Half of Premium Bonds holders earn less than the cash-ISA equivalent.
Could the tax-free draw underperform a simple savings ISA for some goals?
A UK saver weighing low-risk, tax-efficient options needs clear odds and EV.
They also need the full outcome distribution to assess short-, medium- and long-term goals.
Use an EV and odds simulator to compare Premium Bonds with an ISA.
Input your bond holdings, prize-fund rate and reinvestment plan to see EV, median and the prize distribution.
Test the numbers with your own balance and timeframe.
A Monte Carlo run shows the chance of zero, one or multiple wins.
It converts returns to tax-equivalent rates so you can compare fairly with ISAs.
The Premium Bonds Odds & EV Simulator updates NS&I prize-fund rates and exports CSVs.
Premium bonds prize odds & EV simulator: how it works
The simulator models monthly draws and odds per £1 bond.
It treats draws as random events and aggregates many trials to show likely outcomes.
Prize mechanics and entries
One bond number is issued per £1 invested, up to the £50,000 personal limit (2024).
Monthly draws pick winning bond numbers and allocate prizes by NS&I rules.
Odds math and approximations
Expected value is the average prize per bond per year over many draws.
Use a Poisson or binomial model to approximate the chance of zero wins when probabilities are small.
The most frequent error at this point is treating EV as a guaranteed interest rate.
EV is an average across outcomes.
Data sources and updates
The simulator can auto-fetch the current NS&I prize-fund rate and snapshot it for each run.
See NS&I for the rules and fund NS&I Premium Bonds.
The tool stores a timestamped prize-fund series.
This keeps scenarios reproducible across prize-fund changes.
Using the EV and Monte Carlo simulator step-by-step
The simulator runs thousands of random trials to map outcomes.
It reports median, EV and percentiles so the user sees the full shape of possible returns.
Balance, horizon and prize-fund series are mandatory inputs.
Optional inputs include reinvestment rule, number of simulations and RNG seed.
Defaults suit most savers: 10,000 simulations, monthly resolution and an auto-fetched current prize fund.
More simulations tighten percentile estimates.
Outputs and visualisations
The simulator draws a probability density curve, a percentiles table and a bar for prizes.
The bar shows P(0), P(1) and P(>1) prizes per year.
Each visual is labelled and exportable.
A downloadable CSV lists per-run prizes, annualised rates and aggregated statistics.
This supports independent checks by advisers and software.
Simulator flow
1
Input
Balance, years, prize fund, reinvestment
2
Simulate
Run Monte Carlo trials with fixed RNG seed
3
Analyse
Distribution curve, percentiles, P(0)/P(1)/P(n)
4
Compare
Convert to tax-equivalent gross rate and compare with cash ISAs
Provide a concrete CSV schema and a short sample row so advisers and auditors immediately know what is exported.
Example header and single row (comma separated):
run_id,seed,timestamp_iso,starting_balance_gbp,ending_balance_gbp,total_prizes_gbp,number_of_prizes,annualised_return_pct,median_return_pct,percentile_5_pct,percentile_95_pct,prize_fund_series_id,reinvest_flag,num_simulations
r0001,42,2026-04-21T10:15:00Z,10000,10350,350,2,3.50,1.80,0.00,12.40,nsandi_v2026.04.01,true,10000
The CSV should also include a short legend row or a metadata JSON describing units.
State whether annualised_return_pct is gross EV or net after tax assumptions.
Compare tax-equivalent returns: Premium Bonds vs cash ISAs
Comparing EV directly to a cash ISA interest rate misleads unless adjusted for tax.
The simulator converts EV to a tax-equivalent gross rate by marginal tax band.
Use tax rates when comparing: basic rate 20% (2023), higher 40% (2023) and additional 45% (2023).
These bands determine the gross interest that equals a given after-tax outcome.
Use a tax-aware inversion that accounts for the Personal Savings Allowance and regional rates.
To find gross interest G that equals simulated net prizes N, solve this equation:
N = G - tax_rate * max(0, G - PSA)
tax_rate and PSA are set for the saver.
If N ≤ PSA then G = N — no tax is payable in that case.
Otherwise compute G = (N + tax_rate * PSA) / (1 - tax_rate).
Provide worked examples per tax band and note different Scottish bands or non-resident rules when applicable.
Example: if simulated after-tax prizes average £300 on £10,000.
The tax-equivalent gross rate for a basic taxpayer equals 300 ÷ 10,000 ÷ (1 - 0.20).
A clear numeric example helps savers pick between tax-free ISAs and Premium Bonds EV after tax.
Practical comparison table
| Scenario |
Premium Bonds EV (ann.) |
Tax-equiv gross (basic) |
Cash ISA rate |
Recommended |
| £5,000, 1 year |
0.8% (example) |
1.0% |
1.25% |
Cash ISA if priority is median return |
| £50,000, 5 years |
1.8% (example) |
2.25% |
2.0% |
Split allocation worth considering |
When converting prize EV to a tax-equivalent gross rate be explicit about common UK tax features that change the maths.
- The simple inversion gross = net / (1 - marginal_rate) is valid only if all gross interest is taxable.
- It ignores the Personal Savings Allowance (PSA) and regional tax bands.
- A practical approach is to let N be simulated annual net prize cash and PSA the saver’s remaining PSA.
- Find gross G that satisfies N = G - tax_rate * max(0, G - PSA).
- Example: simulated N = £300 on a £10,000 balance.
- For a basic-rate saver PSA = £1,000 and tax_rate = 20%.
- Any gross G ≤ £1,000 is tax-free so G = N if N ≤ PSA.
- Here G = £300 gives a gross rate 3.0%, not 3.75% from the simple inversion.
- For higher-rate savers with smaller PSA (e.g. £500) the algebra differs and Scottish bands change the tax_rate used.
- Non-UK residents may face different withholding rules.
Always compute gross-equivalent rates by solving the formula N = G - tax_rate * max(0, G - PSA).
Use the appropriate tax_rate and PSA for the individual.
Advanced scenarios, reinvestment and pitfalls to avoid
Reinvestment of prizes changes both mean and variance.
The simulator models both cashing prizes out and using them to buy new bonds.
A common case: a saver reinvests small prizes and sees compounding.
The distribution remains skewed.
The most frequent mistakes are:
- Ignoring the £50,000 limit (2024).
- Assuming EV equals a guaranteed return.
- Failing to re-run scenarios after prize-fund changes.
Reinvestment and compounding
If prizes buy new bonds, the effective balance grows and so does exposure.
The simulator compounds prizes monthly when reinvestment is selected.
This works well in theory.
In practice reinvestment raises variance and lengthens time to predictable median returns for small balances.
Limits, joint accounts and rules
Each person can hold up to £50,000 in Premium Bonds (2024).
Joint accounts split entries between named holders per NS&I rules.
Transfers into ISAs follow ISA rules and may affect tax treatment.
Always check HMRC guidance for ISA transfers and allowances: HMRC tax rates.
Common errors to avoid
Do not present EV as a guaranteed interest figure when advising savers.
EV describes long-run averages across many possible futures.
Do not ignore the probability of long winless stretches, especially for balances below £10,000.
Small savers often face a high P(0) in a year.
Methodology, reproducibility and transparency
All formulas, RNG seeds and data snapshots are published with the simulator for reproducibility.
The tool uses semantic versioning and stores a changelog.
EV per year equals prize-fund rate times balance.
For monthly modelling, EV splits by month and aggregates annually.
The simulator shows the step calculation used.
Exact formulations should define variables and context. If p is the chance that a single £1 bond wins in one month, and T is the number of months, then P(no wins) = (1 - p)^{T}.
For B independent bonds, wins across B·T bond-month trials follow a binomial distribution.
Here trials = B·T and success probability = p.
When λ = B·T·p is much less than one, the Poisson approximation P(0) ≈ exp(-λ) is accurate.
The simulator should report which convention it uses: per-bond per-month p or per-draw p.
It should use exact binomial formulas when B·T remains small enough for exact arithmetic.
Data sources and versioning
Prize-fund rates are fetched from NS&I public data with timestamped snapshots.
Each scenario records the prize-fund series used and the retrieval date.
The simulator logs versions using date stamps like v2026.04.21 and keeps archived CSV outputs for audit and comparison.
To make the methodology verifiable, show the algebra that converts a published annual prize-fund rate into per-bond probabilities.
Also show per-draw probabilities and an EV breakdown by prize tier.
- Start with r = annual prize-fund rate and let V = expected prize value conditional on a win.
- For monthly modelling the expected prize per bond per month is r/12.
- The monthly probability that a single £1 bond wins any prize is p_month = (r/12) / V.
- Once p_month is known, λ = B × T × p_month and probabilities follow binomial or Poisson approximations.
- Example: if r = 1.00% and V = £250, expected per bond per month = 0.01/12 = £0.0008333.
- p_month ≈ 0.0008333/250 ≈ 3.33×10−6, so B = 1,000 gives λ ≈ 0.04 expected wins per year and P(0 wins) ≈ exp(−0.04) ≈ 96%.
- Breaking EV into tiers uses p_i = (r × w_i)/v_i where w_i are normalized weights for each prize tier.
- Summing v_i·p_i recovers r and gives the per-tier contribution to EV.
Practical decision matrix and what to do next
A simple decision matrix helps decide whether to choose a cash ISA, Premium Bonds or a split approach.
The matrix uses balance, horizon and tax band.
Short horizon and need for predictable cash favour cash ISAs.
Long horizon and appetite for lottery-style upside may favour Premium Bonds for some savers.
Suggested next step: run the simulator with your exact balance and tax band.
Then review median, the 5th percentile and P(0) before deciding.
Recommended comparison checklist
Run a baseline simulation with the current prize fund.
Then run a pessimistic and an optimistic prize-fund scenario.
Check median and 5th percentile outcomes.
If median or 5th percentile is below the cash ISA equivalent for your tax band then favour the ISA.
Otherwise consider splitting the holding.
Premium Bonds act as a low-risk source of asymmetric upside rather than a predictable savings vehicle.
They suit savers who value the chance of a large prize and accept frequent zero returns.
They are less suitable when predictable median returns matter, especially for goals under five years.
Use the simulator to check if median and 5th percentile meet your goal.
Premium Bonds are not suitable for savers who need guaranteed short-term interest. They are also unsuitable for non-UK residents with different tax treatment or for those who cannot lock funds. Imminent withdrawals make them inappropriate.
Before the FAQ, try the simulator with your real numbers and export a CSV to share with an adviser.
Questions frequently asked
What are my odds of winning with £50,000?
Odds per £1 depend on current prize-fund rate and pool size.
Use the simulator preloaded with £50,000 to see P(0), median and EV for your horizon.
The simulator shows exact annual probabilities, plus percentiles over multi-year runs.
The user sees both chance and expected cash.
Is Premium Bonds EV better than a cash ISA?
Answer: it depends on prize-fund and tax band.
Run a 10,000 trial simulation for £5,000 over one year to compare median and EV with cash ISA rates.
Small balances often show a high probability of zero prizes in a year.
That shifts preference toward predictable ISA yields for short goals.
How often do people win top prizes?
Top prize frequency depends on the total eligible bond pool and prize structure.
The simulator estimates expected top-prize frequency using pool size and draw rates.
Large prizes remain rare and rare wins lift the mean.
This explains why EV can exceed the median.
Can I hold premium bonds inside an ISA?
No. Premium Bonds cannot sit inside a cash ISA.
They remain separate and prize money is tax-free regardless of ISA status.
This separation matters when comparing tax-equivalent returns with ISAs for taxable savers.
Are premium bond prizes tax-free forever?
Yes. Prize money is tax-free in the UK and does not count as taxable income.
That makes direct comparisons with taxable interest require a tax-equivalent conversion.
Keep in mind that tax rules can change, so re-run scenarios if rules shift.
What happens if the prize fund rate falls?
A lower prize-fund rate reduces EV and compresses percentile outcomes.
Re-run simulations with the lower rate to see impact on median and tail outcomes.
The simulator saves historical snapshots so you can compare past and hypothetical future rates.
Your next step
Run focused simulations for 1, 5 and 10 year horizons.
Enable reinvestment if desired and export the CSV for review.
Use tax-equivalent conversion to compare with cash ISA rates before moving money.
