Suppose you have some money to invest - for simplicity, $1 - and you are plannin

Question

Suppose you have some money to invest - for simplicity, $1 - and you are planning to put a fraction w into a stock market mutual fund and the rest, 1 - w, into a bond mutual fund. Suppose that $1 invested in a stock, fund yields R_s after 1 year and that $1 invested in a bond fund yields R_b, suppose that R_s is random with mean 0.08 (8%) and standard deviation 0.07, and suppose that R_b is random with mean 0.05 (5%) and standard deviation 0.04. The correlation between R_s and R_b is 0.25. If you place a fraction w of your money in the stock fund and the rest, 1 - w, in the bond fund, then the return on your investment is R = wR_s + (1 - w)R_b. Suppose that w = 0.5. Compute the mean and standard deviation of Suppose that w = 0.75. Compute the mean and standard deviation of What value of w makes the mean of R as large as possible? What the standard deviation of R for this value of w? (Harder) What is the value of w that minimizes the standard dev tion of R? (Show using a graph, algebra, or calculus.)

Explanation / Answer

R = wRs + (1-w)Rb

Mean_R = w*Mean(Rs) + (1-w)*Mean(Rb)

Stddev_R = sqrt(w^2*(stddev_Rs)^2 + (1-w)^2 * (stddev_Rb)^2 + 2*w*(1-w)*stddev_Rs*stddev_Rb*r)

Mean(Rs) = 0.08, Mean(Rb) = 0.05

stddev_Rs = 0.07, stddev_Rb = 0.04, r = 0.25

a.

w = 0.5

R = 0.5*0.08 + 0.5*0.05 = 0.065

Stddev_R = sqrt(0.5^2 * 0.07^2 + 0.5^2*0.04^2 + 2*0.5*0.5*0.25*0.07*0.04) = sqrt(0.001975) = 0.0444

b.

w = 0.75

R = 0.75*0.08 + 0.25*0.05 = 0.0725

Stddev_R = sqrt(0.75^2 * 0.07^2 + 0.25^2*0.04^2 + 2*0.75*0.25*0.25*0.07*0.04) = sqrt(0.003118) = 0.0558

c.

Mean_R = w(Rs - Rb) + Rb

So, Mean_R is maximized when w=1 as Rs > Rb

Stdev_R = stddev_Rs = 0.07

d.

Var_R = w^2*(stddev_Rs)^2 + (1-w)^2 * (stddev_Rb)^2 + 2*w*(1-w)*stddev_Rs*stddev_Rb*r

First derivative wrt w should be 0

d(Var_R)/dw = 2w*(stddev_Rs)^2 -2(1-w)*(stddev_Rb)^2 + 2*(1-2w)*stddev_Rs*stddev_Rb*r) = 0

Second derivative should be positive:

d2(Var_R)/dw2 = 2*(Stddev_Rs - Stddev_Rb)^2

So, w*( (stddev_Rs)^2 + (stddev_Rb)^2 - 2*stddev_Rs * stdddev_Rb * r) = stddev_Rb^2 - stddev_Rs*stddev_Rb*r

So, w = stddev_Rb(stddev_Rb - stddev_Rs*r)/(stddev_Rs - stddev_Rb)^2

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.