Reference Bases And Modified Bases

Why Compare Valuation Bases?

A reserve is calculated using a set of valuation assumptions, not in isolation. Those assumptions can include interest, transition probabilities, benefits, premiums and expenses, together with the timing of each cash flow. A reference basis supplies the baseline assumptions for a comparison. A modified basis changes one or more of them. The purpose of comparing the two is to identify not just whether a reserve changes, but which assumption changed and at what duration its effect enters.

The word “modified” does not, by itself, identify a particular method. Here it means that the assumptions used to value the contract differ from those on the reference basis. The reference basis might represent a specified valuation framework; it need not be the insurer’s actual experience or pricing basis. Likewise, a modified premium or benefit in a valuation basis need not mean that the policyholder’s contractual cash flow has changed. The comparison must state exactly which cash flows and assumptions each basis uses.

Definition: A valuation basis is the collection of assumptions and cash-flow definitions used to calculate a reserve. A reference basis is the baseline collection used for comparison. A modified basis is the collection after one or more specified assumptions have been changed. A reserve difference is meaningful only when the timing conventions, state definitions and reserve sign convention are also stated.

The sign convention used throughout this tutorial defines a reserve as the present value of future benefits and expenses less the present value of future premiums. A reserve can therefore be negative when future premiums exceed future outgo on the chosen basis. That algebraic result is not, on its own, a statement about whether a negative reserve is permitted under a particular regulatory or contractual framework.

A Discrete-Time Comparison Framework

First consider a one-state annual model with two possible outcomes during each policy year: death, with probability $q_t$, or survival, with probability $p_t=1-q_t$. Assume an effective annual interest rate $i$, with $v=1/(1+i)$. Let $V_t$ be the reserve immediately before the premium and expense at time $t$. The premium $P_t$ is received and expense $e_t$ is paid at time $t$, regardless of whether death occurs later in that year. A death benefit $B_t$ is paid at the end of the year if death occurs during it. A survivor enters the next year with reserve $V_{t+1}$. Let $V_n=0$ at the end of the contract.

V_t=e_t-P_t+v\left(q_tB_t+p_tV_{t+1}\right).

This equation applies to the stated annual timing convention. If premiums are payable only while in a particular state, expenses are paid only on a transition, or benefits are payable at a different time, the cash-flow terms and their discounting must reflect those conditions. In particular, changing the timing while keeping the same algebra can produce a wrong reserve.

Use superscripts $R$ and $M$ for the reference and modified bases, respectively. Define the reserve difference as $\Delta V_t=V_t^M-V_t^R$, and define the difference in any assumption $x$ as $\Delta x_t=x_t^M-x_t^R$. When both bases use the same $i$, $q_t$ and $p_t$, subtraction of their reserve equations gives:

\Delta V_t = \Delta e_t-\Delta P_t +v\left(q_t\Delta B_t+p_t\Delta V_{t+1}\right).

This comparison isolates the direct changes in current expenses, premiums and benefits, as well as the effect carried back from the next reserve. It is not a general additive formula when interest or transition assumptions also differ. In that case, each basis should first be valued using its own complete assumptions; the resulting reserves can then be compared.

Why the Recursion Works

The reserve at time $t$ must cover the net cash flow at that time and the expected discounted value of what follows. At time $t$, the net outgo is $e_t-P_t$: an expense increases the reserve needed, while a premium reduces it. At the end of the year, death occurs with probability $q_t$, producing benefit $B_t$; survival occurs with probability $p_t$, leaving reserve $V_{t+1}$. The expected end-of-year amount is therefore $q_tB_t+p_tV_{t+1}$. Discounting it by $v$ and adding the time-$t$ net outgo gives the recursion.

Subtracting the reference equation from the modified equation gives the difference formula above because $v$, $q_t$ and $p_t$ are common to both bases. This also explains the duration pattern: a changed assumption at time $t$ can affect $V_t$ directly, while a change that first occurs later affects $V_t$ through the survival-weighted continuation reserve. For a change first occurring at duration $t+1$, the one-year carry-back, when the transition assumptions are common, is $v p_t\Delta V_{t+1}$.

List the basis differences.
Compare interest, transition assumptions, premiums, expenses, benefits and their payment timing separately.

Mark the first affected duration.
A change to a time-$t$ cash flow enters the reserve equation at duration $t$; a later change enters earlier reserves through continuation values.

Value each basis using its own assumptions.
Work backward from the terminal reserve, using the same state and timing definitions for both bases.

Check signs and timing.
Higher expenses increase current reserve; higher premiums reduce it; a changed future reserve is weighted by the probability of reaching that future duration and discounted appropriately.

Worked Comparisons by Assumption

Worked Example: A Modified Valuation Premium

A hypothetical two-year policy has annual death probability $q=0.10$ and survival probability $p=0.90$ in each year. The annual effective interest rate is $i=0$, so $v=1$. A death benefit of $1,000 is paid at each year-end on both bases. There are no expenses. A premium is paid at time 0 and time 1, before the year’s risk. The reference valuation premium is $100 at each time; the modified valuation premium is $90 at each time. Find the reserves at durations 1 and 0 on both bases.

Method. Apply the annual recursion backward, starting with $V_2=0$. The only assumption changed is the valuation premium, so the death probabilities, interest and benefit are identical on both bases.

Reference basis at duration 1. The year-1 premium is received at time 1. The expected year-end benefit is $0.10(1000)=100$. Therefore,

V_1^R=0-100+1\left(0.10(1000)+0.90(0)\right)=0.

At duration 0, the corresponding calculation is

V_0^R=0-100+1\left(0.10(1000)+0.90(0)\right)=0.

Modified basis at duration 1. The year-1 benefit expectation remains $100, but the premium is now $90:

V_1^M=0-90+1\left(0.10(1000)+0.90(0)\right)=10.

At duration 0, the same-year net amount is $-90+100=10$, and the surviving policy carries the $10 duration-1 reserve with probability 0.90:

V_0^M=0-90+1\left(0.10(1000)+0.90(10)\right) =-90+100+9=19.

Thus $\Delta V_1=10$ and $\Delta V_0=19$. The $10 premium reduction at time 1 raises $V_1$ by $10; the same reduction at time 0 adds another $10 directly to $V_0$, while the future $10 reserve contributes $0.90(10)=9$. The result checks against the difference recursion: $\Delta V_0=10+0.90(10)=19$. The common mistake is to carry the duration-1 change back as $10 without multiplying by the probability of survival.

Worked Example: An Expense Change at a Later Duration

Use the same two-year mortality and interest assumptions as in the preceding example: $q=0.10$, $p=0.90$, and $i=0$. A hypothetical policy pays a $1,000 death benefit at each year-end and receives a $100 premium at time 0 and time 1. Reference expenses are $10 at time 0 and $20 at time 1. Modified expenses are $10 at time 0 and $35 at time 1. Find $V_1$ and $V_0$ on each basis.

Method. Work backward from $V_2=0$. The duration-0 expense is unchanged; the only difference is the $15 increase in the time-1 expense.

On the reference basis, at duration 1:

V_1^R=20-100+1\left(0.10(1000)+0.90(0)\right) =20-100+100=20.

At duration 0, the net outgo at time 0 is $10-100=-90$. The expected benefit is $100, and survival carries the duration-1 reserve:

V_0^R=10-100+1\left(0.10(1000)+0.90(20)\right) =-90+100+18=28.

On the modified basis, the duration-1 expense is $35:

V_1^M=35-100+1\left(0.10(1000)+0.90(0)\right) =35-100+100=35.

The duration-0 expense and premium are unchanged, so

V_0^M=10-100+1\left(0.10(1000)+0.90(35)\right) =-90+100+31.5=41.5.

The reserve differences are $\Delta V_1=35-20=15$ and $\Delta V_0=41.5-28=13.5$. The later expense increase enters $V_1$ dollar for dollar and reaches $V_0$ only if the policy survives: $0.90(15)=13.5$. The common mistake is to add the $15 expense change directly to both reserves, ignoring the chance that the contract terminates during the first year.

Worked Example: A Benefit Change That Starts in Year 2

A hypothetical two-year policy has $q=0.10$, $p=0.90$, and $i=0$ in both years. It has no expenses and receives a $100 premium at time 0 and time 1. The year-1 death benefit is $1,000 on both bases. At the end of year 2, the reference benefit is $1,000, while the modified valuation benefit is $900. Find the reserves at durations 1 and 0.

Method. Begin at duration 1, where the modified benefit first differs. Then carry the reserve difference back through year 1.

On the reference basis,

V_1^R=0-100+1\left(0.10(1000)+0.90(0)\right)=0,

and

V_0^R=0-100+1\left(0.10(1000)+0.90(0)\right)=0.

On the modified basis, the year-2 benefit expectation is $0.10(900)=90$, so

V_1^M=0-100+1\left(0.10(900)+0.90(0)\right)=-10.

The year-1 benefit is unchanged, but the survivor reserve at duration 1 is now $-10$. Consequently,

V_0^M=0-100+1\left(0.10(1000)+0.90(-10)\right) =-100+100-9=-9.

Thus $\Delta V_1=-10$ and $\Delta V_0=-9=0.90(-10)$. Reducing the later benefit reduces the reserve, and the earlier effect is survival-weighted. A common mistake is to apply the $100 benefit reduction at duration 0 as though it were payable in year 1; it is paid only if death occurs in year 2.

Worked Example: A Change in the Interest Assumption

Consider a hypothetical contract with certain survival to time 2, no premiums, no expenses and a guaranteed $121 benefit payable at time 2. On the reference basis, $i=0$. On the modified basis, $i=0.10$ effective annually. Find the reserves at times 1 and 0 on each basis.

Method. With certain survival and no intervening cash flows, each reserve is the discounted value of the guaranteed future benefit. The reference basis gives $V_1^R=121$ and $V_0^R=121$.

On the modified basis, $v=1/1.10$. At time 1,

V_1^M=\frac{121}{1.10}=110.

At time 0,

V_0^M=\frac{121}{(1.10)^2} =\frac{121}{1.21}=100.

The modified reserves are therefore $110 at time 1 and $100 at time 0, compared with $121 at each time on the reference basis. The calculation uses the modified interest rate for both years. The common mistake is to discount once at time 0 even though the benefit is payable two years later. No rounding is needed for these amounts.

Reading the Comparison in Multi-State Models

In a multi-state contract, the same comparison principle applies, but each state has its own reserve and outgoing transitions. For a continuous-time model, let $V_j^b(t)$ denote the reserve in state $j$ at time $t$ on basis $b$. Let $\delta^b(t)$ be the force of interest, $\mu_{jk}^b(t)$ the transition intensity from state $j$ to state $k$, $c_j^b(t)$ the premium inflow rate, $e_j^b(t)$ the expense outflow rate, and $b_{jk}^b(t)$ the transition benefit. Under this cash-flow sign convention, the Thiele equation on that basis is

\frac{dV_j^b(t)}{dt} = \delta^b(t)V_j^b(t)+c_j^b(t)-e_j^b(t) -\sum_{k\ne j}\mu_{jk}^b(t) \left(b_{jk}^b(t)+V_k^b(t)-V_j^b(t)\right).

This equation assumes the stated transition benefits and continuous premium and expense rates; lump sums at other times must be represented with the appropriate boundary or jump conditions. It shows why a basis comparison in a disability-income or long-term-care model must identify the state and transition affected. Changing an expense rate in a care state directly changes the reserve equation in that state. Changing a recovery intensity can also change how likely the policy is to move to another state and what reserve is needed there. Such a transition change is not captured by simply adding an expense difference to a reserve.

When interest and transition assumptions are common, an isolated cash-flow change can often be traced cleanly through the continuation reserves. When the assumptions themselves differ, their effects interact: a different transition intensity changes both the probability of paying a transition benefit and the probability of entering a state with a particular reserve. The reliable comparison is to calculate each basis consistently, not to apply a single adjustment without checking its assumptions.

Common Traps and Practical Checks

Issue	Incorrect approach	Reliable check
Premium sign	Adding a higher premium to the reserve as though it were an outgo.	With the stated convention, premiums enter as $-P_t$; expenses enter as $+e_t$.
Future changes	Carrying a later reserve difference back dollar for dollar.	Apply the appropriate discount and probability of reaching the later duration.
Benefit timing	Applying a later-duration benefit change to the current year’s death probability.	Match each benefit to the transition and duration on which it is payable.
Interest changes	Using the reference discount factor for a modified-basis reserve.	Use each basis’s own interest assumption throughout its calculation.
Negative results	Changing the sign because a reserve “should” be positive.	Recheck the cash-flow timing and algebra; the stated prospective net-outgo convention can produce a negative reserve.

A useful final audit is to ask, for every difference between bases: What changed? When does it first enter? Which states or transitions are affected? How can the policy reach the affected duration? What discounting applies? These questions prevent a correct-looking numerical adjustment from being attached to the wrong duration.

Key takeaway: Under common interest and transition assumptions, a changed premium, expense or benefit enters at its own cash-flow duration and can affect earlier reserves through discounted continuation values. If interest, transition assumptions or timing also change, value each basis with its own complete assumptions before interpreting the difference.

Check Your Understanding

Use the stated annual reserve convention: $V_t=e_t-P_t+v(q_tB_t+p_tV_{t+1})$. Treat each scenario independently, and report reserves in dollars.

A one-year hypothetical policy has $q=0.20$, $p=0.80$, $i=0$, death benefit $500 at year-end, and expense $5 at time 0. The reference premium at time 0 is $100 and the modified premium is $90. The terminal reserve is zero. Find the reserve on each basis. (Answer: Reference $5; modified $15.)
A two-year policy has $q=0.25$, $p=0.75$, $i=0$, no benefits or premiums, and terminal reserve zero. At time 1, the expense is $40 on the reference basis and $52 on the modified basis. There is no expense at time 0. Find the difference $\Delta V_1$ and $\Delta V_0$. (Answer: $\Delta V_1=$12$; $\Delta V_0=$9$.)
A two-year policy has $q=0.05$, $p=0.95$, $i=0$, no premiums or expenses, and zero terminal reserve. The time-0 benefit is identical on both bases. The time-1 death benefit is $800 on the reference basis and $600 on the modified basis. Find $\Delta V_1$ and $\Delta V_0$. (Answer: $\Delta V_1=-$10$; $\Delta V_0=-$9.50$.)
A hypothetical contract survives with certainty to time 2 and pays $242 at time 2, with no other cash flows. The reference interest rate is $i=0$; the modified rate is $i=0.10$ effective annually. Find $V_0$ on each basis and the modified-minus-reference difference. (Answer: Reference $242; modified $200; difference $-$42$.)
In a one-year model with common $q=0.10$, $p=0.90$, and $i=0.05$, the only change is a $20 increase in the year-end death benefit. There is no premium or expense difference, and the terminal reserve is unchanged. Find the reserve difference at time 0. (Answer: $v q(20)=20(0.10)/1.05$, or $1.90 to the nearest cent.)

What You'll Learn