Investor utility is represented by: U = E(R) - A2 . There are two investors with different risk aversion coefficients (A). Angela has a risk

Investor utility is represented by: U = E(R) - ½Aσ2 . There are two investors with different risk aversion coefficients (A). Angela has a risk aversion coefficient of 6 and Boris has a risk aversion coefficient of 2. Investors are able to short-sell each industry throughout the report. Investors are unable to borrow or lend at the risk-free rate except for part 5 of the report. Please use sample variance and standard deviation, rather than population variance and standard deviation. The expected returns per month to be used throughout the report are in the following table.

 

 

Date	COND	CONS	ENGY	HC	INDU	IT	MATL	TELE	UTIL	FIN	RE
31/01/2017	-0.0430	-0.0104	0.0072	0.0456	-0.0441	-0.0418	0.0477	-0.0168	0.0108	-0.0205	-0.0460
28/02/2017	0.0151	0.0602	-0.0213	0.0390	0.0341	0.0185	-0.0319	-0.0314	0.0307	0.0413	0.0433
31/03/2017	0.0473	0.0547	0.0433	0.0560	0.0435	0.0363	0.0022	0.0028	0.0628	0.0389	0.0095
28/04/2017	0.0070	-0.0253	-0.0055	0.0340	0.0409	0.0352	-0.0025	-0.0948	0.0308	0.0191	0.0249
31/05/2017	-0.0032	-0.0044	0.0152	-0.0238	0.0469	0.0022	-0.0024	0.0338	0.0100	-0.0771	-0.0075
30/06/2017	0.0087	-0.0243	-0.0684	0.0608	-0.0025	0.0188	0.0063	-0.0166	-0.0269	0.0167	-0.0413
31/07/2017	-0.0063	0.0097	0.0062	-0.0750	-0.0274	-0.0049	0.0348	-0.0421	-0.0528	0.0125	-0.0017
31/08/2017	-0.0154	0.0517	0.0518	0.0025	0.0457	0.0099	0.0452	-0.0720	0.0322	-0.0212	0.0137
29/09/2017	-0.0006	-0.0187	0.0112	0.0227	-0.0111	0.0060	-0.0138	-0.0446	-0.0364	0.0111	0.0119
31/10/2017	0.0544	0.0534	0.0641	0.0548	0.0418	0.0844	0.0444	0.0235	0.0496	0.0326	0.0132
30/11/2017	0.0087	0.0324	0.0433	0.0296	0.0221	0.0450	0.0201	-0.0158	0.0298	0.0000	0.0474
29/12/2017	0.0368	0.0167	0.0652	-0.0048	-0.0095	0.0294	0.0611	0.0546	-0.0448	0.0042	0.0038
31/01/2018	-0.0020	0.0021	-0.0028	0.0311	-0.0201	0.0249	0.0046	0.0077	-0.0435	-0.0074	-0.0323
28/02/2018	-0.0119	0.0210	-0.0375	0.0696	-0.0037	0.0166	0.0037	-0.0620	-0.0159	0.0064	-0.0187
30/03/2018	-0.0226	-0.0135	-0.0249	-0.0332	-0.0114	-0.0219	-0.0424	-0.0620	-0.0075	-0.0585	-0.0013
30/04/2018	0.0295	0.0574	0.1072	0.0719	0.0308	0.0276	0.0744	0.0194	0.0216	0.0011	0.0410
31/05/2018	0.0527	0.0000	0.0041	0.0545	0.0124	0.0203	0.0197	-0.1005	0.0103	-0.0006	0.0301
29/06/2018	0.0209	0.0594	0.0774	0.0250	0.0064	0.0590	0.0165	-0.0549	0.0591	0.0399	0.0256
31/07/2018	0.0175	-0.0088	0.0118	0.0223	0.0316	-0.0112	-0.0025	0.0759	-0.0143	0.0204	0.0099
31/08/2018	0.0359	0.0264	-0.0122	0.1042	0.0326	0.1221	-0.0486	0.1302	0.0061	0.0005	0.0261
28/09/2018	-0.0383	-0.0267	0.0427	-0.0730	-0.0215	-0.0065	0.0406	0.0266	-0.0310	-0.0222	-0.0186
31/10/2018	-0.0834	-0.0490	-0.1028	-0.0725	-0.0529	-0.1137	-0.0518	-0.0744	-0.0403	-0.0602	-0.0380
30/11/2018	-0.0453	-0.0292	-0.1068	-0.0397	-0.0060	0.0099	-0.0475	-0.0314	-0.0176	0.0141	-0.0259
31/12/2018	-0.0194	0.0120	-0.0220	0.0272	-0.0029	-0.0408	0.0513	-0.0505	0.0277	-0.0313	0.0107
31/01/2019	0.0428	0.0273	0.1146	0.0391	0.0338	0.0882	0.0697	0.0766	0.0592	-0.0025	0.0585
28/02/2019	0.0669	-0.0139	0.0784	0.0133	0.0644	0.0748	0.0640	0.0435	0.0403	0.0907	0.0219
29/03/2019	0.0139	0.0368	-0.0415	0.0131	0.0198	0.0255	0.0340	0.0377	0.0132	-0.0262	0.0541
30/04/2019	0.0546	0.0733	0.0155	0.0314	0.0315	0.0735	-0.0205	0.0244	-0.0049	0.0440	-0.0167
31/05/2019	0.0204	-0.0416	-0.0380	0.0354	0.0025	-0.0312	0.0311	0.0715	-0.0062	0.0262	0.0258
28/06/2019	-0.0151	0.0292	0.0217	0.0442	0.0555	0.0105	0.0617	0.0224	0.0308	0.0342	0.0324
31/07/2019	0.0475	0.0959	0.0149	0.0596	0.0335	0.0511	0.0117	0.0276	0.0185	0.0177	0.0319
30/08/2019	0.0064	0.0010	-0.0562	0.0343	-0.0250	0.0067	-0.0734	-0.0321	-0.0224	-0.0258	0.0240
30/09/2019	0.0333	0.0174	0.0446	-0.0215	-0.0013	0.0011	0.0295	-0.0285	0.0201	0.0423	-0.0229
31/10/2019	0.0055	-0.0224	0.0031	0.0727	0.0289	-0.0319	-0.0176	-0.0139	0.0123	-0.0288	0.0177
29/11/2019	0.0422	0.0810	0.0729	0.0881	0.0397	0.1055	0.0444	0.0722	-0.0053	-0.0201	0.0233
31/12/2019	-0.0221	-0.0775	-0.0138	-0.0266	-0.0305	-0.0389	0.0182	-0.0546	0.0078	-0.0156	-0.0448
31/01/2020	0.0427	0.0781	0.0057	0.1201	0.0186	0.1024	0.0163	0.0765	0.0098	0.0474	0.0600
28/02/2020	-0.0874	-0.0740	-0.1742	-0.0395	-0.0841	-0.1631	-0.1169	-0.0878	-0.0396	-0.0492	-0.0449
31/03/2020	-0.2642	-0.0355	-0.3764	-0.0564	-0.2301	-0.1758	-0.1325	-0.1536	-0.0674	-0.2770	-0.3558
30/04/2020	0.1644	0.0259	0.2521	0.0487	0.1282	0.2184	0.1437	0.0484	0.0323	0.0289	0.1412
29/05/2020	0.0690	-0.0049	0.0474	-0.0510	0.0381	0.1431	0.0839	0.0852	0.0305	0.0535	0.0706
30/06/2020	0.0506	0.0475	-0.0212	0.0339	-0.0157	0.0370	0.0217	-0.0032	0.0085	0.0434	-0.0145
31/07/2020	0.0288	0.0323	-0.0626	-0.0384	-0.0380	0.0484	0.0591	0.0350	-0.0023	-0.0102	-0.0010
31/08/2020	0.0971	-0.0031	0.0342	0.0402	0.0462	0.1515	0.0116	-0.0377	-0.0480	0.0135	0.0761
30/09/2020	-0.0242	-0.0658	-0.1074	0.0082	-0.0032	-0.0639	-0.0286	-0.0211	-0.0332	-0.0613	-0.0134
30/10/2020	0.0107	0.0456	-0.0097	0.0096	-0.0348	0.0858	-0.0105	-0.0073	-0.0149	0.0627	0.0029
30/11/2020	0.0843	-0.0066	0.2824	0.0289	0.1198	0.0414	0.0766	0.1435	0.0147	0.1612	0.1323
31/12/2020	0.0139	0.0221	-0.0069	-0.0469	-0.0218	0.0857	0.0877	-0.0068	-0.0538	-0.0050	0.0012
29/01/2021	0.0478	0.0202	0.0096	-0.0170	-0.0307	0.0020	-0.0100	0.0255	-0.0009	0.0226	-0.0428
26/02/2021	-0.0262	-0.0450	0.0250	-0.0282	-0.0184	-0.0802	0.0706	0.0038	-0.0802	0.0515	-0.0146
31/03/2021	0.0668	0.0274	-0.0022	0.0243	0.0344	-0.0267	-0.0307	0.0587	0.0681	0.0429	0.0556
30/04/2021	0.0344	-0.0237	-0.0468	0.0287	0.0415	0.0979	0.0748	0.0304	-0.0116	0.0320	0.0278
31/05/2021	0.0316	0.0237	-0.0154	0.0347	-0.0112	-0.0910	0.0179	0.0206	-0.0662	0.0565	0.0147
30/06/2021	0.0453	0.0543	0.0415	0.0214	0.0310	0.1236	0.0008	0.0563	0.0345	-0.0018	0.0481
30/07/2021	-0.0055	0.0150	-0.0241	0.0111	0.0409	-0.0636	0.0706	-0.0148	0.0160	-0.0142	0.0085
31/08/2021	0.0334	0.0679	-0.0376	0.0662	0.0252	0.1619	-0.0692	0.0550	0.0097	0.0492	0.0604
30/09/2021	-0.0042	-0.0358	0.1500	-0.0476	-0.0041	-0.0370	-0.0926	0.0128	0.0250	0.0155	-0.0215
29/10/2021	0.0060	-0.0231	-0.0189	0.0087	-0.0318	0.0243	0.0015	-0.0087	-0.0063	0.0083	0.0037
30/11/2021	-0.0147	0.0413	-0.0776	0.0148	0.0054	-0.0111	0.0598	0.0466	0.0388	-0.0690	0.0371
31/12/2021	0.0106	-0.0228	0.0218	-0.0221	0.0385	-0.0579	0.0618	0.0071	0.0792	0.0426	0.0495
31/01/2022	-0.0919	-0.0958	0.0752	-0.1213	-0.0764	-0.1709	0.0040	-0.0796	0.0256	-0.0660	-0.0945
28/02/2022	-0.0501	0.0536	0.0843	-0.0024	0.0077	-0.0684	0.0514	-0.0213	0.0341	0.0300	0.0175
31/03/2022	0.0385	0.0444	0.1008	0.0254	0.0473	0.1175	0.0913	0.0496	0.0759	0.0851	0.0168
29/04/2022	-0.0347	0.0336	0.0248	0.0233	0.0300	-0.0986	-0.0397	-0.0023	0.0933	0.0004	0.0099
31/05/2022	-0.0553	-0.0655	-0.0070	-0.0119	-0.0099	-0.0795	-0.0030	-0.0674	-0.0019	-0.0225	-0.0874
30/06/2022	-0.0746	0.0022	-0.0059	-0.0324	-0.0479	-0.1116	-0.1286	-0.0393	-0.0679	-0.1191	-0.1057

 

Question 4. Calculate the optimal portfolio for both investors that consists of all eleven industries. Compare this to the other portfolios in terms of diversification benefits. What do you observe? Contrast the differences in what you observe between the two investors. 

 

Question 5. Now consider the case where both Boris and Angela can invest in a risk-free asset. The riskfree rate is 0.5% per month. Compare the diversification benefit between the five industry optimal portfolio in part 3 and the eleven industry optimal portfolio in part 4. How does the existence of the risk-free rate affect your conclusion regarding diversification benefits? Are diversification benefits increased or reduced if the investors can borrow or lend at the riskfree rate? 

Industry Consumer Discretionary Consumer Staples Energy Health Care Industrials Information Technology Materials Telecom Utilities Financials Real Estate Abbreviation COND CONS ENGY HC INDU IT MATL TELE UTIL FIN RE Expected Return 0.75% 0.85% 0.78% 0.92% 0.72% 0.85% 0.91% 0.65% 0.69% 0.68% 0.70%