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A basis reverting to its mean

Expected valueEasy

A futures basis follows $X_{t} = 2 + 0.75 X_{t - 1} + ϵ_{t}$ , where $ϵ_{t}$ has mean $0$ and is independent of the past. Today the basis is $20$ . What is its expected value $2$ days from now? Give an exact decimal.

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Hint

Find the long-run mean, then shrink today's distance from it once per day.

Solution

The mean is $\frac{2}{1 - 0.75} = 8$ , and today's distance is $20 - 8 = 12$ .
Over $2$ days the distance is multiplied by $0.75 \times 0.75 = 0.5625$ .
The expected value is $8 + 0.5625 \times 12 = 14.75$ .

Answer: $14.75$

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