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Can anybody, in the most intuitive way possible explain why there is an FVA DVA overlap, specifically why DVA and FBA are similar?

Note my mathematical ability is only to bachelor degree level, so go easy on me.

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The most intuitive explanation I have found is to consider two different options for a bank in need of funds, they could either:

  1. Emit a bond, let's say a zero-coupon bond with nominal 1 which expires in one year, or
  2. Create a synthetic bond using an uncollateralized derivative where we receive some up-front premium and pay 1 in one year.

Once the initial transaction is complete we're left with a single payment in one years time, regardless of the option we chose. Now, what is our perceived value of this liability?

In option 1 we would look at the prevailing bond price, this is the price at which we could buy back our debt and hence it represents the economic value. This bond price can traditionally be seen as the risk-free discounted price, minus some adjustment to account for the credit risk.

In option 2, without the use of XVA's, we would simply value it using risk-free discounting.

If we introduce DVA as $(1-R)\int_0^Td(t)ene(t)pd(t)dt$ where $d$ is the discount factor, $R$ is the recovery rate, $T$ is the payment date, $ene$ is the expected negative exposure and $pd$ is the probability of us defaulting, then we should arrive at the same value as option 1.

If we further introduce FVA as $\int_0^Td(t)ee(t)fs(t)dt$ where $ee$ is the expected exposure (which in this case is always negative) and $fs$ as the funding spread, then we will arrive at a different value than option 1, which I find very hard to explain. In particular, we would seemingly prefer this over option 1 as we experience an additional funding benefit, but I can't see how or why the bank is better off in this scenario.

I don't have a firm view on how one should model it, but if they are simply added together I believe that there is certainly an overlap. Anyways, I hope that this gives you an idea of the problem!

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