# Portfolio Management

Portfolio Management

- As a portfolio manager you analyze 3 components (A, B, C) of the EUROSTOXX 50 index. You have estimated the expected returns of these components for next year as: Ra = 12%, Rb = 6% and Rc = 8,5% and the corresponding variance / covariance matrix as:

0,09 | 0,0175 | 0,041 |

0,0175 | 0,0625 | 0,0225 |

0,041 | 0,0225 | 0,04 |

Where Cov (Ra, Rb) = 0,0175

- Calculate Sharpe ratio of each component, where the annualized risk-free rate is 1%. Then comment on the results and on the use of Sharpe index as a performance measure.

- You plan to invest in a portfolio P1 defined by the following weights: xa = 30%, xb = 50%, xc = 20%. According to the risk/return tradeoff compare your portfolio with the alternative proposal of your colleagues to invest in equally weighted portfolio of the three components (P2). Justify your decisions with appropriate calculations.

- Now you are considering investing in a stock portfolio of only two of the three components A, B, C (short positions are not allowed). Discuss whether each combination give the benefit of the diversification effect in terms of lower variance (Hint: compare the ratio of any two component volatilities with the corresponding correlation coefficient).

- By applying the Share single index model, you have computed the following parameters of the three components (A, B, C) and the benchmark (M) that is the EUROSTOXX 50

αi | βi | Qi | |

A | -0,025 | 0,76 | 0,15 |

B | -0,014 | 1,3 | 0,07 |

C | 0,030 | 0,97 | 0,11 |

M | 0,055 | 1,0 | 0,12 |

Where: α_{M }= R_{M }, Q_{i} = σ^{2}_{ε}_{i }, Q_{M }= σ^{2}_{M}

Calculate the expected returns of portfolios P1 and P2 defined in point b.

*You plan to invest in a portfolio with a beta of 1with respect to the benchmark*EUROSTOXX 50.The*portfolio is*composed of only two of the stocks A and B analyzed in Question d). Calculate the variance and determine the optimum weights for a portfolio with a combination of stocks A and B only. [Hint: given two risky assets, the weights to be invested in each security are obtained easily from the formula used to calculate the beta of the portfolio.]

- In your role as a portfolio manager you are asked to make valuations of a number of corporate bonds. Consider the following bonds:

Bond A: corporate bond issued by company X, annual coupon of 3% (paid semi-annually), expiry in 3 years, redemption price at 100,00, YTM = 2,75%.

Bond B: corporate bond issued by bank Y, annual coupon of 2% (paid annually), expiry in 5 years, redemption price at 100,00, YTM = 1,45%.

Bond C: corporate bond issued by bank Z, zero coupon bond), expiry in 7 years, redemption price at 100,00, YTM = 4,00%.

- For the bonds given above complete the following table

Price | Duration | Convexity | |

Bond A | 2,89 | 9,47 | |

Bond B | 102,634 | 27,679 | |

Bond C | 75,992 | 7 |

- For bond B calculate the relative price change (in %) following a yield curve parallel shift of +25 basis points, specifying both the price change due to the price duration only and also due to the price convexity (Note: if you have not answered question a) assume the duration of bond B as 4,75)
- Suppose you create an equally-weighted portfolio with the three bonds given in the table. Calculate the portfolio convexity. Note: if you have not answered question a) use the following data

Duration | Convexity | |

Bond A | 2,7 | 9,5 |

Bond B | 4,75 | 30,0 |

Bond C | 7,0 | 50,0 |