Hey Chegg, I need just your help with the tangency portfolio below.

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:

Expected Return	Standard Deviation
Stock fund (S)	15%	32%
Bond fund (B)	9	23

The correlation between the fund returns is .15

1.Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of 0% to 100% in increments of 20%. What expected return and standard deviation does your graph show for the minimum-variance portfolio?

The parameters of the opportunity set are:

E (r_s) = 15%, E (r_B) = 9%, (Standard Deviation) STD_S = 32%, STD_B = 23%, coefficient correlation (p) = 0.15, r_f = 5.5%

From the standard deviations and the correlation coefficient, we generate the covariance matrix [note that Cov(r_s, r_B) = pSTD_SSTD_B]:

	Bond	Stocks
Bonds	STDB * STDB	p * STDB * STDS
Stocks	p * STDB * STDS	STDS * STDS

	Bond	Stocks
Bonds	529.0(23^2)	(.15 x 32 x 23) = 110.4
Stocks	110.4	1024.0 (32^2)

The minimum-variance portfolio proportions are:

_WMin(_S) = STD_B² - Cov(r_s,r_B) / STD_s² + STD_B² - 2Cov(r_s,r_B)

_WMin(_S) = 23² - .15 x 23 x 32 / 32² + 23² - 2(.15 x 23 x 32)

W_Min(_S) = 529 - 110.4 / 1,024 + 529 - (2 x 110.4) = .3142

_WMin(_B) = 1 W_Min(_S) => 1 - .3142 = .6858

The mean and standard deviation of the minimum variance portfolio are:

E(r_Min) = (.3142 x 15%) + (.6858 x 9%) = 10.89%

_STDMin= [W_S² * STD_S² + W_B² * STD_B² + 2 * W_S * W_B * Cov(r_S, r_B)]^1/2

STD_Min = [(.3142² * 1024) + (.6858² * 529) + (2 * .3142 * .6858 * 110.4)]1/2

STD_Min = 19.94%

Example for 20% in stock and 80% in bond:

E(r_Min) = (.20 x 15%) + (.80 x 9%)

E(r_Min) = (0.20 x 0.15) + (0.80 x 0.09)

E(r_Min) = 0.03 + 0.072 = 0.102 x 100 = 10.2%

_STDMin²= [W_S² * STD_S² + W_B² * STD_B² + 2 * W_S * W_B * Cov(r_S, r_B)]^1/2

= [(.20² x 32²) + (.80² x 23²) + (2 x .2 x .8 x .15 x 32 x 23)]

= 40.96 + 338.56 + 35.33 = 414.85

_STDMin= Square Root of 414.85 = 20.37%

% in stocks	% in bonds	Exp. Return	Std. dev.	Sharpe Ratio
0.0000	1.0000	0.0900	0.2300	0.1522
0.2000	0.8000	0.1020	0.2037	0.2308
0.3142	0.6858	0.1089	0.1994	0.2701	Minimum Variance Portfolio
0.4000	0.6000	0.1140	0.2018	0.2924
0.6000	0.4000	0.1260	0.2250	0.3155
0.6466	0.3534	0.1288	0.2334	0.3162	Tangency Portfolio
0.8000	0.2000	0.1380	0.2668	0.3111
1.0000	0.0000	0.1500	0.3200	0.2969

I know the problem seems very long, but I have done all the steps for you to follow. I need help figuring out how to Tabulate and draw the investment opportunity set of the two risky funds using investment proportions for the stock fund of 0% to 100% in increments of 20%

It is shown on the table above, but I don't understand how the professor figured out those calculations. I would really appreciate if you can help me understand the table above with the tangency portfolio.