Throughout this unit assume a constant effective annual interest rate \(i\) with discount factor \(v = 1/(1+i)\) and force of interest \(\delta = \ln(1+i)\). For a benefit \(b_t\) paid at time \(t\), the present value is \(b_t \cdot v^t = b_t e^{-\delta t}\). The actuarial present value (APV) is the expected present value, taking the random time of death \(T = T(x)\) into account.
The APV is
\[ \bar A_x = E[Z] = \int_0^{\infty} v^t \cdot {}_tp_x \cdot \mu_{x+t}\,dt. \]where the leading superscript 2 means "evaluated at double the force of interest." Hence
\[ \mathrm{Var}[Z] = {}^{2}\!\bar A_x - (\bar A_x)^2. \]If \(\mu_x = \mu\) for all \(x\),
\[ \bar A_x = \frac{\mu}{\mu + \delta}, \qquad {}^{2}\!\bar A_x = \frac{\mu}{\mu + 2\delta}. \]Constant force \(\mu = 0.04\), force of interest \(\delta = 0.06\). Compute \(\bar A_x\) and its standard deviation.
\(\bar A_x = 0.04 / (0.04 + 0.06) = 0.40\).
\({}^{2}\!\bar A_x = 0.04 / (0.04 + 0.12) = 0.25\).
\(\mathrm{Var}[Z] = 0.25 - 0.16 = 0.09\); SD = 0.30.
So for a unit benefit, the APV is ₹0.40 and the SD of the present value is ₹0.30 — meaning individual realisations vary widely around the mean.
The death benefit is assumed paid at the end of the year of death, hence the exponent \(k+1\).
Under the Uniform Distribution of Deaths (UDD) assumption,
\[ \bar A_x = \frac{i}{\delta}\,A_x. \]The factor \(i/\delta > 1\) reflects that paying the benefit at moment of death (on average half a year sooner than end of year) is more costly.
Given \(A_{40} = 0.16132\), \(i = 0.06\). Find \(\bar A_{40}\).
\(\delta = \ln 1.06 = 0.05827\); \(i/\delta = 0.06/0.05827 = 1.02971\).
\(\bar A_{40} = 1.02971 \times 0.16132 = 0.16611\).
Benefit 1 paid at moment of death if death occurs within \(n\) years; nothing otherwise.
\[ \bar A^1_{x:\overline n|} = \int_0^{n} v^t\,{}_tp_x\,\mu_{x+t}\,dt, \] \[ A^1_{x:\overline n|} = \sum_{k=0}^{n-1} v^{k+1}\,{}_kp_x\,q_{x+k}. \]The superscript "1" over the age means "first event" — payment is contingent on death.
Benefit 1 paid at end of \(n\) years if and only if the life survives.
\[ A_{x:\overline n|}^{\;\;1} = {}_nE_x = v^n\,{}_np_x. \]The superscript "1" is over \(n\), meaning the payment is contingent on the term (survival).
Pays 1 at moment of death if death occurs within \(n\) years; pays 1 at time \(n\) if the life survives. Hence:
\[ \bar A_{x:\overline n|} = \bar A^1_{x:\overline n|} + {}_nE_x, \] \[ A_{x:\overline n|} = A^1_{x:\overline n|} + {}_nE_x. \]This is the most commonly sold contract: it guarantees a payout regardless of outcome.
Pays 1 at moment of death, but only if death occurs after \(m\) years.
\[ {}_{m|}\bar A_x = \int_m^{\infty} v^t\,{}_tp_x\,\mu_{x+t}\,dt = {}_mE_x \cdot \bar A_{x+m}. \]Constant force \(\mu = 0.03\), \(\delta = 0.05\), term \(n = 20\). Compute \(\bar A^1_{x:\overline{20}|}\).
\[ \bar A^1_{x:\overline{20}|} = \int_0^{20} e^{-0.05 t} \cdot e^{-0.03 t} \cdot 0.03\,dt = 0.03 \int_0^{20} e^{-0.08 t}\,dt = \frac{0.03}{0.08}(1 - e^{-1.6}) = 0.375 \times 0.7981 = 0.2993. \]
So a 1-unit term insurance for 20 years costs ₹0.2993 in APV under these assumptions.
From Example 1, term piece = 0.2993. Pure-endowment piece:
\({}_{20}E_x = e^{-0.05 \times 20} \cdot e^{-0.03 \times 20} = e^{-1.6} = 0.2019\).
Endowment APV \(\bar A_{x:\overline{20}|} = 0.2993 + 0.2019 = 0.5012\).
About 40% of the endowment cost (0.2019 of 0.5012) pays for the survival payout and 60% (0.2993) for the death payout.
Death benefit is \(k+1\) at the end of year \(k+1\):
\[ (IA)_x = \sum_{k=0}^{\infty}(k+1)\,v^{k+1}\,{}_kp_x\,q_{x+k}. \]Death benefit is \(t\) at moment of death:
\[ (\bar{IA})_x = \int_0^{\infty} t\,v^t\,{}_tp_x\,\mu_{x+t}\,dt. \]Death benefit is \(n - k\) (start at \(n\), decrease by 1 each year):
\[ (DA)^1_{x:\overline n|} = \sum_{k=0}^{n-1}(n - k)\,v^{k+1}\,{}_kp_x\,q_{x+k}. \]Used to provide a mortgage-protection benefit that matches a declining loan balance.
An insurer offers a 30-year increasing whole-life policy: ₹1 lakh in year 1, ₹2 lakh in year 2, etc. The APV scales linearly with \((IA)_x\), so the premium for the increasing policy is \((IA)_x / A_x\) times the level-benefit premium.
A buyer takes a ₹40 lakh, 20-year home loan with level monthly EMI. A 20-year decreasing-term policy with starting sum assured 40 lakhs and annual reduction of 2 lakhs gives mortgage protection: if the borrower dies in year \(k+1\), the family receives \(40 - 2k\) lakhs. This straight line is only a rough match to the loan. A level-EMI loan is repaid slowly at first and quickly near the end, so in the middle years the cover is below the debt. At 8% a year, the balance after 10 years is about ₹27.6 lakh against a cover of ₹20 lakh. Mortgage-protection policies therefore often reduce the cover along the loan's own repayment schedule instead.
Recursive identities reduce the work of computing APVs from tabulated \(q_x\) values.
Reads as: pay 1 at end of year if you die this year (prob \(q_x\)), or carry forward \(A_{x+1}\) if you survive (prob \(p_x\)).
From a life table \(q_{60} = 0.012\), \(A_{61} = 0.45\), \(i = 0.06\). Find \(A_{60}\).
\(A_{60} = (1/1.06)(0.012) + (1/1.06)(0.988)(0.45) = 0.01132 + 0.41943 = 0.4308\).
Recursion makes it easy to fill an entire column of \(A_x\) by starting at the limiting age \(\omega\), where nobody is alive. At the last age with survivors, \(\omega - 1\), death within the year is certain, so \(A_{\omega-1} = v\). Then work backwards: \(A_{\omega-2} = v q_{\omega-2} + v p_{\omega-2} A_{\omega-1}\), and so on. This is precisely how commutation columns and modern actuarial software build the table.
The joint-life status fails on the first death of either life. Under independence:
\[ {}_tp_{xy} = {}_tp_x \cdot {}_tp_y. \]Whole-life joint insurance pays 1 at the first death:
\[ A_{xy} = \sum_{k=0}^{\infty} v^{k+1}\,{}_kp_{xy}\,q_{x+k:y+k}. \]The last-survivor status fails on the second death. The key identity:
\[ A_{\overline{xy}} = A_x + A_y - A_{xy}. \]This is exactly the inclusion-exclusion principle applied to two random lifetimes.
For a couple aged \((x, y) = (65, 60)\): \(A_x = 0.40\), \(A_y = 0.28\), \(A_{xy} = 0.46\).
Last-survivor whole-life: \(A_{\overline{xy}} = 0.40 + 0.28 - 0.46 = 0.22\).
Interpretation: paying ₹1 at the second death is cheaper (₹0.22) than the first death (₹0.46) because the payout is delayed until both lives die.
A "joint and last-survivor" annuity that continues to the surviving spouse is priced using \(\ddot a_{\overline{xy}}\); the corresponding insurance protecting the contract owner against simultaneous deaths uses \(A_{xy}\). The identity above lets the insurer split aggregated portfolio risk into elementary single-life pieces.