Valuing Care Benefits with Upgrades and Recovery

Why care severity changes the valuation

A long-term care benefit may pay different amounts at different care levels. A claimant may begin in moderate care, move to severe care, or recover. Death ends future care payments. Valuing the contract therefore requires more than multiplying a benefit by the expected time in care: the value of the future benefit depends on which state is reached and when.

This tutorial values benefits from a specified care state under a continuous-time Markov model. It focuses on the link between the current state’s payment rate and the values of states that can follow it. The transition rates are assumed constant while the claimant occupies each state. If rates vary by duration, age, or calendar time, the valuation must account for that variation rather than use the constant-rate equations unchanged.

Definition: Let $V_i$ be the expected present value, at the instant a person enters care state $i$, of all future benefits under the stated model. Let $b_i$ be the continuous benefit payment rate while in state $i$, $\delta$ the force of interest, and $q_{ij}$ the transition intensity from state $i$ to state $j$. Write $q_i=\sum_{j\ne i}q_{ij}$ for the total intensity of leaving state $i$. Unless otherwise stated, recovery and death states have value zero: there are no further payments and no re-entry after recovery.

State-value equations

Suppose a claimant in state $i$ receives benefits continuously at rate $b_i$. During a short interval, the claimant may remain in that state or move to another state. An upgrade can lead to a state with a higher benefit rate, so its value is reflected in the value of the destination state. Recovery and death generally lead to states with no future care benefit in the model.

If a lump sum $c_{ij}$ is also paid immediately on transition from $i$ to $j$, include it as a transition benefit. For constant intensities and a constant force of interest, the state-value equation is:

(\delta+q_i)V_i=b_i+\sum_{j\ne i}q_{ij}(V_j+c_{ij}).

When there is no transition lump sum, set $c_{ij}=0$. For a recovery or death destination with no further benefits, set its value to zero. Then only transitions to states with nonzero values contribute to the sum.

List the possible destinations.
For each care state, identify every upgrade, recovery, and death transition, with its intensity.

Write the payment amounts.
Record the continuous benefit rate in each care state and any lump sum payable on a transition.

Form and solve the equations.
Use $\delta+q_i$ on the left side. When upgrades move only to more severe states, solve from the most severe state back toward the initial state.

Check units and assumptions.
Intensities and $\delta$ must use the same time unit; benefit rates are payments per that time unit. Confirm that the recovery and death values match the assumed contract.

Why the equation holds

Consider a short interval of length $h$ while the person is in state $i$. Under constant transition intensities, the probability of a transition to $j$ in that interval is $q_{ij}h+o(h)$, and the probability of remaining in $i$ is $1-q_i h+o(h)$. The continuous payment over the interval is $b_i h+o(h)$. Discounting over the interval multiplies future values by $e^{-\delta h}=1-\delta h+o(h)$.

The value at the start of the interval is therefore the payment during the interval plus the discounted expected value after it. Substituting these short-interval expressions, subtracting $V_i$, dividing by $h$, and taking the limit as $h$ tends to zero gives:

0=b_i+\sum_{j\ne i}q_{ij}(V_j+c_{ij})-(\delta+q_i)V_i.

Rearranging gives the state-value equation above. This argument relies on the stated constant force and transition intensities, and on benefits being paid continuously at the specified rates. A benefit payable only at the end of a month or year requires a payment-timing adjustment.

Worked Example: Checking a Care-State Payment Rate

A hypothetical policy pays continuously while a claimant is in one care state. Its annual benefit rate is calculated as 15 covered service units per year at $31 per unit, plus a $100 annual allowance. The force of interest is $\delta=0.04$ per year. Recovery and death intensities are 8 and 5.96 per year, respectively. There are no upgrade payments or benefits after leaving the care state. Find the value on entry to care.

Method. First calculate the total continuous benefit rate. The exit intensity is the sum of the recovery and death intensities:

b=15(31)+100=465+100=565\text{ dollars per year}, \qquad q=8+5.96=13.96\text{ per year}.

Since both destinations have value zero, the state-value equation is $(\delta+q)V=b$. Thus:

(0.04+13.96)V=565, \qquad 14V=565, \qquad V=\frac{565}{14}=40.3571429.

The value is $40.36, rounded to the nearest cent. The check is that $14(40.3571429)$ is approximately 565. The deliberately high exit intensities make this a compact arithmetic illustration, not a claim about typical care durations. A common mistake is to calculate $15(31)+100$ as 255; the correct total is $465+100=565$, and the value must be based on 565.

Valuing an upgrade followed by recovery or death

When one care state can upgrade to another, the current state’s value includes the intensity of upgrading multiplied by the value reached after the upgrade. The upgrade intensity is not itself a probability. It enters the continuous-time valuation equation alongside the other exit intensities.

Worked Example: Moderate Care Upgrading to Severe Care

A hypothetical long-term care contract pays $2,000 per year continuously in moderate care and $7,000 per year continuously in severe care. From moderate care, the claimant upgrades to severe care at intensity $0.25$, recovers at intensity $0.15$, and dies at intensity $0.10$. From severe care, recovery and death intensities are $0.20$ and $0.30$. The force of interest is $\delta=0.10$. Recovery and death stop benefits, with no re-entry. Find both care-state values.

Method. All rates are per year. Severe care has no further upgrade, and both exit destinations have value zero. Therefore:

(0.10+0.20+0.30)V_S=7, \qquad 0.60V_S=7, \qquad V_S=\frac{7}{0.60}=11.6666667.

Amounts in this calculation are in thousands of dollars, so $V_S$ is $11,666.67. In moderate care the total exit intensity is $0.25+0.15+0.10=0.50$, and only the upgrade destination has a nonzero value. Thus:

(0.10+0.50)V_M=2+0.25V_S, \qquad 0.60V_M=2+0.25(11.6666667)=4.9166667,

V_M=\frac{4.9166667}{0.60}=8.1944445\text{ thousand dollars}.

The moderate-care value is $8,194.44, and the severe-care value is $11,666.67, each rounded to the nearest cent. The check for moderate care is $0.60(8.1944445)\approx 2+0.25(11.6666667)=4.9166667$. The common unit trap is to report 8.19444 as dollars. It is in thousands of dollars, so it must be converted to $8,194.44.

Several care levels and transition payments

With a sequence of upgrades, the same equation applies at each care level. If transitions move only from less severe to more severe care, the equations can be solved in reverse order: value the most severe state first, then use that result to value the less severe states. Recovery and death contribute to the total exit intensity even when their destination values are zero.

If an upgrade also triggers a lump-sum payment, add that payment to the value associated with the destination transition. In particular, a transition at intensity $q_{ij}$ with a lump sum $c_{ij}$ contributes $q_{ij}(V_j+c_{ij})$ to the right side. It is not a continuous annual payment and must not be added to $b_i$.

Worked Example: Three Care Levels with an Upgrade Bonus

A hypothetical policy pays continuously at $1,200 per year in level L, $3,000 per year in level M, and $6,000 per year in level S. From L, the upgrade intensity to M is $0.30$, recovery intensity is $0.20$, and death intensity is $0.10$. From M, the upgrade intensity to S is $0.20$, recovery intensity is $0.25$, and death intensity is $0.15$. From S, recovery and death intensities are $0.30$ and $0.20$. An additional $500 is paid immediately on the upgrade from L to M. The force of interest is $0.05$, and there are no benefits after recovery or death. Find the value in each care state.

Method. Work in dollars throughout. For level S, the total exit intensity is $0.30+0.20=0.50$, so:

(0.05+0.50)V_S=6{,}000, \qquad V_S=\frac{6{,}000}{0.55}=10{,}909.0909.

For level M, the total exit intensity is $0.20+0.25+0.15=0.60$. The upgrade has no lump sum, so:

(0.05+0.60)V_M=3{,}000+0.20V_S =3{,}000+0.20(10{,}909.0909) =5{,}181.8182,

V_M=\frac{5{,}181.8182}{0.65}=7{,}972.0280.

For level L, total exit intensity is $0.30+0.20+0.10=0.60$. On upgrade, the value received is the level M value plus the $500 transition payment. Therefore:

(0.05+0.60)V_L =1{,}200+0.30(V_M+500) =1{,}200+0.30(7{,}972.0280+500) =3{,}741.6084,

V_L=\frac{3{,}741.6084}{0.65}=5{,}756.3206.

The values are $V_S=$$10,909.09, $V_M=$$7,972.03, and $V_L=$$5,756.32, rounded to the nearest cent. Substitution checks the last equation: $0.65(5{,}756.3206)\approx3{,}741.6084$. The transition bonus is included only on the L-to-M upgrade; adding it to the annual payment rate would incorrectly treat a one-time payment as a continuous benefit.

Common exam traps

Omitting an exit intensity from the denominator. The left side uses $\delta+q_i$, where $q_i$ includes every way of leaving the current state, including recovery and death.
Giving a zero value to an upgrade destination. An upgrade is not a termination if benefits continue in the more severe state; include the destination value.
Confusing a transition benefit with a payment rate. A lump sum paid on upgrade belongs in the transition term, not in the continuous benefit rate.
Mixing units. If benefits are stated in thousands of dollars, resulting state values are also in thousands until converted.
Assuming recovery always has value zero. It is zero only under a model with no further benefits or re-entry after recovery. If coverage resumes after recovery, recovery must lead to a state whose future value is modeled rather than set to zero.
Using the constant-rate equation outside its assumptions. If transition intensities or discounting vary with time or care duration, a constant-coefficient state equation alone does not generally give the value.

Key takeaway: For each care state, add its continuous benefit rate to the intensity-weighted values of its destinations, then divide by the force of interest plus the total exit intensity. This gives the state value when rates are constant and payment timing matches the model.

Check Your Understanding

For each problem, assume continuous benefit payments, constant transition intensities, and zero value after recovery or death unless another care state is specified.

A care state pays $2,500 per year. The force of interest is $0.03$, and recovery and death intensities are $0.12$ and $0.08$. Find the value on entry. (Answer: $10,869.57)
Moderate care pays $1,500 per year and upgrades to severe care at intensity $0.40$; its recovery and death intensities are $0.20$ and $0.10$. Severe care pays $5,000 per year and has recovery and death intensities $0.30$ and $0.20$. If $\delta=0.05$, find the moderate-care value. (Answer: $6,848.48)
Moderate care pays $2,000 per year. Its upgrade, recovery, and death intensities are $0.25$, $0.15$, and $0.10$. Severe care pays $7,000 per year and has recovery and death intensities $0.20$ and $0.30$. If $\delta=0.10$ and upgrades lead to severe care, find the moderate-care value. (Answer: $8,194.44)
A state pays $1,000 per year and upgrades to a second care state at intensity $0.20$, with an immediate $500 upgrade payment. Its recovery and death intensities are each $0.10$. The second state pays $4,000 per year and has recovery and death intensities each $0.20$. If $\delta=0.05$, find the value in the first state. (Answer: $6,395.06)

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