Posted on 12 September 2020
Updated 19 Aug 2023 to due to updated odds and rate for September 2023
I've got some premium bonds, and wanted a better idea of the distribution of returns than the MSE Calculator provides. Unlike the MSE Calculator, I am quite likely not to update this as often as the rates change. It's based on the odds and rate given here and the estimated prize distribution here for September 2023. See below for the details of the calculation.
The calculator shows a cumulative probability of each prize by sampling from the distribution of total prize money from our bonds over all of the months. It takes 20000 samples to form the distribution displayed. To calculate each sample from the distribution of total prize money from our bonds over the number of months specified, we first work out how many bonds will win a prize, based on the current published odds of winning a prize, and then draw this many prizes from the distribution of prizes.
To calculate the number of prize wins, we draw from the Binomial Distribution. For extremely large numbers of bond draws (in particular if the number of bond draws is greater than , where is the probability of a prize), we use the Normal approximation to the Binomial Distribution for performance reasons.
Once we have the number of prize wins, we draw each win from the distribution of prizes. I have found that the published expected distribution of prizes can be incorrect when the rate changes. I observed this in August 2023, where NS&I used the August rate rather than the September rate to compute the estimated September distribution, meaning the estimated distribution combined with the September odds did not give the September interest rate. See this post for more details and the calculations, and corrected values. To ensure I'm robust to incorrect distributions published by NS&I, I now compute the distribution of prizes according to the published rules governing the distribution of prizes, ensuring we have a distribution that has the correct overall interest rate. If the rate doesn't change in a month, this will match the published prize distribution, which I check whenever I update.