Investment and Portfolio Management

This report discusses Investment and Portfolio Management with the help of two stocks, Apple Inc. and Exxon Mobil Corporation. The weights of each stock were calculated to determine the optimum portfolio. The expected return and standard deviation were calculated for both portfolios. Diversification helps in reducing the risk of the portfolio by investing in different stocks from different sectors.

Running head: INVESTMENT AND PORTFOLIO MANAGEMENT
Investment and Portfolio Management
Name of the Student:
Name of the University:
Author’s Note:

1INVESTMENT AND PORTFOLIO MANAGEMENT
Table of Contents
Introduction................................................................................................................................2
Discussion..................................................................................................................................2
Annual Mean Return, Standard Deviation and Correlation...................................................2
Investment Opportunity Set...................................................................................................3
Optimal Risky Portfolio.........................................................................................................5
Risk and Return of Optimal Portfolio....................................................................................6
Minimum Variance Portfolio.................................................................................................7
Risk and Return of Minimum Variance Portfolio..................................................................7
Diversification........................................................................................................................8
Conclusion..................................................................................................................................9
Reference..................................................................................................................................11

2INVESTMENT AND PORTFOLIO MANAGEMENT
Introduction
The stock price analysis for the Exon Mobil and Apple Stock Incorporation was taken
into consideration for the purpose of Investment Analysis and Portfolio Construction. Five-
years of monthly data was taken into consideration for the purpose of the analysis of the
stocks. Risk and return are some of the important characteristics of investment and the same
has been taken into consideration while deciding the optimum weights of each of the security
in the portfolio (Calvo, Ivorra and Liern 2016).
Discussion
Annual Mean Return, Standard Deviation and Correlation
The annual mean for both the stocks were evaluated by taking the monthly return
generated by each of the stocks and annualizing the same to get the annualised return of the
same. The standard deviation of each of the stock was derived by taking a square root of the
annual standard deviation of both the stocks (Hoesli and MacGregor 2014). The annual mean
return for the Apple Incorporation was around 22.47% and the standard deviation for the
stock was around 23.24%. The annual mean return for the Exxon Mobil Company was
around 2.53% and the standard deviation for the stock was around 14.28%. The correlation of
the stock were around 0.04476 (Finance.yahoo.com 2019).
CALCULATING THE RETURN
Apple Inc. Exxon Mobil Corp
Month Price Return Price Return
Monthly Mean 1.87% 0.21%
Monthly Variance 0.45% 0.17%
Monthly Standard Dev. 6.71% 4.12%
Annual Mean 22.47% 2.53%
Annual Variance 5.40% 2.04%
Annual Standard Dev. 23.24% 14.28%

3INVESTMENT AND PORTFOLIO MANAGEMENT
Investment Opportunity Set
The investment proportions for the two stocks were prepared for the two stocks from
0% to 100% using interval of 5%. The investment opportunity set for the stocks could be well
prepared with the help of the stock giving the maximum return in contrast to the risk taken by
the stock. The investment opportunity curve was made where the optimum investment
opportunity in the stock could be found with the help of risk/return formula. While preparing
the investment opportunity set the Apple stock was denoted as “S” and Exon Mobil was
denoted as “B” (Chandra 2017).
E(Rs) E(Rʙ) σs σʙ ρsʙ Rƒ
22.47% 2.53% 23.24% 14.28% 4.48% 2.44%
Ws Wʙ
= 1-Ws
Expected return E(Rp)
=Ws*E(Rs)+Wʙ*E(Rʙ)
Variance
(σ^p)
Standard
Deviation
(σp)
Risk/Return
0% 100% 2.53% 0.0204 14.28% 0.17716
5% 95% 3.53% 0.0186 13.62% 0.25894
10% 90% 4.52% 0.0171 13.07% 0.34614
15% 85% 5.52% 0.0160 12.64% 0.43671
20% 80% 6.52% 0.0153 12.35% 0.52775
25% 75% 7.52% 0.0149 12.20% 0.61575
26% 74% 7.81% 0.0149 12.19% 0.64049
30% 70% 8.51% 0.0149 12.21% 0.69715
35% 65% 9.51% 0.0153 12.37% 0.76898
40% 60% 10.51% 0.0160 12.67% 0.82940
45% 55% 11.50% 0.0172 13.10% 0.87785
50% 50% 12.50% 0.0187 13.66% 0.91491
55% 45% 13.50% 0.0205 14.33% 0.94190
60% 40% 14.49% 0.0228 15.09% 0.96050
65% 35% 15.49% 0.0254 15.93% 0.97238
70% 30% 16.49% 0.0284 16.84% 0.97909
75% 25% 17.49% 0.0317 17.81% 0.98191
80% 20% 18.48% 0.0354 18.82% 0.98189
85% 15% 19.48% 0.0395 19.88% 0.97982
90% 10% 20.48% 0.0440 20.97% 0.97632
95% 5% 21.47% 0.0488 22.09% 0.97185
100% 0% 22.47% 0.0540 23.24% 0.96675
105.0% -5.0% 23.47% 0.0596 24.41% 0.96126
106.5% -6.5% 23.76% 0.0613 24.77%

4INVESTMENT AND PORTFOLIO MANAGEMENT
0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00%
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
2.53%
3.53%
4.52%
5.52%
6.52%
7.52%
7.81%
8.51%
9.51%
10.51%
11.50%
12.50%
13.50%
14.49%
15.49%
16.49%
17.49%
18.48%
19.48%
20.48%
21.47%
22.47%
23.47%23.76%
Series2
CML
Market
Optimal Risky Portfolio
The optimal risky portfolio could be calculated with the help of the return and risk of
each of the security is calculated for determining the optimal risky portfolio. The weight of
the Apple stock was denoted by Stock 1and the weight of the Exon Mobil Company was
denoted by Stock 2 (Klingebiel and Rammer 2014). The weight on optimal risky portfolio for
the stock were calculated with the formula:
Wʙ (Stock 2)
=
[E(Rʙ) - Rƒ] σs^2 - [E(Rs) - Rƒ] σʙσsρʙs
[E(Rʙ) - Rƒ] σs^2 + [E(Rs) - Rƒ] σʙ^2 - [E(Rʙ) - Rƒ + E (Rs) - Rƒ] σʙσsρʙs
The weight of Stock 2 (Exxon) was calculated to be around -6.49% and the weight of
the weight of Stock 1 (Apple Inc.) was taken to be around 106.49%. The risk free rate was
taken at 2.44% that was the five-years Treasury bill rate of the US which has been taken from
the Bloomberg Data (Bloomberg.com 2019).
Stocks Weights
Stock 1 (Apple Incorporation) 106.49%

5INVESTMENT AND PORTFOLIO MANAGEMENT
Stock 2 (Exxon Mobil) -6.49%
Optimal Risky Portfolio =
Sp = (E(Rp) - Rƒ) / σp Sharp Ratio 0.862056101
Wʙ (Stock 2) = [E(Rʙ) - Rƒ] σs^2 - [E(Rs) - Rƒ] σʙσsρʙs
[E(Rʙ) - Rƒ] σs^2 + [E(Rs) - Rƒ] σʙ^2 - [E(Rʙ) - Rƒ + E (Rs) - Rƒ] σʙσsρʙs
Weight of B -6.49%
Weight of S (Stock 1) (1- Weight of B) 106.49%
E (Rp) 23.76%
σp= 24.74%
Risk and Return of Optimal Portfolio
The expected return from the portfolio was calculated with the help of the weights of
each of the security in the portfolio and the return generated by each of the stocks (Kevin
2015). The return for the optimal risky portfolio was calculated with the help of the formula:
Expected Return (E(Rp)) = Weight of Stock 1* Return of Stock 1+ Weight of Stock 2*
Return of Stock 2.
The return for the portfolio was calculated to be around 23.76%, which was derived
by taking the optimum weightage given to each of the security in the portfolio
(Finance.yahoo.com 2019). The risk or the standard deviation of the portfolio was derived to
be around 24.74% (Damodaran 2016).

6INVESTMENT AND PORTFOLIO MANAGEMENT
Minimum Variance Portfolio
The weights of each of the stock in the minimum variance portfolio could be
calculated with the help of the formula:
Ws (min) = (σʙ²- σʙ*σs*ρ) / ( σs²+ σʙ²- 2σʙ*σs*ρ)
The weight of apple stock was determined to be around 26.5% which is Stock 1 and
the weight of the stock 2 (Exxon Mobil) was taken at 73.5% (Bloomberg.com 2019). The
weightage of the stock was given at such a stage where the investor with a least amount of
risk can generate the highest amount of return for a stock (DeFusco et al. 2015).
Risk and Return of Minimum Variance Portfolio
The Minimum variance portfolio shows the diversified portfolio which consist of well
diversified portfolio which when hedged together results in lowest possible risk for the rate of
expected return. The return for the minimum variance portfolio was calculated with the help
of the optimum weights of the securities and the constituent risk. The expected return for the
stock could be calculated with the help of the formula:
Expected Return (E(Rp)) = Weight of Stock 1* Return of Stock 1+ Weight of Stock 2*
Return of Stock 2.
The return for the stock was calculated to be 7.81% for the minimum variance
portfolio stating that the portfolio can earn as high as 7.81% by taking the lowest amount of
risk (Bloomberg.com 2019). The standard deviation for the stock was calculated with the
formula:
Standard Deviation (S.D): The standard deviation for the portfolio could be calculated with
the help of the risk of each security in the portfolio and the weightage of each of the stock in

7INVESTMENT AND PORTFOLIO MANAGEMENT
the portfolio. The correlation of the stock was to be taken at the 4.476% (Finance.yahoo.com
2019). The standrad deviation for the stocks were calculated with the help of weight of each
of the security and the risk of each of the security.
Portfolio Standard Deviation: w2A*σ2(RA) + w2B*σ2(RB) + 2*(wA)*(wB)*Cov(RA, RB)
The standard deviation for the stock was calculated to be around 12.40%, which states
that this is the minimum risk that can be taken by the stocks for getting the highest amount of
return. Thus, the investor can earn the best with the application of minimum variance
portfolio that is by investing around 73.5% of the total fund in the Apple Inc. Stock and
26.5% in the Exxon Mobil Company Stock (Finance.yahoo.com 2019).
Ws (min) = (σʙ²- σʙ*σs*ρ) / ( σs²+ σʙ²- 2σʙ*σs*ρ) 26.5%
WB = (1- Weight of Apple Stock) 73.5%
E (Rp)= Ws (min) * E(Rs) + (1-Ws (min))*E(Rʙ) 7.81%
σp= 12.40%
Minimum variance portfolio =
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Diversification
Diversification plays an important role in the context of portfolio management where
taking two or more stocks in the portfolio would be helping the risk and return modification.
Diversification is important in modifying the return and risk of the portfolio thereby reducing
the concentration risk of the portfolio. In the above report both the stocks were analysed
based on the return and risk generated by each of the stock and the implication of each of the
stock in the portfolio. The inclusion of various stocks in a portfolio is a risk management
strategy to yield higher and better returns (Howard 2014).
The return generated from the Apple Incorporation via the optimal risky portfolio was
much more as the theory behind the same is that the return from the portfolio should be
maximized and that was the key reason for giving the Apple stock a weightage of 106.49%

8INVESTMENT AND PORTFOLIO MANAGEMENT
and -6.49% to Exxon Mobil. The standard deviation of the portfolio was around 24.74%. The
reason behind the same can be given to the poor performance by the Exxon Mobil where the
company gave a return less than 2.44% ,which was the risk free rate of return. Return
generated by Exxon Mobil Company was just 2.53% which was found to be inconsistent
(Finance.yahoo.com 2019). The sharp ratio of the minimum variance portfolio was around
0.42 which shows the excess unit of return generated by the portfolio for a single unit of risk
taken.
It is crucial to note however on the other hand the minimum variance portfolio takes
the concept of the higher risk by taking a minimum amount of risk in the given set of
investment opportunity. The same was computed with the help of the various investment
intervals and the best interval class was selected which gave the highest amount of return by
taking the least amount of risk at the same time. The return generated by the minimum
variance portfolio was around 7.81% and the correspondence standard deviation of the
portfolio was taken at 12.19%. The sharp ratio of the portfolio was around 0.85 which
comparatively is far more better and efficient than the minimum variance portfolio.
Conclusion
In the above report both the stocks Apple Inc. and Exxon Mobil were analysed based
on the return and risk generated by each of the stock and the implication of each of the stock
in the portfolio. The optimal weights in a stock was determined based on the minimum
variance portfolio and the optimal risky portfolio where optimal risky portfolio was found to
generate more return for the investors. On the other hand side the minimum variance
portfolio considered the best allocation mix of securities according to the inherent risk and
return associated with the same.

9INVESTMENT AND PORTFOLIO MANAGEMENT
Reference
Bloomberg.com. (2019). Treasury Yield. [online] Available at:
https://www.bloomberg.com/markets/rates-bonds/government-bonds/us [Accessed 12 Mar.
2019].
Calvo, C., Ivorra, C. and Liern, V., 2016. Fuzzy portfolio selection with non-financial goals:
exploring the efficient frontier. Annals of Operations Research, 245(1-2), pp.31-46.
Chandra, P., 2017. Investment analysis and portfolio management. McGraw-Hill Education.
Damodaran, A., 2016. Damodaran on valuation: security analysis for investment and
corporate finance (Vol. 324). John Wiley & Sons.
DeFusco, R.A., McLeavey, D.W., Pinto, J.E., Anson, M.J. and Runkle, D.E.,
2015. Quantitative investment analysis. John Wiley & Sons.
Finance.yahoo.com. (2019). Apple Share Price. [online] Available at:
https://finance.yahoo.com/quote/AAPL/history?p=AAPL [Accessed 12 Mar. 2019].
Finance.yahoo.com. (2019). Exxon Mobil Share Price. [online] Available at:
https://finance.yahoo.com/quote/XOM/history?p=XOM&.tsrc=fin-srch [Accessed 12 Mar.
2019].
Hoesli, M. and MacGregor, B.D., 2014. Property investment: principles and practice of
portfolio management. Routledge.
Howard, C.T., 2014. Behavioral portfolio management. Journal of Behavioral Finance &
Economics, Forthcoming.
Kevin, S., 2015. Security analysis and portfolio management. PHI Learning Pvt. Ltd..

10INVESTMENT AND PORTFOLIO MANAGEMENT
Klingebiel, R. and Rammer, C., 2014. Resource allocation strategy for innovation portfolio
management. Strategic Management Journal, 35(2), pp.246-268.