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Correlated strategies combined

StatisticsHard

Daily P&L of strategy A is normal with mean $3$ and standard deviation $5$ . Daily P&L of strategy B is normal with mean $4$ and standard deviation $8$ . They are jointly normal with correlation $- 0.5$ . What is the $z$ -score of a combined daily P&L of $- 14$ ?

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Hint

For the sum, add the covariance term $2 \times ρ \times σ A \times σ B$ to the variances.

Solution

The mean of the combined P&L is $3 + 4 = 7$ .
The variance is $25 + 64 + 2 \times (- 0.5) \times 5 \times 8 = 49$ .
So the standard deviation of the combined P&L is $7$ .
The $z$ -score is $(- 14 - 7) \div 7$ , which equals $- 3$ .

Answer: $- 3$

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