A BRIEF INTRODUCTION TO COPULAS FOR FINANCIAL AFFAIRS (PART 1)

So, this time around I am going to deploy a couple of examples about COPULAS in real life and for practical financial purposes by reviewing some ideas belonging to Dr. David Li and his groundbreaking works on the matter. Furthermore, the main idea behind the concept states the possibility to link 2 ensembles or 2 time series. How? CORRELATION. Which values? UP and DOWN PROBABILITIES or SUCCESS and DEFAULT PROBABILITIES.

Why? A Conventional Statistical Approach assumes correlation = 0 in any case or if correlation is available such a value is not taken into account (This is, the 2 or more sets linked are thoroughly independent among themselves. There is not a pattern at all!) Therefore, through this focus, one never knows the accurate joint probability for a given scenario in particular.

The following example with 2 traded benchmark indexes: Emerging Markets Index (EMI) has a historical probability of 0.634020618556701 to succeed and a probability of 0.365979381443299 to fail. While Domestic Markets Index (DMI) has a historical probability of 0.81340206185567 to succeed and a probability of 0.18659793814433 to fail. The value for the PEARSON PRODUCT between EMI and DMI is = 0.955177393671544. The portfolio shares show EMI with a 40% weight and DMI with a 60% weight. How much is the joint probability for a success on both indexes? How much is the joint probability for a failure on both indexes? How much is the probability for a failure on an index and a succes on the remaining one?

Tricky and capscious taks but not impossible to solve! The investor has a long position on EMI and a short position on DMI.

First, the PEARSON PRODUCT (lineal correlation) has to be turned into a GAUSSIAN PRODUCT (normal correlation). The formula tell us: 6*ASIN( 0.955177393671544/2)/ ¶ = 0.950934469164048. Hence, we have to build a probability space by taking into account all of the odds and weights.

So thus,the eventual outcomes should be: 0.950934469164048*(0.634020618556701*0.4*0.81340206185567*0.6)+0.950934469164048*(0.365979381443299*0.4*0.18659793814433*0.6) +0.0490655308359519 *(0.634020618556701*0.4*0.18659793814433*0.6) +0.0490655308359519 *(0.365979381443299*0.4*0.81340206185567*0.6) = 0,138182689990975 This is what is worth the probability space (a nummerical value around 0 and 1).

Now we can answer the questions: 0.950934469164048 *(0.634020618556701*0.4*0.81340206185567*0.6) / 0,138182689990975= 0,851759211870039 +0.950934469164048*(0.365979381443299*0.4*0.18659793814433*0.6) / 0,138182689990975 = 0,964549484951555 This is the joint probability for a success with one asset and a default with the remaining one. Rememeber, we are taking short position on DMI.

0.0490655308359519*(0.634020618556701*0.4*0.18659793814433*0.6)/0,138182689990975 = 0,0100819449748797 For a success on both assets.

+0.0490655308359519*(0.365979381443299*0.4*0.81340206185567*0.6) /0,138182689990975 = 0,0253685700735654 For a failure on both assets.

The framework above is quite useful for HEDGING AND DIVERSIFICATION, the latter is different because you have to build 4 (correlation = 0) or 8 scenarios. All scenarios have the same chances to happen regardless of what position is held (LONG OR SHORT). When correlation is negative, such a value is implying different movements as a rule. Two scenarios are possible when correlation equals accurately -1 or 1