Deriving the Single-State Thiele Equation

Introduction

A reserve changes over time because the existing fund earns interest, premiums and expenses flow continuously, and a death may trigger a benefit while ending the obligation to hold a reserve for a living policyholder. The single-state Thiele equation expresses these effects as a local balance: it gives the instantaneous rate of change of the reserve, conditional on the policyholder being alive.

The derivation is a useful discipline in its own right. Rather than memorizing signs, start with a short interval, state which cash flows occur, and compare the fund available with the expected amount required at the end of that interval. The result is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model and Cash-Flow Conventions

Definition: Consider a policy that remains in one active state until death. At time $t$, $V(t)$ is the reserve conditional on the insured being alive at $t$. The force of interest is $\delta(t)$, the mortality intensity is $\mu(t)$, the continuous premium rate while alive is $\pi(t)$, and the continuous expense rate while alive is $e(t)$. A death at time $t$ generates a benefit outgo $b(t)$. If a claim expense is paid at death, it may be included in $b(t)$.

All rates are per unit of time, and their time unit must agree. For example, if $t$ is measured in years, $\delta$, $\mu$, $\pi$, and $e$ must be annual rates. The quantities $\pi$ and $e$ are dollar amounts per year, while $V$ and $b$ are dollar amounts. Premiums increase the fund available; expenses and the death benefit are outgo.

The model assumes that death is the only transition out of the active state, that death occurs with intensity $\mu(t)$, and that the reserve and rate functions are sufficiently regular for a local differential equation to exist. In a short interval of length $h$, conditional on being alive at $t$, the probability of death is $\mu(t)h+o(h)$, and the probability of remaining alive is $1-\mu(t)h+o(h)$. Here $o(h)$ denotes a quantity whose ratio to $h$ tends to zero as $h$ tends to zero.

This is not the appropriate one-state model if there are other transitions, such as disability, recovery, or a state-dependent premium cessation. Those require a balance that includes the other transition intensities and their associated payments. The derivation below concerns only the specified active-to-death model.

Deriving the Equation From a Short-Interval Balance

Take a short interval from $t$ to $t+h$. If the policyholder is alive at $t$, the reserve earns interest, premiums are received, and expenses are paid during the interval. To first order in $h$, the fund available from the reserve and these continuous cash flows is $V(t)+[\delta(t)V(t)+\pi(t)-e(t)]h$.

At the end of the interval, the required amount depends on whether death occurred. If the policyholder survives, the required reserve is $V(t+h)$. If death occurs, the benefit $b(t)$ is paid instead. Since death within this short interval has probability $\mu(t)h+o(h)$, the expected required amount is, to first order, $(1-\mu(t)h)V(t+h)+\mu(t)h b(t)$. The value $b(t)$ can be used at time $t$ in this first-order expression: differences between the benefit at the exact time of death and at $t$ contribute only higher-order terms under the stated regularity assumptions.

Write the balance.
Match the fund accumulated over the interval to the expected amount required at its end, retaining terms through first order in $h$.

Expand the future reserve.
Use differentiability to write $V(t+h)=V(t)+V'(t)h+o(h)$.

Compare the coefficients of $h$.
Cancel the common current reserve, divide by $h$, and let $h$ tend to zero.

The one-step balance, with omitted terms of smaller order than $h$, is

V(t)+\bigl[\delta(t)V(t)+\pi(t)-e(t)\bigr]h = \bigl[1-\mu(t)h\bigr]V(t+h)+\mu(t)h\,b(t)+o(h).

Substitute the expansion of $V(t+h)$ on the right. Products such as $\mu(t)hV'(t)h$ have order $h^2$, so they vanish after division by $h$ and passage to the limit. The right-hand side becomes $V(t)+[V'(t)-\mu(t)V(t)+\mu(t)b(t)]h+o(h)$. Equating the first-order terms with those on the left gives the equation.

Theorem: Under the single-state assumptions above, the reserve satisfies the single-state Thiele differential equation

\frac{dV(t)}{dt} = \delta(t)V(t)+\pi(t)-e(t)-\mu(t)\bigl[b(t)-V(t)\bigr].

The mortality term is $-\mu(t)[b(t)-V(t)]$, not simply $-\mu(t)b(t)$. On death, the insurer pays $b(t)$, but it also no longer needs to hold the reserve $V(t)$ for a surviving policy. The net amount associated with death is therefore the benefit less the reserve released.

Worked Applications of the Local Balance

Worked Example: Checking a Reserve Derivative With a Short Interval

A hypothetical term policy has a reserve of $20,000 at time $t$. The annual force of interest is $\delta=0.04$, the annual mortality intensity is $\mu=0.01$, the continuous premium rate is $1,500 per year, the continuous expense rate is $300 per year, and the death benefit is $100,000. Find the reserve derivative at $t$, then check the one-step balance using $h=0.001$ years.

Method. Apply the derived equation, keeping every rate on an annual basis. The net continuous premium and expense rate is $1{,}500-300=1{,}200$ dollars per year. The mortality amount is the intensity times the benefit less the reserve: $0.01(100{,}000-20{,}000)=800$ dollars per year.

V'(t)=0.04(20{,}000)+1{,}500-300-0.01(100{,}000-20{,}000) =800+1{,}200-800 =\$1{,}200\text{ per year}.

For the short-interval check, the fund supplied by the current reserve, interest, premiums, and expenses is, to first order, $20{,}000+[0.04(20{,}000)+1{,}500-300](0.001)$. The rate inside the brackets is $800+1{,}200=2{,}000$ dollars per year, so this amount is $20,002.

The derivative implies $V(t+h)\approx20{,}000+1{,}200(0.001)=20{,}001.20$. The death probability to first order is $0.01(0.001)=0.00001$; the survival probability is $0.99999$. Thus the expected required amount using these approximations is $0.99999(20{,}001.20)+0.00001(100{,}000)$. The survival term is $20{,}001.20-0.200012=20{,}000.999988$, and the death term is $1$. Their sum is $20,001.999988, which agrees with $20,002 to first order. The difference, $0.000012, is a higher-order effect from using a finite interval.

Common mistake avoided. The death probability is $0.00001$, not $0.001$: the interval length must be multiplied by the annual intensity. Using $0.001$ would treat the intensity as a probability over the interval and overstate the expected death payment by a factor of 100.

Worked Example: A Reserve Increasing Without Premiums

For a hypothetical policy after premiums have ceased, suppose the reserve is $48,000, the annual force of interest is $0.03$, the annual mortality intensity is $0.02$, the continuous expense rate is $400 per year, and the death benefit is $90,000. Determine the instantaneous reserve change while the insured is alive.

Method. Set the premium rate to zero, retain the ongoing expense as an outgo, and use the death benefit less the reserve as the net death amount. The interest contribution is $0.03(48{,}000)=1{,}440$ dollars per year. The continuous expense contribution is $-400$ dollars per year. The mortality contribution is $-0.02(90{,}000-48{,}000)=-0.02(42{,}000)=-840$ dollars per year.

V'(t)=1{,}440-400-840=\$200\text{ per year}.

The reserve is increasing at that instant even though there is no premium income: interest exceeds the combined expense and net mortality outgo by $200 per year. This conclusion is local; it does not assert that the derivative remains constant at later times.

Common mistake avoided. Treating the death benefit as the entire mortality cost would give $-0.02(90{,}000)=-1{,}800$ dollars per year and a derivative of $-760$ dollars per year. That calculation ignores the $48,000 reserve no longer required following death. The correct mortality term is $-0.02(90{,}000-48{,}000)$, giving a derivative of $200 per year.

Worked Example: Including a Death-Claim Expense in the Outgo

A hypothetical policy has a reserve of $25,000. Its annual force of interest is $0.02$, its annual mortality intensity is $0.04$, its continuous premium rate is $1,000 per year, and its continuous maintenance expense is $250 per year. A death claim pays $75,000, and an additional $500 claim expense is paid at death. Find the instantaneous reserve derivative.

Method. Combine the death benefit and the claim expense into the total death outgo $b=75{,}000+500=75{,}500$. The interest contribution is $0.02(25{,}000)=500$ dollars per year. The net continuous income is $1{,}000-250=750$ dollars per year. The net amount associated with death is $75{,}500-25{,}000=50{,}500$ dollars, so its contribution to the derivative is $-0.04(50{,}500)=-2{,}020$ dollars per year.

V'(t)=500+750-2{,}020=-\$770\text{ per year}.

The negative derivative means the reserve is decreasing at this instant, conditional on survival. If the death expense were omitted, the mortality contribution would be $-0.04(75{,}000-25{,}000)=-2{,}000$, producing $-$750$ per year instead. The $20 difference is $0.04(500)$, the intensity-weighted death expense.

Common mistake avoided. Do not subtract a claim expense twice. In this calculation it is included once in $b$; if the problem instead defines the stated death outgo as already including claim expenses, use that total directly.

Sign Checks and Scope

A reliable way to check the equation is to ask what each event does to the reserve fund over a short interval. Interest and premiums increase the fund, while continuous expenses reduce it. Death generates a payment $b(t)$, but also removes the need for the survivor reserve $V(t)$. These directions yield the equation's four terms:

Key takeaway: For a policy in one active state with death as its only transition, continuous premium income $\pi(t)$, continuous expenses $e(t)$, death outgo $b(t)$, force of interest $\delta(t)$, and mortality intensity $\mu(t)$, the local reserve balance is $\frac{dV(t)}{dt}=\delta(t)V(t)+\pi(t)-e(t)-\mu(t)[b(t)-V(t)]$.

This equation gives a differential relation, not a reserve value by itself. A particular reserve trajectory also requires a boundary or terminal condition, which is a separate part of reserve calculation. Nor should this equation be used unchanged when another state transition or transition payment is present; then the one-step balance must include the probability and cash flow for each possible transition.

The most frequent sign error is writing the mortality term as $-\mu(t)b(t)$ and stopping there. That treats the death payment as an outgo but forgets the reserve released upon death. An equally serious error is using $+\mu(t)[b(t)-V(t)]$, which reverses the effect of net death outgo. Returning to the interval balance resolves both errors: the insurer must have enough for the survivor reserve or the death benefit, weighted by their respective probabilities.

Check Your Understanding

For each problem, use annual rates and report the reserve derivative in dollars per year unless otherwise stated.

A policy has $V(t)=\$32{,}000$, $\delta=0.025$, $\mu=0.015$, $\pi=\$900$ per year, $e=\$180$ per year, and $b=\$72{,}000$. Find $V'(t)$. (Answer: $\$920$ per year.)
After premiums stop, a policy has $V(t)=\$60{,}000$, $\delta=0.04$, $\mu=0.01$, $e=\$600$ per year, and $b=\$110{,}000$. Calculate $V'(t)$, stating why premium income is absent. (Answer: $\$1{,}300$ per year; the premium rate is zero.)
A policy has $V(t)=\$20{,}000$, $\delta=0.03$, $\mu=0.025$, a premium rate of $1,800 per year, and a continuous expense rate of $300 per year. The death benefit is $70,000. Find the derivative and the net death amount used. (Answer: $V'(t)=\$850$ per year; net death amount $=\$50{,}000$.)
At a particular time, $V(t)=\$40{,}000$, $\delta=0.02$, $\pi-e=\$500$ per year, and $V'(t)=-\$700$ per year. If the death benefit is $90,000, infer the mortality intensity. (Answer: $\mu=0.0400$ per year.)
For a short-interval check, let $V(t)=\$15{,}000$, $\delta=0.02$, $\mu=0.01$, $\pi=\$800$ per year, $e=\$200$ per year, $b=\$55{,}000$, and $h=0.001$ years. First find $V'(t)$, then compare the first-order fund amount with the expected required amount using $V(t+h)\approx V(t)+V'(t)h$ and death probability $\mu h$. (Answer: $V'(t)=\$500$ per year; the fund amount is $15,000.90 and the expected required amount is $15,000.899995, agreeing to first order.)

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