Means, Variances, and Covariances of Present Value Random Variables

Problem Preview: You are given $A_{40} = 0.12$, ${}^{2}A_{40} = 0.032$, $d = 0.05$. The variance of the annuity-due PV is $(0.032 - 0.0144)/0.0025 = 7.04$. Half the candidates forget to square $d$ in the denominator and get 0.352 instead.

HIGH-FREQUENCY: The net single premium $A_x$ is the most tested quantity in Topics 9 and 10.

$$\bar{A}_x = \int_0^{\infty} v^t \cdot {}_{t}p_x \cdot \mu_{x+t}\,dt, \quad A_x = \sum_{k=0}^{\infty} v^{k+1} \cdot {}_{k}p_x \cdot q_{x+k}$$

Under constant force: $\bar{A}_x = \mu/(\mu+\delta)$, $\bar{a}_x = 1/(\mu+\delta)$.

HIGH-FREQUENCY: Second moment at double the force of interest, mortality unchanged.

$$\text{Var}(v^{T_x}) = {}^{2}\bar{A}_x - (\bar{A}_x)^2$$

where ${}^{2}\bar{A}_x$ is $\bar{A}_x$ computed at $2\delta$ (or $v^2$). The rule: replace $v$ by $v^2$, or $\delta$ by $2\delta$. Mortality does not change.

KEY: Second moment at double the force of interest, same mortality. Doubling $\mu$ or $q$ instead of $\delta$ is wrong and produces an incorrect variance.

Common mistakes

• Forgetting to double the force, not the mortality. ${}^{2}A_x$ replaces $\delta$ with $2\delta$. Mortality stays the same. Doubling $\mu$ or $q$ is wrong.
• Squaring the wrong thing. Variance = ${}^{2}A_x - (A_x)^2$. Not $({}^{2}A_x)^2 - (A_x)^2$. Not ${}^{2}A_x - A_x$.
• Using $d^2$ instead of $\delta^2$ for continuous annuity. Due uses $d^2$. Continuous uses $\delta^2$. Mixing these up changes the answer.

Bottom line

• $A_x = \sum v^{k+1} {}_{k}p_x q_{x+k}$ (annual); $\bar{A}_x = \int v^t {}_{t}p_x \mu_{x+t}\,dt$ (continuous)
• Variance $= {}^{2}A_x - (A_x)^2$; the second moment keeps the same mortality but uses double the force of interest
• Annuity variance = insurance variance / $d^2$ (due) or $\delta^2$ (continuous)
• UDD: $A_x^{(m)} = (i/i^{(m)}) A_x$ and $\bar{A}_x = (i/\delta)A_x$; the m-thly ratio exceeds one for positive interest and $m>1$, with a zero-interest limit of one.

Exam shortcut

When given $A_x$ and ${}^{2}A_x$, write the variance formula immediately: ${}^{2}A - A^2$. For annuity variance, divide by $d^2$. These two steps solve a large fraction of variance problems. "Second moment = same mortality, double delta." "Variance of annuity = variance of insurance / d-squared." Check positive interest and more than one period per year before using a UDD multiplier greater than one.

The full lesson (about 3,425 words, 23 min read) adds 7 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 9b

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