1+1 doesn't equal 2

In the world of risk, statistics, finance and many other fields of endeavour, 1+1 does not equal 2.

It's a little bit complicated but as 1+1 does not equal 2 in the realm of risk and while executives believe that 2 is the answer, our banking system will inevitably continue to fail us.
  
1+1 does not equal two

'What lunacy is this' I can hear everyone say but for risk people who know their game well, will appreciate that (one apple) + (one apple) which are from different apple trees do not always equal the true value of two apples. I don't know what they teach in schools these days but this is a fact.

Let me explain what I am talking about here before you cast me into the realm of people who wear either white coats or straight-jackets, sometimes both and I find this latter coterie curiously interesting, hopeful perhaps.

Anyway, banter aside; if we were to take two assets with their prices being normally distributed in variability, we have to accept that we cannot simply add up the change in price of each asset to understand the true value of net exposure across both trades.

Using the normal distribution as a model for price variance is a big assumption in itself because such deviations in price may fit any probability distribution function and the normal distribution can be a major oversight. Nonetheless and for simplicity sake, if you were to compare floating prices in this manner and you added them up, you might just find the result is more than likely to be wrong ... Why is that?

Simply because of a condition known as Subadditivity. 
  
Let's be real, landing on exactly 2 in value for (1 of asset 'a') + (1 of asset 'b') is a rather narrow range between an infinite array of possibilities in the realm of 1.xxxxx9 and 1.xxxxxx1. 

Remember these things are floating, they have variability, they are not static and their variability can change in a correlated manner.
  

Subadditivity | Causal Capital (click to enlarge)

The issue isn't the maths or the models, it is actually the interpretation of both these things together that fails us when we measure risk and there are too many managers in banks who aren't even aware of subadditivity let alone its implications.

There are also some good risk analysts out there who understand this condition but they often fail to accept its predicament when considering the broader context of a portfolio of different floating trades netting together. Each trade price is randomly moving as the market changes, however there are internal correlations between one position and another which change the portfolios overall value in the backdrop of market volatility.

  

The value of an asset then is more than simply its market price alone. When it is booked on a balance sheet and combined with other instruments which are floating in value, it can add to our pleasure or plain in a greater way than simply being additive.

In many cases during the Global Financial Crisis, banks kept being caught out long on a non-performing CDO and also short on the appreciating reference CDS. In a sane world, the CDS would normally be booked to cover the risk or to insure against a loss on the CDO, not as an instrument of profit in its own right.

Then of course when a trader is hedging, they may find themselves on both sides of a deal but if they are not capturing correlation in either case and at an aggregated level, they may find their losses are larger than the two positions added together.

  

Two of the biggest enabling precursors for keeping the subadditivity disorder alive in banks are; (1) Ignorantly structured deals and (2) Risk systems in silos that aren't aggregated with correlation. The finance industry as a whole is laden with both these problems.

So if 1+1 does not equal 2, we can expect value = price = cost to be false as well. Fair value accounting is never going to be fair in the true sense (fair being how it is personally experienced) and we have to accept this truth.

  

Value, Price, Cost are not the same

A year or so ago when I was working for the counterparty risk function of an investment bank, the finance department used to often call our unit up on the phone and question why the MTM values in the credit systems for a set of trades did not match the prices in their systems. Something must be wrong surely but in the world of risk, credit people value default in their calculations and finance people price a deal as if it is to be held to maturity.

People are generally unable to reconcile the differences between value, price and cost especially when buyers, sellers and the "structurer" have alternative motives. This issue runs so deep that it actually fueled the Global Financial Crisis (GFC) and some people in this world blame the failure of the financial system in part on the collapse of the process around fair valuation of assets.

Mark-to-market accounting rules have turned a large problem into a humongous one. A vast majority of mortgages, corporate bonds, and structured debts are still performing. But because the market is frozen, the prices of these assets have fallen below their true value.

Fair value accounting | The Harvard Business Review

During the GFC the issue of "pricing-in" prevailing human sentiment on market securities reached such bitter proportions that in the end, lobbyists from various economic bodies in the US demanded the suspension of mark-to-market accounting. Mark-to-market accounting is perhaps better known as SFAS 157 and more information can be found on it here [Link]. The public debate itself actually nullified the entire purpose under SFAS 157 either way because it generated so much confusion about whether to implement a full suspension or an alteration of SFAS 157 that even today, there are still plenty of investors out there who truly don't know whether SFAS 157 exists anymore.

Has anyone bothered to mention that SFAS 157 (which everyone believes was suspended so that means mark to market is dead) does not force fair value measurement of securities but instead provides guidance on how banks should value these securities.

Fair value is alive and well | Seeking Alpha

Either way, it would be questionable to say that humans are that great at valuing assets, especially after reading all of this. Why is this so?
   

Netting | Causal Capital (click to enlarge)

The fundamental problem is that value, price and cost are not truly the same thing for different people and forget about markets being a zero sum game, they are not.

In our diagram above, we show the value of our asset is different when we net it across the complex operation of a balance sheet, rather than what the asset would be worth to us in the market place alone. Whether we like it or not, when default probability rises, prices will fall and we can expect knock effects across many alternate asset classes we are holding.

Poorly implemented fair value accounting rules aren't going to help but there is actually no way out of this systemic loop unless we prepare ourselves for a correct "risk price" in advance and we consider the netting effect of subadditivity on that risk number as well.