Question

3. Consider Table 2. Table 2 Stock Expected Return 2 12% 6% Standard Deviation 20% 10% 0.20 Correlation Coefficient (a) Consider Table 2. Compute the expected return and standard deviation of return of an equally-weighted (b) Consider Table 2. Solve for the composition, expected return and standard deviation of the minimum (c) Consider Table 2. Sketch the set of portfolios comprised of stocks 1 and 2. Be sure to include the portfolios (d) Consider Table 2. Suppose that a risk-free asset is available and offers a certain return of 2%. Solve for the portfolio of stocks 1 and 2. variance portfolio. whose risk and return are calculated in parts (a) and (b) of this question. composition, the expected return and standard deviation of the tangency portfolio. Sketch the set of portfolios comprised of the risk-free asset and the tangency portfolio.

Answer 1

(a) Expected Return and Standard Deviation of Portfolio with equal weights:

• Expected Return of Portfolio,
  • $ER(P) = W_{A}*ER(A) + W_{B}*ER(B)$
  • ER (P) = [ 50%*12%] + [ 50%*6%]
  • ER (P) = 9%.
• Standard Deviation of Portfolio $(\sigma _{P})$,
  • $\sigma _{P}^{2} = W_{A}^{2}\sigma _{A}^{2}+W_{B}^{2}\sigma _{B}^{2}+2W_{A}W_{B}\sigma _{A}\sigma _{B}R_{AB}$
  • $\sigma _{P}^{2}=$ (0.5)²(20)² + (0.5)²(10)² + 2(0.5)(0.5)(20)(10)(0.2)
  • $\sigma _{P}^{2}=$ 100 + 25 + 20
  • $\sigma _{P}^{2}=$ 145
  • $\sigma _{P}=\sqrt{145}$
  • $\sigma _{P}=12.04%$%.

(b) Minimum Variance Portfolio:

• Weight/Composition of a minimum variance portfolio is,
  • $W_{A}=[\sigma _{B}^{2}-r_{AB}\sigma _{A}\sigma _{B}] \div [\sigma _{A}^{2}+\sigma _{B}^{2}-2 r_{AB}\sigma _{A}\sigma _{B}]$
  • $W_{A}=[(10)^{2}-(0.2)(20)(10)]\div [(20^{2})+(10^{2})-2(0.2)(20)(10)]$
  • $W_{A}=$ 60/420
  • $W_{A}=$ 0.1429 (or)
  • $W_{A}=$ 14%.
  • $\therefore W_{B}=1-W_{A}$
  • $W_{B}=$1 - 0.14 = 86%.

• Expected Return of Minimum Variance Portfolio,
  • $ER(P) = W_{A}*ER(A) + W_{B}*ER(B)$
  • ER(P) = (0.14)(12%) + (0.86)6%.
  • ER(P) = 1.68% + 5.16%
  • ER(P) = 6.84%.
• Standard Deviation of Portfolio $(\sigma _{P})$,
  • $\sigma _{P}^{2} = W_{A}^{2}\sigma _{A}^{2}+W_{B}^{2}\sigma _{B}^{2}+2W_{A}W_{B}\sigma _{A}\sigma _{B}R_{AB}$
  • $\sigma _{P}^{2}=$ (0.14)²(20)² + (0.86)²(10)² + 2(0.14)(0.86)(20)(10)(0.2)
  • $\sigma _{P}^{2}=$ 7.84 + 73.96 + 9.632
  • $\sigma _{P}^{2}=$ 91.432
  • $\sigma _{P}=\sqrt{91.432}$
  • $\sigma _{P}=9.56%$%.

(c)

p-2 a.56 standard deviation (fr ド

(d) Tangency Portfolio (TP): Tangency portfolio is the portfolio which lies on the frontier line, Usually it is the portfolio which has highest Sharpe's ratio. A portfolio with a higher Sharpe ratio is considered to be superior in relative to its peer portfolios.

(i) Sharpe ratio of above both portfolio's is as follows,

• Portfolio-1 Sharpe's ratio  $=[ER(p)-R_{f}]\div \sigma _{p}$
  • Sharpe's ratio = (9-2)/12.04
  • Sharpe's ratio = 0.5814 (or) 58.14%
• Portfolio-2 Sharpe's ratio  $=[ER(p)-R_{f}]\div \sigma _{p}$
  • Sharpe's ratio = (6.84-2)/9.56
  • Sharpe's ratio = 0.5063 (or) 50.63%
• Therefore, Portfolio-1 is the Tangency Portfolio.

(ii)

• Weight of Tangency Portfolio,
  • $W_{A}=[(\sigma _{B}^{2})(ER_{A})-(\sigma _{P})(ER_{B})]\div[(\sigma _{B}^{2})(ER_{A})-(\sigma _{P})(ER_{B})+(\sigma _{A}^{2})(ER_{B})-(\sigma _{P})(ER_{A})]$
  • Where,
    • ER(A) = Return of stock in excess of Risk free Return = 12%-2% = 10%
    • ER(B) = 6%-2% = 4%
    • $\sigma _{A}=20%$%
    • $\sigma _{A}=10%$%
    • $\sigma _{AB} (or)\sigma _{P}=12.04%$%.
  • $W_{A}=[(10^{2})(10)-(12.04)(4)]\div [(10^{2})(10)-(12.04)(4)+(10^{2})(4)-(12.04)(10)]$
  • $W_{A}=$ 951.84 / 1231.44
  • $W_{A}=0.7729 (or) 77%$%
  • $\therefore W_{B}=1-W_{A}$
  • $W_{B}=1-0.77$
  • $W_{B}= 23%$%.

• Expected Return of Tangency Portfolio,
  • $ER(TP) = W_{A}*ER(A) + W_{B}*ER(B)$
  • ER (TP) = (0.77)(12%) + (0.23)(6%)
  • ER (TP) = 9.24% + 1.38%
  • ER(TP) = 10.62%
• Standard Deviation of Tangency Portfolio $(\sigma _{P})$,
  • $\sigma _{P}^{2} = W_{A}^{2}\sigma _{A}^{2}+W_{B}^{2}\sigma _{B}^{2}+2W_{A}W_{B}\sigma _{A}\sigma _{B}R_{AB}$
  • $\sigma _{P}^{2}=$ (0.77)²(20)² + (0.23)²(10)² + 2(0.77)(0.23)(20)(10)(0.2)
  • $\sigma _{P}^{2}=$ 237.16 + 5.29 + 14.168
  • $\sigma _{P}^{2}=$ 256.62
  • $\sigma _{P}=\sqrt{256.62}$
  • $\sigma _{P}=16.02%$%.
• 
  0 6 minPasto CP-リ CP.2) anda aton p