Sensitivity of Reserves to Transition Intensities

Why Reserve Sensitivity Matters

A multi-state reserve depends on the rates at which a life moves between states. If incidence rises, a disability or care reserve may increase; if recovery becomes more likely, it may decrease. Sensitivity analysis quantifies this response while holding the other valuation assumptions fixed. It helps identify which assumptions drive a reserve and whether a proposed change has a material effect. The methods here are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

The calculation is not just a matter of changing one intensity in a completed reserve. Differentiate the entire system of Thiele equations: the reserve values themselves depend on the intensity, and the intensity may also appear directly as a multiplier on the transition cash flow.

Thiele Equations and the Sensitivity System

Consider a policy with transient states indexed by $i$, valuation time $t$, and projection horizon $T$. Let $V_i(t;\theta)$ be the reserve at time $t$ conditional on being in state $i$, where $\theta$ is a parameter that affects one or more transition intensities. Write $\mu_{ij}(t;\theta)$ for the intensity of transition from state $i$ to state $j$. Let $b_{ij}(t;\theta)$ be the benefit paid at that transition, and let $c_i(t;\theta)$ be the net rate of continuous outgo in state $i$. Thus benefits and expenses are positive outgo, while premiums received are negative outgo. Assume the force of interest is $\delta(t;\theta)$.

Definition: The reserve sensitivity in state $i$ is $W_i(t)=\dfrac{\partial V_i(t;\theta)}{\partial\theta}$. It measures the change in reserve per unit change in the parameter $\theta$, with the other model assumptions held fixed.

Under the usual continuous-time Markov model and smooth dependence on $\theta$, the Thiele equation for state $i$ is

V_i'=\delta V_i+c_i+\sum_{j\ne i}\mu_{ij}(V_i-V_j-b_{ij}).

A positive transition benefit $b_{ij}$ is subtracted inside the transition term because it is paid when the transition occurs. The reserve after the transition, $V_j$, is also part of the resulting liability. With terminal reserve $G_i(\theta)$, the boundary condition is $V_i(T;\theta)=G_i(\theta)$.

Differentiate this equation with respect to $\theta$. A prime on $V_i$ denotes differentiation with respect to time; $W_i'$ is likewise a time derivative, whereas $\partial V_i/\partial\theta$ is a parameter derivative. These are different operations.

Theorem: If $\delta$, $c_i$, transition intensities, benefits, and terminal reserves are differentiable in $\theta$, then

W_i'=\delta W_i+\frac{\partial\delta}{\partial\theta}V_i+ \frac{\partial c_i}{\partial\theta} +\sum_{j\ne i}\mu_{ij}\left(W_i-W_j-\frac{\partial b_{ij}}{\partial\theta}\right) +\sum_{j\ne i}\frac{\partial\mu_{ij}}{\partial\theta}(V_i-V_j-b_{ij}), \qquad W_i(T)=\frac{\partial G_i}{\partial\theta}.

If the interest basis, cash flows, and terminal reserves do not depend directly on $\theta$, the corresponding partial derivatives are zero. The remaining terms show two distinct effects: existing transition intensities carry the reserve sensitivities $W_i-W_j$, while a change in an intensity multiplies the current transition loss $V_i-V_j-b_{ij}$.

Why Differentiation Produces These Terms

The transition contribution in the Thiele equation is a product of an intensity and a reserve difference, $\mu_{ij}(V_i-V_j-b_{ij})$. Differentiating that product gives one term for the change in the reserves and benefits, and another for the change in the intensity:

\frac{\partial}{\partial\theta} \left[\mu_{ij}(V_i-V_j-b_{ij})\right] = \mu_{ij}\left(W_i-W_j-\frac{\partial b_{ij}}{\partial\theta}\right) +\frac{\partial\mu_{ij}}{\partial\theta}(V_i-V_j-b_{ij}).

This is the product rule, applied transition by transition. The other terms follow by differentiating $\delta V_i+c_i$. Since the original reserve has a terminal condition at $T$, differentiating that condition gives $W_i(T)=\partial G_i/\partial\theta$. In particular, if the terminal reserve is fixed independently of $\theta$, the sensitivity is zero at $T$. The resulting sensitivity system is solved backward in time, like the reserve system.

Worked Calculations

Worked Example: Incidence Intensity and a Transition Payment

A hypothetical policy has an active state $A$ and an absorbing disabled state $D$. The only transition is $A\to D$, at constant intensity $\theta=0.20$ per year. A benefit of $B=\$1{,}000$ is paid at that transition. There are no premiums, continuous outgo, or other benefits. The horizon is $T=2$ years, the force of interest is $\delta=0$, and both terminal reserves are zero. Find $V_A(0)$ and its sensitivity to $\theta$. Method: solve the one-way Thiele equation and differentiate its closed-form solution. This avoids omitting the direct effect of incidence on the transition payment.

In state $D$, there are no future payments, so $V_D(t)=0$. In state $A$, Thiele's equation is

V_A'(t)=\theta\bigl(V_A(t)-B\bigr),\qquad V_A(T)=0.

Solving backward gives $V_A(t)=B\left(1-e^{-\theta(T-t)}\right)$. Therefore, at time zero,

V_A(0)=1000(1-e^{-0.20(2)}) =1000(1-e^{-0.4}) =1000(1-0.6703200) =\$329.68.

Differentiate with respect to $\theta$, treating $B$ and $T$ as fixed:

W_A(0)=\frac{\partial V_A(0)}{\partial\theta} =BT e^{-\theta T} =1000(2)e^{-0.4} =\$1{,}340.64

per unit change in the annual transition intensity. The sensitivity is positive: increasing incidence makes the transition payment more likely. As a check, $P(\text{transition by }2)=1-e^{-0.4}=0.3296800$, so the reserve is exactly $\$1{,}000$ times that probability. The sensitivity is $\$2{,}000e^{-0.4}$, consistent with differentiating that probability.

Worked Example: Recovery Intensity and a Terminal Disability Payment

A hypothetical two-state model has healthy state $H$ and disabled state $D$. The intensities are $\alpha=0.50$ per year for $H\to D$ and $\beta=0.50$ per year for $D\to H$. There are no benefits before the end of one year. At $T=1$, a payment of $\$1{,}000$ is made if the policyholder is disabled; it is zero otherwise. Assume $\delta=0$. Find the reserve from $H$ at time zero and its sensitivity to $\beta$. Method: solve for the probability of being disabled at the terminal time, then differentiate. This avoids treating the recovery intensity as though it affected incidence directly.

Let $p(t)$ be the probability of being disabled $t$ years after starting healthy. Over a short interval, incidence increases this probability at rate $\alpha(1-p)$, while recovery decreases it at rate $\beta p$. Hence

p'(t)=\alpha-(\alpha+\beta)p(t),\qquad p(0)=0.

Writing $\lambda=\alpha+\beta$, the solution is $p(T)=\dfrac{\alpha}{\lambda}(1-e^{-\lambda T})$. Here $\lambda=1.00$ per year, so

V_H(0)=1000p(1) =1000\left[\frac{0.50}{1.00}(1-e^{-1})\right] =1000(0.3160603) =\$316.06.

To differentiate, hold $\alpha$ and $T$ fixed. Since $\partial\lambda/\partial\beta=1$,

\frac{\partial V_H(0)}{\partial\beta} =1000\alpha\left[-\frac{1-e^{-\lambda T}}{\lambda^2} +\frac{T e^{-\lambda T}}{\lambda}\right].

Substituting $\alpha=0.50$, $\lambda=1.00$, and $T=1$ gives

1000(0.50)\left[-(1-e^{-1})+e^{-1}\right] =500(-1+2e^{-1}) =500(-0.2642411) =-\$132.12.

Thus increasing recovery reduces the reserve for this terminal disabled-state benefit. The units are dollars per unit change in the annual recovery intensity. As a check, the sign is negative because recovery makes it less likely that the policyholder remains disabled at the terminal date.

Worked Example: Sensitivity of a Care-Level Reserve Equation

At a particular time, a hypothetical long-term-care model has state $X$ and destination states $Y$ and $Z$. The force of interest is $\delta=0.02$ per year; $\mu_{XY}=0.10$ and $\mu_{XZ}=0.20$ per year. For a parameter $\theta$, the intensity derivatives are $\partial\mu_{XY}/\partial\theta=1$ and $\partial\mu_{XZ}/\partial\theta=2$. At this time, $V_X=\$200$, $V_Y=\$120$, $V_Z=\$80$, $b_{XY}=\$10$, and $b_{XZ}=0$; the sensitivity values are $W_X=5$, $W_Y=3$, and $W_Z=1$. Assume all other terms are independent of $\theta$. Find $W_X'$. Method: evaluate the differentiated Thiele equation term by term. This avoids dropping either a reserve-difference term or a direct intensity-derivative term.

First calculate the transition reserve differences:

V_X-V_Y-b_{XY}=200-120-10=70,\qquad V_X-V_Z-b_{XZ}=200-80-0=120.

The differentiated equation in state $X$ is

W_X'=\delta W_X+\mu_{XY}(W_X-W_Y)+\mu_{XZ}(W_X-W_Z) +\frac{\partial\mu_{XY}}{\partial\theta}(V_X-V_Y-b_{XY}) +\frac{\partial\mu_{XZ}}{\partial\theta}(V_X-V_Z-b_{XZ}).

Substitute each value separately:

$$ W_X'=0.02(5)+0.10(5-3)+0.20(5-1)+1(70)+2(120) =0.10+0.20+0.80+70+240 =311.10. $$

Therefore $W_X'=311.10$ reserve units per year per unit change in $\theta$, at the stated time and state. The positive direct terms reflect the positive reserve differences for the affected transitions. As an arithmetic check, the five contributions sum to $0.10+0.20+0.80+70+240=311.10$.

Interpreting and Checking a Sensitivity

A sensitivity is local: it describes the first-order change in a reserve for a small change in a model parameter. If $W_i(t)=\$400$ per unit of $\theta$, then a small increase $\Delta\theta=0.01$ suggests a reserve increase of about $\$4$, provided the other assumptions are unchanged and the response is locally close to linear. The approximation is not an exact revaluation for a large parameter change.

The sign depends on the transition loss $V_i-V_j-b_{ij}$, not simply on whether the intensity increases. For an increasing intensity, a positive transition loss contributes a positive direct term; a negative transition loss contributes a negative one. For a recovery transition, the difference may be negative if the destination state carries a smaller reserve, but a recovery benefit or other transition payment can change that conclusion.

Check the parameter: state which intensity or linked set of intensities depends on $\theta$, and calculate each $\partial\mu_{ij}/\partial\theta$.
Check the cash-flow sign: use $V_i-V_j-b_{ij}$ for positive transition outgo under the convention used here.
Check the boundary: a fixed terminal reserve has zero sensitivity, but a parameter-dependent terminal amount does not.
Check the units: a derivative with respect to an annual intensity is per unit change in that annual intensity; a time derivative of the sensitivity adds “per year.”

A common exam trap is to differentiate only the reserve terms and omit $(\partial\mu_{ij}/\partial\theta)(V_i-V_j-b_{ij})$. Another is to report $W_i'$, a time derivative, as though it were $W_i$, the reserve sensitivity itself. A third is an arithmetic slip in summing the direct terms: in the last example, the correct total is $311.10$, not $310.70$. Keeping the contributions in separate lines makes both the calculus and the arithmetic easier to check.

Key takeaway: Differentiate the full Thiele equation using the product rule, apply the differentiated terminal condition, and solve the sensitivity system backward. The direct effect of an intensity change is weighted by the current transition reserve difference $V_i-V_j-b_{ij}$.

Check Your Understanding

For each problem, assume the stated basis is fixed except for the named transition-intensity parameter. Round money to the nearest cent.

A one-way model has a transition intensity $\theta=0.25$ per year from state $A$ to an absorbing state, a payment of $\$600$ on transition, no other cash flows, zero interest, and a two-year horizon with zero terminal reserves. Find $V_A(0)$ and $\partial V_A(0)/\partial\theta$. (Answer: $V_A(0)=\$236.08$; sensitivity $=\$727.84$ per unit of annual intensity.)
A policy starts healthy. Its incidence intensity is $\alpha=0.40$ per year, its recovery intensity is $\beta=0.60$ per year, and it pays $\$500$ at the end of one year only if disabled. There is no interest or other cash flow. Find the initial reserve and its sensitivity to $\beta$. (Answer: reserve $=\$126.42$; sensitivity $=-\$52.85$ per unit of annual recovery intensity.)
At a given time in a three-state model, $\delta=0.03$, $W_A=4$, $W_B=1$, $W_C=2$, $\mu_{AB}=0.20$, and $\mu_{AC}=0.10$. The parameter derivatives of those intensities are 1.5 and 0.5. The transition reserve differences $V_A-V_B-b_{AB}$ and $V_A-V_C-b_{AC}$ are 50 and 20. Find $W_A'$. (Answer: $85.92$ reserve units per year per unit of the parameter.)
For a transition $R\to S$, the force of interest is $0.01$, $W_R=10$, $W_S=4$, and the intensity is $0.08$. The parameter derivative of the intensity is 1, and $V_R-V_S-b_{RS}=30$. With no other transition or parameter-dependent cash-flow terms, find $W_R'$. (Answer: $0.01(10)+0.08(10-4)+30=30.58$ reserve units per year per unit of the parameter.)
At a time in state $A$, $\delta=0.02$, $W_A=5$, and there are two outgoing transitions with intensities 0.10 and 0.20. Their destination sensitivities are 3 and 1. The reserve differences including transition benefits are 70 and 120. The parameter derivatives of the two intensities are 1 and 2. Find $W_A'$. (Answer: $0.02(5)+0.10(5-3)+0.20(5-1)+1(70)+2(120)=311.10$ reserve units per year per unit of the parameter.)

What You'll Learn