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Finance ‹ What I Learned Today

# What I Learned Today

No frills, just learn

## 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...

## Optimizing: the Minimum Variance (or Maximum Return) Portfolio example

The aim is the optimization either by maximizing total return (or minimizing risk), in keeping with the constraints of normalization for a given risk (or return).

Thus Lagrange method is perfectly suited for the job.
For minimizing risk, respect to a desired total return , one must consider the function of total risk

and minimize the lagrangian with constrain over the desired return and sum to one of weights.

Thus, provided that the constraints have non-zero derivatives,

by the fact that  is a non--degenerate constant vector (otherwise fixing the Total Risk is given), the Lagrangian is given by

with  the vector of lagrange multipliers.

Thus the minimization (maximization) is given by finding the extremant of the Lagrangian,

That is solved by considering the following linear system, that considering a symmetric covariance matrix  is given by

where  and  in the definition of the matrix are linear vectors of equal dimension.

Finally considering the Hessian for  operator to find the convexity.

Analogously could be done for a maximum return portfolio or relaxing the attribute for the desired best compromise.

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