Question

1. You run an investment advice consulting operation. Vanguard sends you a booklet about their mutual fund investing options. They think you may be interested in a fund that invests only in natural resource companies (primarily oil and gas), a fund that invests in only socially responsible companies or an index fund that mimics the S&P 500. Ten year performance information is as follows:

Natural Resource Fund: Expected Return = 16.2% with standard deviation of 30.2%

Socially Responsible Fund: Expected Return = 4.8% with a standard deviation of 1.5%

S&P 500 Index Fund: Expected Return = 6.1% with a standard deviation of 18.7%

The booklet states that the correlation between the Socially Responsible Fund and the S&P 500 Fund is -.46. The correlation between the S&P 500 and the Natural Resource Fund is .68. The correlation between the Socially Responsible Fund and the Natural Resource Fund is -.07.

Determine the covariance between each pair of funds.

Analyze a set of possible portfolios of pairs of investments. First, analyze portfolios that are half one fund and half another (Example: 50% in S&P 500, 50% in Natural Resource). Repeat this for each 50-50 combination of funds. Second, consider portfolios that are 75% one fund and 25% another.

Answer 1

Basic Data:

Stock	Return	SD	Correlation
Natural Resource Fund (1)	16.20%	30.20%
Socially Responsible Fund (2)	4.80%	1.50%
S%P 500 Index Fund (3)	6.10%	18.70%
Correlation between (2) and (3)			-0.46
Correlation between (3) and (1)			0.68
Correlation between (1) and (2)			-0.07

Covariances:
CorrelationAB = CovarianceAB / [SD of A x SD of B]
CovarianceAB = CorrelationAB x [SD of A x SD of B]

So, Covariance between Natural Resource Fund and Socially Responsible Fund =	-0.0317%
Covariance between Natural Resource Fund and S&P 500 Index Fund	3.8402%
Covariance between Socially Responsible Fund and S&P 500 Index Fund	-0.1290%

Analysis of various combinations:

Combined Return = Return of A x Weight of A + Return of B x Weight of B
Combined Risk = Square root of [(SD_A x W_A)² + (SD_B x W_B)² + (2.SD_A.SD_B.W_A.W_B.Corr_AB)], where SD_A is Standard Deviation of A, and W_A is Weight of A, and the same reading applies for B as well.

Combination	Return	SD
50% Natural Resource Fund & 50% Socially Resp. Fund	10.50%	15.07%
50% Natural Resource Fund & 50% S&P 500 Index Fund	11.15%	22.53%
50% Socially Responsible Fund & 50% S&P 500 Index Fund	5.45%	9.03%
25% Natural Resource Fund, 75% Socially Responsible Fund	7.65%	7.56%
75% Natural Resource Fund, 25% Socially Responsible Fund	13.35%	22.63%
25% Natural Resource Fund, 75% S&P 500 Index Fund	7.65%	19.94%
75% Natural Resource Fund, 25% S&P 500 Index Fund	13.68%	26.06%
25% Socially Responsible Fund, 75% S&P 500 Index Fund	5.78%	13.86%
75% Socially Responsible Fund, 25% S&P 500 Index Fund	5.13%	4.28%

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