Question

Consider the following two assets. Returns on asset 1 has a mean of ui and standard deviation of 01. Returns on asset 2 has a mean of j2 and standard deviation of 02. The correlation coefficient p1,2 measures how the two assets' returns are correlated, and it takes on values between -1 and +1. An investor puts W1 fraction of her wealth into stock 1, and W2 =1-W1 fraction of her wealth into stock 2. 1. Using the equation on the standard deviation of portfolio returns, argue that the portfolio risk is in- creasing in P1,2 2. When P1,2 = 1, argue using mathematical formulas that the portfolio standard deviation is equal to the weighted average of the standard deviation of the individual stocks in the portfolio. (Hint, Square of Summation). 3. Now suppose P1,2 = -1. If the investor wants to minimize her risk in investing in this portfolio, how should she choose W1 and W2 = 1 – Wį? (Hint, Square of Difference).

Answer 1

UM ap= sland and deviation of portfolio. According to formula, op" = (W, xo)² + (W2 X0 28² 2 x W x W 2 ** \XT 2x1,2 Plze [1,1] when 91,2=-1 Tp²=( Wixri)²+(W 2 XC2)² + 2 x W1 xW2 X 1X02*-1 op =(Wirt, - W28272 op = Wixal - W202 when P1,2=1 Tp= Wixol + W202 op will increase from (wirl-W202) to ( Woeit Wero) as Piz incease from 1 to I ATMINTISMUS Hence portfolio risk will increane with increasing P1,2. 2) when P1,2=1 op 2 = (W1X07)²+(W2x02) "+2 Wiol *W 2X02x) oco+ (a tb )2 =a² +62 + Rab On Coмр сая. Г) ,

op2-(Wixo, t Were) op = WirtWern Hence portfolio std deviation is equal to the weighlage auerage of std cleuiation of the individual stocks in the portfolio 3) P1,2=-1 *. if we | up = Wit-W202 & WI+W2=1 It we know op we can solve for Wi&W 2 Also for risk free portfolio ,op=0 WITI =W202 W2=1-Wi WITH=CI-wilo2 WICT+82)=02 WIE T2 W2= oitT2 THT2

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