hhdl

The basic assumption is that the true cost to a loyalty/reward program of a redemption is the difference in the provider's net cash profit/loss between the universe(s) where the redemption happens and the universe(s) where the redemption doesn't happen. Both of those are of course, impossible to know with certainty, so thinking in terms of probabilities is quite useful. If in the absence of the redemption, the flight/hotel would have sold out (or the relevant class of seats/rooms would have), the difference is basically what the last seat/room would have sold for. Otherwise, the cost of the redemption is arguably some likely-small premium over the marginal cost of the seat/room (in the airline case: fuel to move a person of typical weight the distance of the flight, the labor cost of whatever extra portion of labor is required (FA, cleaning), and the wholesale cost of whatever amenities were provided; in the hotel case, it's the small amount of electricity an occupied room draws over a vacant room, a housekeeper to clean the room, and things like wholesale cost of a complimentary breakfast).

In the formula provided, p is basically the probability that the latter case holds and (1 - p) is the probability that the former case holds (one can make things a lot more complex and maybe improve things by considering more than 2 possible outcomes). m is the marginal cost plus some small premium, and r is the price the last seat/room would fetch in cash. c is an internal valuation of a point; it's almost certainly more than what "providers" (the businesses of flying people in metal tubes or renting rooms by the night) "pay" the loyalty program for points, and a lot less than the cash purchase rate for points and less than what transfer partners effectively pay for points, and it's also greater than the value at which the points are carried as a liability on the balance sheet (if nothing else, because of points lost through account expirations, etc.). In Hilton's case, I'd spitball that the provider purchase rate is 0.1 cent per point (though this is insignificant compared to the other fees properties pay HLT), the internal valuation is something like 0.2 cents per point, and transfer partners like AmEx effectively buy points at 0.25 cents/point (considering that AmEx's swipe fees at airlines and restaurants are probably something like 2%, you can see how the Aspire is profitable for AmEx even if almost all of the annual fee goes to HLT to buy Diamond status).

In the airline case, of course, m is basically fun-bucks as are the notional purchases of points from flying. An airline has broad latitude to make the people-in-metal-tubes business look good at the expense of the reward/loyalty program or vice versa; to the extent that a hotel program's properties are owned by the program, the same latitude exists. The further-and-further the provider is from the program, the more I'd expect the program to be close to my formula: imagine Oneworld launching an alliance-wide FF program and it would probably somewhat resemble Hilton's by being largely revenue+segment (vs. distance) based for earn and dynamic for burn.

As noted, if cash prices are being adjusted by a yield management system to minimize p and manage to do so reasonably well, the rewards will basically become a straight percentage-of-spend (especially to the extent that points are awarded based largely on spend). However, if the cash price were constrained from falling to increase utilization, e.g. because discounting too much would ruin the image of flying first/business or staying in Bora Bora, then this formula suggests that the apparent redemption rate will look really appealing (and that there will be more award redemptions than cash purchases). Comparing typical redemption rates for a Ritzstoria resort to any number of Courtgardens provides some evidence here: it wouldn't really surprise me if a lot of Doubletrees in the US get more from HLT per award stay than the Conrad Koh Samui.