Question

please do it step by step with the use of formulas , thank you..

The following are monthly percentage price changes for four market indexes.

Month DJ SP500

1 0.03 0.02 2 0.07 0.06 3 −0.02 −0.01 4 0.01 0.03 5 0.05 0.04 6 −0.06 −0.04

Russ500 Nicky

0.04 0.04 0.10 −0.02 −0.04 0.07 0.03 0.02 0.11 0.02 −0.08 0.06

(17 points) Compute the following.

a. Average monthly rate of return for each index

b. Standard deviation for each index

c. Covariance between the rates of return for the following indexes:

DJ– SP 500 SP500–Russ500 SP500–Nicky Russ500–Nicky

d. The correlation coefficients for the same four combinations

Answer 1

Month	DJ	SP500	Russ500	Nicky
1	0.03	0.02	0.04	0.04
2	0.07	0.06	0.10	-0.02
3	-0.02	-0.01	-0.04	0.07
4	0.01	0.03	0.03	0.02
5	0.05	0.04	0.11	0.02
6	-0.06	0.04	-0.08	0.06

a) Average monthly rate of return for each index

Average monthly rate of return of DJ = (0.03+0.07-0.02+0.01+0.05-0.06)/6 = 0.0133

Average monthly rate of return of SP500 = (0.02+0.06-0.01+0.03+0.04+0.04)/6 = 0.0300

Average monthly rate of return of Russ500 = (0.04+0.10-0.04+0.03+0.11-0.08)/6 = 0.0267

Average monthly rate of return of DJ = (0.04-0.02+0.07+0.02+0.02+0.06)/6 = 0.0317

b) Standard deviation for each index

As the distribution is Sample, Stock’s Standard Deviation = √{1/(n-1)*∑(R_i-R_mean)²}

where, R_i is expected return of index i

                R_mean is arithmetic mean of return

                n is number of observation

Std dev of DJ:

Expected Return (%)R(i)	R(mean)	R(i)-R(mean)	{R(i)-R(mean)}2
0.03	0.0133	0.0167	0.00028
0.07	0.0133	0.0567	0.00321
-0.02	0.0133	-0.0333	0.00111
0.01	0.0133	-0.0033	0.00001
0.05	0.0133	0.0367	0.00134
-0.06	0.0133	-0.0733	0.00538

Stock’s Standard Deviation = Square root of{sum [{R(i)-R(mean)}²] / (n-1)} = Square root (0.0113/5) Stock’s Standard Deviation of DJ = 0.0476

Std dev of SP500:

Expected Return (%)R(i)	R(mean)	R(i)-R(mean)	{R(i)-R(mean)}2
0.02	0.0300	-0.0100	0.00010
0.06	0.0300	0.0300	0.00090
-0.01	0.0300	-0.0400	0.00160
0.03	0.0300	0.0000	0.00000
0.04	0.0300	0.0100	0.00010
0.04	0.0300	0.0100	0.00010

Stock’s Standard Deviation = Square root of sum [{R(i)-R(mean)}²] / (n-1) = Square root of (0.0028)/5 Stock’s Standard Deviation of SP500 = 0.0237

Std dev of RUSS500:

Expected Return (%)R(i)	R(mean)	R(i)-R(mean)	{R(i)-R(mean)}2
0.04	0.0267	0.0133	0.00018
0.10	0.0267	0.0733	0.00538
-0.04	0.0267	-0.0667	0.00444
0.03	0.0267	0.0033	0.00001
0.11	0.0267	0.0833	0.00694
-0.08	0.0267	-0.1067	0.01138

Stock’s Standard Deviation = Square root of sum [{R(i)-R(mean)}²] / (n-1) = Square root of (0.0283)/5 Stock’s Standard Deviation of Russ500 = 0.0753

Std dev of Nicky:

Expected Return (%)R(i)	R(mean)	R(i)-R(mean)	{R(i)-R(mean)}2
0.04	0.0317	0.0083	0.00007
-0.02	0.0317	-0.0517	0.00267
0.07	0.0317	0.0383	0.00147
0.02	0.0317	-0.0117	0.00014
0.02	0.0317	-0.0117	0.00014
0.06	0.0317	0.0283	0.00080

Stock’s Standard Deviation = Square root of sum [{R(i)-R(mean)}²] / (n-1) = Square root of (0.0053)/5 Stock’s Standard Deviation of Nicky = 0.0325

c) Covariance between the rates of return for the following indexes:

Cov (i,j) = √[1/(n-1)*∑{(X_i-X_mean)* (Y_j-Y_mean)}]

where, X_i is expected return of index i

               X_mean is arithmetic mean of return of index i

Y_i is expected return of index j

               Y_mean is arithmetic mean of return of index j

               n is number of observation

DJ– SP 500

Expected Return (%) X(i)	X(mean)	X(i) - X(mean)	Expected Return (%) Y(j)	Y(mean)	Y(j) - Y(mean)	{X(i) - X(mean)}*{Y(j) - Y(mean)}
0.03	0.0133	0.01667	0.02	0.0300	-0.01000	-0.000167
0.07	0.0133	0.05667	0.06	0.0300	0.03000	0.001700
-0.02	0.0133	-0.03333	-0.01	0.0300	-0.04000	0.001333
0.01	0.0133	-0.00333	0.03	0.0300	0.00000	0.000000
0.05	0.0133	0.03667	0.04	0.0300	0.01000	0.000367
-0.06	0.0133	-0.07333	0.04	0.0300	0.01000	-0.000733

Cov(DJ, SP500) = sum [{X(i)-X(mean)}*{Y(j)-Y(mean)}] / (n-1) = (0.0025)/5

Cov(DJ, SP500) = 0.0005

SP500–Russ500

Expected Return (%) X(i)	X(mean)	X(i) - X(mean)	Expected Return (%) Y(j)	Y(mean)	Y(j) - Y(mean)	{X(i) - X(mean)}*{Y(j) - Y(mean)}
0.02	0.0300	-0.01000	0.04	0.0267	0.01333	-0.000133
0.06	0.0300	0.03000	0.10	0.0267	0.07333	0.002200
-0.01	0.0300	-0.04000	-0.04	0.0267	-0.06667	0.002667
0.03	0.0300	0.00000	0.03	0.0267	0.00333	0.000000
0.04	0.0300	0.01000	0.11	0.0267	0.08333	0.000833
0.04	0.0300	0.01000	-0.08	0.0267	-0.10667	-0.001067

Cov(SP500–Russ500) = sum [{X(i)-X(mean)}*{Y(j)-Y(mean)}] / (n-1) = (0.0045)/5

Cov(SP500–Russ500) = 0.0009

SP500–Nicky

Expected Return (%) X(i)

X(mean)

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