Many firms like to group a client's accounts together, to provide a consolidated report that presents the client with the "overall picture" of what they hold. This is commonly done in the brokerage arena, as well as by others. So, how does one calculate performance for such an arrangement?
Well, what question are you trying to answer? I believe it's "how am I (the client) doing, overall?
And if so, then the answer is quite simple: a money-weighted return. Ideally, we'd use the internal rate of return to do this, though Modified Dietz would work as an approximation to the IRR.