The phrasal verb 'take out' meaning 'arrange for a financial or other service' can be followed by either of the prepositions 'with' or 'from'. We might tend to use 'from' if we think of the arrangement as one-way (e.g. the bank provides the loan to the borrower) or 'with' if we think of it as two-way. In fact most such arrangements are two-way, because the bank agrees to lend you money on condition that you pay it back, either all at once at a later date, or in instalments. Likewise, if you take out an insurance policy, the insurance provider agrees to provide you with a service (the insurance cover) and you agree to pay a sum of money called a 'premium', again either all at once or in stages.

The phrasal verb 'take out' meaning 'arrange for a financial or other service' can be followed by either of the prepositions 'with' or 'from'. 

We might tend to use 'from' if we think of the arrangement as one-way (e.g. the bank provides the loan to the borrower) or 'with' if we think of it as two-way. In fact most such arrangements are two-way, because the bank agrees to lend you money on condition that you pay it back, either all at once at a later date, or in instalments. Likewise, if you take out an insurance policy, the insurance provider agrees to provide you with a service (the insurance cover) and you agree to pay a sum of money called a 'premium', again either all at once or in stages.