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.

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You are on the correct path. The relationship comes from the fact that:

$$cov(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]$$

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.

Applying the covariance formula above will yield the result.

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