Question

7. Using historical data to measure portfolio risk and correlation coefficient

Peter is an investor who believes that past variability of stocks is a reasonably good estimate of future risk associated with the stocks. Peter works on creating a new portfolio and has already purchased stock A. Now he considers two other stocks, B and C. Peter collected data on the historic rates of return for all three stocks, which are presented in the following table. Complete the table by calculating standard deviations for each stock:

Year	Stock A	Stock B	Stock C
2013	40%	-5%	35%
2014	-10%	40%	-5%
2015	35%	-10%	-10%
2016	-5%	35%	40%
Average return	%	%	%
Estimated standard deviation

Suppose Peter can only afford to complement stock A by adding just one of the two other stocks, either stock B or stock C. Complete the following table by computing correlation coefficients between stocks A and B and between stocks A and C, and calculate average returns and standard deviation for the two potential portfolios, AB and AC:

	Stocks A and B	Stocks A and C
Correlation coefficient
Average return	%	%
Standard deviation

Suppose Peter has to choose between two portfolios, AB and AC. Peter will be better off choosing   .

Which of the following statements about portfolio diversifications are correct? Check all that apply.

Returns on stocks in the same industry are more closely correlated than on stocks in different industries.

Diversification can reduce risk but not eliminate it.

Portfolios that include stocks of only big companies minimize risk.

Correlation between returns on stocks of small companies is smaller than returns on stocks of big companies.

Answer 1

Answer 2

1.

average return = sum of all returns / number of terms

For stock A,

average return = (40 - 10 +35 - 5) / 4 = 15%

Standard deviation=

data	data-mean	(data - mean)2
40	25	625
-10	-25	625
35	20	400
-5	-20	400

2 Σ-)- 2050 2 Σ-) 205026.1406 4-1 σ n- 1

For Stock B and Stock C, since the values are same (just the order is different) So mean and Standard deviation will remain same.

MEAN=15%

SD=26.14

2.

(a) STOCK A AND B

CORRELATION COEFFICIENT:-

X	Y	X⋅Y	X⋅X	Y⋅Y
40	-5	-200	1600	25
-10	40	-400	100	1600
35	-10	-350	1225	100
-5	35	-175	25	1225

∑X=60 , ∑Y=60 , ∑X⋅Y=−1125 , ∑X2=2950 , ∑Y2=2950

ΣΧΥ- Σx. ΣΥ η. Τ VΗΣΧ-ΣxΣΥ-Συ] 4. 1125 60 60 --0.9878 Τ |4- 2950 - 602 | V4- 2950 602

AVERAGE RETURN-

Since weight of stock A and B is same (i.e. 0.5 each) So average return is the average of stock A and B i.e. 15%

STANDARD DEVIATION of PORTFOLIO:-

Op (WA2Ag2 o22wAwBooBPAB)1/ 2 +

here W_A = W_B = 0.5

SD of A = SD of B = 26.1406

Rho is the correlation coefficient calculated above = -0.9878

So SD of portfolio comes out to be 2.04

Similarly for portfolio A and C

X	Y	X⋅Y	X⋅X	Y⋅Y
40	35	1400	1600	1225
-10	-5	50	100	25
35	-10	-350	1225	100
-5	40	-200	25	1600

∑X=60 , ∑Y=60 , ∑X⋅Y=900 , ∑X2=2950 , ∑Y2=2950

n. ΣΧΥ-ΣΧ. ΣΥ Τ VIΣΧ-(Σx.ΣΥ- Σ ηi] 4-900 60 60 Τ 4.2050-602 4-2950- 60|

AVERAGE RETURN-

Since weight of stock A and B is same (i.e. 0.5 each) So average return is the average of stock A and B i.e. 15%

STANDARD DEVIATION of PORTFOLIO:-

As above, it is 18.48

3.

Portfolio AB, it has lower standard deviation and negative correlation coefficient

4.

Diversification can reduce risk but not eliminate it
Returns on stocks in the same industry are more closely correlated than on stocks in different industries