Short answer
Most cash buyers start with what the house would sell for after repairs, subtract repair costs, closing and holding costs, and their profit. What is left is the offer. Because the math depends on repair estimates and resale costs, two buyers can offer very different numbers for the same house.
The basic formula
There is no single legal formula, but most investors work backward from the resale price. They estimate what the house would sell for once it is fixed up, often called the after-repair value. They look at recent sales of similar homes nearby to get that number.
From there they subtract the cost of repairs, the cost of buying and selling, the carrying costs while they own it (taxes, insurance, utilities, interest if they borrowed), and the profit they need to make the deal worthwhile. What remains is the most they can pay. A careful buyer shows you each line. A vague buyer gives you one number and no explanation.
Why the same house gets different offers
Repair estimates are the biggest swing. One buyer sees a roof that needs patching. Another sees a full replacement. Buyers also differ on how quickly they think they can resell, how much cash they have tied up, and how much profit they require.
Your own side matters too. A lien, a tenant, an estate that has not been opened, or a title problem can lower an offer because the buyer takes on time and legal cost to fix it. The fastest way to see the spread is to collect more than one offer and compare the terms, not just the headline price.
What to ask any buyer
Ask for the offer in writing. Ask which repairs they assumed and what resale price they used. Ask who pays transfer tax, title charges and any other closing costs, and get the answer in the agreement. Ask how long they need to close and whether the offer changes after they inspect the house.
That last question matters. Some buyers make a high opening offer, then cut it after the inspection. A written offer that states the conditions lets you spot that early. The FTC advises getting terms in writing and being wary of pressure to act fast.
How this compares to listing
A listing price is a request. A cash offer is a price you can usually bank on. Those are different things, so do not compare them straight across.
If the house is in good shape, an agent can often get more by marketing it to buyers who will pay retail. If it needs major work or you need to move quickly, a cash offer can leave you better off after you count repairs, carrying costs and months on the market. Kaizen Home Buyers gives a cash offer within 24 hours, buys as-is, and charges no agent commission. Run your own numbers against it and against a listing.
A quick example of the logic
Here is the logic with made-up round numbers, for illustration only. Say a buyer believes the house would sell for a certain amount once repaired. They subtract an estimate for repairs, then the costs of buying and later selling, then the months of taxes and insurance they will carry, then the profit they want. Whatever is left is their ceiling.
Change one input and the offer moves. A buyer who assumes a cheaper repair scope can pay more. A buyer who expects a slow resale will pay less. This is why a single low offer tells you little. It tells you what that buyer's assumptions produce.