After some further research, I found a fix for this, on the following webpage.

Unable to sign Outlook 2010 VBAProject

The fix is to add an extra step (underlined below).

----------
- Create certificate "MyName" using selfcert.exe.
- Export certificate "MyName" from personal store to .cer file.
- Import certificate "MyName" from .cer file to trusted store.

- Go to Outlook > Developer > Visual Basic > Tools > Digital Signature > Sign As; click on Choose button. At this point, the certificate I created is displayed (there is no option to specify a store or certificate; the one I created is displayed automatically), and when I click on the link "Click here to view certificate properties", it appears to be the trusted one ("This certificate is intended for the following purposes: Ensures software came from software publisher ... ", etc.).

- Click on OK. It then says "The VBA project is currently signed as Certificate Name: MyName"

- Click on OK, then click on the Save icon (or File > Save, or Ctrl + S). (If this step is skipped, the digital signature is not saved.)

- Go to Outlook > Developer > Macro Security and select option "Notifications for digitally signed macros, all other macros disabled."

- Click on OK, close VBA, close Outlook, click on "Yes" when prompted "Do you want to save the VBA project 'VbaProject.OTM'?".

- Re-open Outlook.
----------

Now the macro will run without any security prompts.