Question

State Probability Return on Stock A Return on Stock B

1 0.10 10% 8%

2 0.20 13% 7%

3 0.20 12% 6%

4 0.30 14% 9%

5 0.20 15% 8%

Which of the following portfolio(s) is(are) on the efficient frontier?
A. The portfolio with 20 percent in A and 80 percent in B.
B. The portfolio with 15 percent in A and 85 percent in B.
C. The portfolio with 26 percent in A and 74 percent in B.
D. The portfolio with 10 percent in A and 90 percent in B.
E. A and B are both on the efficient frontier.

Here is the answer, but I don't understand how they got it:

The Portfolio's E(Rp), sp, Reward/volatility ratios are 20A/80B: 8.8%, 1.05%, 8.38; 15A/85B: 8.53%, 1.06%, 8.07; 26A/74B: 9.13%, 1.05%, 8.70; 10A/90B: 8.25%, 1.07%, 7.73. The portfolio with 26% in A and 74% in B dominates all of the other portfolios by the mean-variance criterion.

Answer 1

A) Probability Return on Stock A Return on Stock B

	Probability (P)	Return on stock A RA(%)	Return on stock B RB(%)	Probability Return on stock A (P*RA)	Probability Return on stock B (P*RB)
1	0.10	10	8	1.0	0.80
2	0.20	13	7	2.6	1.40
3	0.20	12	6	2.4	1.20
4	0.30	14	9	4.2	2.70
5	0.20	15	8	3.0	1.60

Probability Return on Stock A = 13.2% (1+2.6+2.4+4.2+3)

Probability Return on Stock B = 7.7% (.8+1.4+1.2+2.7+1.6)

B)Step 1: Calculation of standard deviation of each stock

- Stock A

	Probability(P)	Rate of Return(%)	Deviation (DA)	PDA2
1	0.1	10	-3.2	1.0240
2	0.2	13	-0.2	0.0080
3	0.2	12	-1.2	0.2880
4	0.3	14	0.8	0.1920
5	0.2	15	1.8	0.6480

Variance = 2.1600 (1.0240+.0080+.2880+.1920+.6480)

Standard Deviation (SDA)= Variance

= 2.1600

= 1.4697

- Stock B

	Probability(P)	Rate of Return(%)	Deviation (DB)	PDB2
1	0.1	8	0.3	0.0090
2	0.2	7	-0.7	0.0980
3	0.2	6	-1.7	0.5780
4	0.3	9	1.3	0.5070
5	0.2	8	0.3	0.0180

Variance = 1.2100 (.0090+.0980+.5780+.5070+.0180)

Standard Deviation (SDB) = Variance

= 1.2100

= 1.1

Step 2: Calculation of Correlation between stock A and B

	Probability(P)	Deviation (DA)	Deviation (DB)	P*DA*DB
1	0.1	-3.2	0.3	-0.096
2	0.2	-0.2	-0.7	0.028
3	0.2	-1.2	-1.7	0.408
4	0.3	0.8	1.3	0.312
5	0.2	1.8	0.3	0.108

Covariance = 0.76 (-.096+.028+.408+.312+.108)

Correlation = Covariance / (SDA*SDB)

= .76/ (1.4697 * 1.1)

= .47

Step 3: Calculation of Expected return and standard deviation on each portfolio

Portfolio	Return	Risk (working note)
20% A and 80% B	8.8% (.2*13.2 + .8*7.7)	1.05%
15% A and 85% B	8.53% (.15*13.2 + .85*7.7)	1.06%
26% A and 74% B	9.13% (.26*13.2 + .74*7.7)	1.05%
10% A and 90% B	8.25% (.1*13.2 + .9*7.7)	1.07%

working note: Calculation of risk

Use equation (WA*SDA)² + (WB*SDB)² + (2*WA*WB*SDA*SDB*Corr)

-20% A and 80% B

(.2*1.4697)² + (.8*1.1)² + (2*.2*.8*1.4697*1.1*.47)

= 1.1039

= 1.05

-15% A and 85% B

(.15*1.4697)² + (.85*1.1)² + (2*.15*.85*1.4697*1.1*.47)

= 1.1166

= 1.06

-26% A and 74% B

(.26*1.4697)² + (.74*1.1)² + (2*.26*.74*1.4697*1.1*.47)

= 1.101

= 1.05

-10% A and 90% B

(.1*1.4697)² + (.9*1.1)² + (2*.1*.9*1.4697*1.1*.47)

= 1.1385

= 1.07

Step 4: Analysis

If we look at on return basis P3(26% A and 74% B) has high return.

If we look at on risk basis both P1(20% A and 80% B) and P3(26% A and 74% B) has the lower risk of 1.05%. Out of the two, P3 has the high return.Hence the portfolio with 26% in A and 74% in B dominates all of the other portfolios