You need to realize that as an attending anesthesiologist you can't think of yourself as an employee but rather as a business. Employees just have to show up on time, work hard, be a team player, and not piss people off and they will be fine. As an anesthesiologist you need to worry about factors beyond your control that will impact your bottom line such as losing your contract b/c the ceo wants to go w/somoene cheaper, drop in volume, change in payor mix etc. In order to combat that you need to think like a business in terms of cash flow and boom and bust months. When apple makes 5 billion this year in profit will they spend all 5 billion? No they spend some on R&D, some on marketing, and put some away for when their business tanks.
That's how you have to think, realize that you will likely have plenty of work over the next 20 years and you will make a decent salary but there will be some ups and downs that are out of your hands which you need to be prepared for. First off, figure out your expenses not just mortgage and house stuff, also factor in health insurance, disablity, car, gas etc b/c that's all stuff you're gonna need. Then make sure you have at least 6 months worth, preferably a year. This is money that should be liquid and invested ultraconservatively I.E. put in the bank and take that 1% interest b/c it's your emergency fund and you never want to risk it. Think about it as your companies severance package b/c you own your own company and you need a nice severance package if you were to be let go. Secondly, make sure to max out your retirement b/c that's what you're gonna need to get out of the rat race eventually and be totally independent.
Make sure you keep your contacts up to date. Be on good terms w/the your coworkers (surgeons, nurses, administrators) and anyone else you can. When it comes time to look for a new job, they will be the best way to find something and a well placed call from one of these folks will go a long way to getting you to the front of the line. Even if your group loses a hospital contract if these people like you, they will rally for you and you will likely be able to stick around when the new group comes in if you so choose.
Keep your skills up to date, nothing is worse than being out of a job and not having the skills to impress another group in town. This means keeping up on CMEs, going to conferences and learning new skills. Weak in regional? Attend a conference, learn ultrasound and the latest blokcs. Currently working at an ASC? Pick up a shift or 2 at a local hospital to keep up your OB skills and take care of sick pts. Always remember that you are 3 months away from searching for a new job and keeping your skills current is the best way to ensure future employment.
When all is said and done realize that you are in a better position than 95% of other people in this world. You have a specific skill set that is very highly compensated and will always be in demand but you are subjected to the boom and bust cycle of business. As long as you realize that your job security has more to do with you and less to do with your hospital you will be find plenty of work and thus job security.