Many people argue that Ancient Coin is better than spellthief’s edge. But is it really?
Ancient Coin’s passive grants user 2 gold per minion death in lane.
Because there are 6 minions in a wave every 30 seconds, and 7 minions every 1minute and 30 seconds.
30 seconds – 6 minions – 12 gold
60 seconds – 12 minions – 24 gold
90 seconds – 19 minions – 38 gold
120 seconds – 25 minions – 50 gold
150 seconds – 31 minions – 62 gold
180 seconds – 38 minions – 76 gold
So, we can safely establish that, every 3 minutes the owner will get an estimate of 76 gold from its passive.
Now lets take a took at Spellthief’s Edge.
From its GP 10 Passive alone, it grants the owner 36 gold already.
Now, the auto-attack passive will be very dependent on the champion, EX: Sona, Lulu, Nami, the poke-oriented supports will be more effective.
Now, spellthiefs passive procs for every champion, every 10 seconds. Meaning you can auto-attack one of your opponents, get gold, auto attack the other, and get gold.
This means that every 10 seconds, at max you can get 8 gold.
And 3 minutes later, you can see yourself with a 144 gold wealth.
Combined, it is a possible 180 gold income per 3 minutes, or effectively an 10 gold every 10 seconds income. However; I highly doubt the enemy laners will let you auto-attack them every 10 seconds. So how many auto-attacks must you land within 3 minutes for spellthief’s to be better?
11.
10 auto-attacks will result in the same income, 11 auto-attacks for a higher income.
That is saying, every 16 seconds, you land an auto-attack on the enemy will result you a higher income than the other item. Are auto-attacks that hard to land? Assuming you aren’t a melee support, or going against a Cait/Jinx or something, this shouldn’t be too hard.
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So Medallion Vs. Frostfang?
Honestly, Its probably still going to be the same.
Frostfang still holds a +2GP10 lead.
Gold income from auto-attacks and spells are doubled, as well as the income from dead minions for Medallion.
However; because now with frostfang, spells are included, it should be easier for you to get income.
Also, lets take into consideration that when walking back to lane, from death or recalling. Is the Ancient coin granting you gold? No.
You know what is? Spellthiefs Edge.
Walking back to lane with Medallion happy with 2GP 10?
Frostfang = 4GP10.
League of Legends 
Gold Coin is more useful late game than early game. The effect gives your WHOLE team a boost of speed. Plus, AP mages and ADC’s will be clearing out waves of minions in one hit. Lots of gold.
Spellthief falls off late game. The active only slows 1 person and slightly damages them. The Gold Coin active gives the whole team a boost. Catching up. Letting you kite an enemy too.
Also the other eff of spellthief is useless. What lulu has time to autoattack during lategame/teamfights?
I am talking ONLY about the gold income.
However; if you want to take the stats into account…
The Ancient Coin does not provide any offensive utility. And it’s passive only occurs if your partner in lane last hits the minion.
Spellthief’s Edge is gold efficient even without it’s gold generating passive while Ancient Coin is not.
Also, with champions like Sona, Nami, and Lulu’s utility all scale with AP. The AP that the Talisman’s line does not provide.
Champions like those three listed above have slows and speed buffs that scale with AP.
Kiting is much more effective when the enemy is slowed rather than yourself buffed. A 50% slow Vs. a 40% buff.
In ‘Lategame’ team fights, an AOE slow is more effective to peel for your ADC.
Also, in this ‘lategame’ you speak of where everybody has completed full 6 items? Every extra damage your team can output is wholeheartedly welcome.
I must admit, champions like Leona or Taric or champions of the sort probably welcome the speedbuff to initiate fighters rather than a ranged slow. But Twin Shadows is an easy substitute. For poke oriented supports to gain gold.
Also, after you have proceeded down the Spellthief line one item, you can gain gold from spells too. And I bet that Lulu will be using her spells.
adc doesn’t have to last hit for you to get money from ancient coin
My bad, mistaken it for the mastery.
The Coin is pretty good late game in terms of gold. Your mage/adc will be blowing up waves with their abilities and pushing lanes. That will get you lots of gold.
Just saying, getting 11 hits within the first 3 minutes is going to be pretty tough for most supports. Way better on some, but only a few.