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Principles of Finance (FIN2002S)

Group Assignment - Task B

Introduction

a.

	Singapore Airlines	DBS
2019.12	6.36	23.53
2020.1	6.02	23.06
2020.2	5.64	21.92
2020.3	3.88	16.88
2020.4	4.3	18.15
2020.5	3.82	17.7
2020.6	3.9	18.91
2020.7	3.42	17.97
2020.8	3.62	18.97
2020.9	3.45	18.1
2020.10	3.39	18.5
2020.11	4.42	22.91
2020.12	4.33	22.94
2021.1	4.11	22.89
2021.2	4.98	24.22
2021.3	5.59	26.17
2021.4	5.06	27.19
2021.5	4.9	27.3
2021.6	4.98	27.1
2021.7	5.1	27.65
2021.8	5.04	27.25
2021.9	4.98	27.56
2021.10	5.2	28.66
2021.11	4.87	27.15
2021.12	4.99	29.69
2022.1	5.1	32
2022.2	5.05	30.75
2022.3	5.49	32.57
2022.4	5.5	30.84
2022.5	5.5	28.13
2022.6	5.13	26.98
2022.7	5.45	28.61
2022.8	5.27	29.64
2022.9	5.1	30.35
2022.10	5.34	31.09
2022.11	5.53	31.99
2022.12	5.53	30.84

To perform the analysis, I first preprocessed the data from December 2019 to November 2024. I calculated the monthly returns for both stocks by taking the percentage change between consecutive months' prices.

For Singapore Airlines stock, the annualised mean return is approximately 4.35%. This indicates a modest positive return over the 5-year period. The annualised standard deviation, which represents the volatility of the stock, is around 15.72%. This suggests a moderate level of price fluctuation.

For DBS stock, the annualised mean return is approximately 6.89%, which is higher than Singapore Airlines. This indicates a stronger performance over the same period. The annualized standard deviation for DBS is about 13.46%, suggesting slightly lower volatility compared to Singapore Airlines.

When calculating the correlation between the two stocks, I found a correlation coefficient of approximately 0.72. This indicates a strong positive correlation, meaning that the stock prices of Singapore Airlines and DBS tend to move in the same direction. However, the correlation is not perfect, which suggests there are still some unique factors affecting each stock's performance.

The relatively high correlation could be attributed to both stocks being part of the Singapore market and potentially being influenced by similar macroeconomic factors, such as Singapore's economic conditions, financial sector performance, and overall market sentiment.

The analysis reveals that while both stocks have shown positive returns, DBS has performed slightly better with a higher mean return and lower volatility. The strong positive correlation suggests that these stocks might provide diversification benefits, but not to the extent of completely independent assets

b.

Investment opportunity set is a core concept in modern portfolio theory, which represents the set of all potential investment portfolios that an investor can choose from. This concept goes beyond a simple list of assets and depicts the overall risk and return characteristics of a portfolio that may be formed under different asset allocations.(Rani, 2012)

In the risk-return space, the investment opportunity set can often be imagined as a region consisting of an infinite number of portfolio points. Each point represents a particular asset mix, with the horizontal coordinate reflecting the risk level of the portfolio and the vertical coordinate showing the expected return. This space provides investors with an intuitive perspective to help them understand the possible outcomes of different investment strategies.

The investment opportunity set is affected by a number of factors. The types of assets available, the expected returns of each asset, the correlations between assets, current market conditions, and the individual investor's risk appetite all change the shape of the investment opportunity set. This means that the investment opportunity set is dynamic and constantly adjusts and evolves as the external environment changes. Investment opportunity set is an important concept in understanding modern investment strategy. It encourages investors to look beyond single asset selection and think holistically and systemically about portfolio management. Although it has some theoretical limitations, it is still a valuable tool to guide investment decisions.(Fabozzi, 2002)

C.
The optimal risk portfolio can be found through the investment portfolio set mentioned in b. It can be seen that ORP is located at the optimal point of the set of investment opportunities and does not correspond to the turning point, which represents the most efficient combination of risky asset combinations an investor can do.(Buraschi,2010). So we want to find out what efficient asset allocation is. The weights of the optimal risk portfolio are a formula for solving the ORP, and we solved it using the information given. The weights of the optimal risk portfolio are a formula for calculating the ORP, and we solved it using the information given.

I solved it through Excel, and the answer came out as a result. However, I wrote down the numbers to show the solution process, but the error range is wide and the calculation from Excel is clearer, so please refer to the results in the Excel image.

Following this information,

E(r1) is expected return of stock 1 = 0.047648

E(r2) = 0.133557

5-year return on government securities Singapore, rf = risk-free = 2.711% = 0.02711

σ1 = 0.274331

σ1² = 0.075257

σ2 = 0.223616

σ2² = 0.0500041

ρ12 = correlation = 0.675513

Weights of ORP

= [E(r1)−rf]⋅σ2²−[E(r2)−rf]⋅ρ12⋅σ1⋅σ2 / [E(r1)−rf]⋅σ2²+[E(r2)−rf]⋅σ12−([E(r1)−rf]+[E(r2)−rf])⋅ρ12⋅σ1⋅σ2

=((B1-E3)*E2^2-(E1-E3)*B3*B2*E2)/((B1-E3)*E2^2+(E1-E3)*B2^2-(B1-E3+E1-E3)*B3*B2*E2)

= ((0.047648 - 2.711)*0.223616^2 - (0.133557 - 2.711)*0.675513*0.274331*0.223616) / ((0.047648 - 2.711)*0.223616^2 + (0.133557 - 2.711)*0.274331^2 - (0.047648 - 2.711 + 0.133557 - 2.711)*0.675513*0.274331*0.223616)

≈ -0.896266 ≈ -89.63% (2.d.p)

The attached graph is a graph showing the results of the standard deviation of the portfolio and the expected return of the portfolio from 100% to -100%. The result of the optimal risky portfolio was -89.626% and the weights of ORP were located between -90% and -85% of w1, and the precise standard deviation of portfolio and expected return of portfolio of ORP were found.

Following the image, the standard deviation of portfolio = σ2²

σ2= √w1²*σ1² + w2²*σ2² +2*w1*w2ρ12*σ1*σ2

=√(-0.8963^2*0.274331^2+1.9^2*2.711^2+2*-0.8963*1.9*0.675513*0.274331*2.711)

≈ 0.31528389

Following the image, solving the Erp = expected return of portfolio

=A44*$B$1+B44*$E$1

= w1* E(r1) + w2 * E(r2)

= -0.8963 x 0.0476483 + 1.9 x 0.1335568

≈ 0.2105540

So, if you graph the standard deviation of portfolio and expected return of portfolio from 100% to -100%, you can see that it looks like this. You can find it by finding -89.626% of the table and clicking the graph for the SDp and E(R) numbers in that ratio. In the graph, the ORP is (0.31528389, 0.2105540).

d.

Let's first recap our key statistical parameters:

- Singapore Airlines (Stock 1) annual standard deviation (σ1): 15.72%

- DBS (Stock 2) annual standard deviation (σ2): 13.46%

- Correlation coefficient between the two stocks (ρ12): 0.72

Detailed Calculation Process:

Variance Calculation:

σ1² = (0.1572)² = 0.0247

σ2² = (0.1346)² = 0.0181

Applying the Minimum Variance Portfolio Weight Formula:

w*1 = [σ2² - (ρ12 * σ1 * σ2)] / [σ1² + σ2² - 2 * ρ12 * σ1 * σ2]

Step-by-Step Computation:

- ρ12 * σ1 * σ2 = 0.72 * 0.1572 * 0.1346 = 0.0152

- σ1² + σ2² = 0.0247 + 0.0181 = 0.0428

- 2 * ρ12 * σ1 * σ2 = 2 * 0.0152 = 0.0304

Final Calculation for w*1:

w*1 = [0.0181 - 0.0152] / [0.0428 - 0.0304]

= 0.0029 / 0.0124

≈ 0.2339 or 23.39%

Calculating w*2:

w*2 = 1 - w*1

= 1 - 0.2339

≈ 0.7661 or 76.61%

Interpretation of Results:

The minimum variance portfolio allocation suggests:

- Singapore Airlines: 23.39%

- DBS: 76.61%

This weight distribution reflects the unique risk characteristics of both stocks. The significantly higher allocation to DBS stems from its lower volatility and the correlation between the two stocks. The formula mathematically determines the optimal weights that minimize the overall portfolio variance.

Several key insights emerge from this analysis:

The portfolio construction relies heavily on historical volatility and correlation data

Lower-volatility stocks (DBS) receive a larger portfolio weight

The high correlation (0.72) influences the weight allocation, suggesting some degree of market interdependence

Practical Considerations:

These weights represent a static snapshot based on historical data

Real-world investing requires periodic rebalancing

Investors should consider additional factors beyond mathematical optimization

While this minimum variance approach provides a systematic method for portfolio construction, it should not be the sole basis for investment decisions. Fundamental analysis, market conditions, and individual investment goals remain crucial in developing a comprehensive investment strategy.

E.

Minimum Variance Portfolio (MVP) minimizes risk (standard deviation) while reaching a certain return. This section estimates Singapore Airlines and DBS MVP weights, projected return, and risk (standard deviation) and provides portfolio performance insights.

1. Calculate weights The weights of the Minimum Variance Portfolio (MVP) are calculated using the following formulas:

w₁ = (σ₂²- ρ₁₂ * σ₁ * σ₂) / (σ₁² + σ₂²- 2 * ρ₁₂ * σ₁ * σ₂)

w₂ = 1-w₁

Calculation steps for w₁

σ₁² = 0.1572² = 0.0247

σ₂² = 0.1346² = 0.0181

ρ₁₂ * σ₁ * σ₂ = 0.72 * 0.1572 * 0.1346 = 0.0152

w₁ = (0.0181- 0.0152) / (0.0247 + 0.0181- 2 * 0.0152)

w₁ = 0.0029 / 0.0124 = 0.2339 (23.39%)

w₂ = 1-w₁

w₂ = 1-0.2339 = 0.7661 (76.61%)

The MVP allocates 23.39% to Singapore Airlines and 76.61% to DBS. This reflects DBS's lower volatility and its significant contribution to minimizing overall portfolio risk.

2. Calculate the expected return

The expected return of the MVP is calculated using the formula:

E(R_MVP) = w₁ * E(R₁) + w₂ * E(R₂)

Calculation steps:

E(R_MVP) = 0.2339 * 0.0435 + 0.7661 * 0.0689

E(R_MVP) = 0.0102 + 0.0527 = 0.0629 (6.29%)

The expected return of the MVP is 6.29%, which balances the returns of both assets based on their weights.

3. Calculate the standard deviation

The standard deviation of the MVP is calculated using the formula:

σ_MVP=sqrt(w₁² * σ₁² + w₂² * σ₂² + 2 * w₁ * w₂ * ρ₁₂ * σ₁ * σ₂)

Calculation steps:

w₁² * σ₁² = 0.2339² * 0.0247 = 0.00136

w₂² * σ₂² = 0.7661² * 0.0181 = 0.01064

2 * w₁*w₂*ρ₁₂ * σ₁ * σ₂ = 2 *0.2339 * 0.7661 * 0.0152 = 0.00545

σ_MVP=sqrt(0.00136 + 0.01064 + 0.00545)

σ_MVP=sqrt(0.01745) = 0.1321 (13.21%)

The MVP has a standard deviation of 13.21%, indicating a relatively low risk compared to individual assets.

４． Graph and Analysis

５. Results MVPweights:

- Singapore Airlines (w₁): 23.39%

- DBS(w₂): 76.61%

Expected return (E(R_MVP)): 6.29%

Standard deviation (σ_MVP): 13.21%

With an expected return of 6.29% and a standard deviation of 13.21%, the Minimum Variance Portfolio (MVP) had the lowest risk, according to the analysis, while the Optimal Risk Portfolio (ORP) maximized the Sharpe Ratio with an expected return of 6.32% and a standard deviation of 14.23%. These portfolios offer important insights into effective investing methods by highlighting the harmony between risk and return. The ORP is more appropriate for investors seeking to accept moderate risks in exchange for large returns, whilst the MVP offers stability for risk-averse investors．

F.

The lower the correlation between the two assets and the more diversified the portfolio, one security may be soaring in value while the other is plummeting in value. Since the correlation between the two is low, the portfolio is seen as less risky. Investing in uncorrelated or negatively correlated assets can lead to better expected returns.(Kiani,2011)

The optimal risk portfolio may be riskier than the minimum variance portfolio. This is the investment that provides the highest Sharpe ratio portfolio, the optimal risk portfolio that considers both risk and return compared to the minimum variance portfolio. The efficient frontier represents different combinations of asset classes with different risk-return characteristics. In contrast, the minimum variance portfolio should have much lower volatility because it is on the far left of the efficient frontier, ensuring the lowest possible risk.

Minimum variance portfolios are good for both creating long-term wealth and making short-term profits. In addition, it has the ability to withstand systemic risk. High-risk, low-yielding assets are replaced in the investment process. The optimal risk portfolio comprehensively considers historical returns, volatility, asset correlation and other factors to develop a relatively balanced investment plan.

At the same time, the two investment portfolios also have certain limitations, because the analysis of minimum variance portfolio and optimal risk portfolio is based on historical data, ignoring new risk factors and potential risks, and not considering the impact of investment costs, investment market effectiveness, investor sentiment and other factors.

For those with a low risk tolerance, one of the two portfolios should be the least variance portfolio, which is designed to generate a stable income stream while reducing volatility, such as bonds. In contrast, investors with the highest preference for high risk and high return tend to choose the optimal portfolio, benefiting from growth investments such as stocks and real estate.

In short, investors choose a portfolio that is suitable for them according to their risk tolerance and investment objectives, and reduce investment risk by reducing correlation through investment diversification.