I often get asked two similar but different questions by clients about equity:

1. What happens if the appraisal comes in above the purchase price?
2. You CAN indeed increase the contract price within the appraised value, and request a seller credit to cover your closing costs. Let's say you wrote an offer at $200,000, and the appraisal comes back at $205,000. You could increase your offer to $205,000 and request a $5,000 credit to cover closing costs.
3. Sellers don’t always love doing this, but it is possible! 
4. Even if the seller doesn't agree to this plan, you have “instant equity”! If your appraisal is one of 50% that have been coming in high recently, then at the least you can have a mini celebration.
5. What happens if property values DROP after you close on a new purchase?
6. Nothing. Keep making your monthly payments and you’re completely fine.
7. However, remember since 2000, property values have gone up annually on average 3.57% per year (8.38% on average the last 5 years).
8. Even if there is a small dip, quality assets have always gained value over the long haul

By the way:

We want to be your ally in getting offers acccepted! We have successfully helped clients get accepted offers in competition by offering a quick closing.

the first image below was a 15 day closing start to finish, second was 14 days, the third was 13!

It is certainly a brisk experience but if you are competing against 5-25 other offers, certainly can help!