Your client would like to form a portfolio consisting of the following four pharmaceutical companies - BioNTech (BNTX), Moderna (MRNA), Gilead Sciences (GILD), and Abbott Labs (ABT). You estimated the forward looking return covariances (VCV) and expected returns(ER) of the four stocks, all in annual terms. These are set out below:

VCV						ER
	BNTX	MRNA	GILD	ABT
BNTX	0.3	0.2	0.1	0.2		0.12
MRNA	0.2	0.4	0.1	0.3		0.15
GILD	0.1	0.1	0.2	0.1		0.10
ABT	0.2	0.3	0.1	0.5		0.10

Assume that the risk-free rate is 3%per annum and that your client can short sell any stocks. Your client does not trust numerical optimizers such as Excel Solver and demands analytical solutions.

your client asked you to calculate the lowest standard deviations to achieve the target rates of expected return, 0.10 and 0.15, by forming portfolios of the four stocks. That is, fill in the following table. Show your workings.

Target portfolio expected return	0.1	0.15
Lowest SD to achieve the target

[12 Marks]