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.

$${\bar{A}}_x={\int }_0^{\infty }{v}^t\cdot {}_t{p}_x\cdot {\mu }_{x+t}dt,A_x=\sum _{k=0}^{\infty }{v}^{k+1}\cdot {}_k{p}_x\cdot q_{x+k}$$

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 ratio is always > 1

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." "UDD multiplier: i over i-superscript-m, always slightly above 1."

The full lesson (about 2,901 words, 19 min read) adds 6 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 9b

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