Question

4. Consider two stocks with returns ?? and ?? with the following properties. ?? takes values -10 and +20 with probabilities 1/2. ?? takes value -20 with probability 1/2 and +40 with probability 1/2. ????(??,??) = ? (some number between -1 and 1). Answer the following questions

   1. (a) Express C??(??,??) as a function of ?

   2. (b) Calculate the expected return of a portfolio that contains share ? of stock ? and

      share 1 − ? of stock ?. Your answer should be a function of ?

   3. (c) Calculate the variance of the portfolio from part ? (Hint: returns are now potentially dependent)

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4. (d) What value of ?* minimizes the variance of the portfolio? Your answer should be a function of ?, denoted by ?*(?).

5. (e) For what range of values for ? is your ?*(?) 1? What is the solution to the above problem if ? is outside of that range? (Hint: draw a graph and nd ?* ∈ [0, 1] that minimizes variance)

6. (f) Is ?*(?) increasing or decreasing? (Hint: take the derivative with respect to ?)

7. (g) Which ? would the investor prefer to have, positive or negative? What is the intuition

   for that result? (Hint: how does expected return of the portfolio change with ??)

Answer 1

And As per geven PERA=x} = ( data in question, xs-10,20 ; PERB = x} = ( Å x = -20, 40 Now & conhelation coefficient to them is Cora CRA, RB) = ? ECRA): E XPOW= (-10+20) Ź = 5. VCRA) = { EPCX) -[Extew]?: (100+400) { +526. - 250 - 23 Similarly, ECRB) = { repoy = (-2otho) € = 10. VCRB) = Es pou) -Ezreion] [ww+1600){ - [lo]? loop ico 225. NOL go CONCRA, RB). Coach CRARB) = COUCRARB) VCRA JUCRB - - COVERARB V225. 900 = COUCRARB) - 750 2. (b) het Then, x = QA+-) B RX = QRA+ (1-2) RB So E CRX = OECRA) + (-2) E (RB) - 25 +(-) 10 = 20 - 50

Q VCR) = - rcaRA +(1-2) RB ) 2² VERA) + (1-d) 2 VCRB) + 22(1-2) CONCRA, RB) R250x2 + (- 900 + 2C ) 15 . (1925 - 15008) a² + (1500k - 1800 &+ 900 = In order to minimia VCRX). We differentials VCRX) whicha, Then, dv(Rx7 C =0 S 2[1125-15002]2+ [15wR -1800] = 0 = 1500 h -1800 [125-1501]