How do I evaluate the CR for this homebrew Phoenix creature?

Question

The determination of CR of simple and standard creature is quite simple, and it suffices to follow the DMG book. But now that I am trying to create a complex creature, with high CR, I don't know how to properly evaluate the CR. Here is my homebrew block stat:

Phoenix

Large elemental, neutral

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Armor Class: 23 (natural armor)
Hit Points: 149 (23d10 + 23)
Speed: 30 ft., fly 90 ft.

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STR	DEX	CON	INT	WIS	CHA
14 (+2)	30 (+10)	13 (+1)	30 (+10)	20 (+5)	13 (+1)

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Saving Throws : Wisdom +13
Skills : Perception +13
Damage Immunities : poison, fire; bludgeoning, piercing, and slashing from nonmagical weapons
Condition Immunities : poisoned
Senses : darkvision 120 ft., passive Perception 23
Languages : Common, Primordial
Challenge : 23 (50,000 XP)

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Legendary Resistance (3/Day). If the phoenix fails a saving throw, it can choose to succeed instead.

Innate Spellcasting. The phoenix spellcasting ability is Intelligence (spell save DC 26, +18 to hit with spell attacks). The phoenix can innately cast the following spells, requiring no material components:

At will: fire bolt, greater invisibility (self only), fireball, wall of fire, flame strike, fire shield
1/day: globe of invulnerability, fire storm, meteor swarm

Fire Aura. At the start of each of the phoenix turns, each creature within 5 ft. of it takes 10 (3d6) fire damage, and flammable objects in the aura that aren’t being worn or carried ignite. A creature that touches the phoenix or hits it with a melee attack while within 5 ft. of it takes 10 (3d6) fire damage.

Flyby. The phoenix doesn't provoke an opportunity attack when it flies out of an enemy's reach.

Actions

Multiattack. The phoenix makes three attacks: one with its beak and two with its claws.

Claw. Melee Weapon Attack: +10 to hit, reach 5 ft., one target. Hit: 11 (2d8 + 2) slashing damage plus 27 (5d10) fire damage.

Beak. Melee Weapon Attack: +10 to hit, reach 10 ft., one target. Hit: 15 (3d8 + 2) piercing damage plus 27 (5d10) fire damage.

Fire Tail (Recharge 5-6). Each creature within 30 ft. of the phoenix must make a DC 26 Dexterity saving throw, taking 38 (7d10) fire damage plus 38 (7d10) radiant damage on a failed save, or half as much damage on a successful one.

Legendary Actions

The phoenix can take 3 legendary actions, choosing from the options below. Only one legendary action option can be used at a time and only at the end of another creature's turn. The phoenix regains spent legendary actions at the start of its turn.

Attack. The phoenix makes one claw attack or beak attack.
Heal Self (Costs 3 Actions). The phoenix magically regains 11 (2d8 + 2) hit points.
Blinding Light (Costs 3 Actions). Each creature within 30 ft. of the phoenix must succeed on a DC 21 Constitution saving throw or be blinded until the end of its next turn.

My question: which CR is correct here? Is there an easy way to calculate it while taking into account innate spellcasting and legendary actions? Thank you.

EDIT

I followed the suggestions adjusting the phoenix parameters. Here is my final result, for the curiouses:

Answer 1

Numerically this looks right, but it may not play that way

Regarding legendary actions, the trick to this is to focus on the overall damage output per round. That is the measurement that the DMG uses. It does not really care if that comes from an attack routine with lots of attacks, or from legendary actions, or from a big spell.

Assuming a typical adventuring party of at least 3 characters, with 3 legendary actions the Phoenix will be able to use all of them.

So the calculation becomes, following the Steps in DMG p. 274 ff:

Raw Rating

1. Defensive Challenge Rating: 149 hp = CR6; AC 23 (8 above the 15 normal for CR6): +4 CR. Overall Defensive CR = CR 10.

2. Offensive Challenge Rating: Assuming the multiattack for 42 + 38 + 38 + 3 legendary actions (beak attacks) x 42 = 244, CR 21; at +10 to hit 2 lower than normal to hit for CR 13: -1 CR. Overall Offensive CR 20.

Average Challenge Rating: (10 + 20) / 2 = CR 15

Modifications from special abilities

HP Multiplier for multiple damage resistances (slashing, piercing, bludgeoning from non-magical weapons) give a theoretical x 1.25 hp multiplier for the CR 10-16 range. The immunity to poison gives another x 1.5.

Legendary Resistance: Adds 30 hp per use, +90 hp.

Flying: no effect, as CR is over 10.

Flyby: no change.

Spellcasting: this depends on the damage output. We average the damage over the first 3 turns. See below.

Superior Invisibility (from Greater Invisibility) would give it +2 AC, but it consumes one round of actions, and it may also be questionable how well it would work with the Phoenix giving off a lot of heat. (Apart from, what fun is an awesome fiery Phoenix, if you cannot see it?)

Fire Aura: we'll increase the damage by 21 per round, as in Elemental body. (Assuming adding both active and passive damage once.)

Adjusted Damage Calculation

• The highest it can do is likely meteor swarm, for 140 damage in one round to a target, and it probably can hit at least 3 characters with it due to the very large area, instead of 2 like for breath weapon. That would be 420 points in round one. To this add 126 from legendary beak attacks, total = 546.
• It can use its Fire Tail in round 2, for a total of 14d10, assuming 2 targets hit like for a breath weapon, this is worth 154 damage, plus 126 from legendary attacks, total = 280.
• Fire Storm, assuming it also hits 3 targets due to a large area, would be 116 damage for round 3. Plus 126 from legendary attacks, total = 242. This us lower than the base attack routine of 244, so we stick with that.
• Other spells like Fireball, or Flame Strike, if we assume it hits 2 targets, will deal 56 damage, plus legendary attacks, total = 182. This is worse than any of the above, so we won't use it; most fights only last 3-4 rounds anyways.

This on average gives you a damage output of 357 per round, plus 21 from Elemental body = 378.

With the adjusted hp and damage we get:

1. Defensive Challenge Rating: 149 x 1.25 x 1.5 + 90 = 369 hp = CR20; AC 23 (4 above the 19 normal for CR20): +2 CR. Overall Defensive CR 22.

2. Offensive Challenge Rating: Damage output 378, CR 26 if we extrapolate from CR 23 with 17 each step; at +10 to hit -2 adjustment to CR. Overall Offensive CR 24.

Average Challenge Rating: (22 + 24) / 2 = CR 23, matching your number exactly.

Summary

So, in a vacuum this looks quite close to the CR 23 you got, but I suspect this creature is a far cry from a real CR 23 threat.

Most of the offensive CR comes not from physical damage, but from added fire damage. Without it, the normal damage per round would be a paltry 82. And if the PCs are not stupid you can expect them to pack protection from energy or absorb elements when they know they are facing a Phoenix. At this CR, one trick ponies have a hard time.

That is assuming they do not just stick it into a force cage, and kill it with missile fire. There is no save, and as it has no access to teleportation, it's a sitting duck then. (Well, a duck that can throw a meteor swarm and fireballs, but still).

To be honest, at these levels it would be somewhat surprising if the PCs have no magical weapons, and if so, the resistance will not really make it tougher to beat the Phoenix. Poison also sucks generally, and unless the PCs have a special poison magic item, the immunity in reality will not make it tougher to beat, either.

Most of the defensive CR comes from pumping up the hp with probably meaningless resistance and immunity, and I think there is a high chance that with its relatively puny 149 normal hp, this Phoenix will just get slaughtered by a high level group. But do not take my word for it, follow the guidance in the DMG, p. 275:

Creating a monster isn't just a number-crunching exercise. The guidelines in this chapter can help you create monsters, but the only way to know whether a monster is fun is to playtest it. After seeing your monster in action, you might want to adjust the challenge rating up or down based on your experiences.

Answer 2

CR calculation is always a bit of a black art.

Using the DM's guide suggests the Phoenix is somewhere around CR 21, though I'm not including the spellcasting and blinding ability there. So 23 is probably about correct.

However, the CR guide is pretty unreliable. Its an average of a creature's Defensive CR and Offensive CR and for the Phoenix the Defensive CR works out at around 20. Except this is only because the 'Damage immunities" doubles the effective HP. But by the time players face a Phoenix they are almost certainly going to have magical weapons and attacks!

Realistically this puts the effective hit points back down to 179 (increased now only due to legendary resistances). This would lower the defensive CR down to only 11.

Personally I think you really have to take your player's characters abilities into account and 'rule-of-thumb' it. But considering this Defensive CR problem, I reckon this would lower the Phoenix's overall CR to around 16 to 18, bearing in mind that my previous adjustment makes the Offensive CR considerably higher than the Defensive CR, which could make any battle quite 'swingy'.