This article provides an overview of actuarial and financial subjects related to Long Term Actuarial Mathematics, following the syllabus of LTAM exam given by the SOA.
1. Survival distributions
1.a. Survival/death probabilities
For someone age \((x)\), let’s \(T_x\) his (or her) future life time (or time-to-death) and \(K_x = \left\lfloor T_x \right\rfloor\) his (or her) curtate future lifetime, which measures the number of completed future years by \(x\) prior to death. While \(T_x\) is a continuous variable, \(K_x\) is a discrete one.
We define the survival function of \((x)\) at time \(t\) as the probability that \((x)\) survives \(t\) years. \[S_x(t) = P(T_x > t) = P(T_0 > x+t \mid T_0 > x) = \frac{S_0(x+t)}{S_0(x)}\] The first equality is the definition of survival function. The second one tells us that the chance that someone age \((x)\) survives \(t\) years is equal to the chance that someone who’s just born survives \(x+t\) years, given that he or she survives at least \(x\) years for sure.
Some trivial remarks: \(x\) survives certaintly at the considered moment: \(S_x(0) = 1\). Sadly, he or she can’t live forever: \(S_x(\infty) = 0\). Finally, \(S_x(.)\) is a non-increasing function, since it’s more likey for someone to survives 1 year than 10 years.
We’ll now introduce some actuarial notations that will be frequently used later in this course:
- The probability that \((x)\) survives \(t\) years: \(\space_tp_x = P(T_x >t) = S_x(t)\)
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The probability that \((x)\) dies within \(t\) years: \(\space_tq_x = P(T_x \le t) = F_x(t)\)
- The probability that \((x)\) survives \(u\) years and then dies within the following \(t\) years: \(\space_{u \mid t}q_x = P(u < T_x \le t+u) = P(T_x \le u+t \mid T_x > u) \space P(T_x > u)= \space_t q_{x+u} \space_up_x = \space_up_x - \space_{u+t}p_x = \space_{u+t}q_x - \space_{u}q_x\)
Remarks : \(\space_tp_x + \space_tq_x = 1\). When \(t=1\), we usually drop the \(t\) in the notations: \(p_x = \space_1p_x \), \(q_x = \space_1q_x \) and \(\space_{1\mid t}q_x = \space_{\mid t}q_x \). According to these notations, \(P(K_x = k) = P(k \le T_x < k+1) = \space_kp_x q_{x+k} = \space_{\mid k}q_x \).
In practice, these functions are hard to be expressed explicitely. Theirs values at a specific point \(t\) are usually derived from a life table which is a table of survivals number \((l_x)\) for different age \((x)\), with an arbitrary initial number \(l_0\).
\[\space_tp_x = \frac{l_{x+t}}{l_x}\] \[\space_tq_x = \frac{l_x - l_{x+t}}{l_x}=\frac{\space_td_x}{l_x}\] \[\space_{t\mid u}q_x = \frac{l_{x+u} - l_{x+u+t}}{l_x}=\frac{\space_td_{x+u}}{l_x}\]
1.b. Force of mortality
Loosely speaking, it measures the instantaneous death rate \[ \mu_{x+t} = \frac{f_x(t)}{S_x(t)} = \frac{-\frac{d}{dt} \space_tp_x}{\space_tp_x} \] Recalling that \(\frac{d}{dt}ln(X_t) = \frac{dX_t/dt}{Xt}\), with \(X_t = \space_tp_x\) and taking the integral on both sides, one can express the survival function as: \(\space_np_x = exp(-\int_0^n \mu_{x+t}dt )\). There are some other useful formulas where the force of mortality helps connect survival/death probabilities:
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The probability that \((x)\) survives \(t\) years is the “sum of all probabilities” that \(x\) survives \(s\) years and then suddenly dies, for all \(s \ge t\): \(\space_tp_x = \int_t^\infty \space_sp_x \mu_{x+s} \).
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The probability that \((x)\) dies within \(t\) years is the “sum of all probabilities” that \(x\) survives \(s\) years and then suddenly dies, for all \(s \le t\): \(\space_tq_x = \int_0^t \space_sp_x \mu_{x+s} \).
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The probability that \((x)\) survives \(u\) years and then dies within \(t\) following years is the “sum of all probabilities” that \(x\) survives \(u+s\) years and then suddenly dies, for all \(s \le t\): \(\space_{u\mid t}q_x = \int_u^{u+t} \space_sp_x \mu_{x+s} \).
1.c. Moments
Since we’ve made a distinction between the complete future file time \(T_x\) and the curtate future life time, we also consider the complete expectation and the curtate expectation.
Complete expectation
Using the definition of \(T_x\) and applying integration by parts, we have: \[ \mathring{e_x} = E[T_x] = \int_0^\infty t \space_tp_x \mu_{x+t}dt = \int_0^\infty \space_tp_x dt \]
\[ E[T_x^2] = \int_0^\infty t^2 \space_tp_x \mu_{x+t}dt = \int_0^\infty 2t \space_tp_x dt \]
\[Var(T_x) = E[T_x^2] - (\mathring{e_x})^2 \]
Curtate expectation
The discrete version for \(K_x\): \[e_x = E[K_x] = \sum_{k=0}^\infty k \space_{k\mid} q_{x} = \sum_{k=0}^\infty k \space_{k}p_x q_{x+k} = \sum_{k=1}^\infty \space_kp_x\]
\[E[K_x^2] = \sum_{k=0}^\infty k^2 \space_{k\mid} q_{x} = \sum_{k=0}^\infty k^2 \space_{k}p_x q_{x+k} = \sum_{k=1}^\infty (2k-1) \space_kp_x\]
\[Var(K_x) = E[K_x^2] - (e_x)^2 \]
Remark: The complete expectation is greater than the curtate expectation by 0.5 year: \(\mathring{e_x} \approx e_x + 0.5\). This result is not surprising and easy to remember (for example, if \(T_x\) is uniformely-distributed between 10 years (excluded) and 11 years (excluded), its mean \(\mathring{e_x}\) will be 10.5 while \(e_x = 10\)).
Sometimes, we have to work with temporary expectations \(\mathring{e_{x:\overline{n\mid}:}}\) and \(e_{x:\overline{n\mid}:} \). Their formulas are very similar to \(\mathring{e_x} \) and \(e_x\):
\[\mathring{e_{x:\overline{n\mid}:}} = \int_0^n t \space_tp_x \mu_{x+t} dt + n \space_n p_x = \int_0^n \space_tp_x dt \]
\[e_{x:\overline{n\mid}:} = \sum_{k=0}^{n-1} k \space_{k\mid} q_{x} + n \space_np_x = \sum_{k=1}^n \space_kp_x\]
The main difference from the complete expectations is the upper limit of \(n\) years in the calculation. If \((x)\) survives \(n\) years, the complete (or curtate) future life time chosen for calculation is \(n\), no matter when exactly \(x\) will die.
There are also some very useful recursive formulas that are worth remembering:
\[ \mathring{e_x} = \mathring{e_{x:\overline{n\mid}}} + \space_np_x \space \mathring{e}_{x+n} \]
\[ \mathring{e_{x:\overline{m+n\mid}}} = \mathring{e_{x:\overline{m\mid}}} + \space_mp_x \space \mathring{e}_{x+m : \overline{n \mid}} \] with \(n\) an arbitrary positive value. We have two similar formulas for \(e_x\).
1.d. Some mortality laws
We’ll begin with the two most basic mortality laws: the constant force of mortality and the uniform distribution.
Constant force of mortality
As its name states, \(\mu_x = \mu , \space \forall x \).
As a consequence,\( \space_tp_x = e^{-\mu t} \); \( \mathring{e_x} = \frac{1}{\mu} \) and \( \mathring{e_{x:\overline{n\mid}:}} = \frac{1 - e^{-\mu n}}{ \mu} \)
Uniform distribution
The mortality is uniformally distributed over the time, so that the number of survives age \((x)\) will be a (decreasing) linear function of \(x\): \(l_x = k (w-x)\). Note that \(w\) is the limit age, since there’s no survivals at age \((w)\): \(l_w = 0\).
Thus, \(\space_tp_x = \frac{l_{x+t}}{l_x} =\frac{w-x-t}{w-x} \) and \(\frac{d}{dt}\space_tp_x = \frac{-1}{w-x}\).
It follows that \( \mu_{x+t} = \dfrac{-\frac{d}{dt}\space_tp_x}{\space_tp_x} =\frac{1}{w-x-t}\). In particular, \(\mu_x = \frac{1}{w-x}\).
Note that \(\space_tq_x = \space_{u \mid t}q_x = \frac{t}{w-x}\). This formula illustrates the uniform property of the mortality for \(x\). The fact that \((x)\) survives \(u\) more years doesn’t change his or her probability of dying within the following \(t\) years.
Another interesting formula is \(\mathring{e_x} = \frac{w-x}{2}\). Intuitively, if the mortality is uniformely distributed for somone age \(x\), given that he or she could at best live up to age \(w\), his or her average number of survival years from now is \(\frac{w-x}{2}\).
What about \(\mathring{e_{x:n\mid}} \)? Well, if \((x)\) survives \(n\) years, it’s \(n\), otherwise it’s the mean of \([0,n]\), which is \(\frac{n}{2}\). So, \(\mathring{e_{x:n\mid}} = \space_np_x (n) + \space_nq_x \big( \frac{n}{2} \big) \).
We’ll now look at some other famous mortality laws, communly used in actuarial mathematics.
Beta distribution
\[l_x = k(w-x)^\alpha\] It’s an upgraded version of the uniform distribution. Or, more accurately, the uniform distribution is a special case of the beta distribution, with \(\alpha = 1\). We can show that \(\mu_{x+t}=\frac{\alpha}{w-(x+t)}\), \(\mu_{x}=\frac{\alpha}{w-x}\), \(\space_tp_x = \big( \frac{w-(x+t)}{w-x} \big)^\alpha \) and \(\mathring{e_x} = \frac{w-x}{\alpha + 1}\).
Gompetz’s law: \(\mu_x = Bc^x\) with \(B > 0\) and \(c > 1\).
Makeham’s law: \(\mu_x = A + Bc^x\) with \(0 < -A \le B \) and \(c > 1\).
1.e. Fractional ages
We talk about fractional ages when dealing with \(0 < t < 1\). There are two commun approaches, based on two assumptions: constant force of mortality or uniform distribution. Please note that these assumptions are made for \(0 < t < 1\), so don’t make confusions with mortality laws in section 1.d.
Constant force of mortality \( \rightarrow \) exponential interpolation \(l_{x+t} = (l_x)^{1-t}(l_{x+1})^t\). \[\space_tp_x = (p_x)^t\] \[\mu_{x+t}= \mu_x =-ln(p_x)\] \[f_x(t) = \space_tp_x \mu_{x+t} = e^{-t\mu_x} \mu_x \]
UDD \( \rightarrow \) linear interpolation \(l_{x+t} = (l_x)(1-t) + (l_{x+1})(t)\). \[\space_tq_x = t\space q_x\] \[\mu_{x+t}= \frac{-\frac{d}{dt}\space_tp_x}{\space_tp_x} = \frac{q_x}{1 - t \space q_x}\] \[f_x(t) = \space_tp_x \mu_{x+t} = q_x\]
1.f. Selected age and ultimate mortality
The age at which an insured is selected is denoted as \([ x ]\) and has an impact on his or her mortality. Insurer cannot apply a same mortality table to two same-age insured selected at different age. We denote \(q_{[x]+t}\) the select mortality with \(x\) the select age and \(t\) the number of years after selection. As stated earlier, \(q_{[x]+t} \ne q_{[x+\epsilon]+t-\epsilon}\) with \(\epsilon \ne 0\). After a certain number of years \(T\) from the selection, called select period, the select age \([x]\) won’t impact the mortality anymore: \(q_{[x]+t} = q_{x+t} , \forall t \ge T \).
The mortality table for different select age is constructed according to this principle, and is communly read from the left to the right, until reaching the select period, and then continue downwards.
The first chapter has given us some important mathematical concepts and notations that are useful for the following. We’ll now attack the core of this LTAM course: insurance contracts and annuities.
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