Exposures are not additive: case in point

If we have a portfolio of vanilla trades (say, swaps), we can calculate EAD on each trade individually (an example is discussed here). Naturally, it is tempting to say that the exposure on the portfolio is the sum of the exposures on individual trades. That’s very wrong because exposures are not additive.

Here is an example. Suppose we have a netting agreement with a certain counterparty, and at the moment we have a single trade with them: a semi-annual receiver swap of 6 month EURIBOR versus 4.08% fixed from 19 June 2009 to 19 June 2015. Below is the potential exposure profile of the swap at 95% confidence.


We see that the peak of the potential exposure profile is around 38. We want to add another trade to this portfolio: a semi-annual payer swap of 6 month EURIBOR versus 4.5% fixed from 19 June 2011 to 19 June 2017. Firstly, let us see the exposure profile of the new swap alone:


The peak exposure is around 34.

If we simply add the two peak exposures, we get 38+34 = 72. However, if we calculate exposure profile of the two netted swaps with a Monte Carlo simulation, we will see that the total exposure of the netted portfolio is much lower:


The peak is around 14, which is lower than each of the two swaps separately. Indeed, under the netting agreement the swaps partially compensate each other, so adding the second swap actually lowers our exposure to the counterparty.

Why would we have two trades that cancel each other out? It often happens in practice. Say, the receiver swap was done a while ago, and it still has a few years to go. Right now a trader needs a payer swap, so this swap is added to the portfolio.

To summarise: we have seen that the sum of exposures on individual trades can be very far from the total exposure on these trades covered by a netting agreement. The exposure profiles above are screenshots of MarketSimulator.

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5 Responses to Exposures are not additive: case in point

  1. Manuel says:

    Hey Cyril,
    I didn’t know you had a blog.

    While it is naive to expect exposures to be additive for the reasons you have exposed, I think it is reasonable to expect exposures to be subadditive, i.e., the exposure of 2 nettable trades together, can never be higher than the sum of the exposures of the 2 trades separately.

    Quantiles don’t satisfy this property, you probably know the example. Consider a trade that pays 1 with probability 4% and 0 with probability 96%. It’s exposure (95% confidence) is zero.

    Now consider 2 such trades. The sum of their exposures as separate trades is still zero. However, if you consider them as one whole (and assuming independence), you get:

    with probability 0.96*0.96=92.16% it pays zero
    with probability 2*0.96*0.04 = 7.68% it pays 1
    with probability 0.04*0.04 = 0.16% it pays 2

    The 95%ile is now 1.

    I got this example from Richard’s paper on tail exposure as an example why quantiles are not a good measure of exposure.

  2. alluve says:

    Hi Manuel,

    Thanks for the non-subadditivity example, it’s a nice one.

    Indeed, it’s naive to expect exposures to be additive, but I am surprised at how many people believe that the sum of the individual exposures is not very far off the total exposure. It takes a lot of effort to actually convince people that Monte Carlo is much better than just taking a PFE from a table, multiplying it by the notional, and adding up the results for all trades. People use some tricks to take netting into account, but those tricks were designed to work only under a few particular conditions. Naturally, no-one remembers anymore what those conditions were.

    And yes, I do agree that tail exposure is a better measure than quantile.

  3. Manuel says:

    Oh I’m surprised too. All but the most simple of portfolios are best done with MC. Otherwise you have to choose between having no idea of what you’re doing or being way too conservative.

  4. govind pratap chand says:

    Hi Cyril,

    Though I am a rookie in the world of quant finance, I would like to give some feedback on this,

    It is true that PFE is non additive, but you forgot to mention PFE is time based profile and based on the same level of interest rates (simulated or stressed) at a particular time, the MTMs or the exposure ( from both side of the involved parties) can be used to cancelled out in a netted portfolio .
    assumption : vanillas swaps portfolio only

    Also can you give an intuitive explanation of a scenario when EPE profile can be over the PFE profile.

    • Alluve says:

      Consider a portfolio which MtM is negative in 96% of your scenarios and positive in 4% of them (say, a long option which is far out of the money). Its peak exposure at 95% confidence level is zero; its expected exposure will be a positive number, thus EPE > PFE.

      As regards your previous point, I am not sure I follow your meaning. I do agree that MtMs are additive, but I am afraid it’s not the point that you were making.

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