Theory of Levered Investing
Borrowing in order to increase investment exposure is a time-honored and legitimate activity. It's the optimal way to increase your exposure, according to finance theory (which assumes you get a good interest rate...more on this later). In your case it may or may not be a good idea. Based on the information in your post, I believe that in your case it is not a good idea. Consider the following concerns.
In finance, reward comes with risk and in no other way. Investing borrowed money means there is a good (not small) chance that you will lose enough money that you will need to pull significant wealth from your own savings in order to make up the difference. If you are in a position to do this and OK with that possibility, then proceed to to the next concern. If losing a lot of money means financial calamity for you, then this is a bad idea. You haven't described your financial situation so I don't know in which camp you fall. If the idea of losing, say, $100K means complete financial failure for you, then the strategy you have described simply has too much risk.
Make no mistake, just because the market makes money on average does not mean it will make money, or as much money as you expect, over your horizon. It may lose money, perhaps a lot of money. Make sure this idea is very clear in your mind before taking action.
Your post implies that you think you can reliably get 10%-12% on an investment. This is not the case. There are many years in which a reasonable portfolio makes this much or more, but on average you will earn less. No ones knows the true long-term market risk premium, but it is definitely less than 10%. A better guess would be 6.5% plus whatever the risk-free rate is (currently about 0%). Buying "riskier" investments means deviating from the optimal portfolio, meaning you took on more risk than is justified by how much extra money you expect to make. I never encourage people to invest based on optimistic or unrealistic goals. If anything, you should be conservative about how you expect things to go. And remember, these are averages. Any portfolio that earns 10%-12% also has a very good chance of losing 25% or more.
People who sell or give advice on investments frequently get you charged up by pointing at times and investments that have done very well. Unfortunately, we never know whether the investments and time period in which we are investing will be a good one, a bad one, or an unexciting one. The reality of investing is...well, more realistic than what you have described.
I can't imagine how you could borrow that much money and only have an annual payment of $2000 as you imply--that must be a mistake. No individual borrows at a rate significantly below 1%. It sounds like it's not a collateralized loan of any kind, so unless you are some kind of prime-loan customer, your interest rate will be significant. Subtract whatever rate you actually pay from 6.5% to get a rough idea of how much you will make if things go as well as they do on average. You will pay the interest whether times are good or bad.
If your rate is typical of noncollateralized personal loans, there's a good chance you will lose money on average using the strategy you have described. If you are OK with taking risk with a negative expected return, consider a trip to Las Vegas. It's more exciting.
I'm not one to make people feel guilty for doing things that are legal but of questionable morality. If that's the case and you are OK with it, more power to you. I'm not sure under what pretense you expect to obtain the money, but it sounds like you might be crossing legal lines and committing actual crimes (like fraud). Make sure to check on whether what you intend is a white lie or something that can get you thrown in prison.
For example, if you are proposing obtaining a subsidized education loan and using it for speculation, I could easily see you spending serious time in prison and permanently ruining your life, even if your plan works out. A judge and 12 of your peers are not going to think welfare fraud is a harmless twist of the truth.
I've said a lot of negative things here. This is because I have to guess about your financial situation and it sounds like you may have unrealistic expectations of the safety and generosity of investing. Quite frankly, people for whom borrowing $250K is no big deal don't normally come and ask about it on StackExchange and they definitely don't tend to lie in order to get loans. Also $18K a year doesn't change their quality of life. However, I don't know. If $250K is small relative to your wealth and you need a good way to increase your exposure to the market risk premium, then borrowing and investing may well be a good idea.