<p>Is there an insurence menu consisting of several pairs (deductible, annual premium) where</p>
<ol>
<li><p>there is more than one option</p>
</li>
<li><p>there is no possibility for adverse selection</p>
</li>
<li><p>there are no two options <span class="math-container">$(d_1,ap_1)$</span>, <span class="math-container">$(d_2,ap_2)$</span> where
<span class="math-container">$$
d_1 &gt; d_2, \hskip 10pt ap_1 &gt; ap_2,
$$</span>
making option 1 clearly (&quot;monotonically&quot;) worse than option 2?</p>
</li>
</ol>
<hr />
<p>If there is no such menu, should all insurance companies offer just one policy, and the same one at that, or is it possible that accounting for consumer preferences (and perhaps more importantly, consumer liquidity) is profitable for a company?</p>

CC BY-SA 4.0
2023-09-23T16:28:32.593Z
I just added an explanation to the question.  There are good reasons for both the insurer and the customer to want different deductible options.  A customer who opts for a lower deductible is signaling a disposition or preference to file more claims, and therefore the expected losses (in terms of expected number of claims filed) on that customer are higher.
feetwet
CC BY-SA 4.0
2023-09-23T17:18:20.850Z
Hi @feetwet. I have read your added explanation. I don't think it changes the gist of my answer, which is that it is impossible to have a menu of policies include more than one option, where an option is not monotonically worse than another or the options do not incentivize adverse selection.
Giskard
CC BY-SA 4.0
2023-09-23T18:20:14.693Z
Oh I see what you are saying now.  But what I am describing is a situation where the higher deductibles *are **worse** in expected loss terms*, and given the adverse incentives of lower deductibles the insurer should price them to be more attractive in expected loss terms.  I.e., holding expected loss constant it is easy to price every deductible to be equal in value.  But adding in the adverse incentive of lower deductibles the insurer should price them in the opposite direction.  (But I've raised the question because a profit-seeker seems unlikely to make such a simple mistake.)
feetwet
CC BY-SA 4.0
2023-09-23T07:04:00.840Z
2023-09-23T07:04:00.840Z
Giskard
<p>The retail insurance market isn't really fixed-price.  Insurance companies can and do penalize adverse customers.  <a href="https://www.investopedia.com/articles/pf/08/claim-raise-rates.asp#:%7E:text=The%20greater%20the%20number%20of,rates%20will%20almost%20surely%20rise." rel="nofollow noreferrer">For example</a>,</p>
<blockquote>
<p>[T]he number of insurance claims you file also has a direct impact on your rates. The greater the number of claims filed, the greater the likelihood of a rate hike.</p>
</blockquote>
<p><a href="https://www.iii.org/fact-statistic/facts-statistics-homeowners-and-renters-insurance" rel="nofollow noreferrer">Actual claim rates are about 1/20 years</a>.  So from an underwriting perspective the $5k deductible is closer to the baseline for profitability and the extra premium earned from lower deductibles is excess profit to the insurer because they do not expect (and, in fact, penalize) more frequent claims.</p>

CC BY-SA 4.0
2023-09-26T05:09:03.130Z
So your table made you suspect that the lower deductible is less profitable for the insurer and now the same table has you suspecting the lower deductible is more profitable for the insurer? It's probably safer to say that the table did not imply anything about profitability and thus nothing about adverse selection either.
H2ONaCl
CC BY-SA 4.0
2023-09-25T18:45:20.170Z
2023-09-25T18:45:20.170Z
feetwet
<p>I just saw a rate table for retail home insurance that appears to offer incentives opposite to what insurance should in theory.  Am I missing something in the following analysis?</p>
<p>For the same underlying policy, here is a table showing the deductible and annual premium.  I added a row computing the total-loss break-even claim frequency = (Premium Delta)/(Deductible Delta), where the delta is against the first row (lowest deductible).</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th style="text-align: center;">Deductible</th>
<th style="text-align: center;">Annual Premium</th>
<th style="text-align: center;">Breakeven Claim Frequency</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: center;">1,000</td>
<td style="text-align: center;">1,700</td>
<td style="text-align: center;">Baseline</td>
</tr>
<tr>
<td style="text-align: center;">2,000</td>
<td style="text-align: center;">1,530</td>
<td style="text-align: center;">5.9 years</td>
</tr>
<tr>
<td style="text-align: center;">3,575</td>
<td style="text-align: center;">1,360</td>
<td style="text-align: center;">7.6 years</td>
</tr>
<tr>
<td style="text-align: center;">5,000</td>
<td style="text-align: center;">1,320</td>
<td style="text-align: center;">10.5 years</td>
</tr>
</tbody>
</table>
</div>
<p>So, for example, people who buy the $5k deductible only expect to save money against the lowest deductible if they make claims <em>less frequently</em> than every 10.5 years.  This seems backwards for two reasons:</p>
<ol>
<li>The insurer is encouraging adverse selection: People who are disposed to suffer losses (or rather, to file claims) are rewarded for buying the lower deductible.</li>
<li>The breakeven is even more favorable to the lower deductible buyers than this calculation because the total-loss calculation ignores unattached losses: A loss for less than \$5k won't result in payment to the high-deductible insured, but will to the low-deductible insured.  (For example, a \$4k loss can result in some payout under every deductible but the highest.)</li>
</ol>
<hr />
<p><strong>ETA:</strong> The <em>adverse selection</em> posited here is not in terms of whether the company obtains a particular customer, but rather in terms of the <em>preferences</em> and <em>marginal incentives</em> of each of the customers it does have (which translate into profitability for the insurer).</p>
<p>To elaborate: All insurance has a deductible (a.k.a. loss attachment).  In one sense this is because <em>an insurance policy with no deductible</em> is more appropriately described as a <em>service contract</em>.  But especially in the retail space <strong>deductibles exist to provide incentives for the buyers to avoid losses</strong>.</p>
<p>Further to point #1 above: A higher deductible creates a greater incentive for the buyer to avoid a loss, which makes the marginal cost of providing the insurance lower.  So on that fact alone we should expect insurance to be discounted as deductible increases – the opposite of what I have observed here, unless I have made an error or misunderstanding.</p>

CC BY-SA 4.0
2023-09-23T05:51:05.940Z
This is not adverse selection because there is nothing in your scenario that describes an incentive to deal with this insurance company rather than another. This is not adverse selection because you did not attempt to demonstrate that one of the 4 choices is systematically less profitable for the insurance company than the others.
H2ONaCl
CC BY-SA 4.0
2023-09-23T06:54:44.880Z
@H2ONaCl I did not find your comment useful to this question, I hope both yours and this gets cleaned up.
Giskard
CC BY-SA 4.0
2023-09-23T09:44:10.090Z
@Giskard my understanding of adverse selection is consistent with the one posted at Econlib.org https://www.econlib.org/library/Enc/Insurance.html
H2ONaCl
CC BY-SA 4.0
2023-09-23T09:44:31.090Z
@Giskard A quote... "Recognizing that the identity of the purchaser affects the cost of insurance, insurers must be careful to whom they offer insurance at a particular price. Those high-risk individuals whose knowledge of their risk is better than that of the insurers will step forth to purchase, knowing that they are getting a good deal. This is a process called adverse selection, which means that the mix of purchasers will be adverse to the insurer."
H2ONaCl
CC BY-SA 4.0
2023-09-23T09:54:51.583Z
@Giskard There is no apparent incentive described in the post to draw bad risks to this company over alternative companies so that's why I said this particular form of adverse selection is not described in the post.
H2ONaCl
CC BY-SA 4.0
2023-09-23T09:55:42.560Z
@Giskard The post seems to assume that the lower deductible option is adverse for the insurer but it was not posited that this is less profitable (than the other 3 choices) so it is not clear why this is adverse.
H2ONaCl
CC BY-SA 4.0
2023-09-24T05:10:14.713Z
I looked at your addendum. You said `on that fact alone we should expect insurance to be discounted as deductible increases`. If by "discounted" you mean the insurance premium is less, then you should look at your table again. The higher deductible is accompanied by a premium that is less so I don't understand the source of your concern.
H2ONaCl
CC BY-SA 4.0
2023-09-24T08:55:37.797Z
If you want to say the lower deductible option is less profitable than the other 3, it would be helpful if you just say it explicitly. Then you will be at last making a claim that there is adverse selection for this particular contract. Then follow up by demonstrating why it is less profitable in order to show that it really is adverse.
H2ONaCl
<adverse-selection><insurance>
Is home insurance priced backwards?
feetwet
# Is home insurance priced backwards?

(Asked by: username_0 on 2023-09-22)

I just saw a rate table for retail home insurance that appears to offer incentives opposite to what insurance should in theory. Am I missing something in the following analysis?

For the same underlying policy, here is a table showing the deductible and annual premium. I added a row computing the total-loss break-even claim frequency = (Premium Delta)/(Deductible Delta), where the delta is against the first row (lowest deductible).

| Deductible | Annual Premium | Breakeven Claim Frequency |
| --- | --- | --- |
| 1,000 | 1,700 | Baseline |
| 2,000 | 1,530 | 5.9 years |
| 3,575 | 1,360 | 7.6 years |
| 5,000 | 1,320 | 10.5 years |

So, for example, people who buy the $5k deductible only expect to save money against the lowest deductible if they make claims *less frequently* than every 10.5 years. This seems backwards for two reasons:

1. The insurer is encouraging adverse selection: People who are disposed to suffer losses (or rather, to file claims) are rewarded for buying the lower deductible.
2. The breakeven is even more favorable to the lower deductible buyers than this calculation because the total-loss calculation ignores unattached losses: A loss for less than \$5k won't result in payment to the high-deductible insured, but will to the low-deductible insured. (For example, a \$4k loss can result in some payout under every deductible but the highest.)

---

**ETA:** The *adverse selection* posited here is not in terms of whether the company obtains a particular customer, but rather in terms of the *preferences* and *marginal incentives* of each of the customers it does have (which translate into profitability for the insurer).

To elaborate: All insurance has a deductible (a.k.a. loss attachment). In one sense this is because *an insurance policy with no deductible* is more appropriately described as a *service contract*. But especially in the retail space **deductibles exist to provide incentives for the buyers to avoid losses**.

Further to point #1 above: A higher deductible creates a greater incentive for the buyer to avoid a loss, which makes the marginal cost of providing the insurance lower. So on that fact alone we should expect insurance to be discounted as deductible increases – the opposite of what I have observed here, unless I have made an error or misunderstanding.

**Tags:** adverse-selection, insurance
> **Comments:**
> **username_1**: This is not adverse selection because there is nothing in your scenario that describes an incentive to deal with this insurance company rather than another. This is not adverse selection because you did not attempt to demonstrate that one of the 4 choices is systematically less profitable for the insurance company than the others.  
> **username_2**: @H2ONaCl I did not find your comment useful to this question, I hope both yours and this gets cleaned up.  
> **username_1**: @Giskard my understanding of adverse selection is consistent with the one posted at Econlib.org https://www.econlib.org/library/Enc/Insurance.html  
> **username_1**: @Giskard A quote... "Recognizing that the identity of the purchaser affects the cost of insurance, insurers must be careful to whom they offer insurance at a particular price. Those high-risk individuals whose knowledge of their risk is better than that of the insurers will step forth to purchase, knowing that they are getting a good deal. This is a process called adverse selection, which means that the mix of purchasers will be adverse to the insurer."  
> **username_1**: @Giskard There is no apparent incentive described in the post to draw bad risks to this company over alternative companies so that's why I said this particular form of adverse selection is not described in the post.  
> **username_1**: @Giskard The post seems to assume that the lower deductible option is adverse for the insurer but it was not posited that this is less profitable (than the other 3 choices) so it is not clear why this is adverse.  
> **username_1**: I looked at your addendum. You said `on that fact alone we should expect insurance to be discounted as deductible increases`. If by "discounted" you mean the insurance premium is less, then you should look at your table again. The higher deductible is accompanied by a premium that is less so I don't understand the source of your concern.  
> **username_1**: If you want to say the lower deductible option is less profitable than the other 3, it would be helpful if you just say it explicitly. Then you will be at last making a claim that there is adverse selection for this particular contract. Then follow up by demonstrating why it is less profitable in order to show that it really is adverse.  

---

## 2 answers

### Answer by username_2 on 2023-09-23. Score: 1

Is there an insurence menu consisting of several pairs (deductible, annual premium) where

1. there is more than one option
2. there is no possibility for adverse selection
3. there are no two options $(d\_1,ap\_1)$, $(d\_2,ap\_2)$ where
   $$
   d\_1 > d\_2, \hskip 10pt ap\_1 > ap\_2,
   $$
   making option 1 clearly ("monotonically") worse than option 2?

---

If there is no such menu, should all insurance companies offer just one policy, and the same one at that, or is it possible that accounting for consumer preferences (and perhaps more importantly, consumer liquidity) is profitable for a company?

> **Comments:**
> **username_0**: I just added an explanation to the question.  There are good reasons for both the insurer and the customer to want different deductible options.  A customer who opts for a lower deductible is signaling a disposition or preference to file more claims, and therefore the expected losses (in terms of expected number of claims filed) on that customer are higher.  
> **username_2**: Hi @feetwet. I have read your added explanation. I don't think it changes the gist of my answer, which is that it is impossible to have a menu of policies include more than one option, where an option is not monotonically worse than another or the options do not incentivize adverse selection.  
> **username_0**: Oh I see what you are saying now.  But what I am describing is a situation where the higher deductibles *are **worse** in expected loss terms*, and given the adverse incentives of lower deductibles the insurer should price them to be more attractive in expected loss terms.  I.e., holding expected loss constant it is easy to price every deductible to be equal in value.  But adding in the adverse incentive of lower deductibles the insurer should price them in the opposite direction.  (But I've raised the question because a profit-seeker seems unlikely to make such a simple mistake.)  

---

### Answer by username_0 on 2023-09-25. Score: 0

The retail insurance market isn't really fixed-price. Insurance companies can and do penalize adverse customers. [For example](https://www.investopedia.com/articles/pf/08/claim-raise-rates.asp#:%7E:text=The%20greater%20the%20number%20of,rates%20will%20almost%20surely%20rise.),

> [T]he number of insurance claims you file also has a direct impact on your rates. The greater the number of claims filed, the greater the likelihood of a rate hike.

[Actual claim rates are about 1/20 years](https://www.iii.org/fact-statistic/facts-statistics-homeowners-and-renters-insurance). So from an underwriting perspective the $5k deductible is closer to the baseline for profitability and the extra premium earned from lower deductibles is excess profit to the insurer because they do not expect (and, in fact, penalize) more frequent claims.

> **Comments:**
> **username_1**: So your table made you suspect that the lower deductible is less profitable for the insurer and now the same table has you suspecting the lower deductible is more profitable for the insurer? It's probably safer to say that the table did not imply anything about profitability and thus nothing about adverse selection either.  

---

