David Clayton has written a good article on how to value this deal, and comes to conclusions similar to ours.
I still think he makes a couple of mistakes.
In particular, he uses a volatility figure of 30%, which was Goldman's volatility in the 5 years PRIOR to 2008. Higher volatility means higher value, with Black-Scholes, and so given the roller coaster of the last 12 months, this drastically underestimates the value.
More important is the very use of Black-Scholes for valuing these options. He says that Buffett thinks that long-term options are overvalued by B-S:
And while Buffett has stated that Black-Scholes increasingly overstates the value of options as the time-to-expiration increases, 5 years isn't so long that the calculation would be geometrically wrong. I've also used a somewhat conservative volatility figure.
I have no idea what 'geometrically wrong' means, but in any case, I think he has this idea backward. Buffett did say B-S can overvalue options, but he was referring to his long-term (12-15 year) equity index puts, not call options. The whole problem with B-S is that it basically makes the assumption that the stock basically moves randomly around its current price, with the volatility figure being one standard deviation of a normal distribution around the current value, and that to know an option's value, it suffices to know current price, strike price, interest rates, dividends, and volatility. In particular,no knowledge about the underlying security or its prospects is required.
That is an untenable assumption for anything other than very short-term options (less than a year or so), and for 12-15 years it is ludicrous. If the price is currently $160, it basically says that there is an equal chance, in 5 years, that the price will be $120 and $200. (That is not quite true, but almost, since one should probably use a logarithmic scale, but that is a detail.) The fact that stocks tend to go up is ignored. And retained earnings are ignored.
So for long-term calls, (and 5 years qualifies), B-S should OVERESTIMATE the value of a put (like the ones Buffett has sold), and UNDERESTIMATE the value of a call (like the warrants he acquired from GS).
Anyways, both of those factors would tend to underestimate the value of the warrants we are interested in. But despite that underestimation, Clayton figures that Buffett bought a $5B investment (preferred + warrants) that, at the time of his buying it, was already worth $7.89, based on a B-S evaluation of the preferred and the yields of similar bonds. At the November low, that investment was down to a worth of $5.5B (including the redemption premium), so even then, Buffett was doing fine. At a recent GS share price of $165, the value of the whole investment had skyrocketed to $11.5B.
To summarize, the initial deal allowed Buffett to purchase, for $5B, securities that were already worth 37% more, according to market prices of similar instruments. The total value of that investment sunk to about a 10% gain on the initial investment 2 months later, but 8 months later is up more than 120%, even using the conservative assumptions I criticized initially.
In conclusion, it is hard to argue with Clayton's conclusion, that this may be one of Buffett's best investments ever, at least in absolute terms. And you have to wonder why Goldman wanted to offer such a sweet deal to Buffett, but I'll leave that for another post...