## 1. Introduction and Motivation

- In many countries, the shares of the elderly population are rapidly growing because of increasing life expectancy and low fertility rates.
- Household sizes are progressively reducing, and this causes a lack of assistance and care services provided to old members of the family inside the family itself.
- LTCI covers are rather recent products, and consequently, experience data are scanty; pricing and reserving problems then arise because of difficulties in the choice of appropriate technical bases.
- Significant safety loadings may be charged to policyholders, because of uncertainty in the biometric assumptions.
- High premiums (in particular due to safety loadings) constitute an obstacle to the diffusion of these products and especially stand-alone LTCI covers, which only provide “protection”.

- whether to allow or not for the inception-dependence of some probabilities (namely, the probability of recovery (see also Point 2 below) and the probability of death), which in the case of dependence are functions of both the insured’s current age and the time elapsed since the LTC claim (that is, the senescent disability inception);
- whether to take into account or not the possibility of recovery from LTC states.

- Inception dependence calls for a semi-Markov setting (instead of a Markov setting). Further, detailed statistical evidence is needed in order to estimate probabilities depending on both the attained age and the time elapsed since disability inception. Conversely, adopting a Markov model can lead to biased evaluations for short-duration disability. However, senescent disability is typically chronic, or anyway long lasting, so that significant biases should not be caused by a Markov setting. Further, it is interesting to note that a semi-Markov dependence structure can also be implemented within a Markov setting, via an appropriate redefinition of the state space; an interesting example is provided by the so-called Dutch model for disability annuities, described by [10].
- The prevailing chronic character of the LTC disability suggests disregarding the recovery possibility. Indeed, the frequency of recovery is particularly low if the benefit eligibility is restricted to severe LTC conditions. Of course, this results in a simpler Markov model.

- the assumptions about senescent disability, in terms of the probability of entering the LTC state(s);
- the age-pattern of mortality of people in LTC state(s).

## 2. Long-Term Care Insurance Products

#### 2.1. LTCI Products: A Classification

- products that pay out benefits with a predefined amount (usually, a lifelong annuity benefit); in particular:
- –
- a fixed-amount benefit;
- –
- a degree-related (or graded) benefit, i.e., a benefit, whose amount is graded according to the degree of disability, that is the severity of the disability itself (for example, assessed according to an ADL or IADL scale);

- products that provide reimbursement (usually partial) of nursing and medical expenses, i.e., expense-related benefits;
- care service benefits (for example, provided in the U.S. by the Continuing Care Retirement Communities; see [1] and the references therein).

#### 2.2. Fixed-Amount and Degree-Related Benefits

- the payment of a single premium;
- an immediate life annuity, whose annual benefit may be graded according to the disability severity.

- the accumulation phase, during which periodic premiums are (usually) paid; as an alternative, a single premium can be paid (of course, while the insured is in the healthy state);
- the payout period, during which LTC benefits (usually consisting of a life annuity) are paid in the case of LTC need.

- a lifelong LTC annuity (from the LTC claim on);
- a deferred life annuity (e.g., from age 80), while the insured is not in the LTC disability state;
- a lump sum benefit on death, which can alternatively be given by:
- (a)
- a fixed amount, stated in the policy;
- (b)
- the difference (if positive) between a stated amount and the amount paid as Benefit 1 and/or Benefit 2.

- an income protection cover (IP; see, for example, [1] and the references therein) during the working period, that is during the accumulation period related to LTC benefits;
- an LTC cover during the retirement period.

## 3. The Model

#### 3.1. Multi-State Models for LTCI

- a = active (i.e., healthy);
- i = incapacitated, or invalid (i.e., in the LTC state);
- d = died;

- $a\to i$ = disablement (i.e., entering the LTC state);
- $a\to d$ = death from the active state;
- $i\to d$ = death from the LTC state.

#### 3.2. Biometric Functions

- ${p}_{x}^{aa}=$ probability of being healthy at age $x+1$;
- ${p}_{x}^{ai}=$ probability of being an invalid at age $x+1$;
- ${q}_{x}^{aa}=$ probability of dying before age $x+1$ from state a;
- ${q}_{x}^{ai}=$ probability of dying before age $x+1$ from state i;
- ${q}_{x}^{a}=$ probability of dying before age $x+1$;
- ${w}_{x}=$ probability of becoming an invalid (disablement) before age $x+1$.

- ${p}_{x}^{i}=$ probability of being alive (and an invalid) at age $x+1$;
- ${q}_{x}^{i}=$ probability of dying before age $x+1$.

#### 3.3. Actuarial Values

- Actuarial value, for a healthy individual age x (i.e., in state a), of a life annuity providing a benefit of one monetary unit per annum, payable at the policy anniversaries, while the individual is disabled (i.e., in state i):$${a}_{x}^{ai}=\sum _{j=1}^{+\infty}{}_{j-1}{p}_{x}^{aa}\phantom{\rule{0.166667em}{0ex}}{p}_{x+j-1}^{ai}\phantom{\rule{0.166667em}{0ex}}{v}^{j}\phantom{\rule{0.166667em}{0ex}}{\ddot{a}}_{x+j}^{i}$$
- Actuarial value, for a disabled individual age $x+j$, of a life annuity providing a benefit of one monetary unit per annum, payable at the policy anniversaries, while the individual is disabled:$${\ddot{a}}_{x+j}^{i}=\sum _{h=j}^{+\infty}{v}^{h-j}\phantom{\rule{0.166667em}{0ex}}{}_{h-j}{p}_{x+j}^{i}$$
- Actuarial value, for a disabled individual age $x+j$, of a temporary life annuity providing a benefit of one monetary unit per annum, payable at the policy anniversaries, while the individual is disabled:$${\ddot{a}}_{x+j:s\rceil}^{i}=\sum _{h=j}^{j+s-1}{v}^{h-j}\phantom{\rule{0.166667em}{0ex}}{}_{h-j}{p}_{x+j}^{i}$$
- Actuarial value, for a healthy individual age x, of a life annuity of one monetary unit per annum, payable at the policy anniversaries, while the individual is healthy:$${\ddot{a}}_{x}^{aa}=\sum _{j=0}^{+\infty}\phantom{\rule{0.166667em}{0ex}}{v}^{j}\phantom{\rule{0.166667em}{0ex}}{}_{j}{p}_{x}^{aa}$$
- Actuarial value, for a healthy individual age x, of a temporary life annuity of one monetary unit per annum, payable at the policy anniversaries, while the individual is healthy:$${\ddot{a}}_{x:r\rceil}^{aa}=\sum _{j=0}^{r-1}\phantom{\rule{0.166667em}{0ex}}{v}^{j}\phantom{\rule{0.166667em}{0ex}}{}_{j}{p}_{x}^{aa}$$
- Actuarial value, for a healthy individual age x, of a deferred life annuity of one monetary unit per annum, payable at the policy anniversaries, while the individual is healthy:$${}_{n|}{\ddot{a}}_{x}^{aa}=\sum _{j=n}^{+\infty}\phantom{\rule{0.166667em}{0ex}}{v}^{j}\phantom{\rule{0.166667em}{0ex}}{}_{j}{p}_{x}^{aa}$$

#### 3.4. Technical Bases

- ${q}_{x}^{aa}$ is given by the first Heligman–Pollard law;
- ${w}_{x}$ is expressed by a specific parametric law;
- ${q}_{x}^{i}={q}_{x}^{aa}+\text{extra-mortality}$ (that is, an additive extra-mortality model is adopted).

- the (remaining) expected lifetime, ${\stackrel{\circ}{e}}_{x}$, at various ages x;
- the Lexis point, ${x}^{\left[\mathrm{L}\right]}$, i.e., the (old) age with the maximum probability of death for a newborn;
- the one-year probability of death, ${q}_{x}$, at various ages x.

- Parameter k expresses the LTC severity category, according to the OPCS scale (see [22]); in particular:
- $0\le k\le 5$ denotes a less severe LTC states, with no impact on mortality;
- $6\le k\le 10$ denotes a more severe LTC states, implying extra-mortality;

In the following calculations, we have assumed $k=8$, that is a severe LTC state (so that the possibility of recovery can be disregarded). Hence, ${q}_{x}^{i}={q}_{x}^{{i}^{\left(8\right)}}$ for all x. - According to [18], we have set $\alpha =0.10$, as we have assumed ${q}^{\left[\mathrm{standard}\right]}={q}_{x}^{aa}$ (that is, the mortality of insured healthy people).

## 4. Premiums

- Product P1: stand-alone LTC cover;
- Product P2: LTC acceleration benefit in a whole-life assurance;
- Product P3: LTC insurance package, including a deferred life annuity and a death benefit; in particular:
- Product P3a: Package a (fixed death benefit);
- Product P3b: Package b (decreasing death benefit);

- Product P4: enhanced pension.

- First, we note that LTCI covers that provide expense reimbursement have been excluded from our analysis. Expense-related benefits can provide a good coverage of LTC needs, but the characteristics of the related insurance products and the relevant actuarial problems significantly differ from the features of the products providing predefined benefits. Indeed, random amounts of LTC expenses imply, on the one hand, specific actuarial models and, on the other, the presence of policy conditions (i.e., fixed-percentage or fixed-amount deductible, limit amount, etc.) similar to those commonly adopted in non-life insurance. Finally, we note that the evaluation of expense-related benefits calls for an important collection of data, which, to some extent, are country specific and, hence, affected by a high degree of heterogeneity.
- As regards the LTCI products providing predefined benefits, we first note that two “extreme” products have been included in the analysis, i.e., the stand-alone cover (Product P1) and the whole-life assurance with LTCI as an acceleration benefit (Product P2). While the former only aims at contributing to cover the LTC needs, the latter has an important savings component and a rather limited LTC coverage.
- The two remaining products either include a life annuity component (Product P2) or are “derived” from a life annuity or pension product, thus aiming at covering the individual longevity risk.
- Although other products are sold on various insurance markets, the products we are addressing represent important LTCI market shares, and at the same time, their simple structures ease the sensitivity analysis and the interpretation of the results.

#### 4.1. Product P1: LTCI as a Stand-Alone Cover

#### 4.2. Product P2: LTCI as an Acceleration Benefit

#### 4.3. Product P3: LTCI in Life Insurance Package

- a life annuity with annual benefit ${b}^{\prime}$, deferred n years, while the individual is healthy;
- an LTC annuity with annual benefit ${b}^{\u2033}$;
- a death benefit C.

- a life annuity with annual benefit ${b}^{\prime}$, deferred n years, while the individual is healthy;
- an LTC annuity with annual benefit ${b}^{\u2033}$;
- a death benefit given by $max\{C-({z}^{\prime}{b}^{\prime}+{z}^{\u2033}{b}^{\u2033}),0\}$, where:
- ${z}^{\prime}$ = number of annual payments ${b}^{\prime}$;
- ${z}^{\u2033}$ = number of annual payments ${b}^{\u2033}$.

- Death in healthy state before time n:$${\Pi}_{x}^{\left(1\right)}=C\sum _{j=1}^{n}{}_{j-1}{p}_{x}^{aa}\phantom{\rule{0.166667em}{0ex}}{q}_{x+j-1}^{aa}\phantom{\rule{0.166667em}{0ex}}{v}^{j}$$In this case, we have: ${z}^{\prime}={z}^{\u2033}=0$.
- Death in healthy state after time n:$${\Pi}_{x}^{\left(2\right)}=\sum _{j=n+1}^{+\infty}{}_{j-1}{p}_{x}^{aa}\phantom{\rule{0.166667em}{0ex}}{q}_{x+j-1}^{aa}\phantom{\rule{0.166667em}{0ex}}max\{C-(j-n)\phantom{\rule{0.166667em}{0ex}}{b}^{\prime},0\}\phantom{\rule{0.166667em}{0ex}}{v}^{j}$$Then: ${z}^{\prime}=j-n,\phantom{\rule{4pt}{0ex}}{z}^{\u2033}=0$.
- Death in LTC state, entered before time n:$$\begin{array}{cc}\hfill {\Pi}_{x}^{\left(3\right)}=& \underset{\text{death}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{first}\phantom{\rule{4.pt}{0ex}}\text{LTC}\phantom{\rule{4.pt}{0ex}}\text{year}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\Rightarrow \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{no}\phantom{\rule{4.pt}{0ex}}\text{LTC}\phantom{\rule{4.pt}{0ex}}\text{benefit}\phantom{\rule{4.pt}{0ex}}\text{paid}}{\underbrace{\sum _{j=1}^{n}{}_{j-1}{p}_{x}^{aa}\phantom{\rule{0.166667em}{0ex}}{q}_{x+j-1}^{ai}\phantom{\rule{0.166667em}{0ex}}C\phantom{\rule{0.166667em}{0ex}}{v}^{j}}}\hfill \\ & +\underset{\text{death}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{second}\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{following}\phantom{\rule{4.pt}{0ex}}\text{LTC}\phantom{\rule{4.pt}{0ex}}\text{years}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\Rightarrow \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{LTC}\phantom{\rule{4.pt}{0ex}}\text{benefits}\phantom{\rule{4.pt}{0ex}}\text{paid}}{\underbrace{\sum _{j=1}^{n}{}_{j-1}{p}_{x}^{aa}\phantom{\rule{0.166667em}{0ex}}{p}_{x+j-1}^{ai}\phantom{\rule{0.166667em}{0ex}}{v}^{j}\left[\sum _{h=1}^{+\infty}{}_{h-1}{p}_{x+j}^{i}\phantom{\rule{4pt}{0ex}}{q}_{x+j+h-1}^{i}\phantom{\rule{0.166667em}{0ex}}max\{C-h\phantom{\rule{0.166667em}{0ex}}{b}^{\u2033},0\}\phantom{\rule{0.166667em}{0ex}}{v}^{h}\right]}}\hfill \end{array}$$In this case: ${z}^{\prime}=0,\phantom{\rule{4pt}{0ex}}{z}^{\u2033}=h$.
- Death in the LTC state, entered after time n:$$\begin{array}{cc}& {\Pi}_{x}^{\left(4\right)}=\phantom{\rule{4pt}{0ex}}{}_{n}{p}_{x}^{aa}\phantom{\rule{0.166667em}{0ex}}{v}^{n}[\underset{\text{death}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{first}\phantom{\rule{4.pt}{0ex}}\text{LTC}\phantom{\rule{4.pt}{0ex}}\text{year}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\Rightarrow \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{no}\phantom{\rule{4.pt}{0ex}}\text{LTC}\phantom{\rule{4.pt}{0ex}}\text{benefit}\phantom{\rule{4.pt}{0ex}}\text{paid}}{\underbrace{\sum _{j=1}^{+\infty}{}_{j-1}{p}_{x+n}^{aa}\phantom{\rule{0.166667em}{0ex}}{q}_{x+n+j-1}^{ai}\phantom{\rule{0.166667em}{0ex}}max\{C-j\phantom{\rule{0.166667em}{0ex}}{b}^{\prime},0\}\phantom{\rule{0.166667em}{0ex}}{v}^{j}}}\hfill \\ & +\underset{\text{death}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{second}\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{following}\phantom{\rule{4.pt}{0ex}}\text{LTC}\phantom{\rule{4.pt}{0ex}}\text{years}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\Rightarrow \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{LTC}\phantom{\rule{4.pt}{0ex}}\text{benefits}\phantom{\rule{4.pt}{0ex}}\text{paid}}{\underbrace{\sum _{j=1}^{+\infty}{}_{j-1}{p}_{x+n}^{aa}\phantom{\rule{0.166667em}{0ex}}{p}_{x+n+j-1}^{ai}\phantom{\rule{0.166667em}{0ex}}{v}^{j}\left[\sum _{h=1}^{+\infty}{}_{h-1}{p}_{x+n+j}^{i}\phantom{\rule{4pt}{0ex}}{q}_{x+n+j+h-1}^{i}\phantom{\rule{0.166667em}{0ex}}max\{C-(j\phantom{\rule{0.166667em}{0ex}}{b}^{\prime}+h\phantom{\rule{0.166667em}{0ex}}{b}^{\u2033}),0\}\phantom{\rule{0.166667em}{0ex}}{v}^{h}\right]}}]\hfill \end{array}$$In this case, we have: ${z}^{\prime}=j,\phantom{\rule{4pt}{0ex}}{z}^{\u2033}=h$.

#### 4.4. Product 4: the Enhanced Pension

## 5. Sensitivity Analysis

- probability of entering the LTC state (i.e., probability of disablement) ${\overline{w}}_{x}\left(\delta \right)$, defined as follows:$${\overline{w}}_{x}\left(\delta \right)=\delta \phantom{\rule{0.166667em}{0ex}}{w}_{x}$$
- extra-mortality of people in the LTC state, defined as follows:$$\overline{\Delta}(x;\lambda )=\lambda \phantom{\rule{0.166667em}{0ex}}\Delta (x,0.10,8)=\frac{\lambda \phantom{\rule{0.166667em}{0ex}}0.06}{1+1.{1}^{50-x}}$$$${q}_{x}^{i}\left(\lambda \right)={q}_{x}^{aa}+\overline{\Delta}(x;\lambda )$$

#### 5.1. Disablement Assumption

#### 5.2. Extra-Mortality Assumption

#### 5.3. Joint Sensitivity Analysis

## 6. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## References

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^{1.}Part of the contents of this paper was presented at the CEPAR Long-Term Care and Longevity Insurance Workshop (hosted by CEPAR and PwC), Sydney, 10 December 2014. A provisional version of the paper has been published in the CEPAR Working Papers Series; see [19].

**Figure 1.**A classification of LTCI products providing predefined benefits (source: [1]).

**Figure 3.**Function ${w}_{x}$ (i.e., probability of entering the LTC state, as a function of the attained age x): males.

**Figure 4.**Functions ${q}_{x}^{aa}$ and ${q}_{x}^{i}$ (i.e., mortality of healthy lives and LTC lives, as functions of the attained age x): males.

a | b | c | d | e | f | g | h |
---|---|---|---|---|---|---|---|

$0.00054$ | $0.01700$ | $0.10100$ | $0.00014$ | $10.72$ | $18.67$ | $2.00532\times {10}^{-6}$ | $1.13025$ |

${\stackrel{\circ}{e}}_{0}$ | ${\stackrel{\circ}{e}}_{40}$ | ${\stackrel{\circ}{e}}_{65}$ | ${x}^{\left[\mathrm{L}\right]}$ | ${q}_{0}$ | ${q}_{40}$ | ${q}_{80}$ |
---|---|---|---|---|---|---|

$85.128$ | $46.133$ | $22.350$ | 90 | $0.00682$ | $0.00029$ | $0.03475$ |

Parameter | Females | Males |
---|---|---|

A | $0.0017$ | $0.0017$ |

B | $1.0934$ | $1.1063$ |

C | $103.6000$ | $93.5111$ |

D | $0.9567$ | $0.6591$ |

E | n.a. | $70.3002$ |

**Table 4.**Product P1 (stand-alone); single premium ${\Pi}_{x}^{\left[\mathrm{P}1\right]}$ and annual level premium ${P}_{x:r\rceil}^{\left[\mathrm{P}1\right]}$; $b=100$.

Age x | Single Premium | Annual Level Premiums | ||
---|---|---|---|---|

$x+r=65$ | $x+r=70$ | $x+r=75$ | ||

40 | $480.4308$ | $\phantom{0}26.77075$ | $24.31464$ | $\phantom{0}22.83546$ |

50 | $513.5436$ | $\phantom{0}43.53108$ | $36.24584$ | $\phantom{0}32.41563$ |

60 | $516.4653$ | $113.69362$ | $64.93099$ | $\phantom{0}49.83906$ |

70 | $473.7323$ | $--$ | $--$ | $109.89082$ |

**Table 5.**Product P2 (whole life assurance with LTC acceleration benefit); single premiums ${\Pi}_{x}^{\left[\mathrm{WLA}\right]}$ and ${\Pi}_{x}^{\left[\mathrm{P}2\left(\mathrm{s}\right)\right]}$; $C=1000$.

Age x | Whole Life No Accel. | Whole Life with Acceleration Benefit | ||||
---|---|---|---|---|---|---|

$s=1$ | $s=2$ | $s=3$ | $s=4$ | $s=5$ | ||

40 | 471.5191 | 565.7242 | 561.1957 | 556.9116 | 552.8608 | 549.0326 |

50 | 560.2152 | 660.9139 | 655.7011 | 650.7873 | 646.1581 | 641.7995 |

60 | 654.6069 | 755.8798 | 750.0631 | 744.6104 | 739.5027 | 734.7218 |

**Table 6.**Product P3a (Package a); single premium ${\Pi}_{x}^{\left[\mathrm{P}3\mathrm{a}(x+n)\right]}$; $C=1000$, ${b}^{\prime}=50,\phantom{\rule{4pt}{0ex}}{b}^{\u2033}=150$.

Age x | $x+n=75$ | $x+n=80$ | $x+n=85$ |
---|---|---|---|

40 | 1007.413 | 0 970.5772 | 0 955.9357 |

50 | 1146.305 | 1098.1236 | 1078.9723 |

60 | 1275.446 | 1206.1263 | 1178.5728 |

70 | 1409.858 | 1285.7893 | 1236.4738 |

**Table 7.**Product 3b (Package b); single premium ${\Pi}_{x}^{\left[\mathrm{P}3\mathrm{b}(x+n)\right]}$; $C=1000$, ${b}^{\prime}=50,\phantom{\rule{4pt}{0ex}}{b}^{\u2033}=150$.

Age x | $x+n=75$ | $x+n=80$ | $x+n=85$ |
---|---|---|---|

40 | 713.8557 | 698.9712 | 694.7115 |

50 | 804.2394 | 784.7703 | 779.1985 |

60 | 883.1407 | 855.1300 | 847.1139 |

70 | 952.4602 | 902.3264 | 887.9789 |

**Table 8.**Product P4 (enhanced pension); reduced benefit ${b}^{\prime}$, in order to obtain a given LTC benefit ${b}^{\u2033}$ ($b=100$).

Age x | ${\Pi}_{x}^{\left[\mathrm{P}4({b}^{\prime},{b}^{\u2033})\right]}={\Pi}_{x}^{\left[\mathrm{SP}\left(b\right)\right]}$ | ${b}^{\u2033}=150$ | ${b}^{\u2033}=200$ | ${b}^{\u2033}=250$ |
---|---|---|---|---|

60 | 1761.478 | 79.259 | 58.517 | 37.776 |

65 | 1522.646 | 75.824 | 51.649 | 27.473 |

70 | 1278.444 | 70.565 | 41.130 | 11.695 |

δ | ${\Pi}_{50}^{\left[\mathrm{P}1\right]}(\delta ,1)$ | ${\rho}_{50}^{\left[\mathrm{P}1\right]}(\delta ,1)$ |
---|---|---|

$0.0$ | $\phantom{00}0.00000$ | $0.0000000$ |

$0.1$ | $\phantom{0}97.44457$ | $0.1897494$ |

$0.2$ | $176.07799$ | $0.3428686$ |

$0.3$ | $241.25240$ | $0.4697798$ |

$0.4$ | $296.47515$ | $0.5773125$ |

$0.5$ | $344.12555$ | $0.6700999$ |

$0.6$ | $385.86840$ | $0.7513839$ |

$0.7$ | $422.90118$ | $0.8234961$ |

$0.8$ | $456.10675$ | $0.8881558$ |

$0.9$ | $486.15044$ | $0.9466585$ |

$1.0$ | $513.54361$ | $1.0000000$ |

$1.1$ | $538.68628$ | $1.0489592$ |

$1.2$ | $561.89632$ | $1.0941550$ |

$1.3$ | $583.42997$ | $1.1360865$ |

$1.4$ | $603.49644$ | $1.1751610$ |

$1.5$ | $622.26854$ | $1.2117151$ |

$1.6$ | $639.89052$ | $1.2460296$ |

$1.7$ | $656.48397$ | $1.2783412$ |

$1.8$ | $672.15229$ | $1.3088514$ |

$1.9$ | $686.98406$ | $1.3377327$ |

$2.0$ | $701.05581$ | $1.3651339$ |

δ | ${\Pi}_{50}^{\left[\mathrm{P}2\left(1\right)\right]}(\delta ,1)$ | ${\rho}_{50}^{\left[\mathrm{P}2\left(1\right)\right]}(\delta ,1)$ | ${\Pi}_{50}^{\left[\mathrm{P}2\left(5\right)\right]}(\delta ,1)$ | ${\rho}_{50}^{\left[\mathrm{P}2\left(5\right)\right]}(\delta ,1)$ |
---|---|---|---|---|

$0.0$ | $492.1453$ | $0.7446436$ | $492.1453$ | $0.7668209$ |

$0.1$ | $522.4302$ | $0.7904664$ | $517.9195$ | $0.8069802$ |

$0.2$ | $547.3508$ | $0.8281727$ | $539.5114$ | $0.8406230$ |

$0.3$ | $568.3981$ | $0.8600184$ | $558.0108$ | $0.8694472$ |

$0.4$ | $586.5416$ | $0.8874705$ | $574.1426$ | $0.8945825$ |

$0.5$ | $602.4415$ | $0.9115280$ | $588.4118$ | $0.9168156$ |

$0.6$ | $616.5641$ | $0.9328964$ | $601.1825$ | $0.9367139$ |

$0.7$ | $629.2483$ | $0.9520882$ | $612.7241$ | $0.9546971$ |

$0.8$ | $640.7467$ | $0.9694859$ | $623.2411$ | $0.9710837$ |

$0.9$ | $651.2520$ | $0.9853810$ | $632.8914$ | $0.9861200$ |

$1.0$ | $660.9139$ | $1.0000000$ | $641.7995$ | $1.0000000$ |

$1.1$ | $669.8509$ | $1.0135223$ | $650.0652$ | $1.0128789$ |

$1.2$ | $678.1584$ | $1.0260919$ | $657.7693$ | $1.0248828$ |

$1.3$ | $685.9139$ | $1.0378264$ | $664.9783$ | $1.0361152$ |

$1.4$ | $693.1814$ | $1.0488226$ | $671.7475$ | $1.0466625$ |

$1.5$ | $700.0145$ | $1.0591615$ | $678.1234$ | $1.0565969$ |

$1.6$ | $706.4581$ | $1.0689111$ | $684.1455$ | $1.0659801$ |

$1.7$ | $712.5507$ | $1.0781294$ | $689.8475$ | $1.0748645$ |

$1.8$ | $718.3251$ | $1.0868664$ | $695.2586$ | $1.0832956$ |

$1.9$ | $723.8097$ | $1.0951649$ | $700.4040$ | $1.0913127$ |

$2.0$ | $729.0293$ | $1.1030626$ | $705.3059$ | $1.0989504$ |

**Table 11.**Products P3a, P3b (insurance packages); $x=50,\phantom{\rule{4pt}{0ex}}C=1\phantom{\rule{0.166667em}{0ex}}000$, ${b}^{\prime}=50,{b}^{\u2033}=150$.

δ | ${\Pi}_{50}^{\left[\mathrm{P}3\mathrm{a}\left(80\right)\right]}(\delta ,1)$ | ${\rho}_{50}^{\left[\mathrm{P}3\mathrm{a}\left(80\right)\right]}(\delta ,1)$ | ${\Pi}_{50}^{\left[\mathrm{P}3\mathrm{b}\left(80\right)\right]}(\delta ,1)$ | ${\rho}_{50}^{\left[\mathrm{P}3\mathrm{b}\left(80\right)\right]}(\delta ,1)$ |
---|---|---|---|---|

$0.0$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}700.5211$ | $0.6379255$ | $524.3054$ | $0.6681005$ |

$0.1$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}762.7792$ | $0.6946205$ | $564.2116$ | $0.7189513$ |

$0.2$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}816.5343$ | $0.7435723$ | $598.8261$ | $0.7630591$ |

$0.3$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}863.9507$ | $0.7867518$ | $629.5434$ | $0.8022009$ |

$0.4$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}906.4564$ | $0.8254594$ | $657.2615$ | $0.8375209$ |

$0.5$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}945.0332$ | $0.8605891$ | $682.5844$ | $0.8697888$ |

$0.6$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}980.3808$ | $0.8927781$ | $705.9351$ | $0.8995436$ |

$0.7$ | $1013.0142$ | $0.9224956$ | $727.6214$ | $0.9271776$ |

$0.8$ | $1043.3239$ | $0.9500969$ | $747.8754$ | $0.9529864$ |

$0.9$ | $1071.6132$ | $0.9758584$ | $766.8772$ | $0.9771996$ |

$1.0$ | $1098.1236$ | $1.0000000$ | $784.7703$ | $1.0000000$ |

$1.1$ | $1123.0514$ | $1.0227003$ | $801.6718$ | $1.0215369$ |

$1.2$ | $1146.5586$ | $1.0441071$ | $817.6790$ | $1.0419342$ |

$1.3$ | $1168.7817$ | $1.0643443$ | $832.8740$ | $1.0612966$ |

$1.4$ | $1189.8365$ | $1.0835178$ | $847.3271$ | $1.0797136$ |

$1.5$ | $1209.8231$ | $1.1017185$ | $861.0993$ | $1.0972629$ |

$1.6$ | $1228.8288$ | $1.1190259$ | $874.2436$ | $1.1140122$ |

$1.7$ | $1246.9299$ | $1.1355096$ | $886.8072$ | $1.1300214$ |

$1.8$ | $1264.1943$ | $1.1512313$ | $898.8317$ | $1.1453438$ |

$1.9$ | $1280.6825$ | $1.1662462$ | $910.3545$ | $1.1600268$ |

$2.0$ | $1296.4487$ | $1.1806036$ | $921.4091$ | $1.1741132$ |

δ | ${b}^{\prime}(\delta ,1)$ | ${\rho}_{x}^{\left[\mathrm{P}4\right]}(\delta ,1)$ |
---|---|---|

$0.0$ | $100.00000$ | $0.7582433$ |

$0.1$ | $\phantom{0}96.96404$ | $0.7819840$ |

$0.2$ | $\phantom{0}94.13166$ | $0.8055136$ |

$0.3$ | $\phantom{0}91.47026$ | $0.8289506$ |

$0.4$ | $\phantom{0}88.95221$ | $0.8524165$ |

$0.5$ | $\phantom{0}86.55461$ | $0.8760288$ |

$0.6$ | $\phantom{0}84.25873$ | $0.8998988$ |

$0.7$ | $\phantom{0}82.04926$ | $0.9241317$ |

$0.8$ | $\phantom{0}79.91365$ | $0.9488283$ |

$0.9$ | $\phantom{0}77.84153$ | $0.9740858$ |

$1.0$ | $\phantom{0}75.82433$ | $1.0000000$ |

$1.1$ | $\phantom{0}73.85486$ | $1.0266668$ |

$1.2$ | $\phantom{0}71.92708$ | $1.0541833$ |

$1.3$ | $\phantom{0}70.03587$ | $1.0826500$ |

$1.4$ | $\phantom{0}68.17685$ | $1.1121713$ |

$1.5$ | $\phantom{0}66.34626$ | $1.1428576$ |

$1.6$ | $\phantom{0}64.54086$ | $1.1748267$ |

$1.7$ | $\phantom{0}62.75783$ | $1.2082052$ |

$1.8$ | $\phantom{0}60.99468$ | $1.2431301$ |

$1.9$ | $\phantom{0}59.24927$ | $1.2797513$ |

$2.0$ | $\phantom{0}57.51967$ | $1.3182330$ |

λ | ${\Pi}_{50}^{\left[\mathrm{P}1\right]}(1,\lambda )$ | ${\rho}_{50}^{\left[\mathrm{P}1\right]}(1,\lambda )$ |
---|---|---|

$0.0$ | $855.7094$ | $1.6662838$ |

$0.1$ | $806.6737$ | $1.5707987$ |

$0.2$ | $761.9567$ | $1.4837234$ |

$0.3$ | $721.0856$ | $1.4041370$ |

$0.4$ | $683.6467$ | $1.3312339$ |

$0.5$ | $649.2769$ | $1.2643073$ |

$0.6$ | $617.6576$ | $1.2027364$ |

$0.7$ | $588.5080$ | $1.1459748$ |

$0.8$ | $561.5807$ | $1.0935405$ |

$0.9$ | $536.6571$ | $1.0450079$ |

$1.0$ | $513.5436$ | $1.0000000$ |

$1.1$ | $492.0686$ | $0.9581828$ |

$1.2$ | $472.0797$ | $0.9192592$ |

$1.3$ | $453.4411$ | $0.8829652$ |

$1.4$ | $436.0319$ | $0.8490650$ |

$1.5$ | $419.7439$ | $0.8173482$ |

$1.6$ | $404.4804$ | $0.7876263$ |

$1.7$ | $390.1547$ | $0.7597305$ |

$1.8$ | $376.6889$ | $0.7335090$ |

$1.9$ | $364.0128$ | $0.7088255$ |

$2.0$ | $352.0634$ | $0.6855570$ |

λ | ${\Pi}_{50}^{\left[\mathrm{P}2\left(1\right)\right]}(1,\lambda )$ | ${\rho}_{50}^{\left[\mathrm{P}2\left(1\right)\right]}(1,\lambda )$ | ${\Pi}_{50}^{\left[\mathrm{P}2\left(5\right)\right]}(1,\lambda )$ | ${\rho}_{50}^{\left[\mathrm{P}2\left(5\right)\right]}(1,\lambda )$ |
---|---|---|---|---|

$0.0$ | $660.9139$ | 1 | $640.3371$ | $0.9977214$ |

$0.1$ | $660.9139$ | 1 | $640.4879$ | $0.9979563$ |

$0.2$ | $660.9139$ | 1 | $640.6376$ | $0.9981896$ |

$0.3$ | $660.9139$ | 1 | $640.7863$ | $0.9984213$ |

$0.4$ | $660.9139$ | 1 | $640.9341$ | $0.9986515$ |

$0.5$ | $660.9139$ | 1 | $641.0808$ | $0.9988801$ |

$0.6$ | $660.9139$ | 1 | $641.2265$ | $0.9991071$ |

$0.7$ | $660.9139$ | 1 | $641.3712$ | $0.9993326$ |

$0.8$ | $660.9139$ | 1 | $641.5150$ | $0.9995566$ |

$0.9$ | $660.9139$ | 1 | $641.6577$ | $0.9997791$ |

$1.0$ | $660.9139$ | 1 | $641.7995$ | $1.0000000$ |

$1.1$ | $660.9139$ | 1 | $641.9404$ | $1.0002194$ |

$1.2$ | $660.9139$ | 1 | $642.0802$ | $1.0004374$ |

$1.3$ | $660.9139$ | 1 | $642.2191$ | $1.0006538$ |

$1.4$ | $660.9139$ | 1 | $642.3571$ | $1.0008688$ |

$1.5$ | $660.9139$ | 1 | $642.4941$ | $1.0010822$ |

$1.6$ | $660.9139$ | 1 | $642.6302$ | $1.0012943$ |

$1.7$ | $660.9139$ | 1 | $642.7653$ | $1.0015048$ |

$1.8$ | $660.9139$ | 1 | $642.8995$ | $1.0017140$ |

$1.9$ | $660.9139$ | 1 | $643.0328$ | $1.0019216$ |

$2.0$ | $660.9139$ | 1 | $643.1652$ | $1.0021279$ |

**Table 15.**Products P3a, P3b (insurance packages); $x=50,\phantom{\rule{4pt}{0ex}}C=1000$, ${b}^{\prime}=50,{b}^{\u2033}=150$.

λ | ${\Pi}_{50}^{\left[\mathrm{P}3\mathrm{a}\left(80\right)\right]}(1,\lambda )$ | ${\rho}_{50}^{\left[\mathrm{P}3\mathrm{a}\left(80\right)\right]}(1,\lambda )$ | ${\Pi}_{50}^{\left[\mathrm{P}3\mathrm{b}\left(80\right)\right]}(1,\lambda )$ | ${\rho}_{50}^{\left[\mathrm{P}3\mathrm{b}\left(80\right)\right]}(1,\lambda )$ |
---|---|---|---|---|

$0.0$ | $1373.1426$ | $1.2504444$ | $1\phantom{\rule{0.166667em}{0ex}}030.1514$ | $1.3126789$ |

$0.1$ | $1333.7360$ | $1.2145591$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}992.0364$ | $1.2641106$ |

$0.2$ | $1297.7979$ | $1.1818323$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}957.9426$ | $1.2206663$ |

$0.3$ | $1264.9490$ | $1.1519186$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}927.4057$ | $1.1817544$ |

$0.4$ | $1234.8573$ | $1.1245157$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}900.0200$ | $1.1468579$ |

$0.5$ | $1207.2314$ | $1.0993584$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}875.4306$ | $1.1155246$ |

$0.6$ | $1181.8156$ | $1.0762136$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}853.3264$ | $1.0873583$ |

$0.7$ | $1158.3843$ | $1.0548760$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}833.4345$ | $1.0620108$ |

$0.8$ | $1136.7389$ | $1.0351648$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}815.5147$ | $1.0391763$ |

$0.9$ | $1116.7039$ | $1.0169200$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}799.3555$ | $1.0185853$ |

$1.0$ | $1098.1236$ | $1.0000000$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}784.7703$ | $1.0000000$ |

$1.1$ | $1080.8603$ | $0.9842793$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}771.5943$ | $0.9832104$ |

$1.2$ | $1064.7915$ | $0.9696463$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}759.6816$ | $0.9680305$ |

$1.3$ | $1049.8081$ | $0.9560017$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}748.9029$ | $0.9542957$ |

$1.4$ | $1035.8128$ | $0.9432570$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}739.1434$ | $0.9418596$ |

$1.5$ | $1022.7189$ | $0.9313331$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}730.3010$ | $0.9305921$ |

$1.6$ | $1010.4485$ | $0.9201591$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}722.2849$ | $0.9203775$ |

$1.7$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}998.9319$ | $0.9096716$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}715.0140$ | $0.9111125$ |

$1.8$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}988.1065$ | $0.8998136$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}708.4160$ | $0.9027050$ |

$1.9$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}977.9161$ | $0.8905337$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}702.4263$ | $0.8950725$ |

$2.0$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}968.3098$ | $0.8817858$ | $\phantom{0}\phantom{\rule{0.166667em}{0ex}}696.9867$ | $0.8881411$ |

λ | ${b}^{\prime}(1,\lambda )$ | ${\rho}_{x}^{\left[\mathrm{P}4\right]}(1,\lambda )$ |
---|---|---|

$0.0$ | $62.34898$ | $1.2161277$ |

$0.1$ | $64.17119$ | $1.1815946$ |

$0.2$ | $65.86125$ | $1.1512738$ |

$0.3$ | $67.43103$ | $1.1244723$ |

$0.4$ | $68.89119$ | $1.1006390$ |

$0.5$ | $70.25128$ | $1.0793302$ |

$0.6$ | $71.51992$ | $1.0601847$ |

$0.7$ | $72.70488$ | $1.0429056$ |

$0.8$ | $73.81315$ | $1.0272469$ |

$0.9$ | $74.85106$ | $1.0130027$ |

$1.0$ | $75.82433$ | $1.0000000$ |

$1.1$ | $76.73813$ | $0.9880920$ |

$1.2$ | $77.59716$ | $0.9771534$ |

$1.3$ | $78.40567$ | $0.9670771$ |

$1.4$ | $79.16755$ | $0.9577704$ |

$1.5$ | $79.88630$ | $0.9491531$ |

$1.6$ | $80.56513$ | $0.9411556$ |

$1.7$ | $81.20698$ | $0.9337169$ |

$1.8$ | $81.81451$ | $0.9267834$ |

$1.9$ | $82.39015$ | $0.9203081$ |

$2.0$ | $82.93615$ | $0.9142494$ |

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).