Recently I've been reading The Mathematics of Poker (2009, Bill Chen and Jerrod Ankenman) and I came across an interesting idea that I thought I'd write about. For me, understanding how to analyse this situation really gets to the heart of how I think about poker. I'd love to spend more time playing and studying poker but it's such a timesink and I don't really have the oodles of free time it would require, but every now and again I'll still open up a poker book and read about something, this is one of those interesting topics I was reading about hopefully you find it as interesting as I do. Calling a shove preflop in heads up The scenario being analysed in the book is a relatively common situation, particularly in online poker where people are more inclined to shove than in real life games. The question is: How should we analyse whether to call when we have a moderately strong hand against an opponent who has gone allin pre flop. Let's set up an example so that we have something concrete to talk about. Say we have pocket Kings pre flop, and our opponent goes allin, how should we decide whether to call? Obviously without knowing our opponent's hand there is no 100% correct answer. There is however one very useful way of analysing the situation. Equity against a range We need to ask ourselves  what cards would are opponent go allin with, and how does my current hand fare against that range? i.e. we need to calculate our equity against our opponent's range. We are adding another layer of stochastic uncertainty on the event, instead of trying to guess what hand our opponent has (which is almost impossible) we are trying to guess what kind of hands he might go allin with (which is very much possible). We then take this extra level of uncertainty and calculate the correct mathematical way to proceed. On the one extreme, let's suppose that based on our read of how our opponent is playing, we might think that they would only go allin with very strong hands, in this case just pocket Aces. We would then be a 4:1 underdog if we call with Ks, and we should definitely fold. (In order to calculate this we could use any standard poker calculator like the following) www.cardplayer.com/pokertools/oddscalculator/texasholdem One the other hand, suppose we know for a fact that our opponent has not looked at their cards at all but has still decided to go all in. In this case we should definitely call. The only cards we will be behind are pocket Aces, which make up a small fraction of the possible hands that our opponent could shove with, and we will be ahead or equal against all other possible hands. Therefore we would have a positive EV when calling. What if our read on our opponent's range is somewhere in between though? What we need to do is calculate our equity against each individual hand in our opponent's range, and then calculate the probability of our opponent having a given hand from that range. That is to say, in order to calculate the conditional expectation, we need to calculate the unconditional expectations against each hand and then multiply by the conditional probability of our opponent having that hand, given our belief about our opponent's range. Numerical Example Let's go back to our numerical example, and suppose that we have pocket Kings, and we put our opponent on either Pocket Aces, Kings, Queens, or Jacks. All of these hands are equally likely, so there is a 25% chance of our opponent having each hand. We can look up our equity against each hand (after you've been playing for a while, you naturally start to memorise hand equities anyway) Our probability of winning is then: $$P(A) * P(beating A) + P(K)*P(beating K) + P(Q)*P(beating Q) + P(J) * P(beating J)$$ Putting in our values: $$ 0.25*0.2 + 0.25*0.5 + 0.25*0.8 + 0.25*0.8 = 0.575.$$ We therefore see we have a positive expectation against this range, and should call. No one actually thinks like this in real games? It is a general misconception that professional poker is a game where players are trying to guess exactly what hand their opponent has, are constantly trying to bluff each other, or trying to pick up on subtle tells or signs that their opponent is or isn't bluffing. The more mundane truth is that poker is ultimately a game of imperfect information, where the better player is the one who can correctly interpret the betting information their opponent is giving them, and can then quickly and accurately make the kind of judgements described above during a game. Obviously poker players are not carrying out these calculations in their head to multiple decimal places in real time, what they will do though is review their hands after a game, calculate exactly what they should have done, and try to build up an intuition as to what the correct answer is, so that in the middle of a game they can quickly make decisions. Software to Analyse this situation Is there an easy software based method way of calculating our equity against a range? After I did a quick google there are a few programs that offer this type of analysis. For example: combonator.com/ www.powerequilab.com/ More interestingly though, I also found the following open source software, that can be adapted to carry out this type of analysis: github.com/zekyll/OMPEval At some point, I might try to use this code to set up a page on this website to let people analyse this situation. Was the Lognormal Distribution misnamed?20/2/2018 I was thinking about this last week at work when I was coding part of a model involving the parameters of a truncated lognormal distribution. The lognormal distribution definitely feels like it was named the wrong way round. What is a Lognormal Distribution?
We say that a Random Variable $X$ has a LogNormal Distribution that is:
$$ X \sim LogN( \mu , { \sigma }^2 ) $$ if: $$ Log (X) \sim N( \mu , { \sigma }^2 ) $$In other words, a Lognormal distribution is a distribution such that the log of the distribution is a normal distribution. It is not, as you might think, a distribution which is the log of the normal distribution. So if $Y \sim N( \mu , {\sigma}^2 ) $ then $Log ( Y ) $ is not a lognormal distribution, instead $ e ^ Y $ is a lognormal distribution. So to create a lognormal distribution, we don't take the log of the normal distribution, we take the exponential! Why does this matter?
Definitions are just definitions after all, and as long as everyone knows how something is defined and there is no ambiguity one definition is usually as good as another. In this case though, defining it in this way does have some ugly and unnatural consequences. For example, if we take the result that the sum of two independent normal distributions is also a normal distribution, i.e.
If: $$ X \sim N( {\mu}_1 , {{\sigma}_1}^2 ) , Y \sim N( {\mu}_2 , {{\sigma}_2}^2 ) $$ Then: $$ X + Y \sim N( {\mu}_1 + {\mu}_2 , {{\sigma}_1}^2 + {{\sigma}_2}^2 ) $$ Then applying this result to the lognormal distribution, we get: If $ X \sim LogN( {\mu}_1 , {{\sigma}_1}^2 ) $ and $ Y \sim LogN( {\mu}_2 , {{\sigma}_2}^2 ) $ assuming independence, Then:$$ XY \sim LogN( {\mu}_1 + {\mu}_2 , {{\sigma}_1}^2 + {{\sigma}_2}^2 ) $$ Maybe this doesn't look too bad to you. But what if I replace $X$ and $Y$ with ${LogN}_1$ and ${LogN}_2$? Then we get: $$ {LogN}_1 * {LogN}_2 \sim LogN( {\mu}_1 + {\mu}_2 , {{\sigma}_1}^2 + {{\sigma}_2}^2 ) $$ This should definitely look wrong to you! Remember that for a standard logarithm: $$ Log (AB) = Log(A) + Log(B) $$ Instead we have an identity that looks much more like an exponential:$$ e^A * e^B = e^{ (A + B ) } $$ And that's precisely because we are dealing with an exponential! The lognormal distribution is simply the exponential of the normal, which is a much more natural way of phrasing it than to say that the lognormal distribution is a distribution such that the logarithm of the distribution is a normal distribution. So we have two reasons why the Lognormal Distbribution should have been called the Exponential Normal Distribution (Or possibly the XNormal Distribution for short). The identity above makes perfect sense when using exponentials, and we would have a naming convention that is much more natural. It is quite simple to calculate the Reinstatement Premium resulting from a loss to an Excess of Loss contract. Therefore, it seems reasonable that we should be able to come up with a simple formula relating the price charged for the Excess of Loss contract to the price charged for the Reinstatement Premium Protection (RPP) cover. I was in a meeting last week with two brokers who were trying to do just this. We had come up with an indicative price for an XoL layer and we were trying to use this to price the equivalent RPP cover. At the time I didn't have an easy way to do it, and when I did a quick Google search nothing came up. Upon further reflection, there are a couple of easy approximate methods we can use. Below I discuss three different methods which can be used to price an RPP cover, two of which do not require any stochastic modelling. Let's quickly review a few definitions, feel free to skip this section if you just want the formula. What is a Reinstatement Premium? A majority of Excess of Loss contracts will have some form of reinstatement premium. This is a payment from the Insurer to the Reinsurer to reinstate the protection in the event some of the limit is eroded. In the London market, most contracts will have either $1$, $2$, or $3$ reinstatements and generally these will be payable at $100 \%$. From the point of view of the insurer, this additional payment comes at the worst possible time, the Reinsured is being asked to fork over another large premium to the Reinsurer just after having suffered a loss. What is a Reinstatement Premium Protection (RPP)? Reinsurers developed a product called a Reinstatement Premium Protection cover (RPP cover). This cover pays the Reinsured's Reinstatement Premium for them, giveing the insurer further indemnification in the event of a loss. Here's an example of how it works in practice: Let's suppose we are considering a $5m$ xs $5m$ Excess of Loss contract, there is one reinstatement at $100 \%$ (written $1$ @ $100 \%$), and the Rate on Line is $25 \%$. The Rate on Line is just the Premium divided by the Limit. So the Premium can be found by multiplying the Limit and the RoL: $$5m* 25 \% = 1.25m$$ So we see that the Insurer will have to pay the Reinsurer $1.25m$ at the start of the contract. Now let's suppose there is a loss of $7m$. The Insurer will recover $2m$ from the Resinsurer, but they will also have to make a payment to cover the reinstatement premium of: $\frac {2m} {5m} * (5m * 25 \% ) = 2m * 25 \% = 0.5m$ to reinstate the cover. So the Insurer will actually have to pay out $5.5m$. The RPP cover, if purchased by the insurer, would pay the additional $0.5m$ on behalf of the insurer, in exchange for a further upfront premium. Now that we know how it works, how would we price the RPP cover? Three methods for pricing an RPP cover Method 1  Full stochastic model If we have priced the original Excess of Loss layer ourselves using a Monte Carlo model, then it should be relatively straight forward to price the RPP cover. We can just look at the expected Reinstatements, and apply a suitable loading for profit and expenses. This loading will probably be broadly in line with the loading that is applied to the expected losses to the Excess of Loss layer, but accounting for the fact that the writer of the RPP cover will not receive any form of Reinstatement for their Reinsurance. What if we do not have a stochastic model set up to price the Excess of Loss layer? What if all we know is the price being charged for the Excess of Loss layer? Method 2  Simple formula Here is a simple formula we can use which gives the price to charge for an RPP, based on just the deposit premium and the Rate on Line, full derivation below: $$RPP = DP * ROL $$ When attempting to price the RPP last week, I did not have a stochastic model set up. We had come up with the pricing just based off the burning cost and a couple of 'commercial adjustments'. The brokers wanted to use this to come up with the price for the RPP cover. The two should be related, as they pay out dependant on the same underlying losses. So what can we say? If we denote the Expected Losses to the layer by $EL$, then the Expected Reinstatement Premium should be: $$EL * ROL $$ To see this is the case, I used the following reasoning; if we had losses in one year equal to the $EL$ (I'm talking about actual losses, not expected losses here), then the Reinstatement Premium for that year would be the proportion of the layer which had been exhausted $\frac {EL} {Limit} $ multiplied by the Deposit Premium $Limit * ROL$ i.e.: $$ RPP = \frac{EL} {Limit} * Limit * ROL = EL * ROL$$ Great! So we have our formula right? The issue now is that we don't know what the $EL$ is. We do however know the $ROL$, does this help? If we let $DP$ denote the deposit premium, which is the amount we initially pay for the Excess of Loss layer and we assume that we are dealing with a working layer, then we can assume that: $$DP = EL * (1 + \text{ Profit and Expense Loading } ) $$ Plugging this into our formula above, we can then conclude that the expected Reinstatement Premiums will be: $$\frac {DP} { \text{ Profit and Expense Loading } } * ROL $$ In order to turn this into a price (which we will denote $RPP$) rather than an expected loss, we then need to load our formula for profit and expenses i.e. $$RPP = \frac {DP} {\text{ Profit and Expense Loading }} * ROL * ( \text{ Profit and Expense Loading } ) $$Which with cancellation gives us: $$RPP = DP * ROL $$ Which is our first very simple formula for the price that should be charged for an RPP. Was there anything we missed out though in our analysis? Method 3  A more complicated formula: There is one subtlety we glossed over in order to get our simple formula. The writer of the Excess of Loss layer will also receive the Reinstatement Premiums during the course of the contract. The writer of the RPP cover on the other hand, will not receive any reinstatement premiums (or anything equivalent to a reinstatement premium). Therefore, when comparing the Premium charged for an Excess of Loss layer against the Premium charged for the equivalent RPP layer, we should actually consider the total expected Premium for the Excess of Loss Layer rather than just the Deposit Premium. What will the additional premium be? We already have a formula for the expected Reinstatement premium: $$EL * ROL $$ Therefore the total expected premium for the Excess of Loss Layer is the Deposit Premium plus the additional Premium: $$ DP + EL * ROL $$ This total expected premium is charged in exchange for an expected loss of $EL$. So at this point we know the Total Expected Premium for the Excess of Loss contract, and we can relate the expected loss to the Excess of Loss layer to the Expected Loss to the RPP contract. i.e. For an expected loss to the RPP of $EL * ROL$, we would actually expect an equivalent premium for the RPP to be: $$ RPP = (DP + EL * ROL) * ROL $$ This formula is already loaded for Profit and Expenses, as it is based on the total premium charged for the Excess of Loss contract. It does however still contain the $EL$ as one of its terms which we do not know. We have two choices at this point. We can either come up with an assumption for the profit and expense loading (which in this hard market might be as little as only be $5 \%  10 \%$ ). And then replace $EL$ with a scaled down $DP$: $$RPP = \frac{DP} {1.075} * ( 1 + ROL) * ROL $$ Or we could simply replace the $EL$ with the $DP$, which is partially justified by the fact that the $EL$ is only used to multiply the $ROL$, and will therefore have a relatively small impact on the result. Giving us the following formula: $$RPP = DP ( 1 + ROL) * ROL $$ Which of the three methods is the best? The full stochastic model is always going to be the most accurate in my opinion. If we do not have access to one though, then out of the two formulas, the more complicated formula we derived should be more accurate (by which I mean more actuarially correct). If I was doing this in practice, I would probably calculate both, to generate some sort of range, but tend towards the second formula. That being said, when I compared the prices that the Brokers had come up with, which is based on what they thought they could actually place in the market, against my formulas, I found that the simple version of the formula was actually closer to the Broker's estimate of how much these contacts could be placed for in the market. Since the simple formula always comes out with a lower price than the more complicated formula, this suggests that there is a tendency for RPPs to be underpriced in the market. This systematic underpricing may be driven by commercial considerations rather than faulty reasoning on the part of market participants. According to the Broker I was discussing these contracts with, a common reason for placing an RPP is to give a Reinsurer who does not currently have a line on the underlying Excess of Loss layer, but who would like to start writing it, a chance to have an involvement in the same risk, without diminishing the signed lines for the existing markets. So let's say that Reinsurer A writes $100 \%$ of the Excess of Loss contract, and Reinsurer B would like to take a line on the contract. The only way to give them a line on the Excess of Loss contract is to reduce the line that Reinsurer A has. The insurer may not wish to do this though if Reinsurer A is keen to maintain their line. So the Insurer may allow Reinsurer B to write the RPP cover instead, and leave Reinsurer A with $100 \%$ of the Excess of Loss contract. This commercial factor may be one of the reasons that traditionally writers of an RPP would be inclined to give favourable terms relative to the Excess of Loss layer so as to encourage the insurer to allow them on to the main programme and to encourage them to allow them to wrte the RPP cover at all. Moral Hazard One point that is quite interesting to note about how these deals are structured is that RPP covers can have quite a significant moral hazard effect on the Insurer. The existence of Reinstatement Premiums is at least partially a mechanism to prevent moral hazard on the part of the Insurer. To see why this is the case, let's go back to our example of the $5m$ xs $5m$ layer. An insurer who purchases this layer is now exposed to the first $5m$ of any loss. But they are indemnified for the portion of the loss above $5m$, up to a limit of $5m$. If the insurer is presented with two risks which are seeking insurance  one with a total sum insured of $10m$, and another with a total sum insured of $6m$, the net retained exposure is the same for both risks from the point of view of the insurer. By including a reinstatement premium as part of the Excess of Loss layer, an therefore ensuring that the insurer has to make a payment any time a loss ceded to the layer, the reinsurer is ensuring that the insurer keeps their financial incentive to not have losses in this range. By purchasing an RPP cover, the insurer is removing their financial interest in losses which are ceded to the layer. There is an interesting conflict of interest in that the RPP cover will almost always be written by a different reinsurer to the Excess of Loss layer. The Reinsurer that is writing the RPP cover is therefore increasing the moral hazard risk whichever Reinsurer has written the Excess of Loss layer. Which will almost always be business written by one of the Reinsurer's competitors! Working Layers and unlimited Reinstatements Another point to note is that this pricing analysis makes a couple of implicit assumptions. The first is that there is a sensible relationship between the expected loss to the layer and the premium charged for the layer. This will normally only be the case for 'working layers'. These are layers to which a reasonable amount of loss activity is expected. If we are dealing with clash or other higher layers, then the pricing of these layers will be more heavily driven by considerations beyond the expected loss to the layer. These might be capital considerations on the part of the Reinsurer, commercial considerations such as Another implicit assumption in this analysis is that the reinstatements offered are unlimited,. If this is not the case, then the statement that the expected reinstatement is $EL * ROL$ no longer holds. If we have limited reinstatements (which is the case in practice most of the time) then we would expect the expected reinstatement to be less than or equal to this. 
AuthorI work as a pricing actuary at a reinsurer in London. Categories
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February 2021
