Policy values for insurance and annuities on joint lives

Exam ALTAM – Advanced Long-Term Actuarial Mathematics · SOA (Society of Actuaries) · 4 min read · Updated: 10 Nov 2026

Welcome to Policy Values for Joint Lives!

In your previous studies, you mastered how to calculate policy values (reserves) for a single life. In actuarial practice, insurance and retirement solutions frequently cover pairs of lives—such as married spouses or business partners. In this chapter, we explore how to determine and manage the policy values for both insurance and annuities on joint lives using both classical status notation and the multiple state model framework.

1. Core Principles of Joint Policy Valuation

As with single-life policies, the prospective Policy Value at time \(t\) (denoted \(_tV\)) represents the expected present value of future benefits and expenses minus the expected present value of future gross or net premiums:

Policy Value = PVFB + PVFE – PVFP

With joint lives \((x)\) and \((y)\), cash flows depend on the status failures and survival of two lives:

Joint Life Status (\(xy\)): Fails upon the first death of \(x\) or \(y\).
Last Survivor Status (\(\overline{xy}\)): Fails upon the second (last) death.

2. Policy Values for Joint Life Insurance (\(xy\))

For a fully discrete joint life insurance paying a sum insured upon the first death, funded by annual net level premiums \(P_{xy}\) payable while both are alive:

\(_tV_{xy} = A_{x+t:y+t} - P_{xy} \cdot \ddot{a}_{x+t:y+t}\)

where \(P_{xy} = \frac{A_{xy}}{\ddot{a}_{xy}}\). If either life dies prior to or at time \(t\), the contract terminates and the policy value becomes 0.

3. Policy Values for Last Survivor Insurance (\(\overline{xy}\))

A last survivor policy pays the benefit only upon the second death. Determining the policy value depends directly on who remains alive at time \(t\):

While both are alive (State 0): If annual premiums \(P_{\overline{xy}}\) are paid while at least one life survives, the reserve is \(_tV^{(0)} = A_{\overline{x+t:y+t}} - P_{\overline{xy}} \cdot \ddot{a}_{\overline{x+t:y+t}}\).
After the death of \(y\) (State 1): The contract continues on \(x\) alone. The prospective reserve becomes \(_tV^{(1)} = A_{x+t} - P \cdot \ddot{a}_{x+t}\) (using whichever premium rules apply post-first-death).
After the death of \(x\) (State 2): The prospective reserve becomes \(_tV^{(2)} = A_{y+t} - P \cdot \ddot{a}_{y+t}\).

Important Caveat on Linear Identities: While expected present values of benefits satisfy \(A_{\overline{xy}} = A_x + A_y - A_{xy}\), the policy values do not generally satisfy \(_tV_{\overline{xy}} = \ _tV_x + \ _tV_y - \ _tV_{xy}\) for premium-paying policies, because the equivalence principle premium \(P_{\overline{xy}} \neq P_x + P_y - P_{xy}\).

4. Policy Values for Joint and Reversionary Annuities

Valuing joint annuities requires tracking payments contingent on status survival:

Joint Life Annuity: Pays 1 at the beginning of each year while both are alive. The prospective policy value at duration \(t\) (for a single premium contract) is simply \(_tV = \ddot{a}_{x+t:y+t}\).
Last Survivor Annuity: Pays 1 while at least one life is alive. While both are alive, \(_tV^{(0)} = \ddot{a}_{\overline{x+t:y+t}} = \ddot{a}_{x+t} + \ddot{a}_{y+t} - \ddot{a}_{x+t:y+t}\). Following the death of \(y\), the reserve immediately drops to the single-life annuity reserve \(_tV^{(1)} = \ddot{a}_{x+t}\).
Reversionary Annuity: Pays an annuity to \(x\) after the death of \(y\), provided \(x\) survives \(y\). While both lives are alive, the expected present value of future benefits is \(a_{x+t|y+t} = \ddot{a}_{x+t} - \ddot{a}_{x+t:y+t}\). If funded by level premiums \(P\) payable while both are alive, the prospective reserve at duration \(t\) is \(_tV^{(0)} = (\ddot{a}_{x+t} - \ddot{a}_{x+t:y+t}) - P \cdot \ddot{a}_{x+t:y+t}\). When \(y\) dies, premiums cease and the reserve experiences a jump to the full single-life annuity value: \(_tV^{(1)} = \ddot{a}_{x+t}\).

5. The Multiple State Model Framework

Joint life contracts are modeled systematically with a four-state continuous- or discrete-time Markov model:

State 0: Both \(x\) and \(y\) are alive.
State 1: \(x\) is alive, \(y\) has died.
State 2: \(y\) is alive, \(x\) has died.
State 3: Both \(x\) and \(y\) have died.

In this framework, \(_tV^{(i)}\) denotes the policy value at time \(t\) conditional on the policy being in state \(i\).

6. Recursion and Thiele’s Differential Equation

Discrete-Time Recursion:

For a policy in State 0 at time \(t\) with annual premium \(P_t\) received at the start of the year and transition benefits \(S_{t+1}^{(0j)}\) payable at the end of the year:

\((_tV^{(0)} + P_t)(1+i) = p_{x+t:y+t}^{00} \cdot \ _{t+1}V^{(0)} + \sum_{j=1}^{3} p_{x+t:y+t}^{0j} \left( S_{t+1}^{(0j)} + \ _{t+1}V^{(j)} \right)\)

where \(_{t+1}V^{(3)} = 0\) since no future benefits follow terminal absorption into State 3.

Continuous-Time Thiele's Differential Equation:

In continuous time, let \(\delta\) be the force of interest, \(P_t^{(0)}\) the rate of premium cash inflow, \(B_t^{(0)}\) the continuous annuity benefit payout rate in State 0, \(S_t^{(0j)}\) the lump-sum benefit paid upon transition from 0 to \(j\), and \(\mu_{x+t:y+t}^{0j}\) the transition intensity. Thiele's equation for state 0 is:

\(\frac{d}{dt} \ _tV^{(0)} = \delta \cdot \ _tV^{(0)} + P_t^{(0)} - B_t^{(0)} - \sum_{j \neq 0} \mu_{x+t:y+t}^{0j} \left( S_t^{(0j)} + \ _tV^{(j)} - \ _tV^{(0)} \right)\)

The term \(\left( S_t^{(0j)} + \ _tV^{(j)} - \ _tV^{(0)} \right)\) is the net amount at risk associated with transition \(0 \to j\).

7. Summary and Key Takeaways

Joint Life (\(xy\)): Cash flows cease at the first death; reserves reflect joint survival probabilities \(_k p_{xy}\).
Last Survivor (\(\overline{xy}\)): Cash flows continue until the second death; reserves must be tracked across distinct states (both alive vs. one survivor).
Annuity Reserves: Reversionary annuities exhibit a substantial jump in policy values upon the death of the primary life, shifting from \(_tV^{(0)}\) to the unreduced single-life reserve \(_tV^{(1)} = \ddot{a}_{x+t}\).
Valuation Tools: Discrete-time policy values advance via Markov transition recursions, while continuous-time policy values follow Thiele's differential equation.

References

Exam ALTAM Syllabus, Spring 2026 — Society of Actuaries

Quick check

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Open each question to check the key ideas from this chapter.

What is the prospective policy value for a contract on joint lives at duration \(t\)?

The prospective policy value at time \(t\) is the expected present value of future benefits and expenses minus the expected present value of future gross or net premiums: \(_tV = \text{PVFB} + \text{PVFE} - \text{PVFP}\).

What is the prospective net premium reserve at duration \(t\) for a fully discrete joint life insurance on \((xy)\) while both lives remain alive?

\(_tV_{xy} = A_{x+t:y+t} - P_{xy} \cdot \ddot{a}_{x+t:y+t}\), where \(P_{xy} = \frac{A_{xy}}{\ddot{a}_{xy}}\).

How does the policy value of a last survivor insurance contract differ when both lives are alive compared to after one life has died?

While both are alive, the reserve is \(_tV^{(0)} = A_{\overline{x+t:y+t}} - P_{\overline{xy}} \cdot \ddot{a}_{\overline{x+t:y+t}}\); once one life dies (e.g., \(y\)), the policy transitions to a single-life contract with reserve \(_tV^{(1)} = A_{x+t} - P \cdot \ddot{a}_{x+t}\).

Why does the linear identity \(A_{\overline{xy}} = A_x + A_y - A_{xy}\) not generally hold for policy values such that \(_tV_{\overline{xy}} = \ _tV_x + \ _tV_y - \ _tV_{xy}\)?

Because the net level premiums do not satisfy a linear relationship; that is, \(P_{\overline{xy}} \neq P_x + P_y - P_{xy}\).

What happens to the policy value of a reversionary annuity paying \(x\) after the death of \(y\) upon the death of \(y\)?

The reserve experiences an upward jump from \(_tV^{(0)} = (\ddot{a}_{x+t} - \ddot{a}_{x+t:y+t}) - P \cdot \ddot{a}_{x+t:y+t}\) to the unreduced single-life annuity reserve \(_tV^{(1)} = \ddot{a}_{x+t}\).

What are the four states standardly used in a multiple state Markov model for joint life contracts?

State 0: both \(x\) and \(y\) are alive; State 1: \(x\) is alive, \(y\) is dead; State 2: \(y\) is alive, \(x\) is dead; State 3: both \(x\) and \(y\) are dead.

What is the discrete-time recursive formula for the policy value in State 0, \(_tV^{(0)}\), for a joint life contract in a multiple state framework?

\((_tV^{(0)} + P_t)(1+i) = p_{x+t:y+t}^{00} \cdot \ _{t+1}V^{(0)} + \sum_{j=1}^{3} p_{x+t:y+t}^{0j} \left( S_{t+1}^{(0j)} + \ _{t+1}V^{(j)} \right)\).

What is Thiele's differential equation for the reserve in State 0, \(_tV^{(0)}\), in a continuous-time joint life multiple state model?

\(\frac{d}{dt} \ _tV^{(0)} = \delta \cdot \ _tV^{(0)} + P_t^{(0)} - B_t^{(0)} - \sum_{j \neq 0} \mu_{x+t:y+t}^{0j} \left( S_t^{(0j)} + \ _tV^{(j)} - \ _tV^{(0)} \right)\).

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