Item Focus: Liandry's Torment

It's time for me to call to question the viability of this item on the grounds that it just doesn't seem that strong for the gold.  Now, mind you, I don't actually KNOW the math behind this, but rather, this post is more of me calculating it out in a way that I hope you--the reader--will be able to understand and apply in future endeavors, if you so happen to get into mathcrafting in League of Legends.
First and foremost, let's examine the facts:

• Liandry's Torment
• Costs: Haunting Guise (Health Crystal (475), Amplifying Tome (435), recipe (575)), Amplifying Tome (435), recipe (980)
• Bases: +70 AP, +200 HP, +15 MPen
• Passive: Spell damage equal to 5% over 3 seconds of enemy's current health for single-target spells or 2.5% over 1.5 seconds otherwise (aoe, sustained damage, etc.).  If movement is hindered in any way, it'll double the damage done.
• Note that this effect does not stack, so if you blow, say, 4 spells in 1 second, the total effect (if it ends in a single-target spell) will be 4 seconds, or 6 1/3% of their current health
• As this is spell-damage, it also factors in magic resistance
• Is not proc'd by the mastery, Spellsword, but it does proc Rylai's, which will apply the double-damage effect.

So, let's analyze first how much better it is than Haunting Guise (which you'd used to get as, say, Cassiopeia or Veigar strictly for the magic penetration).

• Haunting Guise
• Costs: 1485 (1415 less than Liandry's Torment)
• Bases: +25 AP, +200 HP, +15 MPen

So, for 1415 gold, we're buying 45 AP and a passive.  That's 31.44...gold  per AP, in comparison to Blasting Wand with a 21.5 ratio, or a 20 ratio from Needlessly Large Rod.
The logical step here would calculate the gold-worth of the passive by setting Liandry's AP:gold ratio to that of either (or both!), then calculating the total worth of the AP from the item, subtracting that from its cost over Haunting Guise, and that'll give us the "worth" of the passive (and its AP equivalent for the cost).  If we take the Blasting Wand's ratio, we have 967.5 gold, so the remaining passive would cost 447.5 gold, or trade that with ~20.8 AP (for the other, it's 900 gold for the AP and 515 gold/~25.75 AP for the passive).
But what does that MEAN?!  My dear readers, what this leads to is the question of if the passive is better than the AP that it'd provide otherwise, or if it's just more prudent to just buy something else after Haunting Guise (if you do decide to get it).  Because of the several variables here (current health and magic resistances), we'll proceed empirically and expand out, hoping to find some pretty formula that we can make some snazzy general statements concerning the item.  Asymptotes and such.

First, some assumptions.

• I feel like Annie is pretty much the most average champion in the game.  Because of that, I use her kit to mathcraft to.  If people have an issue with this, let me know and I'll do your champion next, I guess.
• We'll assume true damage.  Magic resistance will alter the numbers, but it will just add effective health.  If you really want to, you can alter the numbers, but I don't particularly see any reason to.
• The player's built a pro-as-fuck build: Liandry's first, questions later.  In fact, very later: not only  has (s)he bought it first, at the 50 minute mark, it's the only thing (s)he has.  (S)He is level 18, though.

So, let's look at the numbers at the kit.  It should be mentioned that the first set is the description of damage, the second is what it'd look like with AP from Liandry's.

• Annie
• Q
• 245+.7*AP (single-target) => 245+.7*70 = 294
• W
• 280+.75*AP (aoe) => 280+.75*70 = 332.5
• E
• Irrelevant
• R
• 450+.75*AP (aoe), we'll ignore the consistent damage for the time being => 450+.75*70 = 502.5

So the maximum potential damage would be achieved by casting W, waiting 1.5 seconds, R, waiting another 1.5 seconds, then Q.  That'd be 332.5 +  (current health)*.0083 + (current health)*.0083 + (current health)*.0083 + 502.5 + (current health)*.0083  + (current health)*.0083  + (current health)*.0083 + 294 + (current health)*.0083  + (current health)*.0083  + (current health)*.0083  + (current health)*.0083  + (current health)*.0083  + (current health)*.0083.  It's important to note that current health will go down after each step, so when you calculate this out, it's harder than just assigning one value to health.  It'd be easier in a spreadsheet because copy and pasting would be a life-saver for the 12 (current health)*.83's.

So, let's say that you're against another Annie, with 15 MR (so true damage), with 3,000 health initially.  The next picture will give you an idea of what I did and how by just plugging in the numbers into a spreadsheet:

The second column, mind you, is the current health after that instance's damage.

The damage from Liandry's alone did 203.23, the equivalent of ~92 AP (that's to say, it would do 203 damage if, instead of a Liandry's, you had an item that gave you 92 AP).  Pretty substantial, because that's on top of the 70 base-AP given by the item, and effectively gave you a total of 162 AP.  Alone, that means that you spent 17.9 gold per AP, which makes it more cost-efficient than a standard Rabadon's, who (alone) is at 21.33 gold per AP.

In order for a Needlessly Large Rod  to be cost-efficient after the Haunting Guise, the current health at the start of this combo must be a little over 1900 (I'd say that I worked some voodoo math-magic, but really, I just kept plugging in numbers).  It would do 99 damage from tick, which is the equivalent of 44.9 AP (with the +25 from Haunting Guise).  Mind you, a NLD is 185 gold more than a Liandry's after Guise, and I guess, if you wanted me to, I could get really technical but by this point, my head is practically swimming from multiplication.

In conclusion, here's my final thoughts:

As with all AP items, you should really pick what you need: if you're doing fantastic, absolutely stomping everything and just wrecking shit and they're not building magic resistance and all that, there's less reason to build an expensive mpen item than to build something that'd more effectively increase your potential to continue to carry.  Haunting Guise has always been one of those niche items; if your opponent in mid is building MR-boots, then Sorceror's Shoes are great.  If they build a Chalice of Harmony for more MR, then, yes, build a Haunting Guise.  However, don't feel obliged to build a Liandry's just because you have a Guise, especially early-game, where it just isn't cost-efficient.  An alternative is to go Haunting Guise for the health and mpen, but switch over to a NLR because of its end-items being capable of being far more useful, then revisiting Liandry's if they get way too big.  Because Rylai's procs it and the double-damage, people like Brand and Malzahar can definitely use it in some sort of Doran's opener (it's just one more step in the synergy).