When does a DCF produces a lower valuation than the LBO

Agree - a nit, but I think it should be COE vs. IRR (i.e. a PE firm's COE) to be apples-to-apples. If this is for PE interview prep, I assume this is the answer they are looking for.

I actually disagree with the answer above about declining industry, however, as the scenario provided is not really apples-to-apples across the LBO and DCF scenarios - it's basically arbitrarily making the exit multiple / TV in an LBO scenario to be higher than in a DCF scenario, which is the only reason the value of the business is higher. The interim cash flows in a DCF scenario are more valuable than in an LBO scenario, assuming COE Return hurdle. For TV, you don't have to use perpetuity method in a DCF, you could use a market multiple in your exit value / TV as well or just project out cashflows for a longer period of time (which seems to be what's happening in the LBO scenario). Also, in an LBO scenario in an obviously declining industry, you should model multiple contraction to be intellectually honest.

The way I think about it, an LBO is really just a form of a regular DCF valuation with leverage assumptions that are typically very aggressive, the only difference really being a DCF is seeing what value is implied by a specific discount rate, an LBO is seeing what specific value you need to pay to hit a specific return / discount rate. But really the same thing at the end of the day - you project out the CFs to the equityholder and discount them to determine what you're willing to pay for them. So basically, it's really only the discount rate that matters at the end of the day.