Markowitz's optimal portfolio

Table des matières

1 - Introduction 3

2 - Markowitz’s optimal portfolio 3

2-1. Introduction 3

2-2. Knowledge behind 3

2-3. Model implementation 3

2-4. Model drawbacks: 6

Introduction

Wei LI

Markowitz’s optimal portfolio

Introduction

In this section our goal is to find the optimal portfolio made with the following assets available in the NASDAQ (National Association of Securities Dealers Automated Quotations) market in USA.

Risky asset 1: Google Inc.

Risky asset 2: Helmerich & Payne Inc(HP).

Risky asset 3: Mircrosoft Corporation.

Riskless asset: interest Risk free rate.

The historical data used in this project is available at: http://fr.finance.yahoo.com/, we choose a daily return for a period of 3 years from: 22/02/2010 to 21/02/2013.

Knowledge behind

In order to choose the optimal portfolio, we will use a Markowitz’s analysis. Harry Markowitz, a Noble prizewinner, founded modern portfolio analysis when he demonstrated the nature of portfolio risk and how risk can be minimized. The model is developed in a two-step approach. The first step is to calculate the expected return and standard deviation for individual assets. The second step is to demonstrate that combining the these assets into a portfolio, results in a portfolio standard deviation and coefficient of variation that is lower than the standard deviation and coefficient of variation for either of the assets held in isolation.

Model implementation

First step : Return/Variance Covariance Matrix

Firstly we computed the daily return of these three stocks based on the formulas:

Return = 365 * ln(St/St-1)

Secondly, I calculated the assets covariance in order to obtain the Variance-Covariance Matrix, whose values are shown below:

Ω Google Hp Samsung

Google 34.502 23.0296 12.1645

Hp 23.0296 82.4166 20.9114

Samsung 12.1645 20.9114 22.2917

Step 2: Calculating the efficient frontier equation

We assume that our portfolio is made with three risky assets and one risk-free asset:

(Rp) ̃=∑_(i=1)^3▒〖w_i (r_i ) ̃ 〗+w_4 r_f

For defining so, I needed to compute the portfolio’s expected return and its variance, which are computed, respectively as:

σ_p^2=var((Rp) ̃ )=w^' Ωw

µ_p (w)=E[(Rp) ̃]=w^' µ

The optimal portfolio boils down to the resolution of the following equation:

∑_(i=1)^3▒〖w_i (r_i ) ̃ 〗+w_4 r_f = µ ̅

∑_(i=1)^4▒〖w_i=1〗

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