A pension fund manager is considering three mutual funds. The first is a stock f

Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 5.8%. The probability distribution of the risky funds is as follows: The correlation between the fund returns is 0.18. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio. (Do not round intermediate calculations and round your final answers to 2 decimal places. Omit the "%" sign in your response.) Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rf. The characteristics of two of the stocks are as follows: Calculate the expected rate of return on this risk-free portfolio? (Round your answer to 2 decimal places.) A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are: The correlation between the fund returns is .15. What is the expected return and standard deviation for the minimum-variance portfolio of the two risky funds? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Explanation / Answer

Answer (1)

For Stock Fund - Expected Return Re = 19% Standard Deviation SDe = 48%
For Debt Fund - Edpected Return Rd = 9% Standard Deviation SDd = 42%

Correlation p = 0.18

Risk-free rate on T-Bill Fund = 5.80%

Covariance Cov(e,d) = 0.18 * 48 * 42 = 362.88

Weight of Equity in an optimal portfolio

We = A/B

Where

A= {(Re - Rf) * SDd^2 - (Rd - Rf) * Cov (e,d)}
B = {(Re - Rf) * SDd^2 + (Rd-Rf)* SDe^2 - (Re - Rf + Rd - Rf) * Cov (e,d)}

A = (19-5.8) * 42^2 - (9-5.8) * 362.88 = 23284.8 - 1161.216 = 22123.584
B = (19-5.8) * 42^2 + (9-5.8)*48^2 -(19-5.8+9-5.8)*362.88
= 23284.8 +7372.8 - 5951.232
= 24706.368

Portfolio invested in stock We = 22123.584 / 24706.368 = 0.8955 or 89.55%

Portfolio investedin Debt Wd = 1- 0.8955 = 0.1045 or 10.45%

Expected return Rp = We*Re + Wd * Rd = 0.8955 * 19 + 0.1045*9
= 17.0145 + 0.9405 = 17.955%

Standard Deviation of Portfolio

SD = [We^2 * SDe^2 + Wd^2 * SDd^2 + 2 * We * Wd * Cov (e,d)]^1/2
= [0.8955^2 * 48^2 + 0.1045^2 * 42^2 + 2 * 0.8955 * 0.1045 * 362.88]^1/2
= [1847.624 + 19.263 + 67.916]^1/2
= 1934.803^1/2 = 43.986%

Answer (2)

Stock A = Expected Return Ra = 9% Standard Deviation Sa = 60%
Stock B = Expected Return Rb = 5% Standard Deviation Sb = 40%

Correlation = -1

Let Wa and Wb = 1-Wa are proportions invested in a portfolio of above two stocks. With a negative correlation of -1 , to get a risk-free portfoio the standard deviation of the portfolio should be equal to zero.

Portfolio standard Deviation = Absolute value ( Wa*Sa - Wb Sb)
==> 0 = 60*wa -(1-wa)40 = 60wa + 40wa - 40 ==> 100 Wa = 40 or Wa = 40/100 = 0.40

Wb = 1-0.40 = 0.60

Expected Rate of Return Rp = Wa * Ra + Wb * Rb = 0.40 * 9 + 0.60 * 5%
= 3.6 + 3 = 6.60%

Answer (3)

For Stock Fund - Expected Return Re = 15% Standard Deviation SDe = 32%
For Debt Fund - Edpected Return Rd = 9% Standard Deviation SDd = 23%

Correlation p = 0.15

Risk-free rate on T-Bill Fund = 5.5%

Covriance Cov(s,d) = p * SDe * SDd = 0.15 * 32 * 23 = 110.40

Weight of Equity in a minimum variance portfolio

W(e) = {SDd^2 - Cov (e,d)} / {SDe^2 + SDd^2 - 2 Cov (e, d)}

W(e) = { 23^2 - 110.40} / { 32^2 + 23^2 - 2 * 110.4}

= (529 - 110.40)/ (1024+529-220.8) = 418.6 / 1332.20
= 0.3142

W(d) = 1-W(e) = 1-0.3142 =0.6858

Expected return of Portfoio Rp = We * Re + Wd * Rd = 0.3142 * 15 + 0.6858 * 9
= 4.713+6.1722 = 10.8852%

Standard Deviation of Portfolio = [We^2 * SDe^2 + Wd^2 * SDd^2 + 2 * We * Wd * Cov (e,d)]^1/2

= [0.3142^2*15^2+0.6858^2*9^2+2*0.3142*0.6858*110.40]^1/2
= [0.09872*225 + 0.47032 + 81 + 47.5625]^1/2
= [22.212 + 38.0961 + 47.5776]^1/2
= [107.8857]^1/2
= 10.3868 or 10.387

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

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