In this workout, we're asked to calculate the diluted shares outstanding. We're also told there are no share issuances or repurchases other than those involving options. So, we're told that there's a share price of 7.50, and we've got basic shares outstanding, but then we are told that there are some options outstanding. Let's imagine the CEO has given 1,000 options, and she can exercise her options at a price of five. So, we're going to do this twice. We're going to do the treasury calculation, do a long-handed version here, and then we'll do a quick version underneath. So our first question is, are the options in the money? Well, the CEO, she can exercise at a cost of five. She'll pay five, and she'll get a share for 7.50. These are definitely in the money. She definitely wants to exercise her options. So, the cash raised by the options exercise, i.e. the cash given to the company by the CEO, is five times by 1,000 options, so 5,000. The company now goes out to the public market. It takes that 5,000 and buys shares, which each cost 7.50. So, how many shares can the company purchase? It can purchase 666.7. So, the company then tries to give those shares to the CEO and she says, what? 666.7? That's not what I wanted, I wanted a 1,000. So the number of shares the company has to issue is 1,000. So, it's got 666. It needs to issue 1,000. Thus, it needs to create the difference. It needs to create the difference between 1000 and 666.7. It needs to create 333.3. That is the diluted effect of the options. So, my diluted shares outstanding is the 10,000 basic plus the 333.3 created.

Great. That was the long-winded way. It helps to explain it that way. Let's now do the treasury method, the quick way. This is where we use our max formula. I take the maximum of the share price minus the stripe price of five, all divided by the share price again. So, that gives me 2.50 over 7.50, i.e 1/3 of the new shares are gonna have to be created. I then times that by the number of options, 1,000. I then want the maximum of that and zero. So, if they were out of the money, this would gimme a figure of zero. There's the 333. I add that onto the basic shares outstanding of 10,000. That gets me their 10,333. There is the same figure. Now, let's just check if that max formula does work if they were out of the money. If I go back up again, the share price is 7.50. Let's now have an exercise price of eight. The CEO in this case is not going to want to pay eight in order to get something with only 7.50. So, if I change that to an eight our original method didn't quite work. It gives us this negative. But our quick method, which used the max formula, stops at zero. It doesn't go below that. And our diluted shares outstanding stays at the 10,000.