108156.pdf

90 Mathematics for Finance Exercise 4.8 Apply the Fundamental Theorem of Asset Pricing to find the time 0 and 1 prices of a put ...

5. Portfolio Management...................................... An investment in a risky security always carries the burden of pos ...

92 Mathematics for Finance out to be a convenient measure of risk. Exercise 5.1 Compute the risk Var(K 1 ), Var(K 2 )andVar(K 3 ...

Portfolio Management 93 This illustrates the following general rule: Var(aK)=a^2 Var(K), σaK=|a|σK for any real numbera. Remar ...

94 Mathematics for Finance 5.2 Two Securities We begin a detailed discussion of the relationship between risk and expected retur ...

Portfolio Management 95 The weights are defined by w 1 = x 1 S 1 (0) V(0) ,w 2 = x 2 S 2 (0) V(0) , wherex 1 andx 2 are share ...

96 Mathematics for Finance Exercise 5.4 Compute the value V(1) of a portfolio worth initially V(0) = 100 dollars that consists o ...

Portfolio Management 97 as follows: Scenario ReturnK 1 ReturnK 2 ω 1 12% −4% ω 2 10% 7% Remark 5.2 A similar formula to (5.2) ...

98 Mathematics for Finance Suppose thatw 1 = 60% of available funds is invested in stock 1 and 40% in stock 2. The expected retu ...

Portfolio Management 99 To avoid clutter, we introduce the following notation for the expectation and variance of a portfolio ...

100 Mathematics for Finance Example 5.7 Consider another portfolio with weightsw 1 = 80% andw 2 = 20%, all other things being th ...

Portfolio Management 101 Exercise 5.8 Using the data in Example 5.6, find the weights in a portfolio with expected returnμV= 4 ...

102 Mathematics for Finance Figure 5.1 Typical portfolio lines withρ 12 =−1and1 Suppose thatρ 12 =−1. It follows from the proof ...

Portfolio Management 103 and (5.8) take the form μV=(1−s)μ 1 +sμ 2 , (5.11) σV^2 =(1−s)^2 σ 12 +s^2 σ 22 +2s(1−s)ρ 12 σ 1 σ 2. ...

104 Mathematics for Finance Figure 5.2 The minimum ofσ^2 Vas a function ofs The line on theσ, μplane defined by the parametric e ...

Portfolio Management 105 3) If σσ^12 <ρ 12 ≤1, then there is a portfolio with short selling such that σV<σ 1 , but for e ...

106 Mathematics for Finance Using Theorem 5.5, we compute s 0 ∼=− 1. 1663 ,smin=0. It follows that in the portfolio with minimum ...

Portfolio Management 107 The line on theσ, μplane representing portfolios constructed from one risky and one risk-free securit ...

108 Mathematics for Finance It is well known that the covariance matrix is symmetric and positive definite. The diagonal element ...

Portfolio Management 109 2. To find a portfolio with the smallest variance among all portfolios in the attainable set whose exp ...