Continuous Annuities and Thiele Reserve Dynamics

Continuous Payments as Reserve Outgo

A continuous annuity pays at a specified rate over time rather than as a single amount at discrete dates. In a life-contingent contract, the rate is paid only while the insured is alive. For a reserve calculation, this payment is therefore an ongoing outgo from the active life state, not a lump sum at the start or end of the year.

This distinction matters in Thiele’s equation: continuous benefits and premiums enter as rates, while a death benefit enters through the mortality transition term. The resulting reserve reflects both the payment stream while the insured is alive and the possibility of a benefit at death. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model: Consider a hypothetical contract in force at time $t$, conditional on the insured being alive. The contract ends at time $n$. While alive, it pays a continuous benefit rate $b$ and receives a continuous premium rate $p$. If death occurs before $n$, it pays a lump sum $B$ immediately. If the insured survives to $n$, it pays a maturity benefit $M$. Assume constant force of interest $\delta$, constant force of mortality $\mu$, no expenses, and no other benefits or premiums. The prospective reserve $V(t)$ is the expected present value of future benefits less future premiums, conditional on being alive at $t$.

The rates $b$ and $p$ are amounts per year. Over a short interval of length $h$, the continuous benefit is approximately $bh$, and the premium is approximately $ph$, provided the insured remains alive. The death benefit $B$, in contrast, is a one-time payment if death occurs. This model assumes $\delta$ and $\mu$ are measured per year and are constant over the remaining term.

Thiele’s Equation With a Continuous Annuity

Let $c=b-p$ be the net continuous outgo rate while alive. A positive $c$ means the contract pays more continuously than it receives in premiums. On death, the insurer pays $B$, and the reserve for the terminated contract is zero. The Thiele equation for this single active state is

\frac{dV(t)}{dt} = \delta V(t)-c+\mu\bigl(V(t)-B\bigr) = (\delta+\mu)V(t)-\bigl(c+\mu B\bigr).

The term $\delta V(t)$ reflects interest on the reserve. The term $-c$ reflects the continuous net outgo while the insured is alive. The transition term $\mu(V(t)-B)$ accounts for death: the insurer releases the reserve $V(t)$ but pays the death benefit $B$ at rate $\mu$. A maturity benefit is handled by the terminal condition $V(n)=M$, not as a continuous payment.

Key result: Define $\lambda=\delta+\mu$ and $K=b-p+\mu B$. With constant forces and terminal reserve $V(n)=M$, the reserve at time $t$ is $$ V(t)=\frac{K}{\lambda}\left(1-e^{-\lambda(n-t)}\right)+M e^{-\lambda(n-t)}, \qquad \lambda>0. $$ If $\lambda=0$, the limiting form is $V(t)=K(n-t)+M$. The formula assumes all stated continuous rates apply only while the insured is alive and the maturity benefit is paid only on survival to $n$.

The expression includes the continuous annuity rate in $K$. It does not treat the annuity as a guaranteed payment stream: mortality reduces its expected present value because payments stop at death. In particular, if there is no death benefit and no premium, then $K=b$. If the annuity is accompanied by a death benefit, the additional term $\mu B$ contributes to the reserve’s effective net outgo rate.

Why the Constant-Force Formula Holds

The formula can be justified directly from the prospective definition. Let $T=n-t$ be the remaining term. Under constant mortality, survival for $s$ years from time $t$ has probability $e^{-\mu s}$, and the probability density of death at duration $s$ is $\mu e^{-\mu s}$. Discounting at force $\delta$ multiplies a payment at duration $s$ by $e^{-\delta s}$.

Value the continuous net outgo.
The rate $c=b-p$ is paid while alive. Its expected present value is $c\int_0^T e^{-(\delta+\mu)s}\,ds$, since both discounting and survival are required.

Value the death benefit.
The expected present value of $B$ paid at death is $B\int_0^T \mu e^{-(\delta+\mu)s}\,ds$, using the death density and the discount factor.

Value the maturity benefit.
The benefit $M$ is paid only if the insured survives to $T$, so its expected present value is $M e^{-(\delta+\mu)T}$.

Adding these components and integrating gives

V(t) = \int_0^T e^{-(\delta+\mu)s}(c+\mu B)\,ds +M e^{-(\delta+\mu)T} = \frac{c+\mu B}{\delta+\mu} \left(1-e^{-(\delta+\mu)T}\right) +M e^{-(\delta+\mu)T}.

This is the stated formula because $K=c+\mu B$. Differentiating the formula with respect to $t$, using $T=n-t$, gives $V'(t)=(\delta+\mu)V(t)-K$, which verifies the Thiele equation. At $t=n$, the integral term is zero and the remaining term is $M$, so the terminal condition also holds.

Worked Reserve Calculations

Worked Example: Annuity Payments and a Death Benefit

A hypothetical five-year contract pays a continuous annuity of $\$600$ per year while the insured is alive and $\$10{,}000$ immediately at death during the term. There is no premium or maturity benefit. Assume constant forces $\delta=0.04$ and $\mu=0.02$ per year. Calculate $V(0)$ and $V(2)$.

Method. Use the constant-force prospective reserve formula. Here $b=600$, $p=0$, $B=10{,}000$, $M=0$, and $\lambda=\delta+\mu=0.04+0.02=0.06$. The effective outgo rate is

K=b-p+\mu B=600+0.02(10{,}000)=600+200=800.

At issue, the remaining term is $5$ years:

V(0)=\frac{800}{0.06}\left(1-e^{-0.06(5)}\right) =13{,}333.3333\left(1-e^{-0.30}\right).

Using $e^{-0.30}=0.7408182$, the value is $13{,}333.3333(0.2591818)=\$3{,}455.76$, rounded to the nearest cent.

At time $2$, the remaining term is $5-2=3$ years:

V(2)=\frac{800}{0.06}\left(1-e^{-0.06(3)}\right) =13{,}333.3333\left(1-e^{-0.18}\right) =\$2{,}196.40.

The reserve is lower at time $2$ than at issue because less time remains for the annuity and death benefit to be paid. As a check, the reserve is zero at time $5$, as required by the zero terminal benefit. Common mistake avoided: the death benefit is not included as a continuous payment of $\$10{,}000$ per year; its contribution is $\mu B=200$ per year in the reserve equation.

Worked Example: Maturity Benefit and the Survival-Contingent Annuity

A hypothetical four-year contract pays $\$400$ per year continuously while the insured is alive and $\$1{,}500$ at death during the term. It also pays $\$2{,}000$ at time $4$ if the insured survives. There are no premiums. Assume $\delta=0.02$ and $\mu=0.03$ per year. Find $V(0)$ and confirm the terminal reserve.

Method. Set $b=400$, $p=0$, $B=1{,}500$, $M=2{,}000$, and $\lambda=0.02+0.03=0.05$. The effective outgo rate is

K=b-p+\mu B=400+0.03(1{,}500)=400+45=445.

The four-year annuity and death-benefit component is

\frac{445}{0.05}\left(1-e^{-0.05(4)}\right) =8{,}900(1-e^{-0.20}) =8{,}900(0.1812692469) =\$1{,}613.30.

The maturity benefit component is

2{,}000e^{-0.20} =2{,}000(0.8187307531) =\$1{,}637.46.

Therefore,

V(0)=1{,}613.2963+1{,}637.4615=\$3{,}250.76, \qquad V(4)=M=\$2{,}000.00.

The issue reserve is the sum of the expected present values of the continuous outgo, death benefit, and maturity benefit. Common mistake avoided: valuing the annuity as guaranteed would use only the interest force. The correct annuity component uses $\delta+\mu=0.05$, because each payment requires survival as well as discounting.

Worked Example: Premiums Offset a Continuous Annuity

A hypothetical six-year contract pays a continuous annuity of $\$900$ per year while the insured is alive. It receives continuous premiums of $\$500$ per year while the insured is alive, pays $\$8{,}000$ at death during the term, and pays $\$5{,}000$ at time $6$ on survival. Assume $\delta=0.03$ and $\mu=0.02$ per year. Calculate $V(0)$, $V(3)$, and the instantaneous reserve derivative at time $0$.

Method. The net continuous outgo rate is $b-p=900-500=400$. The mortality contribution is $\mu B=0.02(8{,}000)=160$, so $K=400+160=560$, and $\lambda=0.03+0.02=0.05$.

At time $0$, six years remain:

V(0)=\frac{560}{0.05}(1-e^{-0.30})+5{,}000e^{-0.30} =11{,}200(0.2591817793)+5{,}000(0.7408182207).

The two components are $2{,}902.8359$ and $3{,}704.0911$, giving $V(0)=\$6{,}606.93$.

At time $3$, three years remain:

V(3)=\frac{560}{0.05}(1-e^{-0.15})+5{,}000e^{-0.15} =11{,}200(0.1392920236)+5{,}000(0.8607079764) =\$5{,}863.61.

For the instantaneous direction of change, use the differential equation:

V'(0)=0.05V(0)-560 =0.05(6{,}606.9270)-560 =-\$229.65\text{ per year}.

The negative derivative means the reserve is decreasing at time $0$. The calculation checks the sign convention: although the contract has a maturity benefit, the current reserve and outgo rate determine the local direction of change. Common mistake avoided: subtracting the premium from the death-benefit term would be incorrect; the premium offsets the continuous annuity rate, so $K=(b-p)+\mu B$.

Interpreting the Reserve and Avoiding Traps

The reserve formula and differential equation provide complementary checks. The formula values future cash flows from a specified duration. The differential equation describes how the reserve changes instantaneously as time advances. With the notation above,

V'(t)=(\delta+\mu)V(t)-\bigl(b-p+\mu B\bigr).

Thus, a large reserve does not by itself imply that the reserve is increasing: the net outgo rate $b-p+\mu B$ must also be considered. At a time when $V(t)=K/(\delta+\mu)$, the derivative is zero. Above that level it is positive; below it, it is negative. This statement is local and does not replace the terminal condition or the full prospective valuation.

Do not treat a life-contingent annuity as guaranteed. Its expected payments require survival, so the combined discount and survival factor is $e^{-(\delta+\mu)s}$ under constant forces.
Keep rates and lump sums distinct. The annuity and premium are annual rates; $B$ and $M$ are lump-sum benefits. In the differential equation, the death benefit appears as $\mu B$, not as $B$ alone.
Use the correct terminal condition. If a maturity benefit is payable on survival, then $V(n)=M$. Setting the terminal reserve to zero would omit that benefit.
Check the time remaining. At duration $t$, the formula uses $n-t$, not $n$, for the remaining term.

Key takeaway: Under constant interest and mortality forces, a continuous annuity changes the Thiele equation through its ongoing net payment rate $b-p$. Mortality both stops future annuity payments and creates the expected death-benefit outgo $\mu B$; the maturity benefit is imposed as the terminal reserve.

Check Your Understanding

Use the constant-force model stated above. Round monetary answers to the nearest cent.

A three-year contract pays $\$300$ continuously while alive, has no premium, death benefit, or maturity benefit, and has $\delta=0.04$, $\mu=0.01$. Find $V(0)$. (Answer: $V(0)=\$835.75$)
A five-year contract pays $\$250$ continuously while alive, $\$4{,}000$ at death, and $\$1{,}000$ at maturity on survival. There are no premiums; $\delta=0.03$, $\mu=0.02$. Find $V(0)$. (Answer: $V(0)=\$2{,}238.72$)
A four-year contract pays an annuity of $\$700$ per year and receives premiums of $\$200$ per year, both continuously while alive. It pays $\$2{,}500$ at death and $\$3{,}000$ at maturity on survival. If $\delta=0.01$ and $\mu=0.04$, find $V(0)$. (Answer: $V(0)=\$4{,}631.42$)
A six-year contract has $b=\$500$, $p=\$100$, $B=\$6{,}000$, and $M=\$1{,}500$, with $\delta=0.04$ and $\mu=0.01$. Find $V(2)$, using the four years remaining. (Answer: $V(2)=\$2{,}895.77$)
At a particular duration, $V(t)=\$9{,}000$, $b=\$500$, $p=\$200$, $B=\$10{,}000$, $\delta=0.03$, and $\mu=0.02$. Determine the instantaneous reserve derivative and state whether the reserve is locally increasing or decreasing. (Answer: $V'(t)=-\$50.00$ per year; locally decreasing)

What You'll Learn