# Mango Paper

I’m just going to start a new thread, because I’m sure I won’t be the only one to have questions on this.

After looking at this paper about half a dozen times, I think I finally have the hang of what’s going on here. My question is whether the exhibits in the paper are off - specifically Exhibits 3 and 4 where he shows the risk loads for X and Y after redistributing the differences. Exhibit 3, I get - hey, here’s the deferred risk load. But instead of actually re-distributing that deferred risk load, in Exhibit 4 it looks like it all gets shoved into Account X.

Am I missing something there, or are the final values in Exhibit 4 really incorrect?

I’m not following why you are okay with Table 3, but not Table 4. They are showing the exact same thing, but one is using MS and the other MV, highlighting the sub- and super- additivity of the two methods.

Because … because I don’t. That’s the easy answer.

Is the point of Exhibit 3 to show that when you renew X on Exhibit 4, the full deferred risk load you say you need on 3 just goes to X? If you renewed Y instead, would the full deferred risk load go to Y? I guess I’m asking in 3 when he says “there’s this extra risk load that has to go somewhere” why it doesn’t get split between X and Y somehow (Shapley, covariance share) in Exhibit 3 so that when either is renewed its risk load in 4 doesn’t suddenly change because of the order of adding, which is the entire point of Mango’s paper: that risk loads for accounts shouldn’t be order-dependent.

EDIT: I’ve looked at this more, I think it might finally make a little more sense … but this is still clear as shit. It is certainly not intuitive to what I would expect given the explanation in the paper.

What don’t you understand marginal variance or marginal surplus?

Well, I’ve kind of explained it above a couple times now so I’ll assume it’s just me not giving a good explanation.

I get MS and MV. I get “you start with X, you add Y by either MS or MV and the marginal risk load for Y isn’t what would be if you just looked at X on its own.” I get “if you do it by MS, the marginal risk load is less than what Y would suggest; if you do it by MV, the marginal risk load is more than what Y would suggest.”

So, we go to showing how things change with Shapley value and covariance share. In Exhibit 3 he shows the scenarios for which risk is added 1st/2nd, computes the changes in variance, and says “the Shapley value for X is 21,070,450 and for Y it’s 1,828,509” and then right below it says “oh, we’re computing the risk load for X? Yeah … the Shapley value for X is really 19,619,900” without any explanation, while the Shapley value for Y doesn’t change. He then computes a risk load for each, says “whoa, X + Y is different from (X+Y) by \$100.03.” Then when we renew X in Exhibit 4, it’s “oh yeah … now the Shapley value for X is really 21,070,450” and the risk load gets computed as expected. [The same argument goes for covariance share.]

So … why does he do that (change the Shapley value in Exhibit 3 for X but not for Y when computing the risk load for each)?

Sorry Mr. Hoffman. I have a little tldr itis.

The key aspect you need to know is that Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y). The Shapley value of both together is 2Cov(X,Y).

There are two important situations and I think this is what you are not clear about:

1. Build up - Adding a new risk to company. So originally having X and now adding Y.
2. Renewal - Both X and Y renew at the same time.

The case presented in the text (it’s been months since I have read this) is Renewal for Shapley. This means that we assume both of them are occuring at the same time so
ShapleyX = TotalRiskLoad_x+y - Riskload_x = Var(X+Y) - [Var(Y) + Cov(X,Y)] = Var(X) + Cov(X,Y)
ShapleyY = Var(Y) + Cov(X,Y).

However, if we are talking about build up where Y was added to x then we aren’t applying the shapley value to X. X first got a risk load of Var(X) and now Y gets a risk load of Var(Y) + Cov(X,Y) because it is being added to X. There was no covariance when X initially got the risk load. Thus, it is not renewal additive in this case.

The Covariance Share method just allocates the 2Cov(X,Y) based on the expected losses. So you multiply it by X/(X+Y) or Y/(X+Y).

Or Shapley is equivalent to the Covariance Share method when you assume the weights X/(X+Y) and Y/(X+Y) or 50% each.

If you want to ask about game theory or super/sub-additivity that would be a good review as well.

I get this.

I get this too.

I agree, we aren’t applying Shapley value to X on build-up. Nor should we be applying it to Y, IMO. We just showed earlier in the paper that Y has to get Var(Y) + 2Cov(X,Y) because of marginal variance. [And, we showed it’s not the right amount because order of addition impacts things - which is why we’re trying to figure out what is the right amount.] If you say “yeah, Y only gets Var(Y) + Cov(X,Y)” then there’s another Cov(X,Y) that necessarily has to go somewhere so that the total variance for (X+Y) is accounted for. I’m asking where it goes on build-up.

Look, it might just be me and I’m an idiot on this and don’t understand this and never will - and I’m OK with that. I get how after renewal the sum of the risk loads for X and Y equals the risk load for (X+Y). That’s what Exhibit 4 shows. I just don’t see how Exhibit 3 says “hey, our total risk load is \$1579 - how does that get allocated? It’s \$1353 to X and \$126 to Y and \$100 … ¯_(ツ)_/¯” and we don’t bother with putting that \$100 anywhere until we renew accounts.

Build up is a practical thing. How could X have a risk load using Var(X) + Cov(X,Y) when Y doesn’t exist yet…

When Y exists the Shapley method only assigns Cov(X,Y) in addition to Var(Y).

Yes … I know, When you build up and you only have X, you can’t have a Cov(X,Y) until you have Y. Once you add Y, though, you’re adding Var(Y) + 2Cov(X,Y) to the total account variance. If you’re using Shapley to allocate risk loads, you have to put 1Cov(X,Y) to each of X and Y. Except, Exhibit 3 hangs the risk load for 1Cov(X,Y) out there as some “deferred” piece when it’s really not deferred - it exists, and it should go to X where the exhibit doesn’t really put it there until X renews and …

I’ll just take the L on this.

I think it’s because you only allocate 1Cov(X,Y) to Y even though there is 1Cov(X,Y) still sitting out there…

Can you tell me how the the covariance share values are calculated in Exhibit 3?

Perhaps to start we can be specific - how is the value of 9,920,635 calculated?

Yeah so I’m pretty confused as to why the values in those covariance share columns are so huge. The loss for Y for event 1 is just 200. 200 squared is 40,000 so I’m struggling to find out how the share can possibly even be 79,365.

Ok, I finally figured out where those numbers come from. So the formula for the covariance share for a given event i (let’s assume for X) is 2WiXiYipi*(1-pi) where Wi = Xi/(Xi+Yi). So for whatever reason, Mango didn’t multiply by pi*(1-pi) in the Covariance Share columns. So if you multiply 2 * 25,000 * 200 * 25,000/25,200 you get 9,920,635. Repeat this for all events X and Y, then to get the portion of covariance assigned to X, you multiply pi * (1-pi)*CovShareXi, add down each row, and you will get 2,328,401.

Wow, thanks so much. Pretty sure Mr. Mango is watching from actuary-heaven giving you the thumbs up.