Assuming **continuous rating** the insured is **always worse off with backdating** on a net basis. A net basis assumes products are priced using the equivalence principle, i.e. at expected cost.

**Heuristic**: You are paying for coverage during the deferral period that you know you won’t use. To a second order you are also paying premium that you may not have been required to pay if you had died during the deferral period. Both of these represent costs to the insured. If, instead, the insured opted to buy a policy at their rated age their then expected cash flows from the policy would be zero, by the equivalence principle. Therefore the insured is worse off with backdating **in this situation**.

If rating is discrete then it is possible the insured could be better off because the expected present value at issue for the insured will vary with exact age. I am trying to find a life actuary to talk with to understand how this is actually handled in practice.

Consider a fully continuous whole life policy on \((x)\), a life aged \(x\). Fully continuous means premium is paid continuously and benefits are paid at the moment of death. The age \(x\) need not be an integer. The insurer prices using exact age.

Assume the rate of interest is \(i\), so the force of interest is \(\delta=\log(1+i)\), the discount factor is \(v=(1+i)^{-1}\) and the rate of discount is \(d=iv=1-v\).

The expected present value of a benefit of 1, payable at the moment of death, is \(\bar A_x\) and it is computed as \[ \bar A_x = \int_0^\infty e^{-\delta t}{{}_tp_x}\mu_{x+t} dt \] where \({{}_tp_x}\) is the probability \((x)\) survives \(t\) years, and \(\mu_x\) is the force of mortality.

For those not familiar with life contingencies notation: the force of mortality, aka the hazard rate, is \(\mu_x = -d\log({{}_tp_x})/dt = \lim_{t\downarrow 0} {}_tq_x/t\) where \({}_tq_x=1-{{}_tp_x}\). Therefore \({{}_tp_x}=\exp(-\int_0^t \mu_{x+s}ds)\). Observe that \({{}_tp_x} \mu_{x+t}\) is the density of death at time \(t\). This can be understood by seeing the probability of death for \(t\in[t, t+dt]\) is equal to the probability of surviving to time \(t\), i.e. \({{}_tp_x}\), times the probability of dying in the next instant \(dt\), which is \({}_{dt}q_{x+t}\approx dt\mu_{x+t}\).

The expected present value of premium is \(\bar a_x\), which can be computed as \[ \bar a_x = \frac{1-\bar A_x}{\delta} = \int_0^\infty e^{-\delta t}{{}_tp_x} dt. \]

The premium for a fully continuous whole life issued to \((x)\) is \(\bar P_x:=\bar A_x / \bar a_x\), by the equivalence principle.

Finally let \(\bar A^{1}_{x:\lcroof{n}}\) be the expected present value of an \(n\) year term life policy on \((x)\) with benefit paid at the moment of death, \(\bar a_{x:\lcroof{n}}\) the expected present value of an \(n\) year continuous annuity on \((x)\), and \(\bar a_{\lcroof{n}}\) the present value of an \(n\) year continuous annuity certain.

Let \(B(b)\) represent the insured’s expected present value of benefits less premium paid, evaluated at time \(t=0\) assuming the policy is backdated \(b\) years. Note \(t=0\) represents the date to which the policy is backdated. In practice \(0<b<1\) but we will work with a any \(b>0\). It is implicit the insured is actually alive at time \(b\), i.e. they have achieved age \(x+b\). We want to compute \(B(b)\) and show it is negative.

To be clear, the time line is as follows:

- At time \(t=0\) the insured is aged \(x\)
- At time \(t=b\) the insured is aged \(x+b\) and purchases a policy backdated by \(b\) years to \(t=0\). The rated age on the backdated policy is \(x\).
- \(B(b)\) is the present value of benefits less premium paid on the backdated policy, at time \(t=0\)
- If the insured purchased a policy at age \(x+b\) the present value of premium would equal the present value of benefits by the equivalent principle.
- The insured is better off with backdating if \(B(b)>0\) and worse off if \(B(b)<0\).

By definition of \(B(b)\) we have \[\begin{align*} B(b) &= v^b\bar A_{x+b} - \bar P_x(\bar a_{\lcroof{b}} + v^b\bar a_{x+b}) \end{align*}\] where the first term is the value of the insurance benefit at time of issue (age \(x+b\)), present valued back \(b\) years to time \(t=0\). The second term is the present value of the premium paid. Premium is paid at a rate of \(\bar P_x\) per year. For the first \(b\) years the premium is paid with certainty. Thereafter it becomes a continuous life annuity on \((x+b)\), which again has to be discounted to time zero. From here a calculation shows \[\begin{align*} B(b) &= -\bar P_x\bar a_{\lcroof{b}} + v^b(\bar A_{x+b} - \bar P_x\bar a_{x+b})\\ &= -\bar P_x\bar a_{\lcroof{b}} + v^b(\bar A_{x+b} - \bar P_{x+b}\bar a_{x+b}+ \bar P_{x+b}\bar a_{x+b} - \bar P_x\bar a_{x+b})\\ &= -\bar P_x\bar a_{\lcroof{b}} + v^b(\bar P_{x+b} - \bar P_x)\bar a_{x+b} \end{align*}\] where \(\bar A_{x+b} - \bar P_{x+b}\bar a_{x+b}=0\) because \(\bar P_{x+b}\) is determined using the equivalence principle. The last formula displays the benefit of backdating as a cost from the additional premium paid up-front (the annuity certain term) plus an offsetting benefit equal to the expected present value at \(t=0\) of the difference in premium, \(P_{x+b}-P_x > 0\), paid for life from age \(x+b\). It is not clear from this formula whether the benefit of lower on-going premium will offset the up-front “buy-in” cost.

The buy-in cost \((\bar P_{x+b} - \bar P_x)\bar a_{x+b}=\bar A_{x+b}-\bar P_x \bar a_{x+b}=: {}_b\bar V\) where \({}_b\bar V\) is the policy value at time \(b\). The policy value is defined as the expected present value of future benefits minus expected present value of future premiums per insured alive at time \(b\). Policy values are used by life insurance companies to set reserves for unpaid liabilities and are generally the largest liability item on their balance sheets. Computing reserves is a very important actuarial function. For a whole life policy reserves are always positive, and so this formulation does not help determine the sign of \(B(b)\).

Using the fact that \(\bar a_x = (1-\bar A_x)/\delta\) it is easy to see that \({}_b\bar V=1-\bar a_{x+b}/\bar a_x\) and that \(\bar P_x = (1/\bar a_x)-\delta\). Recall also that \(\bar a_{\lcroof{b}}=(1-v^b)/\delta\). Using these relationships we can write \[\begin{align*}
B(b) &= -\bar P_x\bar a_{\lcroof{b}} + v^b {}_b\bar V \\
&= 1 - \frac{1}{\bar a_x}\left( \bar a_{\lcroof{b}}+v^b\bar a_{x+b} \right).
\end{align*}\] Finally, using the recursion formula (condition on death in first \(b\) years) \(\bar a_x = \bar a_{x:\lcroof{b}} + v^b{}_bp_x\bar a_{x+b}\), and multiplying through by \(\bar a_x\) gives \[\begin{align*}
\bar a_x B(b) &= (\bar a_x - \bar a_{\lcroof{b}}) - v^b\bar a_{x+b} \\
&= (\bar a_{x:\lcroof{b}}-\bar a_{\lcroof{b}}) - v^b{}_bq_x\bar a_{x+b} \\
&< 0
\end{align*}\] where the first term is negative because the first annuity is contingent and the second is certain and the second term is obviously negative. Therefore **backdating represents a net economic cost to the insured for a fully continuous whole life policy**. The two terms in the equation have a natural economic meaning.

- \((\bar a_x - \bar a_{\lcroof{b}})/\bar a_x\) captures the expected difference in net present value from paying premiums for certain during the \(b\) year backdating period (without backdating the insured may have died and not had to pay these premiums). For small \(b\) this term is a second order effect. A calculation shows the term has approximate value \(b^2 \mu_x /2\).
- \(v^b{}_bq_x\bar a_{x+b} /\bar a_x=v^b{}_bq_x(1-{}_b\bar V)\). The term \((1-{}_b\bar V)\) is called the death strain at risk. It represents the unfunded amount, above the policy value, the insurer has to pay in the event the insured dies. Therefore this term computes the lost value to the insured from the fact they are paying for insurance they know they will not need. To a first order approximation, again for small \(b\), this term has value \(b\mu_x v^b\), and is the dominant term in \(B(b)\).

Approximating both terms to a first order \(B(b)\approx -b\mu_xv^b \approx {-}_bq_xv^b = -A^{\, 1}_{x:\lcroof{b}}\) is the value of term life for \(b\) years, reflecting the principal loss to the insured of paying for unusable life insurance. The over-payment of premium (the first term) is a second order effect.

How material are these differences? An illustrative life table is given in “Actuarial Mathematics for Life Contingent Risks” by Dickson, Hardy and Waters as a Makeham mortality law \(\mu_x = A+Bc^x\) where \(A=0.00022\), \(B=2.7\times 10^{-6}\) and \(c=1.124\). Using these values we can compute the tables below. The tables assumes

- Life aged 40 and life aged 75
- Interest rate 5%
- All amounts are shown for a $1 million face value policy, over the whole life of the policy
- \(b\) taking values 1 month through 1 year and then various longer intervals. \(b\) is in years
- Ann shows the cost to the insured from the first term, the difference of annuity values \((\bar a_x - \bar a_{\lcroof{b}})/\bar a_x\)
- \(V\) shows the cost to the insured from the second term, \(v^b{}_bq_x(1-{}_b\bar V)\) coming from the policy value
- Total shows the total cost \(Ann+V\)
- \(\bar A_{x+b}\) shows the value of whole life policy on \((x+b)\)
- \(P_{x+b}-P_x\) shows the difference in annual premiums for the USD 1 million policy
- Buy in shows the amount of the buy in: the catch-up premium for the first \(b\) year(s)
- For reference
- \(10^6A_{40}=121059.21\), \(\bar a_{40}=17.95\) and \(\bar P_{40}=6908.82\).
- \(10^6A_{75}=508676.91\), \(\bar a_{75}=9.81\) and \(\bar P_{75}=53123.19\)

- Pct shows the total cost to the insured as a percentage of the economic value of a non-backdated whole life policy
- As expected, for small values of \(b\) the Ann term is a second order compared to the \(V\) term
- Amounts show the net present value at time \(t=0\) (i.e. the backdated date) of the difference in economics for the insured who backdates over one who does not. As expected all the values are negative
- No expenses or consideration of lapse

\(b\) | Ann | \(V\) | Total | \(A_{x+b}\) | Pct | \(P_{x+b}-P_x\) | Buy In |
---|---|---|---|---|---|---|---|

0.08 | -0.10 | -42.40 | -42.50 | 124506.52 | -0.03 | 29.77 | 574.57 |

0.17 | -0.39 | -84.64 | -85.04 | 124976.20 | -0.07 | 59.69 | 1146.80 |

0.25 | -0.88 | -126.73 | -127.62 | 125447.61 | -0.10 | 89.74 | 1716.71 |

0.33 | -1.57 | -168.67 | -170.24 | 125920.75 | -0.14 | 119.94 | 2284.31 |

0.42 | -2.45 | -210.46 | -212.91 | 126395.63 | -0.17 | 150.28 | 2849.61 |

0.50 | -3.53 | -252.09 | -255.62 | 126872.24 | -0.20 | 180.77 | 3412.62 |

0.58 | -4.80 | -293.58 | -298.38 | 127350.60 | -0.23 | 211.40 | 3973.34 |

0.67 | -6.27 | -334.92 | -341.18 | 127830.71 | -0.27 | 242.18 | 4531.78 |

0.75 | -7.92 | -376.11 | -384.04 | 128312.57 | -0.30 | 273.10 | 5087.96 |

0.83 | -9.77 | -417.16 | -426.94 | 128796.20 | -0.33 | 304.17 | 5641.88 |

0.92 | -11.82 | -458.07 | -469.88 | 129281.58 | -0.36 | 335.39 | 6193.55 |

1.00 | -14.05 | -498.83 | -512.88 | 129768.75 | -0.40 | 366.76 | 6742.99 |

2.00 | -55.68 | -977.44 | -1033.12 | 135754.76 | -0.76 | 755.09 | 13164.88 |

3.00 | -124.21 | -1438.15 | -1562.35 | 142005.84 | -1.10 | 1166.39 | 19280.97 |

4.00 | -219.09 | -1883.15 | -2102.24 | 148531.30 | -1.42 | 1602.20 | 25105.82 |

5.00 | -339.91 | -2314.55 | -2654.46 | 155340.50 | -1.71 | 2064.13 | 30653.29 |

6.00 | -486.41 | -2734.30 | -3220.71 | 162442.81 | -1.98 | 2553.95 | 35936.60 |

7.00 | -658.45 | -3144.28 | -3802.73 | 169847.58 | -2.24 | 3073.55 | 40968.32 |

8.00 | -856.03 | -3546.24 | -4402.27 | 177564.04 | -2.48 | 3624.98 | 45760.43 |

9.00 | -1079.27 | -3941.83 | -5021.10 | 185601.30 | -2.71 | 4210.45 | 50324.35 |

10.00 | -1328.44 | -4332.59 | -5661.04 | 193968.28 | -2.92 | 4832.33 | 54670.94 |

The next table shows the same values for a life aged 75. It illustrates a much more dramatic effect, as expected, because of the higher mortality.

\(b\) | Ann | \(V\) | Total | \(A_{x+b}\) | Pct | \(P_{x+b}-P_x\) | Buy In |
---|---|---|---|---|---|---|---|

0.08 | -6.21 | -1458.31 | -1464.52 | 522678.08 | -0.28 | 303.12 | 4417.95 |

0.17 | -24.84 | -2907.98 | -2932.82 | 524098.86 | -0.56 | 608.28 | 8817.96 |

0.25 | -55.90 | -4349.00 | -4404.90 | 525520.71 | -0.84 | 915.50 | 13200.13 |

0.33 | -99.38 | -5781.38 | -5880.75 | 526943.61 | -1.12 | 1224.80 | 17564.51 |

0.42 | -155.28 | -7205.09 | -7360.37 | 528367.54 | -1.39 | 1536.19 | 21911.19 |

0.50 | -223.61 | -8620.14 | -8843.75 | 529792.48 | -1.67 | 1849.69 | 26240.23 |

0.58 | -304.36 | -10026.53 | -10330.88 | 531218.42 | -1.94 | 2165.31 | 30551.70 |

0.67 | -397.53 | -11424.24 | -11821.77 | 532645.32 | -2.22 | 2483.08 | 34845.68 |

0.75 | -503.13 | -12813.26 | -13316.40 | 534073.17 | -2.49 | 2803.01 | 39122.24 |

0.83 | -621.16 | -14193.60 | -14814.76 | 535501.96 | -2.77 | 3125.11 | 43381.44 |

0.92 | -751.61 | -15565.25 | -16316.86 | 536931.65 | -3.04 | 3449.41 | 47623.36 |

1.00 | -894.49 | -16928.19 | -17822.68 | 538362.23 | -3.31 | 3775.92 | 51848.07 |

2.00 | -3578.60 | -32602.01 | -36180.62 | 555590.79 | -6.51 | 7873.22 | 101227.18 |

3.00 | -8053.24 | -47007.56 | -55060.80 | 572906.87 | -9.61 | 12324.41 | 148254.91 |

4.00 | -14318.44 | -60128.45 | -74446.89 | 590271.19 | -12.61 | 17165.81 | 193043.22 |

5.00 | -22372.23 | -71946.87 | -94319.09 | 607642.65 | -15.52 | 22437.99 | 235698.76 |

6.00 | -32209.40 | -82444.78 | -114654.19 | 624978.61 | -18.35 | 28186.33 | 276323.08 |

7.00 | -43820.09 | -91605.39 | -135425.48 | 642235.22 | -21.09 | 34461.62 | 315012.91 |

8.00 | -57188.20 | -99414.68 | -156602.88 | 659367.80 | -23.75 | 41320.79 | 351860.36 |

9.00 | -72289.75 | -105863.25 | -178153.00 | 676331.23 | -26.34 | 48827.67 | 386953.18 |

10.00 | -89091.04 | -110948.25 | -200039.29 | 693080.43 | -28.86 | 57053.91 | 420374.90 |

Note: computer code in Life-Contingencies.py

In this section we consider a somewhat more realistic view of pricing. Continue to assume pricing is on a net basis, with no expenses or profit provision. However, assume that pricing is based on age-last birthday, rather than exact age. As a result backdating offers the potential for insureds to “save-age”, buying into a lower rate corresponding to a younger age. Because insurance is priced using a level premium for a whole year, an insured who purchases at the beginning of a year will pay a slight economic cost and one who buys at the end of the year will realize a slight economic benefit. Thus backdating offers the potential for an insured to benefit by moving from a beginning of year purchase to an end of year purchase.

Assume the policy is fully discrete, with premiums due at the beginning of each coverage year and benefits paid at the end of the year of death. We consider whole life only.

Assume that the insurance is priced to be correct for an insured exact age \(x+\tau\) for integer age \(x\) and some fixed \(\tau\), \(0\le \tau\le 1\). A natural choice is \(\tau=0.5\). Thus we assume that the premium for age last birthday \(x\) is given by (non-standard notation) \[ P^\tau_x = \frac{A_{x+\tau}}{\ddot a_{x+\tau}}. \]

Using the same Makeham rule used for the continuous model, we can compute the economics, as present value of benefits less present value of premium paid, at time \(t=0\) for an insured exact age \(x+t\), \(0\le t\le 1\). Because of the rule for premium determination, this economic present value is negative for \(t<\tau\) and positive otherwise.

posted 2019-04-15 | tags: life contingencies, actuarial science, life insurance