## Which is the better investment – Premium Bonds or National Lottery?

So here’s a fun (and potentially useful) probability puzzle for you all to ponder. Here in the UK we have an investment option called Premium Bonds. From the premium bonds website:

*“Premium Bonds are an investment where, instead of interest payments, investors have the chance to win tax-free prizes. When someone invests in Premium Bonds they are allocated a series of numbers, one for each £1 invested.”*

There’s a prize draw every month and a range of prizes from 25 pounds right up to 1 million pounds. You can get your money back at any time with no penalty. So, if you invest 1000 pounds then you get 1000 shots at winning a prize every month for as long as you leave the investment alone. More detailed information such as odds etc is available here.

The way I like to think of premium bonds is that they are a bit like the National Lottery (click here for discussion of odds) except that you get your money back after you’ve played. So I was thinking that, instead of buying premium bonds, an alternative investment strategy would be to put all of your money into a high interest cash account (paying N% per year) and use the resulting interest to buy lottery tickets.

Both strategies offer similar security (you can get your principle investment back at any time) and both of them are a bit of fun since they are based on games of chance. Assuming I am to choose one of these strategies, which one is going to offer me the best rate of return over the long term?

Here is an on-line lottery simulator to try out

http://justwebware.com/notalot/notalot.html

I have looked at the Canadian 649 lottery (see http://www.mapleprimes.com/maplesoftblog/97816-Application-Worksheet-Lottery-Simulator ) which is similar to the UK National Lottery. There, for every £1 in you should expect to get £0.18 out. That would turn a 1.5% interest rate account into a 0.27% interest rate account.

If you wanted to use your interest to play every lotto drawing, you’d need to save about £10000 at 1.5%. If you were happy to play just once a month, you could fund that with less than £1000 at 1.5%. While for Premium Bonds, there is just a £100 minimum deposit.

Now, to calculate the expected return on the bonds, the February prize pool of 54.4e6 is calculated from 1.5% interest on the deposit pool. That means a total deposit pool of 54.4e6 / (0.015 / 12) ~= 43.5e9 pounds. To be pessimistic, let us assume you will never win any of the “High Value” prizes. In that case the expected win for each £1 deposited is:

(1000*1089 + 500*3267 + 31986*150 + 25*1746099) / 43.5e9 ~= £0.001176

That is equivalent to a 1.4% interest rate and is quite a bit better than the lottery’s expected 0.27%.

I suspect the bonds pays off so much better since 89% of the money (the interest) put in is paid out to high probability prizes paying £25 to £100. While with the lottery only 45% of the ticket money is used for prizes at all. And little calculation suggests that only about 26% of the ticket money is used for the high probability £10 and ~£75 prizes (match 3 and match 4).