I've been looking to scale up the CR of a few Ancient dragons for one of my settings, so I've been figuring out the offensive and defensive CR of each dragon to determine how my scaled-up dragons should compare. The thing is, by my calculations (as per DMG p. 274-281), most of these dragons (virtually all that are young or older) end up with a higher CR than is listed in the Monster Manual.

For example, a Young Red Dragon, listed at CR 10. By my calculations, its effective AC and HP come out to 22 and 356, respectively, making its defensive CR 21. Its average damage over 3 rounds is about 68, with an attack bonus of +10 and a DC of 17, giving it an offensive CR of 10 or 11. Regardless, its CR comes out as around 16. Ignoring the fire immunity and either rounding down or ignoring the fly speed (or the saving throw bonuses) gives a CR of 10, as written, but I can't figure out how exactly they arrived at their result.

For those who are requesting more complete details on calculations:

• Base AC is 18, +2 from a fly speed and a CR of 10 or lower, +2 from 4
  saving throw bonuses, resulting in 22 effective AC


• Base HP is 178, x2 from fire immunity and a CR from 5-10, resulting
  in 356 effective HP


• 22 effective AC and 356 effective HP gives a DCR of 21


• DMG p. 280 says to assume that a breath weapon hits two creatures and
  that both fail their saving throws, giving 112 damage on turns it
  uses its breath weapon.


• DPR for the dragon's multiattack is 46


• The breath weapon recharges on a 5 or a 6, so it is usable for 1 out
  of every 3 rounds on average.


• Per DMG p. 278, the three round average thus comes out to 68.


• DPR of 68, an attack bonus of +10, and a saving throw DC of 17 gives
  an OCR of 11 or 10, respectively, based on if the attack bonus or
  saving throw is considered more used.

Don't double the HP

Doubling the HP should only be made when it actually presents some hindrance for the party due to their inability to do other types of damage, as described in the subsection from the DMG:

If a monster has resistance or immunity to several damage types - especially bludgeoning, piercing, and slashing damage from nonmagical weapons-and not all the characters in the party possess the means to counteract that resistance or immunity, you need to take these defenses into account when comparing your monster's hit points to its expected challenge rating.

Fire Damage is not exactly a main type of damage - although spellcasters usually have some spells that rely on that, it's not enough to actually grant effective double HP to the monster. The martial fighters - which deal good part of the damage in fights - are still damaging the dragon at normal damage.

This already puts the dragon at a defensive CR of 11 - and note that he is at the lower end of the HP bar (178 where the minimum is 176 for the CR 8 HP). Taking the OCR value of 10, you are already at a final CR of 10.

I would still make a point that the average damage from the DMG is kinda optimistic.

Finally, note that the monsters in the Monster Manual do not necessarily follow the DMG guidelines for CR.

Your calculations are mostly correct, but I think they neglect the one final key to determining any monster's true CR. As detailed in this answer, it is necessary to playtest your monsters and see how they do in action.

WotC likely did a playtest with the dragons and determined that setting the CR as high as indicated by your calculations was not representative of their actual threat.

And upon analysis, this makes sense. Doubling the HP of the dragon because of fire immunity isn't very sensible. By the time characters encounter this enemy, they ought to be able to figure out a means to get around a fire immunity. Furthermore, red dragons' reputation preceeds them, so it's very unlikely that players are going to lose a round and a spell slot trying to afflict the enemy with a fireball.

By simply nixing that doubling, you find that the DCR calculation then works out to become 8 for hit points (low end). Adding the AC impacts from base AC, flight, and saves, nets a final DCR of 11.

The OCR is fairly straightforward. The DPR is 68, so OCR is 10. Accounting for higher than average attack bonus bumps that up to 11. So the final OCR is 11.

Presumably this nets a monster whose CR is 11, but then you have to play it. In play, you would likely realize that as a CR11, this thing is pretty squishy. Only 178 hit points against a 4 person party at levels 6+ is probably going to go down fairly quickly. It's more or less a glass cannon with a bit more glass than cannon, so actual playtest results push the CR down to the range of 10.