Question

Suppose you have some money to invest-for simplicity, $1-and 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 a $1 invested in a stock fund yields R_s, after one year and a $1 invested in a bond fund yields R_b. R_s and R_b are random variables with expected value of 10% and 8% respectively, and standard deviation of 4% and 2% respectively. The correlation between R_s and R_b is 0.70. 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 will be R= wR_s+(1-w)R_b.The risk associated with your investment is measured by the standard deviation.

(a) If you decide to invest 40% of your $1 in stock and the rest in bond, then what is the expected return of your investment? What is its associated risk?

(b) What share of your $1 money should you invest in bond in order to expect a 9.2% return on your investment? For that same share invested in bond, what level of risk is associated with your investment?

(c) What share of your $1 money should you invest in stock mutual fund in order for your investment risk to be 3%? (you can show your work using algebra, or calculus).

*Please explain it step-by-step, and please no reposted answers for the same question. Thank you!

Answer 1

Stock Fund

Expected Return on Stock = 10%

Standard deviation on Stock = 4%

Bond Fund

Expected Return on Bond = 8%

Standard deviation on Bond = 2%

Correlation between stock and bond fund = 0.70

(a)

Expected Return = (0.4*10%) + (0.6*8%)

Expected Return = 8.8%

The standard deviation (risk) is calculated as follows:

Portfolio standard deviation = Square root of { (0.4*0.04)^2 + (0.6*0.02)^2 + 2*0.4*0.6*0.70*0.04*0.02}

Portfolio standard deviation = Square root of { 0.0006688 }

Portfolio standard deviation = 2.59%

(b)

Return on Stock = 10%

Return on Bond = 8%

Assume Investment in stock = x

Thus, Investment in bond = (1-x)

Expected return = x*0.1 + (1-x)*0.08

0.092 = x*0.1 + (1-x)*0.08

0.092 = 0.1x + 0.08 - 0.08x

0.02x = 0.012

x = 60%

Investment in Stock = 60%

Investment in Bond = 1-60% = 40%

We calculate the associated risk (standard deviation) as:

Portfolio standard deviation = Square root of { (0.6*0.04)^2 + (0.4*0.02)^2 + 2*0.6*0.4*0.70*0.04*0.02}

Portfolio standard deviation = Square root of { 0.0009088 }

Portfolio standard deviation = 3.01%

(c)

To find the weight of stock funds, we use the following equation:

Assume weight of stock = w

Thus, weight of bond = 1-w

(0.03)² = Square-root of { (w*0.04)² + [(1-w)*0.02]² + 2*w*(1-w)*0.70*0.04*0.02 }

0.0009 = 0.0016w² + 0.0004 - 0.0004w + 0.0012w - 0.00112w²

0.0009 = 0.00048w² + 0.00072w + 0.0004

Multiplying the equation by 100,000 we get:

90 = 48w² + 72w + 40

48w² + 72w - 50 = 0

Solving for w, we get:

w = 0.45

Thus, (1-w) = 0.55