Introduction:

Due to developments in technology and financial modeling, portfolio management has undergone significant modifications. In this research, we’ll look more closely at how these economic models work and how they’re being used in practice in the context of portfolio management at a subset of FTSE 100 businesses.

Expected return, risk assessment, standard deviation of returns, and correlations between various shares are just some of the fundamentals of portfolio management that you’ll need to understand, implement, and evaluate for this project. This research provides useful insights into portfolio management methods by analyzing real-time share price data from firms included on the FTSE 100 index.

The FTSE 100 businesses selected for this study come from various sectors, making for a well-rounded sample from which to conclude. The data covers a long period and is broken down into weekly share prices. They provide a solid foundation for evaluating interrelationships and trends across time.

The assignment takes a methodical approach by using statistical analysis to determine the anticipated return and amount of risk associated with each share. Furthermore, the article investigates the links between the performance of various stocks. After that, it uses the current risk-free rate of 2% to inform the development of a spreadsheet model that identifies profitable investment portfolios.

One of the key sections of this report will outline the creation and interpretation of the efficient frontier chart, a concept integral to modern portfolio theory. The significance of diversity in reducing risk, as well as the trade-off between risk and return, are also discussed in this paper.

Given these concepts’ theoretical and practical implications, it is essential to acknowledge the limitations of such models. Therefore, the assignment will also critically evaluate the functional restrictions of this approach to portfolio management and the model itself.

By the end of this study, we aim to comprehensively understand these key principles and their application in the context of portfolio management of FTSE 100 companies. This study should be a valuable resource for academia and financial professionals interested in portfolio management strategies and their inherent complexities.

Question 1.

Answer:

Working with the given data of share prices, the expected returns can be calculated as follows:

The weekly expected returns (See Table. 1) for each share have been estimated by taking the mean of the weekly returns over the given period. These figures indicate what the return would be in an average week.

It’s important to note that these figures represent the average return over the period considered, and actual weekly returns can vary significantly.

Similarly, each share’s annual expected returns (See Table. 2) can be calculated by taking the mean of the annual returns over the years.

These figures indicate the average annual return for each share, but actual returns can differ substantially from year to year. In addition, it is essential to stress that these numbers are based on historical results and may not forecast future returns due to the inherent uncertainty of the financial markets.

In conclusion, it is important to exercise caution when interpreting the anticipated weekly and yearly returns, which reflect the average return an investor could expect to earn. Actual returns may be greater or lower than these projections depending on market volatility, business success, and general economic circumstances.

Question 2.

Answer:

The standard deviation of the yearly returns supplies a measurement of the volatility or risk attached to each share. Measures how much share prices fluctuate around an average. Larger standard deviations represent increased volatility and hence risk. To calculate the yearly standard deviation from the weekly one, multiply the latter by the square root of 52, as the question implies.

However, (See Table 3) for Annual standard deviations.

Variation in yearly returns for each share is seen below. Stocks (like AHT.L and SMT.L) with larger standard deviations have more wildly fluctuating annual returns and are riskier. Conversely, smaller standard deviations indicate reduced risk since these stocks’ yearly returns have fluctuated less dramatically.

While the standard deviation is a helpful way to assess past volatility, it is crucial to remember that it is not a foolproof indicator of future danger. Factors like the company’s financial stability, market trends, and the economy at large may all affect the potential reward or loss from investing in a certain stock.

Question 3.

Answer:

The correlation matrix (See Table. 4) shows the correlation coefficient between the weekly returns of each pair of shares. Between -1 and 1, the correlation coefficient looks like this:

• If the value is 1, then the correlation is perfectly positive (i.e., the returns of the two shares move in the same direction).

• When the value is -1, the correlation is perfectly negative (i.e., the returns of the two shares move in opposite directions).

• A correlation value of 0 suggests there is none.

From the matrix, we can see that:

1. The highest correlation is observed between STJ.L and SMT.L (0.829598224), suggesting these shares have returns that move in a similar direction most of the time.

• The lowest correlation is observed between TSCO.L and RR.L (0.545886882), which implies that the returns on these two shares have the least similar direction of movement.

All the correlations are positive, meaning all shares tend to move in the same direction – when one share’s return increases, the other shares’ returns also tend to grow, and vice versa. From an investment perspective, this information can be used for portfolio diversification. Ideally, a portfolio should consist of assets with low correlation to reduce risk. If the assets are highly correlated, they’ll likely decrease in value simultaneously. Still, if they’re less correlated when one support decreases in value, the other might hold its value or even increase, mitigating the loss.

However, remember that correlation does not imply causation and is based on historical data. It should be one of the many factors considered while making investment decisions.

Question 4.

Answer:

Using your data and findings as an example, I will explain the relationship between risk and return and how diversification reduces risk.

Risk and return are the two key considerations for any investor in the context of investment portfolios. They have a trade-off relationship, i.e., If you want a bigger payoff, you must be ready to take on more risk.

In this case, the portfolio’s (See Appendix A) expected return was -6.35%, and the standard deviation, a measure of risk, was calculated as 14.98%. A negative expected return means that, on average, the investment will likely lose value. The standard deviation, or risk, is quite high at 14.98%, indicating a high level of price volatility.

The Sharpe ratio measures the profitability of an investment about its potential loss. The excess return is the average profit above the risk-free rate produced per a given amount of volatility or absolute risk. With a Sharpe ratio of -55.78% (assuming a risk-free rate of 2%), this investment strategy would provide a negative risk-adjusted return. A greater Sharpe ratio indicates a more desirable risk-adjusted return.

Diversification, the process of investing in various assets, reduces a portfolio’s overall risk. It does this by spreading investments across different types of support so that the good performance of others offsets the poor performance of some assets. In the given portfolio, diversification is implemented by investing in various shares. The correlations between these shares are less than 1, meaning they do not move perfectly in sync, providing some risk reduction through diversification.

If you want to maximize your return while minimizing your risk, or vice versa, you should look at the efficient frontier, the set of optimum portfolios that does both. Below the efficient frontier, a portfolio has to offer a higher return relative to its degree of risk to be considered optimum. Portfolios congregating to the right of the efficient frontier also tend to be sub-optimal due to their increased risk relative to the rate of return they provide.

However, it’s important to note that this model makes several assumptions (such as that returns follow a normal distribution and that all buyers have access to this information) that it may only represent certain aspects of reality.

In conclusion, the efficient frontier chart helps illustrate the risk-return trade-off concept and the importance of diversification in portfolio optimization. The optimized portfolio you created represents a combination of weights that provide the most efficient balance between risk and return, considering the parameters and constraints you set.

Question 5.

Answer:

The Markowitz Portfolio Theory and the Efficient Frontier model are powerful financial tools with several practical limitations [1].

1. Assumption of Normal Distribution of Returns: The model assumes that returns are normally distributed. However, financial markets often exhibit skewness and kurtosis (fat tails), indicating that extreme events are more likely than what a normal distribution would predict.

• Estimation Risk: The model’s effectiveness heavily depends on the accuracy of inputs: expected returns, standard deviations, and correlations. Even small changes in these inputs can lead to significant changes in the portfolio weights, potentially leading to over-optimization based on historical data that may not predict future performance.

• Risk-Free Rate: The model assumes a risk-free rate investor can lend and borrow. In practice, there is no truly risk-free asset; even short-term government bonds, often used as a proxy, come with some risk.

• Single-Period Model: The model is based on a single-period investment model. It does not consider the investors’ multi-period investment horizon.

• Transaction Costs: The model does not consider transaction costs. Rebalancing a portfolio to maintain optimal weights can involve significant transaction costs, which could lower the net return.

• Taxes: The model also does not consider the impact of taxes. Tax consequences from buying and selling securities can significantly affect the investor’s net return.

• Investor Risk Preferences: The model assumes that all investors have the same risk preferences and aim to maximize their utility. Investors have unique goals, constraints, and attitudes toward risk.

The model is unrealistic since it presumes investors may lend and borrow a limitless amount of money at a risk-free interest rate, which is different from the actual world due to lending restrictions and credit hazards.

Despite its flaws, the Markowitz Portfolio Theory and the Efficient Frontier continue to play important roles in contemporary financial theory. Even if they don’t fully replicate real-world situations, they give a basic understanding of the risk-return trade-off and the need for diversification. In the end, these resources are used by investors and portfolio managers as part of a larger set of strategies and factors.

Question 6.

Answer:

Value at Risk (VaR) is a statistical metric used to express how much a hazardous asset or portfolio might decline in value over a certain time frame and the associated confidence level. Financial institutions and their overseers widely use VaR to gauge the extent to which they may need further assistance in the event of losses.

For a portfolio, VaR can be calculated using the portfolio’s volatility (standard deviation) and assuming a normal distribution of returns. Taking a 95% confidence level, the VaR is approximately 1.645 standard deviations below the expected return.

To calculate the 95% VaR for a day, use the following formula:

VaR = Portfolio Value * Portfolio’s Standard Deviation * z

Where z represents the confidence level as a z-score, the value of z with a confidence level of 95% is around 1.645.

However, it’s important to note that VaR has several limitations, especially over longer periods.

1. Normal Distribution Assumption: VaR assumes that asset returns follow a normal distribution. As mentioned before, this is only sometimes true. Many assets exhibit “fat tails” where extreme events are more likely than a normal distribution would predict. This means that the real risk can be higher than the VaR indicates.

• No Information Beyond the VaR: VaR gives you the maximum loss that will not be exceeded with a certain confidence level, but it doesn’t tell you anything about the loss if the worst case is surpassed. It doesn’t consider the severity of losses beyond the VaR threshold.

• Not Sub-additive: VaR isn’t sub-additive, meaning the VaR of a portfolio can be greater than the sum of the VaR of its components. This violates the principle of diversification.

• Non-constant Volatility: Financial markets exhibit high and low volatility periods. VaR estimates assuming constant volatility might understate the risk during high-volatility periods and overstate it during low-volatility periods.

So, while VaR can be a useful measure for risk management, other tools should be used to assess risk. Other measures, like Conditional Value at Risk (CVaR), which considers losses beyond the VaR threshold, should also be used for a more comprehensive risk assessment.

Reference:

[1]      “Markowitz Model – What Is It, Assumptions, Diagram, Formula.” https://www.wallstreetmojo.com/markowitz-model/ (accessed May 21, 2023).

Table 1: Weekly Expected Returns

Table 2: Annual Expected Returns

Table 3: Annual Standard Deviation

Table 4: Correlation Analysis