Benefit Escalation During a Claim

Why Escalation Changes a Disability Benefit Value

A disability benefit may remain level after a claim begins, or it may increase as the claim continues. An increase can be specified as a fixed percentage at each anniversary, as continuous growth with claim duration, or by reference to an index such as a hypothetical cost-of-living measure. These benefit patterns are not interchangeable: the contract must specify both when an increase applies and what determines it.

Valuation must also account for the possibility that payments stop when the claim terminates. Consequently, benefit growth does not by itself determine the value. The calculation combines the projected benefit amount with discounting, the probability the claim is still active, payment timing, and any maximum benefit period. The methods below are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $B(t)$ be the benefit amount or continuous benefit rate payable at claim duration $t$, measured from claim commencement. Fixed escalation during a claim specifies how $B(t)$ changes with elapsed claim duration, independently of external index movements. Index-linked escalation specifies that the benefit changes with a named index. For either form, the contract must define the initial amount, adjustment dates or growth rule, and whether the benefit is payable only while the claim remains active.

Continuous Benefits with Continuous Escalation

Suppose a claim has just commenced. Assume the benefit is paid continuously at rate $b_0e^{gt}$ dollars per year while the claim is active, where $b_0$ is the rate at commencement and $g$ is the force of benefit escalation. Assume claim termination has a constant intensity $\lambda$, so the time $T_c$ to termination has an exponential distribution and $P(T_c>t)=e^{-\lambda t}$. Let the force of interest be constant at $\delta$, and let payments stop at duration $n$ even if the claim has not terminated. There are no other benefit contingencies or cash flows in this valuation.

At duration $t$, the rate $b_0e^{gt}$ is paid only if $T_c>t$, an event with probability $e^{-\lambda t}$. Discounting a payment at that time multiplies it by $e^{-\delta t}$. Thus the expected present value (EPV) is the integral of the rate multiplied by its probability of being payable and its discount factor.

\begin{aligned} \operatorname{EPV} &=\int_0^n b_0e^{gt}e^{-\lambda t}e^{-\delta t}\,dt\\ &=b_0\int_0^n e^{-(\delta+\lambda-g)t}\,dt\\ &= \begin{cases} b_0\dfrac{1-e^{-(\delta+\lambda-g)n}}{\delta+\lambda-g}, &\delta+\lambda-g\ne 0,\\[8pt] b_0n,&\delta+\lambda-g=0. \end{cases} \end{aligned}

This derivation shows why the exponent contains all three forces: interest discounting and claim termination reduce expected value, while benefit escalation increases it. The zero-denominator case follows by integrating the constant function $b_0$; it is also the limit of the first expression as $\delta+\lambda-g$ approaches zero. The formula assumes constant forces, continuous payments, and an exponential claim-termination time. It does not apply without adjustment to anniversary-only increases, nonconstant forces, or stochastic index growth.

Worked Example: Continuous Benefit with a Three-Year Maximum

A hypothetical disability contract pays continuously at a rate of $600 per year at claim commencement. The rate escalates continuously at force $g=0.02$. Claim termination has a constant intensity $\lambda=0.08$, and the force of interest is $\delta=0.04$. Payments stop at the end of three years if the claim has not already terminated. Find the EPV, rounded to the nearest cent.

Method and assumptions. The claim is active at time zero; its termination time is exponential with parameter $0.08$ per year; interest and escalation forces are constant; payments are continuous; and there are no payments after duration 3. The combined force in the integrand is $\delta+\lambda-g=0.04+0.08-0.02=0.10$.

Substitute the values into the continuous formula:

\begin{aligned} \operatorname{EPV} &=600\frac{1-e^{-0.10(3)}}{0.10}\\ &=600\frac{1-e^{-0.30}}{0.10}\\ &=600\frac{1-0.7408182207}{0.10}\\ &=600(2.591817793)\\ &=1555.090676. \end{aligned}

Rounded to the nearest cent, the EPV is $1,555.09. As a check, the combined force is positive, so the discounted expected payment rate declines with duration; the result is less than the $1,800 that would be paid over three years if the rate stayed at $600 and there were no discounting or claim termination.

Common mistake avoided: Omitting claim termination would use $\delta-g=0.02$, producing $600(1-e^{-0.06})/0.02=\$1,747.06$. That is not the EPV under the stated model because payments cease at claim termination as well as being discounted.

Anniversary Increases and Payment Timing

Many contracts do not increase benefits continuously. An annual escalation clause may apply an increase on each claim anniversary, while benefits are paid at discrete dates. In that case, project the amount due at each payment date and multiply by the probability the claim is active then and by the appropriate discount factor. Be precise about the first payment: if the first year is paid at the original rate, the payment at the end of year $k$ is $B_0(1+c)^{k-1}$, not $B_0(1+c)^k$, where $c$ is the annual escalation rate.

Discrete valuation rule: If a benefit $B_k$ is payable at the end of year $k$ only when the claim remains active to that date, its expected present value contribution is $B_k\,P(T_c>k)\,v^k$, where $v=1/(1+i)$ and $i$ is the effective annual interest rate. Sum these contributions over the contractual payment dates. This form assumes the specified survival probability correctly represents the claim-continuation rule up to each payment date.

Worked Example: Annual Step Increases with a Claim Limit

A hypothetical policy pays $10,000 at the end of each year for at most three years of a claim. The first payment is at the end of year 1; the annual amount increases by 10% after each completed year, so the year 1, 2, and 3 amounts are $10,000, $11,000, and $12,100. At each year-end, a claim that was active at the start of that year continues through the year with probability $0.90$, independently of earlier years. The effective annual interest rate is $i=0$. Find the EPV.

Method and assumptions. Payments occur only at year-ends, and each is made only if the claim has continued through that payment date. The probability of reaching the payment at year $k$ is $0.90^k$. Since $i=0$, each discount factor is $v^k=1$.

\begin{aligned} \operatorname{EPV} &=10{,}000(0.90)+11{,}000(0.90)^2+12{,}100(0.90)^3\\ &=10{,}000(0.90)+11{,}000(0.81)+12{,}100(0.729)\\ &=9{,}000+8{,}910+8{,}820.90\\ &=26{,}730.90. \end{aligned}

The EPV is $26,730.90. The answer is less than the total of $33,100 payable if the claim certainly lasted three years. Common mistake avoided: multiplying every payment by $0.90$ would use a one-year continuation probability even for later payments. Reaching the second and third year-end requires continuation through two and three successive years, hence probabilities $0.90^2$ and $0.90^3$.

Index-Linked Escalation

An index-linked benefit is determined by an index level rather than solely by elapsed claim duration. For example, a hypothetical contract might set the benefit rate at $b_0 I(t)/I(0)$, where $I(t)$ is the specified index at duration $t$. If the index is assumed to grow deterministically at constant force $h$, then $I(t)/I(0)=e^{ht}$, and the continuous valuation formula applies with $g=h$. This substitution is valid because the index path is deterministic under that assumption.

If the index is stochastic, simply replacing its future value with an average index growth rate is generally not valid. In the continuous model, the relevant expected payment at duration $t$ depends on the expectation of the product of the index-linked benefit and the indicator that the claim is still active. If index growth and claim termination are dependent, their joint behavior must also be modeled. A valuation using a deterministic index assumption should therefore be stated as an assumption, not presented as a general result for stochastic indices.

Worked Example: A Deterministically Indexed Continuous Benefit

A hypothetical long-duration disability claim pays continuously, starting at $1,000 per year. The benefit is linked to an index assumed to grow deterministically at a force of $h=0.03$. Claim termination has constant intensity $\lambda=0.05$, the force of interest is $\delta=0.04$, and the maximum payment duration is four years. Find the EPV, rounded to the nearest cent.

Method and assumptions. The index grows at a constant deterministic force, so the benefit rate is $1{,}000e^{0.03t}$. The claim-termination time is exponential with parameter $0.05$ per year. Payments are continuous and cease at duration 4. The combined force is $\delta+\lambda-h=0.04+0.05-0.03=0.06$.

\begin{aligned} \operatorname{EPV} &=1{,}000\frac{1-e^{-0.06(4)}}{0.06}\\ &=1{,}000\frac{1-e^{-0.24}}{0.06}\\ &=1{,}000\frac{1-0.7866278611}{0.06}\\ &=1{,}000(3.556202315)\\ &=3{,}556.202315. \end{aligned}

Rounded to the nearest cent, the EPV is $3,556.20. This calculation uses index growth in place of fixed claim-duration escalation because the stated index path is deterministic. It would not, by itself, value a contract whose index changes randomly.

Choosing and Checking the Escalation Model

Before calculating, translate the benefit clause into a payment function or a schedule of payment amounts. Then align the payment function with the claim-continuation model and the payment dates. A fixed percentage at each anniversary is usually a stepwise pattern; a force of escalation describes continuous exponential growth. Treating one as the other changes the benefit amounts between anniversaries.

Identify the escalation rule.
Decide whether increases are tied to elapsed claim duration, anniversaries, or an external index. Record the base amount and the first date on which an increase applies.

Identify when payments can occur.
Distinguish a continuous payment rate from payments at specified dates. Apply any maximum benefit period to those same durations.

Apply the claim-termination assumption.
For each payment time, use the probability the claim is still active then. Do not substitute a survival probability for a termination intensity, or vice versa, without the model that relates them.

Discount and sum or integrate.
Use the discount factor for the payment time, then sum discrete expected payments or integrate the continuous expected payment rate.

The main exam trap is to combine quantities with inconsistent timing or units. A force of escalation is expressed per unit time and belongs in a continuous exponential model; an annual percentage increase belongs at the contractual adjustment dates. Also check whether an annual benefit is paid at the start or end of a year and whether it is conditional on remaining disabled to that date. A one-year shift in the first payment or an extra escalation step can materially change the result.

Key takeaway: Value the benefit actually specified by the contract. For continuous deterministic escalation with constant forces, the net exponent is $\delta+\lambda-g$; for discrete anniversary benefits, value each scheduled payment using its own amount, payment-date survival probability, and discount factor.

Check Your Understanding

For each problem, use the stated timing and claim-termination assumptions. Amounts are hypothetical.

A claim pays continuously at a rate of $800 per year at commencement, escalating at force $g=0.01$. The termination intensity is $\lambda=0.06$, the force of interest is $\delta=0.03$, and the maximum duration is two years. Find the EPV, to cents. (Answer: $1,478.56)
A continuous benefit starts at $2,000 per year and escalates at force $g=0.05$. The claim-termination intensity is $0.03$, the force of interest is $0.04$, and the maximum duration is five years. Find the EPV, to cents. (Answer: $9,516.26)
A policy pays at each of the first two year-ends if the claim has continued to that date. The first payment is $5,000, and the second is increased by 4%. The annual continuation probability is $0.80$, independently each year, and $i=0$. Find the EPV. (Answer: $7,328.00)
A continuous claim benefit starts at $1,000 per year and follows a deterministically growing index with force $h=0.02$. The claim-termination intensity is $0.04$, the force of interest is $0.03$, and the maximum duration is three years. Find the EPV, to cents. (Answer: $2,785.84)
A continuous benefit rate of $750 per year escalates at force $g=0.03$. The claim-termination intensity is $0.02$, the force of interest is $0.01$, and the maximum duration is four years. Find the EPV, to cents. (Answer: $3,000.00)

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