Premium Bonds: What's the Real Expected Value?

You buy bonds in £1 units, from a minimum of £25 up to a maximum holding of £50,000. Every £1 bond is entered into a monthly prize draw with odds of 22,000 to 1. Instead of paying interest, NS&I pools what would have been the interest and pays it out as tax-free prizes ranging from £25 to £1 million, with two £1 million jackpots each month and well over six million prizes in total.

The prize fund rate, currently 3.8%, describes the size of that pool relative to the total money invested. If every pound earned its mathematically fair share, a holding would return 3.8% a year. The catch is that prizes are lumpy and indivisible: you cannot win 3.8% of a pound, you either win a £25 prize or you win nothing. That lumpiness is where the average and the typical experience part company. Bonds are also fully backed by HM Treasury, so your capital is completely safe.

The expected value of a Premium Bonds holding is, by construction, the prize fund rate: 3.8% a year. But it helps to see where that comes from. With odds of 22,000 to 1 per bond per month, each £1 has roughly a 1-in-1,833 chance of winning something in a year. The average prize works out at about £70 (the prize pool divided by the number of prizes).

• £1,000 buys 1,000 bonds, giving an expected half a prize per year - one win roughly every two years.
• £10,000 gives an expected 5 to 6 prizes per year.
• £50,000, the maximum, gives an expected 27 prizes per year.

Multiply expected prizes by the average prize and every holding lands back at 3.8%. That is the expected value. The problem is that expected value describes the long-run average across millions of bonds, not what any single saver is likely to see in a given year.

Because the prize distribution is extremely right-skewed. The overwhelming majority of prizes are the smallest, £25, while two enormous £1 million prizes and a scattering of mid-sized ones pull the average upwards. A handful of huge winners collect a large slice of the pool, leaving everyone else below the mean.

This is the difference between the mean and the median. The odds tell you a win is rare for a small holding; the skew tells you that even when you do win, it is almost always the £25 minimum. So the 50th-percentile saver - the one with average luck - earns noticeably less than 3.8%, because the average is being held aloft by jackpot winners they will almost certainly never join. Quoting the mean as if it were the result any single saver should expect is the single most common way Premium Bonds are misunderstood.

With average (median) luck, the picture looks very different from the headline:

• £1,000: you expect about one prize every two years. In a typical year you win either nothing or a single £25 prize, so the realistic return is roughly 0% to 2.5%, and more often than not it is zero.
• £10,000: around five or six small prizes a year, almost all £25, for a typical return of roughly 1% to 1.5%. The headline 3.8% (about £380) assumes a share of bigger prizes most holders never get.
• £50,000: with about 27 prizes a year, the law of large numbers finally starts to work in your favour. Your total drifts closer to the average, typically landing somewhere around 3% to 3.5% - still usually a little under the headline, but far more predictable.

The pattern is the whole lesson: the more bonds you hold, the closer your actual return creeps to the expected value. Small holders are playing a near-lottery; only large holders experience anything like the advertised rate.

Sometimes, and this is where Premium Bonds earn their keep. All prizes are tax-free, which matters depending on your Personal Savings Allowance. Basic-rate taxpayers can already earn £1,000 of ordinary savings interest tax-free each year, higher-rate taxpayers £500, and additional-rate taxpayers nothing.

For a basic-rate taxpayer with modest savings, the tax-free wrapper adds nothing they did not already have, and a normal easy-access account paying around 4.5% beats Premium Bonds on both expected return and certainty. But for a higher-rate taxpayer who has used their £500 allowance, a tax-free 3.8% is equivalent to roughly 6.3% of taxable interest; for an additional-rate taxpayer with no allowance left, it is closer to 6.9%. That gross-equivalent return is genuinely competitive - but only for someone holding near the £50,000 maximum, where their actual return approaches the mean rather than languishing below it.

For a basic-rate taxpayer with a small holding, Premium Bonds are a weak expected-value choice: a lower average return than a standard savings account, and a median return that is frequently zero. The appeal there is purely the fun of the draw and the safety, not the maths.

For a higher or additional-rate taxpayer who has exhausted their Personal Savings Allowance and can hold close to £50,000, the case is genuinely reasonable. The tax-free gross-equivalent return is competitive, the law of large numbers makes the outcome reliable, the money is fully government-backed with no upper protection limit, and the tiny chance of a life-changing prize is a free lottery ticket on top. The honest verdict is that Premium Bonds are not a scam and not a slam-dunk: they are a tax-and-scale play dressed up as a game of chance.