Financial risks faced by investors and financial institutions

ys− ≥ 0,∀ s ∈ Ω
Put-Call Efficient Frontiers
Allows every investor to set his personal standards under of which the portfolio will balance its performance between desirable risk and desirable returns.
Maximize Σ ps U(R(x, rs))
Subject to: R(x, rs)) = Σxirs
Σ xi = 1
x ∈ X
Expected Utility Maximization
Weighted average of the outcome of loss distribution which weights depends on user’s risk Spectral risk measures and therefore enables to link the risk measure to the user’s attitude towards risk, and we might expect that if a user figures higher risk aversion (ceteris paribus) then that user should encounter a higher risk, as given by the value of the S.R.M.
Spectral Risk Measures
Empirical Application
Chapter III
Introduction
Portfolios in this study consist of stocks from DAX (Deutscher Aktienindex) (German Stock Market Index) which is consisting of the 40 major German blue-chip companies trading on the Frankfurt Stock Exchange.
Data Description

• Historical close prices for the index DAX and its stocks on monthly basis.
• Data collected from Thomson Reuters DataStream.
• However, there were only 26 out of 40 stocks on the table during the whole period that we use as our data sample.
• For the rest 14 additional stocks there are no data for the whole period and have been excluded.
• We calculate the returns (155 consecutive returns for every stock and the index) on monthly basis for the whole period (February 2007 - December 2019) as well as the Variance-Covariance Matrix of the respective stocks’ returns.

Puprose of the study
Data Sample
The main aim of this study is to study the most important portfolio optimization models used to mitigate financial risks and construct an optimal portfolio.
Portfolio Optimization

• To create optimal asset allocation, we find the optimal weights for investments in stocks during the examined period. Thus, we solve linear problem of portfolio optimization using the Generic General Algebraic Modeling System (G.A.M.S.).
• Two different types of investors were selected:
• Defensive, we minimize the objective function of the mean variance model, setting as expected target return the achieved MVP point.
• Neutral, we set expected target return.
• The data input is 131 rolling monthly returns (approximately 11 years) of each available stock in these years. In this way, we export the optimal weights under of which we construct optimal portfolios for those investors.