Question

Finance 4310 Class Assignment Investment State I Return (p=0.3) State II Return (p = 0.5) State III Return (p=0.2) 5% 11% 9% 8% -3% Given the above information on two investments A and B, calculate the following statistics: The correlation coefficient between A and B is 0.169. (Note that since the correlation is given, you do not have to do the long calculation for covariance, just use AB = P1003) A. Expected Return for A (Check answer: 8.8%) B. Standard Deviation for A (Check answer: C. Expected Return for B (Check answer: 5.2%) D. Standard Deviation for B (Check answer: 4.19% E. The expected return on a portfolio consisting of 60% A and 40% B. (Check answer: 7.36%) F. The standard deviation of a portfolio consisting of 60% A and 40% B. (Check answer: 2.47%) G. The covariance between A and B (Check answer: 1.84)

Answer 1

A. Expected Return is calculated as per the below formula :-

$\small \textrm{Expected Return =}\sum (X_{i}*P_{i})$

Probability	Returns	Probability * Return
0.3	5%	1.5
0.5	11%	5.5
0.2	9%	1.8
Total		8.8

Thus, Expected Return for A is 8.8%

B. Standard Deviation is calculated as per the following formula :-

$\small \textrm{Standard deviation =}\sqrt{\sum (X_{i}-\bar{X})^{2}*P_{i}}$

were, $\small \bar{X}$ = Expected Return, P_i = Probability

Probability	Returns	Probability * Return	$\small (X_{i}-\bar{X})$	$\small (X_{i}-\bar{X})^{2}$	$\small (X_{i}-\bar{X})^{2}*P_{i}$
0.3	5%	1.5	3.8	14.44	4.332
0.5	11%	5.5	-2.2	4.84	2.42
0.2	9%	1.8	-0.2	0.04	0.008
Total		8.8			6.76

$\small \textrm{Standard Deviation of A = }\sqrt{6.76}$ = 2.6

C. Expected Return of B is calculated as follows :-

Probability	Returns	Probability * Return
0.3	6%	1.8
0.5	8%	4
0.2	-3%	-0.6
Total		5.2

Thus, Expected Return of B is 5.2 %

D. Calculation of Standard Deviation of B :-

Probability	Returns	Probability * Return	$\small (X_{i}-\bar{X})$	$\small (X_{i}-\bar{X})^{2}$	$\small (X_{i}-\bar{X})^{2}*P_{i}$
0.3	6%	1.8	-0.8	0.64	0.192
0.5	8%	4	-2.8	7.84	3.92
0.2	-3%	-0.6	8.2	67.24	13.448
Total		5.2			17.56

$\small \textrm{Standard Deviation of B = }\sqrt{17.56}$ = 4.19

E. Expected Return of Portfolio consisting of 60% of A and 40% of B is

    $\small = W_{a}*\bar{X_{a}} + W_{b}*\bar{X_{b}}$

     = (60%*8.8%) + (40%*5.2%)

     = 7.36 %

F. Standard Deviation of Portfolio consisting of 60% of A and 40% of B is

$\small = \sqrt{(W_{a}^{2}*\sigma^{2}_{a})+(W_{b}^{2}*\sigma^{2}_{b})+(2*W_{a}*\sigma_{a}*W_{b}*\sigma_{b}*Correlation(a,b))}$

$\small = \sqrt{(0.6^{2}*2.6^{2}) + (0.4^{2}*4.19^{2}) +(2*0.6*2.6*0.4*4.19*0.169))}$

= 2.47.

G. Covariance is calculated as follows :-

$\small Covariance = Correlation(a,b)*\sigma _{a}*\sigma _{b}$

                         = 0.169 * 2.6 * 4.19

                         = 1.84.

=