You are hired as an investment consultant by a hedge fund that focuses on the following five US stocks: Exxon Mobile (ticker XOM), Nike (ticker NKE), Amazon (ticker AMZN), Goldman Sachs (ticker GS), and Visa (ticker V). Your task is to submit a professional report for your manager to address the following investment issues. Your analysis is based on implementing the Markowitz approach and monthly returns of the five stocks over the sample period from January 2010 to December 2022, as contained below

1. Discuss the risk return characteristics of the five stocks. Generate a table that reports mean, standard deviation, skewness, Sharpe ratio, and variance and covariance matrix of these five stocks. You also need to prepare a cumulative return plot. Additional Information Express returns, standard deviations, and Sharpe ratios in annual term.

2. Plot and describe the investment opportunity set and efficient frontier of the five stocks with and without the risk-free asset. Clearly identify the weight of each stock in the tangency portfolio as well as the expected return and standard deviation of the tangency portfolio. Assume the risk-free rate is 2% per annum.

3. Investor Jennifer wants to invest only in Amazon because many stock analysts recommended it. You explain to the investor that she will be better off by creating portfolios of the five stocks rather than investing in only Amazon

4. There are two individual investors, Tom and Amy who are interested in investing in your fund. Both investors' utility is represented by: = () 1 2 2 . Tom has a risk aversion coefficient () of 8 and Amy has a risk aversion coefficient of 12. Suppose each investor has $2 million USD to invest, calculate the portfolio allocation (in dollar amount) to the five stocks and risk-free asset you would recommend to the two investors to maximise their utilities, and summarise the expected returns, standard deviations, and Sharpe ratios of these recommended portfolios. Are these portfolios still attainable if investors are not allowed to take short positions?

Additional Information Express returns, standard deviations, and Sharpe ratios in annual term.

Date	XOM	NKE	AMZN	GS	V
29/01/2010	-0.05514008	-0.03511427	-0.06772228	-0.11916605	-0.06208553
26/02/2010	0.01536551	0.06039213	-0.05589667	0.05365791	0.04114348
31/03/2010	0.03046159	0.09127221	0.1467061	0.09133354	0.06742495
30/04/2010	0.01179446	0.03278917	0.00979599	-0.14903598	-0.00878826
28/05/2010	-0.10137226	-0.04650252	-0.08490158	-0.00406338	-0.19555584
30/06/2010	-0.05607012	-0.06300075	-0.12912479	-0.09004568	-0.02359921
30/07/2010	0.04573332	0.09015539	0.07898588	0.1489297	0.03674909
31/08/2010	-0.00217828	-0.04942965	0.05886846	-0.08977599	-0.05787314
30/09/2010	0.0453392	0.14871427	0.25819111	0.05586803	0.0765439
29/10/2010	0.07606404	0.01622165	0.05201833	0.11446952	0.0525182
30/11/2010	0.05278989	0.05758834	0.06155056	-0.02879666	-0.05322422
31/12/2010	0.05117891	-0.00464413	0.02622581	0.07698222	-0.04698715
31/01/2011	0.10339165	-0.03441811	-0.05755556	-0.02699815	-0.00753053
28/02/2011	0.06556766	0.07941313	0.02151612	0.003117	0.04795998
31/03/2011	-0.01636854	-0.14624286	0.03947147	-0.03162775	0.00780287
29/04/2011	0.04576258	0.0874505	0.0870482	-0.04785631	0.06112469
31/05/2011	-0.04591955	0.0258746	0.00449418	-0.06575723	0.03955447
30/06/2011	-0.02503898	0.06915342	0.03965633	-0.05428835	0.03947699
29/07/2011	-0.01953793	0.00188929	0.08817057	0.01412582	0.01519106
31/08/2011	-0.06642441	-0.03882418	-0.03276114	-0.1363266	0.02910915
30/09/2011	-0.0187787	-0.00957876	0.00464619	-0.1864567	-0.02457893
31/10/2011	0.07517554	0.12676875	-0.0125792	0.15864621	0.08796081
30/11/2011	0.03611226	-0.00176438	-0.09938645	-0.1217709	0.04214024
30/12/2011	0.05370462	0.00571847	-0.09979711	-0.05664511	0.04702483
31/01/2012	-0.01203403	0.0790702	0.12328131	0.23266616	-0.00876588
29/02/2012	0.0385718	0.0377921	-0.07585888	0.0360635	0.15848573
30/03/2012	0.002659	0.00815423	0.12699644	0.08016331	0.01400702
30/04/2012	-0.00449679	0.0316304	0.14512864	-0.07413364	0.04220342
31/05/2012	-0.08269631	-0.02976671	-0.08188871	-0.16491537	-0.06147346
29/06/2012	0.08826151	-0.18857461	0.07251891	0.00167193	0.07317708
31/07/2012	0.0149585	0.06345408	0.02167724	0.05257668	0.04400235
31/08/2012	0.01174444	0.04681309	0.06416631	0.05232903	-0.0046487
28/09/2012	0.04753716	-0.02516431	0.02436864	0.07529322	0.04701753
31/10/2012	-0.00306177	-0.0371932	-0.08425177	0.07661857	0.03336309
30/11/2012	-0.02698255	0.06675428	0.08225622	-0.03349946	0.08129153
31/12/2012	-0.01803944	0.06298721	-0.00468164	0.08294419	0.01249078
31/01/2013	0.03951471	0.04748064	0.05831708	0.15914083	0.04176014
28/02/2013	0.00166724	0.01147086	-0.00463281	0.01623153	0.00671266
28/03/2013	0.00625346	0.08354755	0.0084005	-0.01742789	0.07060008
30/04/2013	-0.01242928	0.07778343	-0.0475815	-0.00733936	-0.00812523
31/05/2013	0.02371057	-0.02720124	0.06063597	0.11302796	0.05942059
28/06/2013	-0.00132644	0.03276031	0.03153785	-0.06681886	0.02587853
31/07/2013	0.03763145	-0.01193471	0.08473477	0.08449586	-0.03140906
30/08/2013	-0.06357329	0.00174828	-0.06719338	-0.06949945	-0.01276762
30/09/2013	-0.01284996	0.15631963	0.11267707	0.03996583	0.09563128
31/10/2013	0.04160858	0.04295158	0.16437431	0.01674985	0.029147
29/11/2013	0.05010043	0.04461453	0.0812845	0.05364915	0.03655875
31/12/2013	0.08258444	-0.00328532	0.01313453	0.04924821	0.09446567
31/01/2014	-0.089328	-0.07362667	-0.10055419	-0.0741284	-0.03255793
28/02/2014	0.05143221	0.07810576	0.00950683	0.01754815	0.05064295
31/03/2014	0.01464635	-0.05670501	-0.07107157	-0.01562025	-0.04461362
30/04/2014	0.04842346	-0.01232066	-0.09583335	-0.02459566	-0.06138238
30/05/2014	-0.01162001	0.05757378	0.02768547	0.00337874	0.06228716
30/06/2014	0.00149211	0.00832141	0.03912978	0.04774423	-0.01917793
31/07/2014	-0.01728246	-0.00541594	-0.03630152	0.03242948	0.0014237
29/08/2014	0.0122296	0.02152218	0.08322956	0.0392781	0.00905175
30/09/2014	-0.05439369	0.13558234	-0.04896179	0.02490094	0.00399958
31/10/2014	0.02828279	0.04226462	-0.05266099	0.03497302	0.13150863
28/11/2014	-0.05666424	0.06797891	0.10862314	-0.00515818	0.07140791
31/12/2014	0.0210956	-0.02880451	-0.08354007	0.02876704	0.01553124
30/01/2015	-0.05440778	-0.04056164	0.14235538	-0.1105092	-0.02780323
27/02/2015	0.02070468	0.05582659	0.07229291	0.10428623	0.06621943
31/03/2015	-0.03998194	0.03305189	-0.02120159	-0.0095895	-0.03564182
30/04/2015	0.02788239	-0.01485105	0.13351248	0.04495396	0.00978443
29/05/2015	-0.0164817	0.03146501	0.01766327	0.05304961	0.04163508
30/06/2015	-0.02347418	0.06245695	0.01132257	0.01260968	-0.02227721
31/07/2015	-0.04795671	0.06665437	0.23511261	-0.01781688	0.12196567
31/08/2015	-0.04090394	-0.03011631	-0.0433834	-0.07714439	-0.05203071
30/09/2015	-0.01182881	0.10290829	-0.00194974	-0.0786851	-0.02300139
30/10/2015	0.11284465	0.06554442	0.22272365	0.07907462	0.11369506
30/11/2015	-0.00423005	0.0095398	0.06215044	0.01690669	0.02023718
31/12/2015	-0.04543236	-0.05261565	0.01668175	-0.05152094	-0.01847866
29/01/2016	-0.00128285	-0.00784003	-0.1315155	-0.10358985	-0.03945848
29/02/2016	0.03892104	-0.00677307	-0.05873932	-0.07043822	-0.02631224
31/03/2016	0.04291946	0.00064947	0.07442264	0.04982276	0.05649957
29/04/2016	0.05754283	-0.04115833	0.11109428	0.04541983	0.00993717
31/05/2016	0.01549768	-0.06311499	0.09581702	-0.02425206	0.02382191
30/06/2016	0.05302181	0.0025353	-0.00991987	-0.06834709	-0.06042569
29/07/2016	-0.05109879	0.00543477	0.06035326	0.06885177	0.05231232
31/08/2016	-0.01191678	0.03855855	0.01363975	0.07115427	0.03830875
30/09/2016	0.0016066	-0.08379594	0.08860318	-0.04833	0.02224963
31/10/2016	-0.04537121	-0.0469136	-0.05671733	0.10522726	-0.0022974
30/11/2016	0.05676913	-0.00219212	-0.04969487	0.23395415	-0.06090173
30/12/2016	0.03390606	0.01877376	-0.00093264	0.09193308	0.00905325
31/01/2017	-0.07057393	0.04072398	0.09816366	-0.04230524	0.06011283
28/02/2017	-0.02169507	0.08052927	0.02618157	0.08455429	0.06522795
31/03/2017	0.00848503	-0.02186844	0.0491101	-0.07393371	0.0105754
28/04/2017	-0.00438972	-0.00574196	0.04337087	-0.02577049	0.0264431
31/05/2017	-0.00465403	-0.04367439	0.07527649	-0.052681	0.0457685
30/06/2017	0.00285718	0.11681446	-0.02676398	0.05036448	-0.01522634
31/07/2017	-0.00854704	0.00084744	0.02043391	0.01545745	0.0616336
31/08/2017	-0.03673162	-0.10262486	-0.00726888	-0.00372785	0.04143229
29/09/2017	0.07402072	-0.01817843	-0.01963084	0.0601144	0.01661516
31/10/2017	0.01671133	0.06055937	0.1497166	0.02230277	0.04503996
30/11/2017	0.00851833	0.09874516	0.06466232	0.02437316	0.02550457
29/12/2017	0.00420217	0.03856341	-0.00618656	0.02875139	0.01270095
31/01/2018	0.04375901	0.09064751	0.24063897	0.05153878	0.08954576
28/02/2018	-0.12359685	-0.0174436	0.04242909	-0.01571549	-0.00869356
29/03/2018	-0.01491943	-0.00581824	-0.04304935	-0.04210243	-0.02700504
30/04/2018	0.0420855	0.02934974	0.08207473	-0.05372031	0.06069215
31/05/2018	0.05543406	0.04986114	0.04053941	-0.04888179	0.03191996
29/06/2018	0.01834079	0.11253478	0.04306529	-0.02350805	0.01323436
31/07/2018	-0.01474678	-0.03476401	0.0456759	0.07643825	0.03238964
31/08/2018	-0.00637963	0.07138205	0.13236454	0.00496991	0.07576418
28/09/2018	0.06049643	0.03065699	-0.00482432	-0.05706233	0.02178499
31/10/2018	-0.06280871	-0.11425874	-0.20219171	0.0050392	-0.08155101
30/11/2018	0.00803212	0.00399789	0.05767175	-0.15033054	0.02981502
31/12/2018	-0.14226411	-0.01304584	-0.11134978	-0.12397084	-0.06894364
31/01/2019	0.07464432	0.10439706	0.14431714	0.18533368	0.02326809
28/02/2019	0.08965611	0.0470201	-0.045906	-0.00257562	0.09895564
29/03/2019	0.02239661	-0.01516394	0.08593576	-0.02394505	0.0544829
30/04/2019	-0.0064357	0.0429878	0.08185878	0.07255583	0.05275619
31/05/2019	-0.10762335	-0.11920758	-0.07861329	-0.1096542	-0.01733255
28/06/2019	0.08280346	0.08828101	0.06679177	0.12115733	0.07574537
31/07/2019	-0.02962282	0.02477667	-0.0141791	0.0759042	0.02564101
30/08/2019	-0.06737489	-0.01522723	-0.04847383	-0.06800527	0.01724723
30/09/2019	0.03110393	0.11147927	-0.02273278	0.01628165	-0.04872255
31/10/2019	-0.0430534	-0.04652891	0.02347472	0.02967722	0.0398233
29/11/2019	0.0211632	0.04673361	0.0135873	0.0432093	0.03326621
31/12/2019	0.02421843	0.08364531	0.02612168	0.03876208	0.01837298
31/01/2020	-0.10977358	-0.04945212	0.08706382	0.03401038	0.05891436
28/02/2020	-0.15792017	-0.0693147	-0.06221373	-0.15028389	-0.08498772
31/03/2020	-0.26185843	-0.07428955	0.03502056	-0.23001447	-0.11355634
30/04/2020	0.22386093	0.05366211	0.26890016	0.18649332	0.10923539
29/05/2020	-0.0027975	0.13357423	-0.01278492	0.07807223	0.09411372
30/06/2020	-0.01649439	-0.00537633	0.12956674	0.00575088	-0.01060237
31/07/2020	-0.05903398	-0.00448753	0.14711356	0.00172053	-0.01433972
31/08/2020	-0.03018068	0.14880647	0.09046097	0.04122039	0.11496855
30/09/2020	-0.14046061	0.12199483	-0.08757853	-0.01903643	-0.05670081
30/10/2020	-0.04981074	-0.0434921	-0.03575412	-0.05936213	-0.09131367
30/11/2020	0.19558561	0.12175212	0.04343993	0.21974191	0.15937482
31/12/2020	0.08103856	0.05230144	0.02805832	0.14910221	0.03983831
29/01/2021	0.08782142	-0.05570089	-0.01557601	0.02828874	-0.11649063
26/02/2021	0.23193574	0.01096641	-0.03532843	0.1781539	0.10069858
31/03/2021	0.0268531	-0.01402285	0.00037186	0.02745082	-0.00310751
30/04/2021	0.02525523	-0.00203179	0.12066263	0.06559636	0.10310302
28/05/2021	0.03494055	0.03102859	-0.07047022	0.07122966	-0.02543241
30/06/2021	0.08069219	0.13212661	0.06735498	0.02018711	0.02868458
30/07/2021	-0.08734943	0.08427723	-0.03272226	-0.01225198	0.05375927
31/08/2021	-0.03786693	-0.01489457	0.04303413	0.10838136	-0.06887452
30/09/2021	0.07887013	-0.11842909	-0.05351808	-0.08580206	-0.02771718
29/10/2021	0.09605579	0.15189698	0.02660238	0.09343176	-0.04929291
30/11/2021	-0.0581666	0.01165648	0.03992378	-0.07828721	-0.08322708
31/12/2021	0.02256013	-0.0133834	-0.04925193	0.00934407	0.11838779
31/01/2022	0.24137932	-0.11159772	-0.10282998	-0.07285322	0.04365277
28/02/2022	0.0439705	-0.07780106	0.02667252	-0.0377523	-0.04277757
31/03/2022	0.05317519	-0.01233984	0.06143731	-0.02692725	0.02614293
29/04/2022	0.03220734	-0.07327587	-0.23752514	-0.07455321	-0.03895928
31/05/2022	0.13642229	-0.04691258	-0.03276431	0.07646737	-0.00274484
30/06/2022	-0.10791668	-0.1375263	-0.11645917	-0.09126516	-0.07201771
29/07/2022	0.13183093	0.12446185	0.27059597	0.12244976	0.07730205
31/08/2022	-0.00474573	-0.07370346	-0.06061505	0.00533909	-0.06140679
30/09/2022	-0.08661993	-0.21629868	-0.10862189	-0.11909708	-0.10598365
31/10/2022	0.26915589	0.1150144	-0.09345131	0.17560151	0.1661132
30/11/2022	0.01299521	0.18353477	-0.0575947	0.12812395	0.04967173
30/12/2022	-0.0093407	0.06983316	-0.12989435	-0.11076004	-0.04258067