Premium Loading Changes In Reserve Valuation

Why a Premium Loading Changes a Reserve

A prospective reserve is the expected present value of future benefits and expenses less the expected present value of future valuation premiums. If the benefit and expense assumptions are kept fixed, changing the valuation premium changes only the premium side of that balance. Adding a loading to future valuation premiums therefore reduces the reserve; removing a loading increases it.

The size of the change is not generally the loading multiplied by the number of future premium dates. A future premium is included only if the policy is in a state eligible to pay it, and its contribution is discounted to the valuation date. The reserve effect must therefore reflect both the probability of being eligible at each date and the timing of each payment.

Definition: A valuation-premium loading is a specified addition to, or subtraction from, the premium amount used in a reserve valuation. In this tutorial, benefits, expenses, interest, transition or mortality assumptions, and premium eligibility rules are held fixed. The loading changes the valuation cash flow; it does not, by itself, change the contractual benefits or the policyholder's actual premium.

For an annual model, let the reserve be valued at time $t$, immediately after any premium due at that time. Let $L_j$ be the loading on the valuation premium due $j$ years later, and let $I_j$ indicate that the policy is in a state eligible to pay that premium at that date. The loading can depend on duration or state. With effective annual interest rate $i$, the discount factor is $v=1/(1+i)$.

\Delta V_t = V_t^{\text{loaded}}-V_t^{\text{reference}} = -\sum_{j\geq 1} v^j E_t[I_j L_j].

Here $E_t$ is conditional expectation given the information and policy state at time $t$. The sum covers only future premium dates. If $L_j$ is a constant amount and a premium is due whenever the insured is alive, then $E_t[I_j L_j]=L_j\,{}_jp_x$ for a life aged $x$ at the valuation date. Thus the loading's reserve effect is the negative of its expected present value.

Why the Formula Holds

The prospective reserve is a difference between two expected present values. Write the reference reserve as $V_t^{\text{reference}}=EPV_t(B+E)-EPV_t(P)$, where $B$, $E$, and $P$ denote future benefits, expenses, and valuation premiums. Under the loaded basis, the premium at each future date becomes $P_j+I_jL_j$. The benefits and expenses are unchanged, so subtracting the reference reserve from the loaded reserve cancels their present values and the original premium present values.

Write the two prospective reserves.
Use the same benefit, expense, interest, and decrement assumptions in both valuations.

Subtract the reference reserve.
The benefit and expense terms cancel, as do the reference valuation-premium terms.

Value only the premium difference.
Each extra premium is an additional future receipt to the insurer, so it enters the reserve with a minus sign and is weighted by eligibility and discounting.

\begin{aligned} V_t^{\text{loaded}} &=EPV_t(B+E)-EPV_t(P+I L),\\ V_t^{\text{reference}} &=EPV_t(B+E)-EPV_t(P),\\ \Delta V_t &=-EPV_t(I L) =-\sum_{j\geq 1}v^jE_t[I_jL_j]. \end{aligned}

This argument uses linearity of expectation and applies whether the future premiums are level or vary by duration, and whether eligibility depends on survival alone or on a multiple-state process. It does not apply unchanged if the premium modification also changes benefits, expenses, interest, or the probability model: those effects would no longer cancel.

Worked Examples

Worked Example: A Loading on Two Remaining Life Premiums

A hypothetical life policy is valued immediately after the premium at duration 1. Two later annual premiums remain, each payable only if the insured is alive at its due date. The conditional probability of surviving to the first later date is $0.70$, and the conditional probability of surviving to the second later date is $0.70(0.80)$. The effective annual interest rate is $i=0.05$. A $15 loading is added to each future valuation premium. Find the change in reserve.

Method. Apply the prospective premium-difference formula. The valuation is after the current premium, so neither the current premium nor its loading is included. The first future loading has probability weight $0.70$. The second has probability weight $0.70(0.80)=0.56$.

The discount factor and its square are

v=\frac{1}{1.05}=0.95238095,\qquad v^2=0.90702948.

The reserve change is

\begin{aligned} \Delta V_1 &=-15\left[0.70v+0.70(0.80)v^2\right]\\ &=-15\left[0.70(0.95238095)+0.56(0.90702948)\right]\\ &=-15\left[0.66666667+0.50793651\right]\\ &=-15(1.17460318)\\ &=-17.6190477. \end{aligned}

Rounded to the nearest cent, the loaded-basis reserve is $17.62 lower than the reference-basis reserve. The sign is consistent: a larger future valuation premium reduces the prospective reserve.

Common mistake avoided. Counting two $15 loadings as an immediate, certain $30 reduction ignores both survival and discounting. The correct calculation weights the first payment by $0.70v$ and the second by $0.56v^2$, giving a decrease of $17.62, not $30.

Worked Example: Computing Both Reserves Under a Loaded Basis

A hypothetical three-year contract is valued at time 0, immediately after any premium due then. It pays a $100 death benefit at the end of each policy year in which death occurs, and a $50 maturity benefit at time 3 if the insured survives all three years. Conditional death probabilities for years 1, 2, and 3 are $0.10$, $0.20$, and $0.25$, respectively. The reference valuation premium is $30 at times 1 and 2, payable if the insured is alive at the due date. The loaded basis adds $10 to each of those two premiums. Let $i=0.05$. Find both reserves and their difference. Assume the anniversary premium is paid by those alive at the anniversary; the death benefit for the year just ended is paid to those who died during that year.

Method. First calculate the survival probabilities and benefit probabilities, then value the benefits and reference premiums. Finally, apply the loading formula and check the result by directly valuing the loaded premiums.

The probability of surviving to time 1 is $0.90$. The probability of surviving to time 2 is $0.90(0.80)=0.72$. The probability of surviving all three years is $0.72(0.75)=0.54$. Death probabilities by policy year are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$. The time-3 benefit expectation is therefore $100(0.18)+50(0.54)=18+27=45$.

The required discount factors are $v=1/1.05=0.95238095$, $v^2=0.90702948$, and $v^3=0.86383760$. The expected present value of benefits is

$$ 100(0.10)v+100(0.18)v^2+45v^3 =9.5238095+16.3265306+38.8726919 =64.7230320. $$

The expected present value of reference premiums is

$$ 30(0.90)v+30(0.72)v^2 =25.7142857+19.5918367 =45.3061224. $$

Thus the reference reserve is $64.7230320-45.3061224=19.4169096$, or $19.42 to the nearest cent. The reserve change from the two $10 loadings is

\begin{aligned} \Delta V_0 &=-10(0.90v+0.72v^2)\\ &=-10(0.85714286+0.65306122)\\ &=-15.1020408. \end{aligned}

The loaded reserve is $19.4169096-15.1020408=4.3148688$, or $4.31. As a check, the loaded premiums are $40 and $40, with expected present value $40(0.90)v+40(0.72)v^2=34.2857143+26.1224490=60.4081633$. Subtracting this from the unchanged benefit value, $64.7230320-60.4081633=4.3148687$, confirms the result (the tiny difference is rounding).

Common mistake avoided. The year-2 premium is weighted by the probability of surviving both years, $0.72$, not merely by the year-2 conditional survival probability $0.80$. A premium due at time 2 requires the insured to have reached that date from time 0.

Worked Example: State-Dependent Loadings in a Multiple-State Model

Consider a hypothetical policy with Active (A), Disabled (D), and Dead (X) states. At each annual step, the transition matrix, with rows and columns ordered A, D, X, is

\mathbf{P}= \begin{pmatrix} 0.80&0.15&0.05\\ 0.20&0.70&0.10\\ 0&0&1 \end{pmatrix}.

The policy starts Active. At each of times 1 and 2, a valuation-premium loading of $12 applies in state A, $4 applies in state D, and no premium is payable in state X. The effective annual interest rate is $i=0.04$. Find the reserve change at time 0.

Method. Find the state probabilities at each premium date, calculate the expected loading there, and discount each expected amount. At time 1, the state probabilities are the first row of the transition matrix: $P(A)=0.80$, $P(D)=0.15$, and $P(X)=0.05$. The expected loading is $0.80(12)+0.15(4)=9.60+0.60=10.20$.

To obtain time-2 probabilities, multiply the time-1 row vector by the transition matrix. The probabilities are

\begin{aligned} P(A\text{ at }2)&=0.80(0.80)+0.15(0.20)=0.67,\\ P(D\text{ at }2)&=0.80(0.15)+0.15(0.70)=0.225,\\ P(X\text{ at }2)&=0.80(0.05)+0.15(0.10)+0.05(1)=0.105. \end{aligned}

These probabilities sum to $0.67+0.225+0.105=1$. The expected loading at time 2 is $0.67(12)+0.225(4)=8.04+0.90=8.94$. With $v=1/1.04=0.96153846$ and $v^2=1/1.04^2=0.92455621$, the reserve change is

\begin{aligned} \Delta V_0 &=-10.20v-8.94v^2\\ &=-10.20(0.96153846)-8.94(0.92455621)\\ &=-9.8076923-8.2655325\\ &=-18.0732248. \end{aligned}

The reserve decreases by $18.07 to the nearest cent. As a check, both expected loadings are positive and payable only in non-dead states, so their present value is positive and the reserve change must be negative.

Common mistake avoided. Applying the Active-state loading to every policy still in force would ignore the stated $4 loading in the Disabled state. The expected loading at time 1 is $10.20, not $0.95(12)=\$11.40$; state-specific amounts must be applied before taking the expectation.

Interpreting and Checking the Reserve Effect

The formula is a reserve-difference result, not a complete reserve calculation. It is often efficient to use when the reference reserve is already known: calculate the expected present value of only the changed premium amounts, then adjust the reserve with the appropriate sign. When the base reserve is not known, calculate benefits and reference premiums under the original assumptions, then subtract the loading's expected present value.

Several details determine the correct result:

Valuation timing: A reserve immediately after a premium excludes that payment. A reserve immediately before it may include it if it is due at that instant and payable in the current state.
Eligibility: Use the probability of being in a premium-paying state at the payment date, not simply the probability of being in force under a different definition.
Discounting: A loading due $j$ years after valuation is multiplied by $v^j$. Do not discount from the policy's issue date when the reserve is at a later duration.
Sign: An addition to future valuation premiums makes the reserve lower. A removal of a positive loading makes it higher.
Unchanged basis assumptions: The cancellation argument requires all non-premium assumptions and cash flows to remain fixed. Changing expenses or benefits at the same time requires valuing those changes separately.

Key takeaway: With all other valuation assumptions fixed, the reserve change is the negative expected present value of the future premium loading. Weight each loading by the probability of premium eligibility at its due date, discount it from the reserve valuation date, and include only payments that are still in the future.

This premium-loading calculation is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In exam work, writing the valuation date, payment dates, eligibility condition, and sign before calculating is an effective way to prevent otherwise avoidable errors.

Check Your Understanding

For each question, find the change in reserve caused by the stated premium modification. Unless otherwise specified, value immediately after any premium due at the valuation time. Round money to the nearest cent.

A $8 loading is added to each of three future annual premiums. The probabilities of being eligible at those dates are $0.90$, $0.75$, and $0.60$, respectively, and $i=0.04$. Find the reserve change. (Answer: $-\$16.74$)
A $6 loading is removed from each of two future premiums. The probabilities of eligibility at the two dates are $0.80$ and $0.56$, respectively, and $i=0.05$. Find the reserve change. (Answer: $+\$7.62$)
At two future annual premium dates, state probabilities are Active/Disabled/Dead $=(0.65,0.20,0.15)$ and $(0.50,0.25,0.25)$. Loadings are $10 in Active, $3 in Disabled, and $0 in Dead at each date. If $i=0$, find the reserve change. (Answer: $-\$12.85$)
The reference valuation premiums are $40 at time 1 and $30 at time 2. A 10% loading is added to each. Eligibility probabilities are $0.90$ and $0.70$, and $i=0.05$. Find the reserve change. (Answer: $-\$5.33$)
A reserve is valued immediately before a $5 loading is due at time 1. One further $5 loading is due at time 2 if the policy remains eligible; conditional on eligibility at time 1, the probability of remaining eligible to time 2 is $0.80$. Assume eligibility at time 1 is certain and $i=0.05$. Find the reserve change, including the payment due immediately. (Answer: $-\$8.81$)

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