Chapter 24, Problem 22PS

a

Summary Introduction

To evaluate: The portfolio’s performance and plot the monthly values of alpha plus residual return.

Introduction:

Alpha: Symbolically, alpha is denoted as ‘a’. It is supposed to be the difference amount that is determined after deducting the actual return from the expected return. The selection of the portfolio is made when the alpha of a portfolio is the highest.

Treynor ratio: It is named after its creator - Jack Treynor, who is an American economist. Treynor ratio tries to explore the amount of return which is generated in excess. This also is calculated for risk per unit.

Answer to Problem 22PS

The consumer industry has a record of good performance than the manufacturing. Even then its performance can be judged as below the benchmark.

Explanation of Solution

The information given to us is as follows:

Month	Consumer	Manufacturing	Risk-free rate	Rm-RF	Market return	SMB	HML
2012-08	1.84	1.82	0.9	2.58	3.48	0.5	0.79
2012-09	1.71	2.44	0.7	2.69	3.39	0.72	1.36
2012-10	-0.78	-0.82	1.1	-1.7	-0.6	-1.31	3.43
2012-11	2.23	0.01	1.3	0.71	2.01	0.55	-1.4
2012-12	-0.87	1.8	0.4	1.09	1.49	1.53	3.56
2013-01	5.25	6.5	0.5	5.65	6.15	0.57	0.68
2013-02	1.67	1.09	0.8	1.27	2.07	0.07	-0.04
2013-03	4.78	2.88	0.8	4.07	4.87	0.54	0.08
2013-04	3.16	0.7	0.5	1.57	2.07	-2.2	-0.05
2013-05	1.12	1.51	0.2	2.64	2.84	2.3	0.91
2013-06	-0.07	-1.71	0.3	-1.12	-0.82	1.44	-0.48
2013-07	5.1	5.44	0.2	5.63	5.83	1.26	0.88
2013-08	-3.86	-2.29	0.4	-2.65	-2.25	0.06	-2.17
2013-09	3.74	3.4	0.2	3.74	3.94	2.39	-1.7
2013-10	4.52	4.44	1.1	4.23	5.33	-1.4	1.01
2013-11	2.66	1.52	0.5	3.13	3.63	1	0.24
2013-12	1.12	2.83	0.2	2.7	2.9	-0.57	-0.39
2014-01	-5.95	-4.26	0.2	-3.2	-3	0.83	-1.99
2014-02	4.92	5.13	0.5	4.57	5.07	0.27	-0.85
2014-03	0.68	1.39	0.5	0.39	0.89	-1.04	4.83
2014-04	-0.04	2.96	0.2	-0.13	0.07	-3.91	2.69
2014-05	1.79	0.96	0.3	2.17	2.47	-1.79	-0.37
2014-06	1.55	3	0.2	2.69	2.89	2.83	-0.67
2014-07	-3.26	-4.34	0.2	-2.08	-1.88	-3.93	-0.16
2014-08	5.33	3.97	0.3	4.21	4.51	0.69	-1.11
2014-09	-1.7	-4.55	0.1	-2	-1.9	-3.67	-1.72
2014-10	2.95	0.73	0.2	2.35	2.55	3.08	-0.45
2014-11	6.27	-2.03	0.4	2.41	2.81	-2.3	-3.73
2014-12	0.09	0.36	0.3	-0.22	0.08	2.24	1.15
2015-01	-0.76	-3.11	0.2	-2.95	-2.75	-0.73	-4.73
2015-02	5.58	3.83	0.2	6.11	6.31	1	-0.29
2015-03	-0.47	-1.9	0.2	-1.14	-0.94	2.38	-0.88
2015-04	-1.09	1.76	0.2	0.75	0.95	-2	4.17
2015-05	1.12	-1.15	0.1	1.28	1.38	0.29	-3.53
2015-06	-0.92	-3.06	0	-1.62	-1.62	2.68	-1.42
2015-07	4.18	-3.28	0.3	1.41	1.71	-4.16	-4.77

With the help of given information in the above table, let us calculate the average and standard for column label.

The average has to be calculated using the above formula in each of column.

  $Average=\frac{Sumofterms}{Numberofterms}$

The formula used is to calculate standard deviation is as follows:

  $SD=\sqrt{\frac{{\displaystyle \sum {\left|x-\overline{x}\right|}^2}}{n}}$

Where

S=Sample standard deviation

n=The number of observations

x= The observed mean values of a sample item$\overline{x}$ =The mean of x

The total depicts the sum of the values of 36 months. The formula of average and standard deviation is used to calculate the values for the above said columns.

Particulars	Consumer	Manufacturing	Risk-free rate	Rm-RF	Market return	SMB	HML
Total	53.59	27.97	14.7	51.23	65.93	0.21	-7.12
Average	1.4886	0.7769	0.4083	1.4232	1.8314	0.0058	-0.1978
Standard deviation	2.834	2.900	0.307	2.537	9.918	1.984	2.18

Having the values of average and standard deviation, let us now calculate the Beta, Sharpe ratio, Treynor ratio and Jenson ratio. The formulas to be used are as follows:

  $Beta=\frac{Covariance}{Variance}$

?where:

Covariance=Measure of a stock’s return relative to that of the market Variance=Measure of how the market moves relativeto its mean?

Sharpe ratio:

  $S=\frac{R_p-R_f}{{\sigma }_p}$

Where

S= Sharpe ratio

R_p= Return earned from portfolio

R_f= Risk-free rate of return

s_p=Standard deviation of the portfolio’s return

Treynor ratio:

  $T=\frac{R_p-R_f}{{\beta }_p}$

Where

T= Treynor ratio

R_p= Portfolio return

r_f= Risk free rate of interest

ß_p= Portfolio beta

By using the above said formula, we get the following values.

Particulars	Consumer	Manufacturing	Market return
Beta	0.9711	0.9684	1.0000
Sharpe ratio	0.3811	0.1271	0.5644
Treynor Ratio	-0.3017	-1.0095	1.4231

From the above calculation, it is clear that the consumer industry has a record of good performance than the manufacturing. Even then its performance can be judged as below the benchmark.

b.

Summary Introduction

To compute: The alpha plus residual returns using the Fama-French three factor model.

Introduction:

Alpha: Symbolically, alpha is denoted as ‘a’. It is supposed to be the difference amount that is got from deducting the actual return from the expected return. The selection of the portfolio is made when its alpha is the highest

Treynor ratio: It is named after its creator’s name Jack Treynor who is an American economist. Treynor ratio tries to explore the amount of return which is generated in excess. This also is calculated for risk per unit.

Answer to Problem 22PS

The consumer industry has a record of good performance than the manufacturing. Even then its performance can be judged as below the benchmark.

Explanation of Solution

The information given to us is as follows:

Month	Consumer	Manufacturing	Risk-free rate	Rm-RF	Market return	SMB	HML
2012-08	1.84	1.82	0.9	2.58	3.48	0.5	0.79
2012-09	1.71	2.44	0.7	2.69	3.39	0.72	1.36
2012-10	-0.78	-0.82	1.1	-1.7	-0.6	-1.31	3.43
2012-11	2.23	0.01	1.3	0.71	2.01	0.55	-1.4
2012-12	-0.87	1.8	0.4	1.09	1.49	1.53	3.56
2013-01	5.25	6.5	0.5	5.65	6.15	0.57	0.68
2013-02	1.67	1.09	0.8	1.27	2.07	0.07	-0.04
2013-03	4.78	2.88	0.8	4.07	4.87	0.54	0.08
2013-04	3.16	0.7	0.5	1.57	2.07	-2.2	-0.05
2013-05	1.12	1.51	0.2	2.64	2.84	2.3	0.91
2013-06	-0.07	-1.71	0.3	-1.12	-0.82	1.44	-0.48
2013-07	5.1	5.44	0.2	5.63	5.83	1.26	0.88
2013-08	-3.86	-2.29	0.4	-2.65	-2.25	0.06	-2.17
2013-09	3.74	3.4	0.2	3.74	3.94	2.39	-1.7
2013-10	4.52	4.44	1.1	4.23	5.33	-1.4	1.01
2013-11	2.66	1.52	0.5	3.13	3.63	1	0.24
2013-12	1.12	2.83	0.2	2.7	2.9	-0.57	-0.39
2014-01	-5.95	-4.26	0.2	-3.2	-3	0.83	-1.99
2014-02	4.92	5.13	0.5	4.57	5.07	0.27	-0.85
2014-03	0.68	1.39	0.5	0.39	0.89	-1.04	4.83
2014-04	-0.04	2.96	0.2	-0.13	0.07	-3.91	2.69
2014-05	1.79	0.96	0.3	2.17	2.47	-1.79	-0.37
2014-06	1.55	3	0.2	2.69	2.89	2.83	-0.67
2014-07	-3.26	-4.34	0.2	-2.08	-1.88	-3.93	-0.16
2014-08	5.33	3.97	0.3	4.21	4.51	0.69	-1.11
2014-09	-1.7	-4.55	0.1	-2	-1.9	-3.67	-1.72
2014-10	2.95	0.73	0.2	2.35	2.55	3.08	-0.45
2014-11	6.27	-2.03	0.4	2.41	2.81	-2.3	-3.73
2014-12	0.09	0.36	0.3	-0.22	0.08	2.24	1.15
2015-01	-0.76	-3.11	0.2	-2.95	-2.75	-0.73	-4.73
2015-02	5.58	3.83	0.2	6.11	6.31	1	-0.29
2015-03	-0.47	-1.9	0.2	-1.14	-0.94	2.38	-0.88
2015-04	-1.09	1.76	0.2	0.75	0.95	-2	4.17
2015-05	1.12	-1.15	0.1	1.28	1.38	0.29	-3.53
2015-06	-0.92	-3.06	0	-1.62	-1.62	2.68	-1.42
2015-07	4.18	-3.28	0.3	1.41	1.71	-4.16	-4.77

The Fama-French three factor model suggests the usage of market risk, size of the company and company’s book to market value to calculate the returns. The formula to be used to calculate the required returns is Jenson’s Alpha ratio.

Jensen's Alpha Ratio:

  $Alphaoftheportfolio=r_i-\left[r_f+\beta (r_m-r_f)\right]$

By using the above said formula, we get the following values.

Particulars	Consumer	Manufacturing	Market return
Jenson’s Alpha Ratio	1.3377	0.6300	1.6394

From the above calculation, it is clear that the consumer industry has a record of good performance than the manufacturing. Even then its performance can be judged as below the benchmark.