Portfolio Optimization
A Selection of Stocks
from the Heng Seng Index
17 August 2012
Introduction
A typical investor purchases an asset with the hope that it will generate income or appreciate in the future. Given the plethora of choices in the market, a rational investor would choose an investment with the highest expected return. The Hong Kong Stock Exchange is the sixth largest stock exchange in the world with 1,477 listed companies and a combined market capitalization of HKD 17 trillion. Trading stocks from the Hong Kong stock market has become one of the most popular forms of investment in the region. The question is – how do we find out which asset is the best to invest on? Very often, investors hold a selection, instead of a single stock, in order to maximize their return in the long run. Why is it so? This report contains the explanation on fundamental methodologies employed in selection of stocks and composition of portfolios. In addition, several attempts had been made to construct optimized portfolios with the greatest long-term expected return.

Objective
To construct a portfolio with greatest long-term expected return from the 46 stocks constituting the main board of Hong Kong Stock Exchange. It is assumed that stock prices follow a normal distribution. Hence the calculations of long-term expected returns of single stocks as well as portfolios are based on the historical data, dated 8/31/1981 to 7/31/2011.

Modern Portfolio Theory
Under the current uncertain economic outlook, one should not be difficult to perceive that volatile investments may not result in a good long-term return. And the idea of diversification is widely accepted way of ensuring a good return, also in the long run. This phenomenon can be explained by the Modern Portfolio Theory (MPT). Modern portfolio theory (MPT) is theory which attempts to maximize portfolio expected return for a given amount of portfolio risk, or in other words, to minimize risk for a given...

...PortfolioOptimization Questions
Risk Management
Dr. Castro Fall 2002
Assume you are the manager of a risky portfolio with an expected rate of return of 18 % and a standard deviation of 28%. The T-bill rate is 8%.
1. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected value and standard deviation of the rate of return on hisportfolio?
2. Suppose that your risky portfolio includes the following investments in the given proportions:
Stock A: 25%
Stock B: 32%
Stock C: 43%
What are the investment proportions of your client’s overall portfolio, including the position in T-bills?
3. What is the reward-to-variability ratio (S) of your risky portfolio? Your client’s?
4. Draw the CAL of your portfolio on an expected return-standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund’s CAL.
5. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%.
a. What is the proportion y?
b. What are your client’s investment proportions...

...Portfoliooptimization - a practical approach
Andrzej Palczewski Institute of Applied Mathematics Warsaw University June 29, 2008
1
Introduction
The construction of the best combination of investment instruments (investment portfolio) is a principal goal of investment policy. This is an optimization problem: select the best portfolio from all admissible portfolios. To approach this problem we have to choose the selection criterion ﬁrst. The seminal paper of Markowitz [8] opened a new era in portfoliooptimization. The paper formulated the investment decision problem as a risk-return tradeoﬀ. In its original formulation it was, in fact, a mean-variance optimization with the mean as a measure of return and the variance as a measure of risk. To solve this problem the distribution of random returns of risky assets must be known. In the standard Markowitz formulation returns of these risky assets are assumed to be distributed according to a multidimensional normal distribution N (µ, Σ), where µ is the vector of means and Σ is the covariance matrix. The solution of the optimization problem is then carried on under implicit assumption that we know both µ and Σ. In fact this is not true and the calculation of µ and Σ is an important part of the solution.
of market observations (so called stylized facts) shows that returns deviate...

...
Assignment No 6
Diversification in StockPortfolios
Introduction
Diversification is one of the key components of a successful investment portfolio. Almost all
experts advise the avoidance of concentrating all of your investments in one type. However,
many investors forget about diversification once they see a financially attractive stock and
concentrate all of their assets in it. Other investors make a similar mistake and being influenced
by their emotions fail to listen to their common sense whispering "Diversify".
Many companies have attracted their employees to investing in company stock as part of their
retirement plan through the provided matching of contributions. As a result most investors end
up concentrating their assets in company stock and forgetting about the importance of
diversification. Investing in your company's stock is not something bad. However, you should
own not only your company's stock, because if something bad happens with your company you
risk not only losing your job but all of your assets. Through diversifying your stocks among
different industries you decrease to a great extent the risk of losing your money..
Portfolio Diversification
You can use index funds or exchange traded funds to track a broader...

...Financial Management: FIN 534
Diversification in StockPortfolio
Diversification in StockPortfolio
Background
As a risk averse investor, I am considering investing in one of two economies. The expected return with volatility of all stocks in both economies is the same. In the first economy, all stocks move together, in good times all prices rise together and in bad times they all fall together. In the second economy, stock returns are independent; one stock increasing in price has no effect on the prices of other stocks.
Diversified Portfolio
There are certain guidelines I need to consider when investing and creating a diversified portfolio. First, I must be cognitive of my risk capabilities in order to manage risk within my portfolio. Secondly, Systematic and Nonsystematic are two types of risks that I need to be knowledgeable of also.
Systematic or market risk: This risk pertains to each capital market, which can be volatile. For example, a significant political event could affect several of the assets in my portfolio. It is virtually impossible to protect myself against this type of risk. When the stock market averages fall, most individual stocks fall and when interest rates rise nearly all-individual bonds and preferred shares fall in value....

...Active &Passive Portfolio – Call/Put Options and Futures
This report will document the active traded portfolio held from Friday (July 18th, 2014) until Monday (August 11th, 2014). In this portfolio, the two portfolio managers traded call options and put option for the stocks on the S&P 500, as well as futures contracts in many different asset classes (commodities, currencies, indexes and so on). Trades were made at the end of each week and Monday (August 11, 2014), resulting in four trading days. We decided to divide the portfolio of $500,000 into two of $250,000 dollars, one of those is designed to trade on call/put options and the other will trade only futures.
At the beginning of class we created what they felt like was a “winning” passive portfolio. The belief was that with $500,000 invested in our passive portfolio and $500,000 in our active portfolio that we would turn at least a positive profit. With a portfolio beta of 1.004 and a portfolio forward beta of 1.003 we knew that our returns didn’t have as much potential had our beta been higher but believed that we were the superior investors and would prevail. This belief was shattered when trades were closed and we realized we had actually lost money. Our returns week 1 came out to a total increase of 1.635%. Returns week 2, week 3, and week 4 correspond as...

...Optimization methods in portfolio management
and option hedging ∗
Huyˆn PHAM
e
Laboratoire de Probabilit´s et
e
Mod`les Al´atoires
e
e
CNRS, UMR 7599
Universit´ Paris 7
e
e-mail: pham@math.jussieu.fr
and Institut Universitaire de France
April 24, 2007
Abstract
These lecture notes give an introduction to modern, continuous-time portfolio management and option hedging. We present the stochastic control method toportfoliooptimization, which covers Merton’s pioneering work. The alternative martingale approach is also exposed with a nice application on option hedging with value at risk
criterion.
∗
Lectures for the CIMPA-IMAMIS school on mathematical ﬁnance, Hanoi, April-May 2007.
1
Contents
1 Introduction
3
2 Financial decision-making and preferences
4
3 Dynamic programming and martingale methods : an illustration via a
simple example
7
3.1 Solution via the dynamic programming approach . . . . . . . . . . . . . . .
8
3.2 Solution via the martingale approach . . . . . . . . . . . . . . . . . . . . . .
9
4 Dynamic programming methods for portfoliooptimization in
time
4.1 Dynamic programming and Hamilton-Jacobi-Bellman equation
4.2 Merton’s portfolio selection problem . . . . . . . . . . . . . . .
4.3 Super-replication cost in an uncertain volatility model . . . . .
continuous.......
.......
..........

...diversification benefits of portfolio allocation into account. Modern portfolio theory is the result of his work on portfoliooptimization. Ideally, in a mean-variance optimization model, the complete investment opportunity set, i.e. all assets, should be considered simultaneously. However, in practice, most investors distinguish between different asset classes within their portfolio-allocation frameworks.
In our analysis, we view the process of asset allocation as a four-step exercise like Bodie, Kane and Marcus (2005). It consists of choosing the asset classes under consideration, moving forward to establishing capital market expectations, followed by deriving the efficient frontier until finding the optimal asset mix.
We take the perspective of an asset-only investor in search of the optimal portfolio. An asset-only investor does not take liabilities into account. The investment horizon is 5 - 10 years and the opportunity set consists of twelve asset classes. The investor pursues wealth maximization and no other particular investment goals are considered.
We solve the asset-allocation problem using a mean-variance optimization based on excess returns. The goal is to maximize the Sharpe ratio (risk-adjusted return) of the portfolio, bounded by the restriction that the exposure to any risky asset class is greater than or equal to zero and that...

...Policy statement
• Risks and rewards of investing in the stock market
• How to diversify
• How to buy and sell stocks
Implement the plan by constructing the portfolio
• Different approaches for making the best stock selections – and when to buy or sell them
• How to obtain information and analysis about companies and industries
• How to read and interpret corporate financial and non-financial information
Track the progress of the stocks
• How to track the progress of your stocks
• How to stay informed about any news or actions that may affect the value of your --stocks
Performance evaluation
Mission statement
* Investing initial total amount of $10,000 equally into twelve different stocks
* Diversifying our portfolio to decrease the portfolio risk to acceptable level
* To increase the risk-adjusted return of our stockportfolio
Risk profile
* Setting our cut-loss spot of portfolio in 5%. In addition, the cut-loss spot of individual stock is also set as 5%
* After self-assessment, we conclude that we are moderate in portfolio, which means we don’t mind a little bit of fluctuation in our investment returns, but we would uncomfortable with significant ups and downs
The criterions of stock selection
* Industry
*...