Chapter 1                            Chapter 2                            Chapter 3
      FINANCIAL RISKS FACED BY          MAJOR RISK MEASURES USED IN              EMPIRICAL APPLICATION
      INVESTORS AND FINANCIAL            PORTFOLIO OPTIMIZATION AND 
            INSTITUTIONS                  ASSOCIATED MATHEMATICAL 
                                            FORMULATIONS OF THE 
                                            OPTIMIZATION MODELS
     Introduction                         Introduction                         Introduction
     Risk Categories                      Coherent Risk Measure                Data Description
       Systematic or non-diversifiable    Mean - Variance                      Portfolio Optimization
       Unsystematic or Diversifiable        Beta                               Empirical Results
                                            Sharp Ratio                         Efficient Frontier
      Types of Risks                        Treynor Ratio                       Back Testing
       Liquidity Risk                     Mean Absolute Deviation
       Credit Risk                        Value at Risk (VaR)
       Interest Rate Risk                  Conditional Value at Risk (cVaR)
                                           Stressed VaR and Stressed cVaR
       Inflation Risk                      Expected Shortfall
       Country Risk                        Put - Call Efficient Frontiers
       Operational Risk                    Expected Utility Maximization
       Currency Risk                       Spectral Risk Measures
       Basis Risk
                                     TABLE OF CONTENTS
                                                                                                        Introduction
   The main aim of risk management departments, financial analyst and individual investors is the selection of optimal investments to 
    The main aim of risk management departments, financial analyst and individual investors is the selection of optimal investments to 
   maximize their returns, but at the same time, they desire to eliminate their risk exposures.
    maximize their returns, but at the same time, they desire to eliminate their risk exposures.
                                                                                                                                       •    Portfolio  optimization  is  a  cornerstone  of  modern  finance 
    Definition  of  risk  and  the  measure  of  it,  are  issues  that                                                                    theory, as it is very attractive in the field of decision making 
        financial society is concerned about many years and a variety                                                                       under uncertainty. 
        of interesting research on risk measuring have been published                                                                  •    Financial  crises,  economic  imbalances,  algorithmic  trading 
        throw the 20th century.                                                                                                             and highly volatile movements of asset prices in the recent 
                                                                                                                                            times  have  raised  high  alarms  on  the  management  of 
    Investors, usually construct portfolios including a variety of                                                                         financial risks. 
        asset classes, such as fixed income, cash, real estate, bonds,                                                                 •    Inclusion  of  risk  measures  towards  balancing  optimal 
        stocks,  and  other  financial  assets  each  of  which  will  react                                                                portfolios has become very crucial and equally critical. Varied 
        differently in the event of major systemic changes.                                                                                 mathematical  models  have  emerged  leading  towards 
                                                                                                                                            practical risk-based asset allocation strategies.
         Purpose of                                                To study the most important portfolio optimization models used to mitigate financial 
         Presentation                                                 risks and construct an optimal portfolio. 
                                      Chapter I
               Financial Risks faced by Investors and Financial Institutions
                                    A. Risk Categories
                                                         “Systematic” or “Non-Diversifiable”
     Risk in finance is defined as the 
     difference  between  expected                 Arises from the market, affect the whole market and not 
     outcome  of  a  financial  activity             only an individual item or an individual investor.
     and  the  actual  outcome  that 
     occurs.  As  investors  expose 
     their capitals in potential losses, 
     they  demand  a  reward  for  the 
     risk bearing. The bigger the risk 
     they bear the higher will be the             The risk that each investor is exposed to, as an individual 
     demanded return.                             unit. This part of risk arises from the investors’ decisions for 
                                                       the assets that they hold in their portfolios.
       Risk is divided in two parts:
                                                          “Unsystematic” or “Diversifiable”
                                                              B. Risk Types
  May  emerge  in  portfolio  whose  investments                                                 Refers to the ability that companies, investors, 
  take place at a specific country. The reason is                                                banks,  governments  e.tc  to  meet  their 
  that  investments  are  highly  correlated  and                             Liquidity          payment obligations in time. It varies by the 
                                                           Country Risk          Risk            passage  of  time  depending  on  its  maturity, 
  dependent  on  internal  political  or  financial                                              issuers’   financial   stability,  trend  of  the 
  issues.                                                                                        economy even hopes and rumors.
                                                                                                  In  financial  transactions  arises  from  the 
   Risk  which  results  in  severe  losses  due  to        Operational                           possibility  that  any  counterparty  of  the 
   situations  where  corporate’s  procedures  fail                          Credit Risk          agreement  may  default.  Buyers  have 
                                                             Credit Risk     Credit Risk
   to  work  as  it  would  (process  mistakes,                  Risk                             uncertainty  for  their  future  cash  flows 
   employer practices, workplace safety etc.).                                                    because  there  is  always  a  possibility  of 
                                                                                                  counterparty’s failure to meet its obligations. 
  Resulting  from  engagements  in  financial                                                    Risk that arises from changes in interest rate 
  transactions  which  take  place  in  another              Currency          Interest          yield curves. Financial instruments are closely 
  currency  than  the  official  currency  of  the              Risk          Rate Risk          linked to interest rate curves. 
  country / union the investor is based. 
   Emerges      in    transactions     with    future                                            Risk that can manipulate the power of money. 
   derivatives.  Futures  are  used  for  hedging                                                Inflation usually arises from oversupply in the 
   against  price  movements  in  the  future  but                             Inflation         economy,  macroeconomic  factors  such  as 
   also  for  speculating.  Basis  risk  is  the             Basis Risk           Risk           monetary  policy  of  central  banks  which 
   difference between the spot price and future                                                  controls  the  money  supply,  but  also  from 
   price =(St−Ft). of the asset any time during                                                  other  unexpected factors such as wars and 
   the future’s life.                                                                            political decisions. 
                                                                 Chapter II
                   Major Risk Measures used in Portfolio Optimization and associated 
                         Mathematical Formulations of the Optimization Models (1/3)
                                                                           Mean                   Uses the absolute deviation of the rate of return 
                                                                         Absolute                 of a portfolio instead of the variance to measure 
                                                                         Deviation                the risk.
                                                                                                  Algebraically: MAD = Σ | xi – x | / n
                                                                          Value at                Measures  the  potential  losses  of  a  specific 
                                                                             Risk                 investment or portfolio.
                                                                                                  Algebraically: VaR (x, α) = min {u: F (x, u) 
                                                                            (VaR)                 ≥ 1−a} = min {u: P {R (x, r)̃  ≤ u} ≥ 1 − a}
                                The use of a single risk measure 
                                should not dominate financial risk 
                                management. Each risk measure 
                                offers  its  own  advantages  and       Conditional               Risk  assessment  measure  that  quantifies  the 
                                disadvantages, complementing a            Value at                amount of tail risk an investment has.
                                risk  measure  with  one  other                                   Algebraically: CVaR = (1 / 1 – c) ∫VaR xp(x) 
                                represents  an  effective  way  to          Risk 
                                provide more comprehensive risk            (cVaR)                 dx
                                monitoring.
                                                                          Stressed                Are estimated under the use of a historical data 
                                                                          VaR and                 frame window of a stressed and high volatile 
                                                                            cVaR                  period in prices of underlying assets.