Mean-Variance in Financial Applications: Building Optimal Portfolios

Population and Sample

First, let define and distinguish the terms Population, and Sample
If the data consist of all possible observations of a certain phenomenon, we call it Population
If the data contains only a part of these observations, we call it Sample
The main reason to analyze a sample to reach a conclusion on a population is straightforward

  • Analyzing a population would demand massive amount of time, resources, and money!
    Example: Suppose that we want to know the Mean weekly income of all households in Europe
  • That would be pretty impossible
  • Let draw a sample of 2000 households, and calculate the mean weekly income as follows

The Basic Trade-Off

Unfortunately, it is not possible to achieve both targets

  • Pretty always, higher return corresponds to higher risk
  • Very safe investments give lower return
    Derivative instruments (options,futures,CDS), sovereign bonds of distress country, stock of
    new company
    German Bund, US Treasury Bonds, stock of mature company (e.g., Enel, Eni), bank deposit
    It is called the Trade-Off between Risk and Return
  • It may be considered as the Basic Stone of Finance
    In practice, what we can do, is
  • Maximize the Expected Return, given a certain Risk
  • Minimize the Expected Risk, given a certain Return