It’s a great sales pitch: “Don’t lose your chance to win big with premium bonds”. But new research, exclusive to The Sunday Times, reveals most premium bond holders lose – even those who think they’ve won.

Nearly 24m people, not far from half of Britain’s population, hold premium bonds from National Savings & Investments, the government’s savings arm, with a total of £36 billion invested. Yet no matter how many tax-free bonds you hold, and no matter how long you hold them, most people don’t just win less than a decent savings account, even after tax, they win less than retail price inflation. Meaning that, in real terms, most bond holders are actually losing money and seeing their cash shrink.

Someone with a typical £1,500 in premium bonds has only a one in five chance of beating inflation over a year. And someone with £1,000 invested has a 60% chance of winning nothing.

Cracking the premium bond probability equation has been a pet project. I’ve spent the past few months developing a web calculator that reveals what you’re likely to win, based on the value of your bonds and how long you hold them. The maths is so complex I recruited a post-doctoral cosmology statistician to devise the algorithm and Dr Malwina Luczak, applied financial mathematics lecturer at the London School of Economics, has verified the methodology. The new calculator at moneysavingexpert.com/premiumbonds and the statistics it throws up are shocking.

How do premium bonds work?

Premium bonds are simply a savings account where the interest paid is randomly assigned through a monthly prize draw, and where you can withdraw your cash whenever you want. The minimum holding is £100 (or £50 for monthly standing orders) and the maximum is £30,000. You can opt to have the winnings paid out or to automatically buy more bonds.

Payouts are getting worse

The total prize pool is determined by the interest rate, now 3.8%. This means if you owned every bond in existence, you’d get 3.8% of their total value over a year. It’s easy to assume this means that for every £100 put in, you’d expect £3.80 back, but as I’ll explain later, the way the prizes are distributed mean you’re likely to get much less than that.

First, let’s look at the interest rate. There’s a big gap between it and Bank rate at 5.75%, although the rate will go up this week. Other NS&I products, which have to compete with the banks, pay more, such as its Isa at 6.3%.

What’s more the rate is also lower than inflation at 4.4%. This is a big change from the start of this decade when premium bonds paid out almost double inflation.

Worse than savings accounts

Let’s assume, for the moment, that premium bonds actually deliver 3.8% (which, as I’ll explain later, they don’t). Because premium bonds’ interest rate isn’t taxed, a basic-rate taxpayer would have to earn 4.75% gross from a savings account to get the same amount, whereas the best easy-access account, from ICICI, pays 6.3%. However, a higher-rate taxpayer would need to earn 6.33%, roughly on a par with the best accounts, so for them there is some benefit.

Most people win much less than the interest rate

Even though premium bond rates stack up poorly compared with decent savings rates, the stated interest rate is still misleadingly generous. Your actual payout is likely to be much less, because the rate is skewed by the distribution of prizes.

Let me start with a simple example. Suppose a contest offered a £1m prize and allowed 1m people to buy a ticket costing £1. It could then argue the average winnings per ticket were £1, even though 999,999 people would win nothing.

A similar effect happens with premium bonds. It states the interest rate is 3.8%, but most people will win less than this, because the minuscule number of people who win the big prizes skews the payout average and makes the interest rate look more generous.

This has an enormous impact. For example, put £100 in premium bonds over a year and most people win nothing.

The interest rate predicts a win of £3.80 over a year, but that’s impossible because the smallest prize is £50. Enter the details in my free web calculator and it shows 19 out of 20 people with this amount won’t win a thing, leaving one in 20 winning £50 or more.

Put £1,000 in premium bonds over a year and while the interest rate should predict a £38 win, 61% still win nothing.

The probability is more complex than it appears. NS&I lists on its website enough data to allow anyone with a calculator to work out the chance of one bond winning a £50 prize over a month (1 in 27,000, by the way). Yet to work out the chance of someone with 5,000 bonds winning £500 or more includes countless variables.

To win £500 could involve one £500 prize, or five £100 prizes, or ten £50 prizes, or a combination of them. Then remember there’s a draw each month, so to calculate this over five years means 60 draws and you see why the probability was impenetrable – until now.

There are some bizarre results. For example you’ve more chance of beating a savings account with £1,000 than £5,000. This is because the likelihood of beating a savings account is higher if the amount you need can be made up from an easy combination of prizes.

There is, of course, the chance of winning £1m, but for every £1 premium bond the odds are one in 18 billion – which makes the lottery’s one in 14m chance seem positively generous, although of course with premium bonds you don’t lose the £1 stake.

Who should put money in?

They’re only worth considering if you’re a higher-rate taxpayer who has used up your cash Isa allocation. For those who like a flutter, your cash is protected, but you should only really put in a small sum.

Martin Lewis is a broadcaster and the creator of Moneysavingexpert.com, a free website dedicated to showing people how to save money