Question

Problem 4-07 Given the monthly returns that follow, find the R2, alpha, and beta of the portfolio. Compute the average return differential with and without sign. Do not round intermediate calculations. Round your answers to two decimal places. Portfolio Return 5.8% S&P 500 Return 6.2% -2.3 -2.7 Month January February March April May -1.8 -0.9 2.6 2.0 0.4 -0.2 June -0.8 -0.3 0.4 0.7 1.4 1.1 -0.4 -0.1 July August September October November December -3.1 -3.4 1.7 2.6 0.6 0.3 R2: Alpha: % Beta: Average return difference (with signs): % Average return difference (without signs) %

Answer 1

solution

STEP 1 given that

month	portfolio return	s&p 500 return
jan	5.8%	6.2 %
feb	- 2.3 %	-2.7 %
march	- 1.8 %	-0.9 %
april	2.6 %	2.0 %
may	0.4 %	-0.2 %
june	-0.8 %	-0.3 %
july	0.4 %	0.7 %
aug	1.1 %	1.4 %
sept	-0.4 %	-0.1 %
oct	-3.1 %	-3.4 %
nov	2.6 %	1.7%
dec	0.6 %	0.3%
E(Ri)	0.425	0.39

Step 1 Calculate the Variance of portfolio and S&P

Month	portfolio Return	Ri - E(Ri)	s&p 500 return	Rm -E(Rm)	[(Ri - E(Ri) x(Rm -E(Rm))]
jan	5.8%	5.375	6.2 %	5.81	0.3123%
feb	- 2.3 %	-2.725	-2.7 %	-3.09	0.0842%
march	- 1.8 %	-2.225	-0.9 %	-1.29	0.0287%
april	2.6 %	2.175	2.0 %	1.61	0.04%
may	0.4 %	-0.025	-0.2 %	-0.59	0.0001%
june	-0.8 %	-1.225	-0.3 %	-0.69	0.0085%
july	0.4 %	-0.025	0.7 %	0.31	-0.0000775 %
aug	1.1 %	0.675	1.4 %	1.01	0.0068%
sept	-0.4 %	-0.825	-0.1 %	-0.49	0.0040%
oct	-3.1 %	-3.525	-3.4 %	-3.79	0.1336%
nov	2.6 %	2.175	1.7%	1.31	0.0285%
dec	0.6 %	0.175	0.3%	-0.09	-0.0001575
Average Retyurn					0.6465%

Variance of Portfolio = [(5.375)² +(-2.725)² +(-2.225)²+(2.175)²+(-0.025)²+(-1.225)²+ (-0.025)²+(0.675)²+(-0.825)²+(-3.525)²+(2.175)²+(0.175)²] /12

= 5.4852 %

Variance of s&p 500 return = [(5.81)²+(-3.09)²+(-1.29)²+(1.61)²+(-0.59)²+(-0.69)²+(0.31)²+ (1.01)²+ (-0.49)²+(-3.79)²+(1.31)²+(-0.09)²] /12

= 5.4858 %

Standard Deviation of Portfolio = it is root squre of variance

  $\sigma$_p =$\sqrt{}$ 5.4852

=2.3421

Standard Deviation of s&p 500 return = it is root squre of variance

$\sigma$_m =$\sqrt{}$5.4858

=2.3422

Step 2 : Calculation of CoVariance

COV_pm = 64.65 /12

= 5.3875

Calculation value of R :

R_pm = _{$\frac{COVpm}{\sigma p *\sigma m}$}

= 5.3875 / (2.3421) x(2.3422)

= 5.3875 /5.4857

= 0.9821

STEP 4 value of R² is caluclated as :

R² =(0.9821)²

  ⁼ 0.9645

Therefore the value of R^{2 =}  0.9645

now we calcualted value of Beta

$\beta = \frac{COVpm}{varia. m}$

= 5.3875/5.4858

= 0.9821

Therefore the value of beta is 0.9821

now we calcualted value of alpha

Alpha =E(Rp ⁾ - $\beta$ [E(R_m)]

_{=0.425 -}0.9821( 0.39)

= 0.041981

Therefore the value of alpha = 0.041981

STEP 5 ; computation of Average return differential with sign

month	portfolio return	s&p 500 return	Return Differential
jan	5.8%	6.2 %	-0.4
feb	- 2.3 %	-2.7 %	0.4
march	- 1.8 %	-0.9 %	-0.9
april	2.6 %	2.0 %	0.6
may	0.4 %	-0.2 %	0.6
june	-0.8 %	-0.3 %	-0.5
july	0.4 %	0.7 %	-0.3
aug	1.1 %	1.4 %	-0.3
sept	-0.4 %	-0.1 %	-0.3
oct	-3.1 %	-3.4 %	0.3
nov	2.6 %	1.7%	0.9
dec	0.6 %	0.3%	0.3
Total	0.4

Average = 0.4 /12

= 0.0333 %

step 6 : computation of Average return differential without  sign

month	portfolio return	s&p 500 return	Return Differential
jan	5.8%	6.2 %	-0.4
feb	2.3 %	2.7 %	-0.4
march	1.8 %	0.9 %	0.9
april	2.6 %	2.0 %	0.6
may	0.4 %	0.2 %	0.2
june	0.8 %	0.3 %	0.5
july	0.4 %	0.7 %	-0.3
aug	1.1 %	1.4 %	-0.3
sept	0.4 %	0.1 %	0.3
oct	3.1 %	3.4 %	-0.3
nov	2.6 %	1.7%	0.9
dec	0.6 %	0.3%	0.3
Total	2

Average =2/12

=0.167 %

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