# Calculation Logic/Reasoning: A house is valued at $15K, and it is insured for$8K...

#### KWF

##### Junior Member
A house is valued at $15,000, and it is insured for$8,000 under a policy containing an 80% coinsurance clause. If a fire should causes a $7,500 loss to the house, how much would the owner receive under his policy? Solution:$8,000/(80% * $15,000) *$7,500 = $5,000 What is the reasoning for the above calculation: (amount insured)/(80% * value of house) * (loss by fire ) = amount paid by the insurance company? Why not something like (loss by fire$7,500 ) / (80% * value of house, $15,000) * amount insured,$8,000?

I am trying to understand the reasoning for using the correct calculation.

#### JeffM

##### Elite Member
A house is valued at $15,000, and it is insured for$8,000 under a policy containing an 80% coinsurance clause. If a fire should causes a $7,500 loss to the house, how much would the owner receive under his policy? Solution:$8,000/(80% * $15,000) *$7,500 = $5,000 What is the reasoning for the above calculation: (amount insured)/(80% * value of house) * (loss by fire ) = amount paid by the insurance company? Why not something like (loss by fire$7,500 ) / (80% * value of house, $15,000) * amount insured,$8,000?

I am trying to understand the reasoning for using the correct calculation.
Coinsurance is the percentage of the market value that the insured is obliged to insure. On a 15,000 dollar house with a coinsurance requirement of 80%, that means that the insured is supposed to value the house at 15,000 * 80% = 12,000 or more and to pay premiums for a house with that declared value. With me so far?

If the insured has falsified (deliberately or inadvertently) the declared value and so has been paying premiums to insure a house worth far less than the true value, the ratio of the insurance payment to the actual loss is reduced to the same ratio that the declared value has to the coinsurance value (which is, remember, still less than the true value). In this example, the ratio between the declared value and the coinsurance value equals 8,000 / 12,000, or two thirds. Still with me?

And in this example, the actual loss (meaning the cost to repair or replace) is 7,500. And two thirds of 7500 is indeed 5,000.

$$\displaystyle \text { minimum coinsurance value } = \text { coinsurance percentage } \times \text { market value.}$$

In the example, that is $$\displaystyle 80\% \times 15,000 = 12,000.$$

$$\displaystyle \text {deficiency ratio } = \dfrac{\text {declared value}}{\text {minimum coinsurance value}}.$$

In the example, that is $$\displaystyle \dfrac{8,000}{12,000} \text { or } \dfrac{8,000}{80\% \times 15,000}$$

$$\displaystyle \text {payment due } = \text { loss } \times \text { deficiency ratio.}$$

In the example, that is

$$\displaystyle 7,500 \times \dfrac{8,000}{80\% \times 15,000} = 7,500 \times \dfrac{8,000}{12,000} =$$

$$\displaystyle 7,500 \times \dfrac{2}{3} = \dfrac{15,000}{3} = 5,000.$$

The math is correct. But if you don't understand that it is the result of combining the three formulas I gave you above into a single formula, it appears completely arbitrary.

I hope this helps.

By the way, notice that if you declare anything above the minimum coinsurance value, the payment is 100% of the loss. The whole thing is a way for the insurance company to protect itself from the insuered putting a ridiculously low valuation on the insured property and then demanding payment for losses that reflect the true value.

This was not a very good example. Let's take this one. Again suppose a coinsurance requirement of 80% of market value. Let's say that the market value is 250,000, and there is fire damage that costs 175,000 to repair. In other words, the property is 70% destroyed. But further suppose that the property owner has lowballed the declared value at 75,000 to keep premiums low. If the insurance company does not have the coinsurance requirement, it will have to pay the insured more than 200% of the declared value of the property when the damage is only 70%. No insurance company would stay solvent under those circumstances.

With the coinsurance requirement of 80%, the minimum declared value would be

$$\displaystyle 80\% \times 250,000 = 200,000.$$

$$\displaystyle \text {the deficiency ratio } = \dfrac{75,000}{200,000} = 0.375.$$

Remember that the loss was 175,000 or 70% of the true market value. So what will the insurer pay?

$$\displaystyle 175,000 \times 0.375 = 65,625.$$

That's a lot less than 175,000. But it is 87.5% of the declared value on a loss that was only 70% of the true value.

Feel free to ask more questions if you have any remaining.

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#### Denis

##### Senior Member
No reason really...both same...

a/b*c is same as c/b*a

#### Jomo

##### Elite Member
No reason really...both same...

a/b*c is same as c/b*a
I learn something new from you every day.

#### JeffM

##### Elite Member
No reason really...both same...

a/b*c is same as c/b*a
Hmm. This is of course true, but I don't think it quite resoves the OP's confusion.

#### KWF

##### Junior Member
Coinsurance is the percentage of the market value that the insured is obliged to insure. On a 15,000 dollar house with a coinsurance requirement of 80%, that means that the insured is supposed to value the house at 15,000 * 80% = 12,000 or more and to pay premiums for a house with that declared value. With me so far?

If the insured has falsified (deliberately or inadvertently) the declared value and so has been paying premiums to insure a house worth far less than the true value, the ratio of the insurance payment to the actual loss is reduced to the same ratio that the declared value has to the coinsurance value (which is, remember, still less than the true value). In this example, the ratio between the declared value and the coinsurance value equals 8,000 / 12,000, or two thirds. Still with me?

And in this example, the actual loss (meaning the cost to repair or replace) is 7,500. And two thirds of 7500 is indeed 5,000.

$$\displaystyle \text { minimum coinsurance value } = \text { coinsurance percentage } \times \text { market value.}$$

In the example, that is $$\displaystyle 80\% \times 15,000 = 12,000.$$

$$\displaystyle \text {deficiency ratio } = \dfrac{\text {declared value}}{\text {minimum coinsurance value}}.$$

In the example, that is $$\displaystyle \dfrac{8,000}{12,000} \text { or } \dfrac{8,000}{80\% \times 15,000}$$

$$\displaystyle \text {payment due } = \text { loss } \times \text { deficiency ratio.}$$

In the example, that is

$$\displaystyle 7,500 \times \dfrac{8,000}{80\% \times 15,000} = 7,500 \times \dfrac{8,000}{12,000} =$$

$$\displaystyle 7,500 \times \dfrac{2}{3} = \dfrac{15,000}{3} = 5,000.$$

The math is correct. But if you don't understand that it is the result of combining the three formulas I gave you above into a single formula, it appears completely arbitrary.

I hope this helps.

By the way, notice that if you declare anything above the minimum coinsurance value, the payment is 100% of the loss. The whole thing is a way for the insurance company to protect itself from the insuered putting a ridiculously low valuation on the insured property and then demanding payment for losses that reflect the true value.

This was not a very good example. Let's take this one. Again suppose a coinsurance requirement of 80% of market value. Let's say that the market value is 250,000, and there is fire damage that costs 175,000 to repair. In other words, the property is 70% destroyed. But further suppose that the property owner has lowballed the declared value at 75,000 to keep premiums low. If the insurance company does not have the coinsurance requirement, it will have to pay the insured more than 200% of the declared value of the property when the damage is only 70%. No insurance company would stay solvent under those circumstances.

With the coinsurance requirement of 80%, the minimum declared value would be

$$\displaystyle 80\% \times 250,000 = 200,000.$$

$$\displaystyle \text {the deficiency ratio } = \dfrac{75,000}{200,000} = 0.375.$$

Remember that the loss was 175,000 or 70% of the true market value. So what will the insurer pay?

$$\displaystyle 175,000 \times 0.375 = 65,625.$$

That's a lot less than 175,000. But it is 87.5% of the declared value on a loss that was only 70% of the true value.

Feel free to ask more questions if you have any remaining.

I want to thank you for your reply and the effort to answer my question! I am still confused.

Let's look at it as follows. Ignore the 80% part for now.

Why does the following make sense? (amount insured)/(value of house) * (loss by fire) = amount paid by insurance company.

JeffM indicated that (loss by fire)/(value of house) * (amount insured) = amount paid by insurance company, but I doubt that an insurance company would use this calculation but perhaps one would.

#### JeffM

##### Elite Member
I want to thank you for your reply and the effort to answer my question! I am still confused.

Let's look at it as follows. Ignore the 80% part for now.

Why does the following make sense? (amount insured)/(value of house) * (loss by fire) = amount paid by insurance company.

JeffM indicated that (loss by fire)/(value of house) * (amount insured) = amount paid by insurance company, but I doubt that an insurance company would use this calculation but perhaps one would.
OK. Let's take this in steps.

As denis said, it is a mathematical fact that

$$\displaystyle \dfrac{a}{b} * c = \dfrac{a}{b} * \dfrac{c}{1} = \dfrac{ac}{b} = \dfrac{c}{b} * \dfrac{a}{1} =\dfrac{c}{b} * a.$$

In short, $$\displaystyle \dfrac{a}{b} * c = \dfrac{c}{b} * a.$$

Second, you may be mixed up about order of operations of inline expressions.

$$\displaystyle a / b * c \text { means } \dfrac{a}{b} * c, \text { not } \dfrac{a}{bc}.$$

Inline notation is a minefield of potential confusion. Parentheses are critical.

$$\displaystyle a / (b * c) \text { means } \dfrac{a}{b * c}.$$

You are of course correct that $$\displaystyle \dfrac{a}{bc} \not \equiv \dfrac{c}{ab},$$

but no one is saying that.

Third, let's go back to the logic of coinsurance.

Guy says to insurance company, "Insure my house worth 80,000 against fire." Insurance company says "Ok, that's a premium of 100 a year." The house is completely destroyed by fire, and the guy presents three appraisals from reputable appraisers saying the house is worth 800,000. Guy demands 800,000 in payment because the insurance company insured the house against fire damage. The insurance company says, "Wait a minute. You bought insurance for a house you said was worth 80,000, not 800,000, and you only paid premiums on a house worth 80,000."

What do you think the insurance company should pay? If you answered 800,000, how long will anybody be able to offer insurance?

You might say 80,000, but the insurance company might respond that there was fraud in the inducement and that therefore they owe nothing because the courts should punish fraud as a matter of public polcy.

Now keep the same facts except assume that the guy declared a value of 500,000 rather than 800,000. It is a lot harder to claim fraud. It may have been simple mistake. Most people don't pay an appraiser when they buy an insurance policy. Saying that the insurance company should pay 500,000 when they insured the house for 500,000 doesn't hurt the insurance company: that was the risk that they were paid for.

So without coinsurance, the insurance company pays

$$\displaystyle \text {actual loss} * \dfrac{\text {declared value}}{\text {actual value}}.$$

Does that make sense?

But the insured wants the actual loss if one occurs, not a fraction. Nor does the insured want to pay a premium for a lot more than the house is worth. (Nor does the insurance company want people over-insuring and burning down their insured properties: moral hazard.) So the insurance company offers a deal. You the insured make a reasonable estimate of the value of your house, but don't overinsure it. In return, we shall charge a premium based on that estimate. If it turns out you have a loss and your estimate is in fact reasonable but somewhat low, we shall pay the actual loss. But if your estimate is a way out of line, we are going to go through this convoluted process.

If the coinsurance rate is 80% and your estimate is 80,000, but the value of your house is 800,000, we are going to take 80% of 800,000, which is 640,000. That's what you should have declared the house to be worth. But you said it was worth only 80,000. The degree of deficiency is

$$\displaystyle \dfrac{80,000}{640,000} = 12.5\%.$$

So we will only pay 12.5% of the actual loss.

Let

$$\displaystyle p = \text {payment due;}$$Let

$$\displaystyle a = \text {actual loss on insured property;}$$Let

$$\displaystyle c = \text {coinsurance rate as a decimal, not a percentage;}$$Let

$$\displaystyle d = \text {declared value of insured property; and}$$Let

$$\displaystyle m = \text {market value of insured property.}$$

Then this is what we get

$$\displaystyle \text {If } c * m \ge d, \text { then } p = a, \text { but}$$

$$\displaystyle \text {if } c * m < d, \text { then } p = a * \dfrac{d}{c * m}.$$

The c * m is what the insured should have declared the property to be worth.