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definition ‹ What I Learned Today

# What I Learned Today

No frills, just learn

## C# Lists and Dictionary

In C# there are several key differences to C++. Microsoft Website spells them all: https://msdn.microsoft.com/en-us/magazine/cc301520.aspx

Garbage collector instead of deconstructors providing a different approach to micromenagment of memory and lists, jagged arrays and dynamic memory allocation are already available into the language. It supports also negative index to counting from the last element, similarly to Python.

About lists instead of the usual self-made textbook-example class, there are specific embedded classes for Lists and Dictionaries.

List<type> name = new List<type>(); name.Add(element); name.Remove(element); 

Dictionary<type,type> name = new Dictionary<type,type>(); name.Add(key,element); name.Remove(key,element); 

## Fun with conditional probability

Bayesian concept of probability is linked not with frequency, but with state of information. The processing of this information proceeds in a propositional sense. In fact there are semantic work related to the use of Bayesian statistic as foundation for programming languages (BLOG), First kind logic (MEBN , also look at the fun Of Starships and Klingon) and thus the foundation principles of Network-Centric warfare and commerce.

Property of Baysian calculation of proabability are sometimes difficult to grasp and very counter intuitive, interesting consideration can hide behind apparently trivial formulas that can help to answer questions like: If you know your neighbour has a son born on Tuesday, which is the probability your neighbour has two sons? (more…)

## Value at Risk on expected return

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Value at Risk on expected return is the lower bound for the loss in an investement respect to a probability.
Is a way to estimating losses probabilities in terms of value, instead of standard deviation. E.g. in the case of , means that the investor have a probability  of loosing € or more, investing € on the portfolio .

Considering the system ergodic and thus all the variances of the titles summing up coherently to for a variance of the portfolio distributed within the normal distribution, this is simply calculated considering the quantile (the value of the distribution corresponding to a certain fraction  of probability) .

(!) The  of a portfolio is not a weighted average of the  of its component, because  but is given by the covariance matrix  due to the correlation of assets .

## No-Arbitrage condition

No Arbitrage states the inadmissibility of Free Lunch in applying a structure (metric?) to the prices.

Considering a title with price  and cash flow , or every time , where  it is the discretized time .
Then we have the two No-Arbitrage condition:

• Weak No-Arbitrage : if 
• Strong No-Arbitrage: if  and 

Let's suppose a world where I can borrow money with fixed interest rate , which will be the price  of a product with interest  over one year?
I can borrow money until I expect to be payed back by the revenue A, so the cash flow at and of year would be 0, which is . The price  of the portfolio is given by the price of the product minus the borrowed money at time 0: .
Weak No-Arbitrage fixes then the price  giving .
Same can be applied for selling, giving , thus the meeting of buying and selling is, unsurprisingly .

But, if we use the strong arbitrage condition -satisfied, since we assumed the interest of the product - we get that  for the buyer, and  for the seller (?). So buying and selling cannot be met. And why should they? You gain nothing by selling or buying if you can lend and borrow money indefinitely for a fixed interest rate, thus is better not to do anything at all...

## Investment Portfolio Basics

Considering  the vector of assets i weight in the portfolio (with constrain ) and  the vector of expect returns on the corresponding assets.

Considering  the volatility of the given asset, corresponding to the standard deviation of the normal distribution of returns (and thus  the risk), and  the correlation between assets.

The Total Return is given by:

And the Total Risk is

with  the covariance matrix with elements