# 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