if our portfolio is worth more than £32,000. Any less and we should bunk up with platform B.

A few notes:

• Subtract any fixed fees charged by Platform B from Platform A’s fixed fees before making the calculation.
• If one platform offers fund discounts, or doesn’t stock exactly the same products, then you can factor that in using the portfolio weighted average OCF technique detailed earlier in this piece. Work out your portfolio’s average return in £££s on both platforms and add to the fixed fees above.
• Add on entry and exit charges.

Switching platforms can be mucho hassle, and platforms periodically change their charges. Only switch when it’s seriously worth your while.

### Withholding tax cost comparison

Okay, so we’re tempted by a tracker with a cheap OCF but which has a high withholding tax rate on dividends. We want to pit it against a higher OCF tracker with a lower withholding tax rate.

Find fund one’s dividend yield e.g. 2%. Then multiply the dividend yield by the withholding tax rate. E.g. 30%.

0.02 x 0.3 = 0.006 (the percentage cost of the withholding tax)

Add this cost to the fund’s OCF. E.g. 0.1%.

0.006 + 0.001 = 0.007 x 100 = 0.7% (OCF and withholding tax cost)

Fund two has the same 2% dividend yield, a lower 15% withholding tax, 0.2% OCF.

Going through the same process:

0.02 x 0.15 = 0.003 (the percentage cost of the withholding tax)

0.003 + 0.002 x 100 = 0.5% (OCF and withholding tax cost)

Fund two is 0.2% cheaper than fund one – despite its higher OCF – because it has a lower withholding tax rate.

### Rule of 72

The famed rule of 72 calculates approximately how long it takes to double your money at a given rate of return. Divide 72 by your actual or expected return rate.

For example:

72 / 4% = 18 (years to double your money if you earn 4% p.a.)

For a quick rebalance, multiply your total portfolio value by your target allocation for each asset class.

For example:

Target asset allocation
50% Global equity
50% UK gilts

Total portfolio value = £50,000

0.5 x £50,000 = £25,000 target value for each asset class.

Now subtract the actual value of each fund from the target value (assuming that each fund represents an asset class).

For example:

Global equity tracker
£25,000 – £30,000 = -£5,000
So sell £5,000 of this fund.

UK gilts tracker
£25,000 – £20,000 = +£5,000

### Preference for lotteries

People love a bet. Long odds and big pay-outs are the stuff of dreams. Next time you’re counseling a relative on that lottery ticket purchase, here’s the math:

£1 ticket price
£1,000,000 jackpot
Odds of a win: 0.00001%

The value of the gamble is:

£1,000,000 x 0.0000001 = £0.1 or 10p. The value of your gamble vs the ticket price is -90p.

What if you have the opportunity to invest £1,000 for a 50% chance of making £1,500 vs a 50% chance of losing the lot?

(0.5 x £1,500) – (0.5 x £1,000) = +£250

That’s a better bet than buying a thousand lottery tickets if you can afford the potential loss.

### The final reckoning

Obviously you can use a calculator to do all the work. There’s a Monevator page with a ton of useful financial calculators.

But calculators break. And I think it’s empowering to know how these things work, and to be able to double check the information yourself.

If you have a simpler take on the investor maths above or want to share any other techniques, please tell us in the comments below!

The Accumulator

1. Or 50% or ½ or 0.5. Hmm, landing a maths joke is not easy. []
2. Remembering to take into account the certainty of payment and myriad other factors. []
3. Sort of, in theory with a constant SWR. In practice if one died the other might spend less of that annual withdrawal / allowance, saving a bit and reducing the chance of failure. []
• 1 Deltrotter February 26, 2019, 6:19 pm

Brilliant!

• 2 MrOptimistic February 26, 2019, 6:51 pm

This would make a good appendix for a book. Was just rejoicing that I could join in with the rule of 72 but there it was. All hinges on an approximation for ln(1+X) I hope you realise 🙂

• 3 weenie February 27, 2019, 12:33 am

Probably one of the best posts ever – can’t thank you enough!

• 4 PendleWitch February 27, 2019, 9:52 am

I can do maths … this will be easy … whaaat?! x% of y = y% of x?? Why did nobody tell me this?
Lots of useful stuff, but quite heteronormative. (I’m not sure what that means but it seems to be a bad thing.) 😉
Thanks @TA

• 5 Matthew February 27, 2019, 6:56 pm

I find I can’t rely on brokers’ % figures because for one they record dividend as profit for accumulation funds but not for income funds, second if you ever sell anything, your previous gains/losses are forgotten about. I’m not sure how they deal with a partial sale (ie rebalancing) – do they sell the oldest or newest units first? – as this would change what profit they’re reporting on the remaining units.

If we care enough, keep separate records, but its not a bad thing to not track it because that reinforces good behavior – and really the balance matters more than the recorded profit

• 6 Duncurin February 27, 2019, 9:32 pm

I hadn’t realised that x% of y = y% of x either. At first glance it looks miraculous. But in fact it’s mathematically trivial: x.y/100 = y.x/100.

• 7 TomB February 28, 2019, 11:53 am

40 years to compare two trackers funds seems very unrealistic.

The oldest FTSE 100 tracker fund I can find is the HSBC one (launched in 1994). They slashed fees in 2009 from over 1% to 0.27% in response to Vanguard. Another fee reduction followed in 2015 so it now costs 0.17%. This is still ‘expensive’, but in it’s 25 year history, well short of the 40 you suggest, the gap has narrow substantially. This reflects the fact that mis-priced products have to adapt or will not survive.

You’ve fallen victim to the fallacy of extrapolating. Your underlying point still stands but you’re overstating it.

• 8 Matthew February 28, 2019, 12:03 pm

Apparently the rule of 72.7 is based on assuming a return of 10%:
http://www.moneychimp.com/features/rule72_why.htm

As for continuous compounding, I don’t know why they use e, its not rational, but at least it’s real :p

• 9 MrOptimistic February 28, 2019, 3:21 pm

@Matthew. No, it’s a general rule but relies on the approximation ln(1+X) =X which is only valid if X is much less than 1. Here X is the interest rate or gain or inflation or whatever. 10%=0.1in this sense so beginning to stretch the validity.

• 10 Matthew February 28, 2019, 5:42 pm

@mroptimistic

It said they chose r to be 0.1;
“where K is some number that will make the approximation pretty good for some ranges of r (and pretty lousy for others). We’ll choose K to make the approximation work for a return rate of ten percent:”

So I think they approximated just so they could give us a rule of thumb, if you set R to your rate it should work with anything I think, any deviation from 10% would be not completely accurate (but maybe close enough)

I’m also not sure why it uses ln – why e, and not log10? It doesn’t seem to explain that

• 11 MrOptimistic February 28, 2019, 8:12 pm

Hi Matthew. The approximation only works for ln unless you introduce a factor. Have a go in excel!
https://math.stackexchange.com/questions/2320047/why-is-ln1-x-approx-x-when-x-is-small
Basically Pn = Po(1+I)^n. If Pn is twice Po then 2=(1+i)^n. Take log both sides
ln(2)=nln (1+i)
ln(1+i) ~i-smaller terms in powers of i , so if i much less than 1 can forget smaller terms
So ln2=ni. If i is expressed as percentage must divide it by 100 to plug in here. So 100ln2 = ni.
100ln2 = 69 by my calculations. I think the financial types before calculators approximated it to 72 which has lots of factors ( and also pulled the approximation back a tad.

• 12 chris February 28, 2019, 9:59 pm

Beautiful… one to print out and study…

• 13 Cashflow Cop March 1, 2019, 12:13 am

This is such a brilliant post. I need to set some time aside and play around with some of these formulas. Very sad, I know.

• 14 Algernond March 1, 2019, 1:12 pm

This is marvelous TA.
It will definitely be part of sprog’s financial education before she leaves home!
Thank you.

• 15 The Accumulator March 1, 2019, 3:38 pm

Thanks all, I’m glad some of this is useful. Part of me feared this could be the dullest post ever.

@ TomB – I wasn’t comparing two tracker funds. I compared an active fund with a tracker fund (see the headers). You won’t make big gains switching between competitive and comparable tracker funds, I agree. Well, unless you’re still stuck in that Virgin tracker. But I was specifically comparing an industry standard active fund with a tracker fund. That’s where the real gains lie.

• 16 financialfundi March 1, 2019, 9:45 pm

One of my favourite % values is 2.3% another is 105%
The first represents the increased AGR of an actively managed Vanguard fund relative to the equivalent passively managed Vanguard tracker fund.
The second % value – 105% (more than double the money) – is the resultant increase in the size of one’s retirement pot due to the outperformance of the managed Vanguard fund relative to the tracker over a lifetime of investing.

• 17 Peaco1980 March 5, 2019, 9:07 pm

This website is probably the best I have ever come across. The value of information shared is phenomenal. I’ve learnt so much. Thank you!

• 18 Eelis Vatanen March 7, 2019, 7:05 pm

Interesting! I was always great at math and it never occurred to me that people would need these as a refresher. Thanks I suppose for broadening my perspective!

• 19 Laura March 11, 2019, 11:23 pm

Really useful, thanks.
Thought process: “x% of y = y% of x” ?!?!
What kind of sorcery is this?! Why did no-one tell me this?!

Oh wait, it’s just math.