How should you treat a company's stake in a publicly-traded affiliate when building its EV - book value or market value?

Use the MARKET value of the stake (shares owned x current market price), not the equity-method book value, because the listed price is an observable, more accurate measure of what the holding is worth and could be sold for. Subtract that gross market value from EV as a non-operating asset (its earnings aren't in EBITDA, so the value comes out of EV); subtracting the full market value is the standard default. As a refinement - especially for a large, low-basis stake - you'd haircut for tax on a hypothetical sale and subtract net of tax. For private associates, fall back to carrying value or an implied multiple.