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Question:
Describe the Plotting investment opportunity set for the two stocks.
Answer:

Calculating the annualised mean return, standard deviation, and correlation of Mazda Motor Corporation (MZDAY) and Nissan Motor Co., Ltd. (NSANY):

 

Mazda Motor Corporation

Nissan Motor Co., Ltd.

Date

Adj Close

Return

Adj Close

Return

3/1/2017

7.3200

0.056277

19.9600

0.014227591

2/1/2017

6.9300

-0.05068

19.6800

-0.007564297

1/3/2017

7.3000

-0.10209

19.8300

-0.0085

12/1/2016

8.1300

-0.00733

20.0000

0.054296313

11/1/2016

8.1900

0.001222

18.9700

-0.065977354

10/3/2016

8.1800

0.073491

20.3100

0.034114055

9/1/2016

7.6200

-0.07524

19.6400

0.002040765

8/1/2016

8.2400

0.116531

19.6000

0.012396642

7/1/2016

7.3800

0.111446

19.3600

0.077951118

6/1/2016

6.6400

-0.21606

17.9600

-0.105577739

5/2/2016

8.4700

0.078981

20.0800

0.129358767

4/1/2016

7.8500

0.003836

17.7800

-0.039438088

3/1/2016

7.8200

0.136628

18.5100

0.023217247

2/1/2016

6.8800

-0.24146

18.0900

-0.085439884

1/4/2016

9.0700

-0.12197

19.7800

-0.057646451

12/1/2015

10.3300

-0.01525

20.9900

-0.031826568

11/2/2015

10.4900

0.068228

21.6800

0.045827303

10/1/2015

9.8200

0.247776

20.7300

0.11631664

9/1/2015

7.8700

-0.07953

18.5700

0.031093837

8/3/2015

8.5500

-0.12487

18.0100

-0.06876939

7/1/2015

9.7700

-0.00204

19.3400

-0.075083692

6/1/2015

9.7900

-0.07729

20.9100

0.006255967

5/1/2015

10.6100

0.088205

20.7800

0.001445831

4/1/2015

9.7500

-0.03941

20.7500

0.020659124

3/2/2015

10.1500

-0.05317

20.3300

-0.033285828

2/2/2015

10.7200

0.040777

21.0300

0.235605229

1/2/2015

10.3000

-0.14664

17.0200

-0.025758387

12/1/2014

12.0700

-0.06723

17.4700

-0.061257493

11/3/2014

12.9400

0.115517

18.6100

0.008672086

10/1/2014

11.6000

-0.07643

18.4500

-0.04700413

9/2/2014

12.5600

0.069847

19.3600

0.007808538

8/1/2014

11.7400

-0.0608

19.2100

-0.022889166

7/1/2014

12.5000

0.060445

19.6600

0.036373276

6/2/2014

11.7875

0.081422

18.9700

0.048066243

5/1/2014

10.9000

-0.03582

18.1000

0.047453643

4/1/2014

11.3050

0.01618

17.2800

-0.033016226

3/3/2014

11.1250

-0.07523

17.8700

0.014865248

2/3/2014

12.0300

-0.00021

17.6083

0.041884795

1/2/2014

12.0325

-0.07797

16.9004

0.023214333

12/2/2013

13.0500

0.111348

16.5170

-0.079956204

11/1/2013

11.7425

0.038929

17.9524

-0.09289613

10/1/2013

11.3025

0.007577

19.7908

-0.003958379

9/3/2013

11.2175

0.129658

19.8695

0.019162797

8/1/2013

9.9300

-0.03709

19.4959

-0.051196151

7/1/2013

10.3125

0.036172

20.5479

0.028037407

6/3/2013

9.9525

0.005303

19.9875

-0.064857404

5/1/2013

9.9000

0.157895

21.3737

0.042685796

4/1/2013

8.5500

0.14038

20.4987

0.085937469

3/1/2013

7.4975

-0.00564

18.8765

-0.053721114

2/1/2013

7.5400

0.112915

19.9482

-0.007338513

1/2/2013

6.7750

0.344913

20.0956

0.070717649

12/3/2012

5.0375

0.283439

18.7684

-0.024028559

11/1/2012

3.9250

0.329382

19.2305

0.169856443

10/1/2012

2.9525

-0.08873

16.4383

-0.018203166

9/4/2012

3.2400

0.050243

16.7431

-0.087841505

8/1/2012

3.0850

0.001623

18.3554

-0.006386417

7/2/2012

3.0800

-0.06667

18.4734

-0.011052584

6/1/2012

3.3000

0.033673

18.6799

-0.006276191

5/1/2012

3.1925

-0.22039

18.7979

-0.076328454

4/2/2012

4.0950

-0.064

20.3512

-0.035863907

3/1/2012

4.3750

0.035503

21.1083

0.047317009

2/1/2012

4.2250

0.04064

20.1546

0.090425641

1/3/2012

4.0600

 

18.4833

 

Mazda Motor Corporation (MZDAY)

Average monthly return

     0.0161

Average monthly return

              0.0032

 

Annualised monthly return

     0.1930

Annualised monthly return

              0.0384

 

Monthly Standard Deviation

     0.1172

Monthly Standard Deviation

              0.0643

 

Annualised Standard Deviation

     0.4059

Annualised Standard Deviation

              0.2227

 

correlation

     0.5338

   

Table 1: Annualised return, covariance and annualised standard deviation

(Source: As created by the author)

The above table mainly depicts the annualised retunes, which is provided by Mazda Motor Corporation and Nissan Motor Co Ltd. The annualised return is mainly depicted after multiplying the number of months with the average monthly return of the stock. Moreover, the annualised standard deviation is mainly depicted in the table, which is derived by using the STEDV function in excel. Furthermore, the table also depicts the relevant correlation and covariance of the two stocks with the help of CORREL and COVAR function in excel. The derivation of annualised return, correlation annualised risk and covariance is mainly essential for identifying the optimal portfolio, which could help the investor in maximising its return, while reducing the overall risk. Goswami and Mukherjee (2015) mentioned that derivation of risk from an investment is essential as it helps the investor to diversify its portfolio to match it risk profile. Moreover, investors to maximise the expected return of a predetermined leave of risk mainly conduct the mean variance. With the help of covariance of both the stock investors, identify optimal portfolio, which could increase return and reduce risk.

Plotting investment opportunity set for the two stocks:

w1

w2

Sp

E(rp)

150%

0%

60.89%

28.95%

145%

5%

59.46%

28.18%

140%

10%

58.05%

27.40%

135%

15%

56.65%

26.63%

130%

20%

55.28%

25.86%

125%

25%

53.92%

25.08%

120%

30%

52.58%

24.31%

115%

35%

51.27%

23.54%

110%

40%

49.98%

22.77%

105%

45%

48.72%

21.99%

100%

50%

47.48%

21.22%

95%

55%

46.28%

20.45%

90%

60%

45.11%

19.67%

85%

65%

43.97%

18.90%

80%

70%

42.88%

18.13%

75%

75%

41.82%

17.35%

70%

80%

40.81%

16.58%

65%

85%

39.85%

15.81%

60%

90%

38.94%

15.03%

55%

95%

38.09%

14.26%

50%

100%

37.29%

13.49%

45%

105%

36.56%

12.72%

40%

110%

35.90%

11.94%

35%

115%

35.31%

11.17%

30%

120%

34.79%

10.40%

25%

125%

34.35%

9.62%

20%

130%

33.99%

8.85%

15%

135%

33.72%

8.08%

10%

140%

33.53%

7.30%

5%

145%

33.43%

6.53%

0%

150%

33.41%

5.76%

Table 2: Plotting risk and return from 0% to 150% in portfolio

(Source: As created by the author)

The above table mainly depicts the relevant risk and returns, which is been provided from different weights of investment conducted in Mazda Motor Corporation and Nissan Motor Co Ltd. There weights mainly depict the opportunity to invest and acquire a portfolio with least risk and maximum return. Moreover, the derivation of adequate portfolio weights mainly allows investor to identify the relevant opportunity and make adequate investment decision. Fich, Harford and Tran (2015) mentioned that investors have used the portfolio weights to determine the return and risk from the investment. On the other hand, A?t-Sahalia and Matthys (2015) criticises that portfolio segregation mainly loses its friction during an economic crises, where all the relevant stock will provide negative returns and hamper profitability of the investor. 

E (r1) = 0.19

E (r2) = 0.04

Covariance (r1, r2) = 0.0040

Rf = 0.05

[( 0.19 - 0.05) X 0.0040] - [(0.04 - 0.05) X 0.0040]

[(0.19 - 0.05) X 0.0040] + [(0.04- 0.05 X 0.0040 ) X 62]- [ (0.19-0.04+0.05 ) X 0.004]

 

0.00056 - 0.0004

0.00056 - 0.0004 X 62] - 0.0008

 

0.0006

-0.00272

 

W1 = 0.22

W1 = 0.22

W2 = 1- 0.22

W2 = 0.78

The above table mainly depicts the optimal portfolio weights, which needs to be invested by the investors to increase its chances for attaining higher profits. The derivation of the optimal portfolio is mainly conducted by using return of stock returns, risk free rate and covariance of both the stocks. These valuations mainly help in identifying the optimal portfolio, which could be used by the investors to maximise its profitability. Johannes, Korteweg and Polson (2014) mentioned that investors to identify the relevant stock investment mainly use optimal portfolio formula, which needs to be conducted to maximise return from the investment. However, Bennett and Zitikis (2014) argued that if the risk free rate is relevantly higher than the stock return overall optimal portfolio valuation will be nullified. This mainly indicates that investor wills directly invest in risk free assets, which does not needed any optimal portfolio risk evaluation 

The mean and std deviation of the optimal risky portfolio are:

 

    E (rp) = W1 X 0.19 + W2 X 0.04 = 0.22 x 0.19 + 0.78 x 0.04

              = 0.073

 

  σp = [w12 X 0.19) + w22 X 0.04+ [2 X w1 X w2 X Covariance (r1, r2)]

 

    = 0.05 x 0.19 + 0.61 x 0.04 + 0.001375

    = 0.03492

        Risk = 0.19

The above table states the optimal portfolio return and risk associated with investment. Moreover, the optimal return is 0.073 from investment, whereas it has a risk of 0.19. This is mainly optimal as it could help the investor effectively to improve the current condition of their investment. The derivation of the overall return and risk could effectual help the investor to use the adequate strategy for improvement and develop their portfolio. The determination of return and risk from the portfolio mainly helps in ensuring the minimum amount of shares, which is to be purchased by the investor. Bhuyan et al. (2014) stated that investors by using the portfolio valuation are mainly able to attain higher return by indulging in lower risk. 

0.164781 – 0.0040

0.164781+ 0.049614 – 2 * 0.0040

 

0.05

0.21

 

0.22

W1 = 0.22

W2 = 1- 0.22

W2 = 0.78

The above table mainly depicts the minimum variance portfolio weights, which needs to be maintained by the investor. Moreover, the minimum weights are 0.22 for by Mazda Motor Corporation and 0.78 for Nissan Motor Co Ltd. In addition, this could mainly help the investor in attain minimum variance portfolio, which could reduce the risk from volatility market. Maillet, Tokpavi and Vaucher (2015) mentioned that determination of minimum variance portfolio mainly allows the investor to effective understand investment structure for risk averse strategy. On the other hand, Bodnar and Gupta (2015) argued that accommodation of risk less assets could mainly nullify the significance of minimum portfolio variance. 

Expected return of (MVP)

w1 E (r1) + w2 E (r2)

Expected return of (MVP)

0.22 x 0.19    + 0.78 x 0.04

Expected return of (MVP)

0.04+0.03

Expected return of (MVP)

0.073

 

Expected risk of (MVP)

(0.22^2)*0.16 + (0.78^2)*0.049 + 2* 0.22*0.78*0.004

Expected risk of (MVP)

√0.076

Expected risk of (MVP)

0.276

The minimum variance portfolio (MVP) and optimal risk portfolio (ORP) is effectively depicts the following graph, which could help in generating the maximum return with the least risk associated with the portfolio. MVP is the minimum variance, which is needed by the Thus, with the help of both MVP an ORP investors are mainly able to identify the risk associated with their investment. Xing, Hu and Yang (2014) mentioned that investors only choose the portfolio, which have the minimum risk associated with investment. On the other hand, Yang, Couillet and McKay (2015) criticises that efficient frontier mainly helps in depicting the relevant line, which address both risk and return from the investment. The graphitic representation mainly depicts the upward sloping graph, which includes both risk and return from investment. Moreover, the minimum portfolio variance return is 0.073, which overall risk of the portfolio is 0.76. These values are effectively depicted in the efficient frontiers, which enable the investor to identify the return it could generate by accommodating more risks. Both Minimum variance portfolio (MVP) and optimal portfolio (ORV) is effectively depicted in the efficient frontiers.

It mainly helps in depicting the efficient frontier, which could be used by investees in identify the least investment weights, which reduce risk from investment. Aparicio et al. (2016) mentioned that investors are mainly keen on identifying the minimum risk portfolio, which could be used by companies. The efficient frontier figure mainly depicts both MVP and ORP, where 10.86% return is mainly provided from ORP and 11.17% return is provided from MVP. This indicates the minimum risk portfolio is also increasing the overall return capacity of the investor. Aparicio and Pastor (2014) mentioned the investor to identify the relevant investment opportunity mainly conducts the optimal risky portfolio, which has the least risk.

Discussing in details on diversification with reference to the efficient frontier and comparing the expected return and standard deviation of the optimal risky portfolio to the minimum-variance portfolio:

Efficient frontier is an effective method, which enables the investor to identify the relevant weights of stock, which could be included in their portfolio. The efficient frontier mainly starts from the least risk and rises towards the maximum risk. This rising could mainly help investor to diversify their investment and reduce risk of their portfolio. Yilmaz and Pearson (2016) mentioned that efficient frontier is a region in the graph, which represented by an upward sloping curve. In addition, if the covariance is smaller than the standard deviation then standard deviation of the stock will also be lower.

The expected return and standard deviation of optimal risky portfolio to the minimum-variance portfolio are more or less same. The risk of optimal 
risky portfolio to the minimum-variance portfolio has a difference of 0.01%, whereas the return has a difference of 0.31%. The difference in number if relatively lower and could be ignored. Thus, optimal risky portfolio to the minimum-variance portfolio depicts the same standard deviation and risk (Cui, Li and Li 2015). 

Reference and Bibliography:

A?t-Sahalia, Y. and Matthys, F.H., 2015. Robust portfolio optimization with jumps.

Aparicio, J. and Pastor, J.T., 2014. Closest targets and strong monotonicity on the strongly efficient frontier in DEA. Omega, 44, pp.51-57.

Aparicio, J., Garcia-Nove, E.M., Kapelko, M. and Pastor, J.T., 2016. Graph productivity change measure using the least distance to the pareto-efficient frontier in data envelopment analysis. Omega.

Bennett, C.J. and Zitikis, R., 2014. Estimation of optimal portfolio weights under parameter uncertainty and user-specified constraints: a perturbation method. Journal of Statistical Theory and Practice, 8(3), pp.423-438.

Bhuyan, R., Kuhle, J., Ikromov, N. and Chiemeke, C., 2014. Optimal portfolio allocation among REITs, stocks, and long-term bonds: An empirical analysis of US financial markets. Journal of Mathematical Finance, 2014.

Bodnar, T. and Gupta, A.K., 2015. Robustness of the inference procedures for the global minimum variance portfolio weights in a skew-normal model. The European Journal of Finance, 21(13-14), pp.1176-1194.

Cui, X., Li, D. and Li, X., 2015. MEAN?VARIANCE POLICY FOR DISCRETE?TIME CONE?CONSTRAINED MARKETS: TIME CONSISTENCY IN EFFICIENCY AND THE MINIMUM?VARIANCE SIGNED SUPERMARTINGALE MEASURE. Mathematical Finance.

Fich, E.M., Harford, J. and Tran, A.L., 2015. Motivated monitors: The importance of institutional investors? portfolio weights. Journal of Financial Economics, 118(1), pp.21-48.

Goswami, B. and Mukherjee, I., 2015. Risk-Return Analysis of Different Commodity Futures in Indian Derivative Market. International Journal of Research in Finance and Marketing, 5(6), pp.73-78.

Johannes, M., Korteweg, A. and Polson, N., 2014. Sequential learning, predictability, and optimal portfolio returns. The Journal of Finance, 69(2), pp.611-644.

Longarela, I.R., 2015. A characterization of the SSD-efficient frontier of portfolio weights by means of a set of mixed-integer linear constraints. Management Science, 62(12), pp.3549-3554.

Maillet, B., Tokpavi, S. and Vaucher, B., 2015. Global minimum variance portfolio optimisation under some model risk: A robust regression-based approach. European Journal of Operational Research, 244(1), pp.289-299.

Suen, S.C., 2015, October. A NON-ITERATIVE METHOD OF IDENTIFYING THE COST-EFFECTIVENESS FRONTIER USING NET MONETARY BENEFITS AND ITS GRAPHICAL INTERPRETATION. In 37th Annual Meeting of the Society for Medical Decision Making. Smdm.

Xing, X., Hu, J. and Yang, Y., 2014. Robust minimum variance portfolio with L-infinity constraints. Journal of Banking & Finance, 46, pp.107-117.

Yang, L., Couillet, R. and McKay, M.R., 2015. A robust statistics approach to minimum variance portfolio optimization. IEEE Transactions on Signal Processing, 63(24), pp.6684-6697.

Yilmaz, H. and Pearson, N.D., 2016. Maximum likelihood estimation of covariance matrices with constraints on the efficient frontier. International Journal of Computational Economics and Econometrics, 6(1), pp.71-92.

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My Assignment Help (2021) Plotting Investment Opportunity Set For Mazda Motor Corporation And Nissan Motor Co., Ltd. [Online]. Available from: https://myassignmenthelp.com/free-samples/fin3702b-investment-analysis-and-portfolio-management/mean-return.html
[Accessed 28 February 2024].

My Assignment Help. 'Plotting Investment Opportunity Set For Mazda Motor Corporation And Nissan Motor Co., Ltd.' (My Assignment Help, 2021) <https://myassignmenthelp.com/free-samples/fin3702b-investment-analysis-and-portfolio-management/mean-return.html> accessed 28 February 2024.

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