# The covariance between a whole-life and term assurance proof

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I am looking for the proof for the co-variance between a term assurance and whole life assurance, this is needed to calculate the variance of a deferred assurance, see my attempt and explanation in the attached photo.

+1 vote
answered Jun 26 by (1,850 points)
selected Jun 27 by Natank

$$cov(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]$$
Letting $$X$$ and $$Y$$ be the present values of a whole life and term assurance respectively such that
$$X$$  is defined as $$X = v^{K_{x}+1}$$ when $$K_x = 0, 1, ...$$ and $$0$$ otherwise.
$$Y$$  is defined as $$X = v^{K_{x}+1}$$ when $$K_x < n$$ and $$0$$ otherwise.
Then  $$XY = v^{2(K_{x}+1)}$$ when $$K_x < n$$ and $$0$$ otherwise.