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ElCapitano It sounds to me like you've already decided to purchase the GTX 1080 Ti, regardless of the advice that you receive here. That is completely fine of course.
Your Intel i7-870 is now eight years old. While powerful back it's day, I would now say that it's modern-day CPU equivalent, performance-wise, would be an Intel Pentium G4560. This chip is a dual core hyper-threaded processor that only cost $65 USD on NewEgg.
The user below paired a i5-750 (LGA 1156) CPU with a GTX 1080, and showcased your point: While the bottleneck is present, it was diminished somewhat when he transitioned from the 1080p to 4K resolution game play.
JERMgaming: Will an i5 750 Bottleneck a GTX 1080? - 1080p, 1440p, 4K Benchmarks
https://www.youtube.com/watch?v=TScpVAGNdcI
Kind of funny, but the thing that would frustrate me with that chipset is whether it was limited to only SATA 2 (3.0 GBps), thereby cutting my SSD performance in half. I had trouble locating YouTube performance reviews of the i7-870 with Pascal GPUs. My hope is that the 1080 Ti would be compatible with your current motherboard with the most recent update. If not, then one suggestion is that you could sell your CPU, MB and RAM as a bundle on Ebay and use that money to purchase the beginnings of your future rig. You could get an equivalent Pentium G4560, and Z270 or B250 chipset MB and 8 GB of DDR4 RAM. Then in eight months, when you have some additional money saved up, you could use those funds to update your processor to an i5-7500 or i7-7700K (! B250 business class motherboard does
not allow overclocking).
!
Caution: Kabylake (nor Skylake) CPUs are not compatible with Windows 7, so you shouldn't consider making the switch unless you're on Windows 8.1 or 10. The issue has to do with Microsoft not having native USB 3 support in Windows 7 for those platforms. There is a workaround, but I doubt that you'll want to deal with the hassle of it.