Posted on 12 September 2020

Updated 28 May 2022 to due to updated odds and distribution of prizes

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 given and the estimated prize distribution here for June 2022. See below for the details of the calculation.

Prize Money Interest rate
25th percentile
Median
75th percentile

### The details

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.

If the number of bond draws (number of bonds * months) is greater than $25 * \left(1 - p\right) / p$, where $p$ is the probability of a prize, we use the normal approximation to the binomial distribution to get the number of prizes, rather than using the binomial distribution directly. For these large numbers of bond draws the approximation is extremely good, and performs better than the exact calculation.

Once the number of prizes is determined, it then draws each prize according to the distribution of prizes (with replacement).