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Variance of a mixed book

Market makingMedium

A book holds $2$ units of $A$ , is short $1$ unit of $B$ and holds $1$ unit of $C$ . Per unit, $Var (A) = 9$ , $Var (B) = 16$ , $Var (C) = 25$ , $Cov (A, B) = 6$ , $Cov (A, C) = - 3$ and $Cov (B, C) = 10$ . What is the variance of the value of the book? Give a whole number.

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Hint

Add the squared positions times the variances, then twice each pair's positions times its covariance.

Solution

The diagonal gives $4 \times 9 + 1 \times 16 + 1 \times 25 = 77$ .
The pairs give $2 \times (2 \times (- 1) \times 6 + 2 \times 1 \times (- 3) + (- 1) \times 1 \times 10) = - 56$ .
The total is $77 - 56 = 21$ .

Answer: $21$

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