There are so many things wrong with this statement, it's hard to know where to begin. For one thing you are teaching your children never to take any chances in life! Also, you do take chances, everyday, you just don't realize it. You are lulled into a false sense of security that as long as you are present nothing bad will happen to your child. You FEEL in control. If your child went to ride their bike alone, you would FEEL out of control and therefor anxious. Even statistically your argument makes no sense, do you have any idea how many more children are killed in car accidents every year than by random abductors?!! No, you don't care about facts, just about how you FEEL. You FEEL no anxiety while driving your child to her friends, but you would FEEL anxious if she walked, therefor you ignore what the facts are about which is truly a more dangerous activity and drive her because it makes you FEEL in control.
Let's think about the children we have seen on TV who were abducted right out of their own bedrooms! Does this mean your entire family sleeps together every night? 2 girls I can think of right now, household names, were not even sleeping alone, one with a sister, one with friends at a sleepover.
When is the magic age they can be alone? I would venture to say that if you really want them safe they never be alone ever in their lives! You see there are many more adult women attacked and killed every year by strangers than children! So you better not go out by yourself alone either if you don't want to take any chances. There are actually less than 50 children every year in the entire US who are abducted and killed by strangers, it FEELS like more because the media loves these stories, they sell!
You are truly being counter-productive when you "helicopter" your child. They are not getting life experience in learning to listen to their own instincts and develop some street smarts. If for 18 years they only listen to mom's voice and don't learn to develop and listen to their own they are going to be the most vulnerable young adults when they finally do get on their own. They will either be crippled by fear that you have instilled in them or they will be so glad to be free they will make stupid dangerous decisions. I much prefer to give my kids time during the 18 years I have them to gradually over time develop their own decision making skills and allow them to become confident in their own decisions. You will see my kids riding their bikes through the subdivision alone, you will see them exploring the woods behind our subdivision, you would see them walking to school alone if it wasn't 12 miles away

, you won't see me, I'm in the house, feeling a bit anxious, and knowing that that means I am doing the right thing