4 Premiums/Reserves
4.1 Supplementary Concepts
The random variable \(L\) respresents the loss at issue of an insurance coverage, that is, the present value of the benefits to be paid by the insurer minus the present value of the annuity of premiums to be pasid by the insured. \(L\) may be defined in terms of \(T\) (fully continuous case), \(K\) (fully discrete case), or both (semicontinuous case).
For the fully continuous case and a whole life insurance cover, \(L\) is defined as \(L=v^T-\Pi\,\bar{a}_{\enclose{actuarial}{T}}\), where \(\Pi\) is a level premium payable continuously while \((x)\) is alive. The corresponding expected value is derived below.
\[ \boxed{ \begin{array}{rl} \\ \quad \mathrm{E}[\,L\,] &= \mathrm{E}[\,v^T-\Pi\,\bar{a}_{\enclose{actuarial}{T}}\,] \quad \\ \quad &= \mathrm{E}[\,v^T\,]-\Pi\,\mathrm{E}[\,\bar{a}_{\enclose{actuarial}{T}}\,] \quad \\ \quad &= \bar{A}_x - \Pi\,\bar{a}_x \\ \\ \end{array} } \]
- Analogously, for the fully discrete case and a whole life insurance cover, \(L\) is defined as \(L=v^{K+1}-\Pi\,\ddot{a}_{\enclose{actuarial}{K+1}}\), where \(\Pi\) is a level premium payable at the beginning of each year while \((x)\) is alive. The corresponding expected value is derived below.
\[ \boxed{ \begin{array}{rl} \\ \quad \mathrm{E}[\,L\,] &= \mathrm{E}[\,v^{K+1}-\Pi\,\ddot{a}_{\enclose{actuarial}{K+1}}\,] \quad \\ \quad &= \mathrm{E}[\,v^{K+1}\,]-\Pi\,\mathrm{E}[\,\ddot{a}_{\enclose{actuarial}{K+1}}\,] \quad \\ \quad &= A_x - \Pi\,\ddot{a}_x \\ \\ \end{array} } \]
- The Equivalence Principle is followed whenever \(E[\,L\,]=0\). This means that, the insurer will neither sustain a loss nor a gain at policy issue. Whenever the Equivalence Principle is used, the symbol \(\Pi\) is replaced by the appropriate symbol from the International Actuarial Notation. Thus, for the fully continuous case in #2 above,
\[ \boxed{ \begin{array}{rl} \\ \quad \Pi = \overline{P}(\bar{A}_x) = \dfrac{\bar{A}_x}{\bar{a}_x} \quad \\ \\ \end{array} } \]
and for the fully discrete case,
\[ \boxed{ \begin{array}{rl} \\ \quad \Pi = P_x = \dfrac{{A}_x}{\ddot{a}_x} \quad \\ \\ \end{array} } \] 5. The table below shows the fully discrete version of typical insurance plans, with a benefit of \(1\) payable at the end of the year of death, and annual level premiums payable at the beginning of each year that \((x)\) is alive (unless otherwise indicated).
| Plan | Premium (Equivalence Principle) |
|---|---|
| Whole life insurance | \(P_x=\dfrac{A_x}{\ddot{a}_x}\) |
| \(h\)-year payment whole life insurance | \({_h}P_x=\dfrac{A_x}{\ddot{a}_{x:{{\enclose{actuarial}{h}}}}}\) |
| \(n\)-year term insurance | \(P_{\overset{1}{x}:{{\enclose{actuarial}{n}}}}=\dfrac{A_{\overset{1}{x}:{{\enclose{actuarial}{n}}}}}{\ddot{a}_{x:{{\enclose{actuarial}{n}}}}}\) |
| \(n\)-year endowment insurance | \(P_{x:{{\enclose{actuarial}{n}}}}=\dfrac{A_{x:{{\enclose{actuarial}{n}}}}}{\ddot{a}_{x:{{\enclose{actuarial}{n}}}}}\) |
| \(h\)-year payment, \(n\)-year endowment insurance | \({_h}P_{x:{{\enclose{actuarial}{n}}}}=\dfrac{A_{x:{{\enclose{actuarial}{n}}}}}{\ddot{a}_{x:{{\enclose{actuarial}{h}}}}}\) |
| \(n\)-year deferred annuity | \(P({_n|}\ddot{a}_x)=\dfrac{v^n\,{_n}p_x\,\ddot{a}_{x+n}}{\ddot{a}_{x:{{\enclose{actuarial}{n}}}}}\) |
4.2 Solved Exercises
For a fully discrete whole life insurance issued to \((x)\) let \(L\) be the present value of the loss-at-issue random variable if the premium is calculated using the Equivalence Principle, and let \(L^*\) be the present value of the loss-at-issue random variable if the premium is \(0.015\).
You are given
\(d=0.05\)
\(A_x=0.175\)
\(\mathrm{Var}[\,L\,]=0.022\)
Calculate \(\mathrm{Var}[\,L^*\,]\)
Since \(L\) considers the Equivalence Principle, we will use:
\(\mathrm{Var}[\,L\,]= \dfrac{^2A_x-A^2_x}{\left (1 - A_x \right)^2}\)
We solve the above equation for \({^2}A_x\,\):
\[\begin{align} {^2}A_x &= A^2_x + (1- A_x)^2\,\mathrm{Var}[\,L\,] \\ \\ &= (0.175)^2 + (1 - 0.175)^2(0.022) \\ \\ &= 0.0456 \\ \\ \end{align}\]
Finally, we use the general formula of variance of the loss-at-issue random variable, to calculate \(\mathrm{Var}[\,L^*\,]\,\):
\[\begin{align} \mathrm{Var}[\,L^*\,] &= \left[ 1 + \dfrac{\pi}{d} \right]^2 \left( {^2}A_x - A^2_x \right) \\ \\ &= \left[ 1 + \dfrac{0.015}{0.05} \right]^2 \left( 0.0456 - 0.175^2 \right) \\ \\ &= 0.02531 \end{align}\]
\(\\\)
4.3 Supplementary Exercises
4.1
Let \(_{k|}q_{x}=0.1k\), \(k \in {0, 1, 2, 3, 4}\), and \(i=5\%\).
Calculate \(P_{0}\) assuming the equivalence principle.
4.2
Let \(_{k|}q_{x}=0.1k\), \(k \in {0, 1, 2, 3, 4}\), and \(i=5\%\).
Calculate \(\Pi_{0}\) such that the probability of incurring a financial loss is at most \(0.25\).
4.3
Let \(_{k|}q_{x}=0.79(0.21)^{k}\), \(k \in \mathbb{N}\), and \(i=10\%\).
Calculate the annual net premium under the equivalence principle for an ordinary whole life insurance of \(100\,000\) payable at the end of the year of death, issued at age \(35\).
4.4
Let \(q_{x}=0.01\) \(\forall x\), and \(i=10\%\).
Calculate \(\sigma_{L}\) for an ordinary whole life insurance policy of \(100\,000\) payable at the end of the year of death.
4.5
Let
\(0.0420\) be the annual premium of a 20-year endowment insurance issued at age \(35\);
\(0.0299\) be the annual premium of an ordinary whole life insurance, payable for 20 years, issued at age \(35\);
\(0.6099\) be the single net premium (PNU) of an ordinary whole life insurance issued at age \(55\).
In all the above cases, the insurances are fully discrete and the corresponding premiums are calculated under the equivalence principle.
Calculate the annual net premium of a 20-year term insurance issued at age \(35\).4.6
A fully discrete ordinary whole life insurance of \(500\,000\) with annual premiums payable for 30 years is issued at age \(20\).
If death occurs before age \(60\), all premiums paid are refunded with interest to the insured at the end of the year of death.
Let \(i=5\%\) and \(q_{x}=0.005\), \(\forall x \in \mathbb{N}\).
Calculate the annual net premium.
4.7
Let \(A_{x}=0.4\), \(\forall x \in \mathbb{N}\).
Assuming the equivalence principle, calculate \(\sigma_{L}\) for a fully discrete ordinary whole life insurance policy of \(200\,000\), issued at age \(30\).
4.8
A fully discrete 30-year term insurance of \(10\,000\) is issued at age \(20\).
The insurance also refunds premiums with interest to the insured if death occurs before age \(50\).
The annual net premium is \(317.46\).
Mortality is modeled with De Moivre’s law, where \(\omega=80\).
Calculate the interest rate.
4.9
An 8-year deferred certain annuity-due with annual payments of \(100\) is issued at age \(50\).
Annual premiums are paid during the deferment period and these premiums are refunded with interest to the insured at the end of the year of death, if death occurs during the deferment period.
Let \(i=10\%\) and \(q_{x}=0.01\), \(\forall x \in \mathbb{N}\).
Calculate the death benefit if death occurs during the fourth year.
4.10
Let \(_{k|}q_{x}=0.01\), \(k \in {0, 1, ⋯, 99}\) and \(i=10\%\).
Calculate the minimum annual net premium of a fully discrete ordinary whole life insurance of \(100\,000\) issued at age \(16\), such that the probability of financial loss for the insurer is less than \(30\%\).
- the insurances are fully discrete
- premiums are calculated under the equivalence principle
- calculation assumptions correspond to the Illustrative Life Table
4.11
Compute the reserve at \(k=10\) for an ordinary whole life insurance of \(120\,000\) issued to a person aged \(30\).
4.12
Compute the reserve at \(k=10\) for a 15-year term insurance of \(120\,000\) issued to a person aged \(30\).
4.13
Compute the reserve at \(k=10\) for a 15-year endowment insurance of \(120\,000\) issued to a person aged \(30\).
4.14
Compute the reserve at \(k=10\) for an ordinary whole life insurance of \(120\,000\) with premiums payable for 15 years, issued to a person aged \(30\).