Recap: this feature used to be invoked in "Classic Ancestry" by clicking a link underneath the name of the person on their profile header:

The "View relationship to me" link appeared on trees which had a person's name designated as "Who you are in this tree" on the Tree Settings page.
If the user clicked the link, the relationship to that person would be calculated, and the link would change to a relationship descriptor such as "your great-aunt", which would be "sticky" until you changed the person designated as You. If there was no person designated, the link would not appear.
However, since the transition to the 'New Ancestry' in 2015, the calculator has not been working reliably. On the same profile as the image above, the link to the relationship calculator does not appear. I checked some of the profiles which previously did show a relationship, and the lines with the relationship are not showing. Presumably if the feature was working, they would appear directly under the name of the person and their vital records, just as they did before.
Ancestry has admitted that users were reporting problems on their blog in the Feature Updates listed below (see Relationship to me string is randomly not appearing):
As of the December 11th Feature Update, fixing the Relationship Calculator had not yet appeared on the list of things Ancestry is working on.
Note that in the comments, user Jim Ackerman posted on December 20, 2015 at 8:56 pm:
Just wanted to add that I too am waiting for the relationship
calculator to get fixed. I reported this problem months ago. I hope it
doesn’t take as long to fix this as the phantom hints problem.
and @PolyGeo just posted:
I too am waiting for the relationship calculator to get fixed.
Getting Ancestry.com to show "view relationship to me" on profiles?
Please post comments on the Ancestry blog -- and keep posting them with each new feature update. They will not restore the features that we had in Classic Ancestry unless people demand it.
Message from Crista Cowan in the chatroom after today's (22 Dec 2015) live broadcast of her Barefoot Genealogist talk:
The relationship calculator was fixed and released onto the site
yesterday morning. If you still don't see it, you might be working off
an old cache of your page. Clear cache and cookies and see if that
helps.
On 23 Dec 2015 the "relationship to me" seems to have returned to the profile in the question:
