{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

bkmsol_ch08_09

# bkmsol_ch08_09 - CHAPTER 8 OPTIMAL RISKY PORTFOLIOS 1 The...

This preview shows pages 1–4. Sign up to view the full content.

CHAPTER 8: OPTIMAL RISKY PORTFOLIOS 1. The parameters of the opportunity set are: E(r S ) = 20%, E(r B ) = 12%, σ S = 30%, σ B = 15%, ρ=0.10 From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(r S , r B ) = ρσ S σ B ]: Bonds Stocks Bonds 225 45 Stocks 45 900 The minimum-variance portfolio is computed as follows: w Min (S) = 1739 . 0 ) 45 2 ( 225 900 45 225 ) r , r ( Cov 2 ) r , r ( Cov B S 2 B 2 S B S 2 B = × - + - = - σ + σ - σ w Min (B) = 1 - 0.1739 = 0.8261 The minimum variance portfolio mean and standard deviation are: E(r Min ) = (0.1739 × 20) + (0.8261 × 12) = 13.39% σ Min = 2 / 1 B S B S 2 B 2 B 2 S 2 S )] r , r ( Cov w w 2 w w [ + σ + σ = [(0.1739 2 × 900) + (0.8261 2 × 225) + (2 × 0.1739 × 0.8261 × 45)] 1/2 = 13.92% 2. Proportion in stock fund Proportion in bond fund Expected return Standard Deviation 0.00% 100.00% 12.00% 15.00% 17.39% 82.61% 13.39% 13.92% minimum variance 20.00% 80.00% 13.60% 13.94% 40.00% 60.00% 15.20% 15.70% 45.16% 54.84% 15.61% 16.54% tangency portfolio 60.00% 40.00% 16.80% 19.53% 80.00% 20.00% 18.40% 24.48% 100.00% 0.00% 20.00% 30.00% Graph shown on next page. 9-1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
3. 0.00 5.00 10.00 15.00 20.00 25.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 Tangency Portfolio Minimum Variance Portfolio Efficient frontier of risky assets CML INVESTMENT OPPORTUNITY SET The graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6% and standard deviation approximately 16.5%. 4. The proportion of the optimal risky portfolio invested in the stock fund is given by: ) r , r ( Cov ] r ) r ( E r ) r ( E [ ] r ) r ( E [ ] r ) r ( E [ ) r , r ( Cov ] r ) r ( E [ ] r ) r ( E [ w B S f B f S 2 S f B 2 B f S B S f B 2 B f S S - + - - σ - + σ - - - σ - = 4516 . 0 ] 45 ) 8 12 8 20 [( ] 900 ) 8 12 [( ] 225 ) 8 20 [( ] 45 ) 8 12 [( ] 225 ) 8 20 [( = × - + - - × - + × - × - - × - = w B = 1 - 0.4516 = 0.5484 The mean and standard deviation of the optimal risky portfolio are: E(r P ) = (0.4516 × 20) + (0.5484 × 12) = 15.61% σ p = [(0.4516 2 × 900) + (0.5484 2 × 225) + (2 × 0.4516 × 0.5484 × 45)] 1/2 = 16.54% 5. The reward-to-variability ratio of the optimal CAL is: 9-2
4601 . 0 54 . 16 8 61 . 15 r ) r ( E p f p = - = σ - 6. a.If you require that your portfolio yield an expected return of 14%, then you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is: C C P f p f C 4601 . 0 8 r ) r ( E r ) r ( E σ + = σ σ - + = Setting E(r C ) equal to 14%, we find that the standard deviation of the optimal portfolio is 13.04%. b. To find the proportion invested in the T-bill fund, remember that the mean of the complete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal combination of stocks and bonds (P). Let y be the proportion invested in the portfolio P. The mean of any portfolio along the optimal CAL is: E(r C ) = (l - y)r f + yE(r P ) = r f + y[E(r P ) - r f ] = 8 + y(15.61 - 8) Setting E(r C ) = 14% we find: y = 0.7884 and (1 - y) = 0.2116 (the proportion invested in the T-bill fund). To find the proportions invested in each of the funds, multiply 0.7884 times the respective proportions of stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = 0.7884 × 0.4516 = 0.3560 Proportion of bonds in complete portfolio = 0.7884 × 0.5484 = 0.4324 7. Using only the stock and bond funds to achieve a portfolio expected return of 14%, we must find the appropriate proportion in the stock fund (w S ) and the appropriate proportion in the bond fund (w B = 1 - w S ) as follows: 14 = 20w S + 12(1 - w S ) = 12 + 8w S w S = 0.25 So the proportions are 25% invested in the stock fund and 75% in the bond fund. The standard deviation of this portfolio will be: σ P = [(0.25 2 × 900) + (0.75 2 × 225) + (2 × 0.25 × 0.75 × 45)] 1/2 = 14.13% This is considerably greater than the standard deviation of 13.04% achieved using T-bills

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 23

bkmsol_ch08_09 - CHAPTER 8 OPTIMAL RISKY PORTFOLIOS 1 The...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online