(3) 簡森衡量值(Jensen’s measure) (3 分)

(1) 理論上，要接近完全避險，則該投資人應賣出幾個契約？ (4 分)

(2) 如果投資人賣出 12 份契約，一個月後，那斯達克指數下跌了 180 點，投資人所持之 股票平均每股股價下跌 $5，則該投資人的損益為何？ (4 分)

(1) 夏普衡量值(Sharpe’s measure) (3 分)

(2) 崔納衡量值(Treynor’s measure) (3 分)

(4) 資訊比率值(information ratio) (3 分)

(5) 由哪一個衡量值可顯示：投資組合 P 的績效超越或低於市場投資組合 M 多少？ (2 分) （10 分）

(a) Crystal lives only with two goods, coffee (x1) and chocolate (x2). Suppose that Crystal has the following quasi-linear utility function:u (x1,x2) = α1lh x1 + α2x2.Each day, Crystal spends w dollars on coffee and chocolate. Suppose that the prices of coffee and chocolate are respectively pi and pz. 'That is, the budget constraint of Crystal can be written as:p1x1+p2x2 = w.Suppose that there is an inner solution. Derive x1 (p1, p2, w) and x2 (p1, p2, w), the demand functions of coffee and chocolate of Crystal. That is, solve the following problem:and write x1 and x2 as functions of p1, p2, and I.

(b) u(p1, p2,w), the indirect utility function, gives Crystal's maximal attain-able utility when faced with (Pi, P2), the vector of goods prices, and w, the amount of income:(p1,p2,W) =u[x1(p1,p2,W),x2(p1,p2,w)].Write down the indirect utility function of Crystal. That is, plug in your answers of (a) into the quasi-linear utility function.

(c) Roy's identity is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the demand function to the derivatives of the indirect utility function. Specifcally, demand function for good i can be calculated asCalculate

(a) Suppose that Loretta's instantaneous utility function takes the CRRAform. Compute her coefficient of relative risk aversion:where u' (C) and u" (C) are respectively the first and second derivatives of u (C).