How do I challenge my auto insurance company's appraisal of damages to other car in an at-fault(my) acci ?

I read-ended another car at an intersection. Both of us were stopped at a light. When the light turned green, the other car started moving and immediately slammed on brakes. In the meanwhile, I had taken my foot off brake and hit the car (at < 5mph).

The other party has filed claim for $1350 for damages and also for "unknown" amount of bodily injury. There is not even a mark of any collision on my car (as the speed was low). My insurance company has agreed to their claim related to damages and is waiting for the exact amount for the bodily injury claim.

This has been quite unfair to me and will definitely bump up my premium like crazy.

Is there any way I can challenge my insurance company's appraisal of the damages ? Are there any other alternatives ? What kind of evidence would I need to furnish for the same ?

We had called a "Cop" to report the accident. I wanted him to note that the impact was at < 5mph. However, he refused to do that saying that this would be left to insurance
this appears to be a clear cut case of negligence on your part. the injured party, the one you hit, has the right to present a claim for damages and to be made "whole". the insurance company is acting in your interest by negotiating and settling the claim with minimal cost and problems. if you remove the insurance company from the equation and you tried to handle yourself you may be surprised that the numbers presented by the injured party may seem quite reasonable.
There are so many misconceptions out there regarding liability and insurance settlements. Here's one: if you hit someone in the rear, you are at fault. Not true. How can that be? Simple. What if they were backing up at the same time? And yet you hear that one repeated so often it is almost cliche.

You are at fault if you caused the accident and if this can be legally proven. Sometimes the other party may also be contributorily negligent if they also caused the accident. In some states, this bars them from any recovery. In others, it reduces the amount they are owed by the amount they share in negligence.

However, by the details you provided, you appear to be liable. Thus, this is simply a matter of determining damages. Your insurer handles countless cases like this one and are well acquainted with the usual cost to repair the vehicle as well as the likely injury that may be claimed.

Here's a likely breakdown of the cost to repair the vehicle. You stated there was $1350 in damages. Modern vehicles generally have a bumper cover made of plastic that is for the most part non-repairable. If an impact gouges it, it has to be replaced. It is usually around $250-$300 plus the cost to paint it and put it on the vehicle. Behind that bumper cover there is often an "absorber" (essentially a piece of styrofoam put there to absorb low speed impacts, sparing the car from further damage) and a "reinforcement". The reinforcement is the actual bumper but it is not something you can typically see without taking the bumper cover off. Because this piece is essential to the safety of the car, if it is damaged, it usually required replacement. It is usually in the neighborhood of $150-$200 plus labor to install it. If the insurer had to replace the bumper cover, absorber and reinforcement, $1350 would be about right. There are other parts that may have been affected and some manufacturers have different names for these parts. And parts prices can vary greatly from vehicle to vehicle. Headlamps on modern cars, for example, can vary from $100 each to $1600 each and higher. Crack a pair of those and you could have an astronomical estimate from a low speed impact.

Your insurer will not pay more than is needed to properly repair this vehicle They also won't pay less and you wouldn't want them to. Think about what they are actually doing when they pay that claim. As per the terms of your insurance policy, a legal contract between you and them, they are assuming YOUR liability and paying for the damage YOU caused. This means they are relieving you of YOUR liability. Without insurance, you would have to pay that $1350 yourself.

Or think about this in reverse. If you are hit by someone, do you want your car repaired properly? Put another way, do you want to be able to choose a body shop who will repair the vehicle for what the insurance company is paying? Don't you think the insurance company ought to have a way to estimate damage costs that concurs with what body shops actually charge? Well, they do. Adjusters use software, often the same software used by body shops, to assist them in coming up with accurate parts prices and labor and paint times established by the industry as fair.

Then there's another misconception put out there: you can settle this with the claimant on your own for figures you deem reasonable. You could try but the claimant would not be under any obligation to accept without a court judgement. The insurer will get for you a legal release from liability that is enforceable. They can do this because the amount they put on the release will be reasonable and fair. This is because the adjuster has the training and tools to come up with a fair amount. So unless you also have this training and tools to assist you, demanding the owner of the other vehicle submit estimates to you, taking pictures of the car, etc. would probably accomplish nothing.

As for injuries: can you prove that the other party could not possibly be hurt just because this was less than a 5 mph impact? What if a doctor or a chiropractor finds injury consistent with an automobile accident? You can't prove this, of course. But the adjuster can investigate the accident and evaluate injury claims with their experience and training in mind. Look at these kinds of claims long enough and you get a sense for what is reasonable and what is not. Again, your insurer will pay no more than what is reasonable but they will pay a fair amount to put the claimant back where they were. Again, this is the amount neccessary to relieve you of your liability.

I do understand the concern insureds have for the possibility of premium increases. However, I will admit to a little annoyance when a person is more concerned with their increased insurance expense than the cost for damages they have caused someone else. Here's a sure fire way to keep your insurance rate low: don't run into anyone.
Both of the above people are right. Since you rear-ended someone you are at fault no matter what. If you have a clean driving record up until now, your premium might not go up as much. Some insurance companies also have policies that will "Forgive" an accident every few years.

And with cars now a days it does not take much to do a lot of damage. I got into an accident, when someone hit me. I had about $1500 damage on the car. My whole bumper had to be replaced. It sucked, and neither one of us were going fast.
Something doesn't sound quite right. If you rear-ended then it is YOUR fault and YOUR company has to pay the damages. Doesn't matter if impact was at 5 mph or what, YOU did the deed and should have been paying more attention. And it doesn't take much of a rear-ender today to cause a lot of damage. On the bodily injury claim, your company will determine what's what, but they aren't going to give the boob whom you it hundreds of thousands, so relax.

Answers:

In most states, having a single dollar in bodily injury payout results in an accident being surchargeable (i.e. it can cause your premiums to go up). Most states also have a minimum PD (property damage) amount before an accident is surchargeable. Rest assured that your insurance carrier will do everything it can to pay only what it owes with regard to your accident. The adjusters are trained to discover pre-existing damage and damage that was caused at the body shop; they will scrutinize bodily injury claims that don't make sense.

One thing you need to know is that 30% of your insurance premiums go to paying for insurance fraud - and if you had such a low-impact accident that there is no damage to your vehicle, it's possible the person you hit is exaggerating the severity of his/her injuries. At this point, there's nothing you can do with regard to their claim; it's up to the insurance carrier to determine if there is anything fraudulent about the accident. Before they can deny a claim, they also have to be able to prove beyond a reasonable doubt that the person is committing fraud.

You can get a second opinion on your own damages, but your carrier is responsible for the other party.
First of all, rear ending a car is always going to be to your detriment. However, to answer your question about disputing the amount perhaps I can assist you. Always try to get a couple of estimates of the damage to the other person's car. You will find that the estimates vary. I keep a camera in my car to take pictures for just this reason. Presuming you didn't do that then you need to request a copy of the estimate done by the insurance companies adjuster and at this point you may want to get your own adjuster to review those estimates. So pictures before car is repaired, other estimates, review insurance adjuster's damage and finally your own adjuster would be the step I would go.

Good Luck
As I understand it, you don't dispute you are at fault. What you wish to dispute is the amount of damages. This is why you should keep a disposable camera in your car. Unless you took pictures at the scene of the collision, the only evidence you can present is the police report. It probably doesn't describe any damage in enough detail to help you. You could request an independent inspection of the vehicle. I'm not sure if they are required to cooperate.