Interpreting Intensity Units and Scale

Why Intensity Units Matter

A transition intensity is a rate attached to a particular unit of time. The numerical value alone is not a complete description: \(0.24\) per year and \(0.24\) per month have very different meanings. When a model records time in a different unit from the one used for its intensities, calculations can be wrong even if the underlying assumption has not changed.

This matters in multi-state models because the same transition may be described in annual, monthly, quarterly, or daily time. For example, a hypothetical disability model might report the intensity of moving from an active state to a disabled state per year, while a calculation or software input uses months. The intensity must be converted before it is used with that new time scale.

Definition: If a life is in state \(i\) at time \(t\), the transition intensity from state \(i\) to state \(j\), denoted here by \(\mu_{ij}(t)\), is a local rate of transition per unit of the time scale used for \(t\). Its units are the reciprocal of the time unit. Thus an intensity stated per year has units \(1/\text{year}\), while one stated per month has units \(1/\text{month}\). The numerical value and its time unit together describe the rate.

The intensity definition is based on a limiting rate over a short time interval, conditional on being in the source state at the start of that interval. The earlier material on defining transition intensities and the small-interval intensity approximation supplies that interpretation. Here, the central question is how the rate’s numerical value changes when the unit used to measure time changes.

A unit conversion does not change the underlying transition assumption. It changes the numerical rate so that it remains consistent with the new time scale. As with any rate, record the unit explicitly; a bare number is not enough to determine how it should be used.

Converting Between Common Time Units

Suppose an intensity is \(\lambda\) per year. Using 12 months per year, 4 quarters per year, 52 weeks per year, or 365 days per year, the corresponding intensity per one of those smaller units is found by dividing by the number of such units in a year. For example, an annual intensity converted to a monthly intensity is divided by 12, not multiplied by 12.

\lambda_{\text{per month}}=\frac{\lambda_{\text{per year}}}{12}, \qquad \lambda_{\text{per quarter}}=\frac{\lambda_{\text{per year}}}{4}, \qquad \lambda_{\text{per day}}=\frac{\lambda_{\text{per year}}}{365}.

The reverse conversion multiplies by the number of smaller units in the larger unit. For instance, an intensity of \(0.02\) per month is \(12(0.02)=0.24\) per year under the 12-month convention. State the convention being used when it matters: a model using a 365-day year should not silently substitute a different number of days.

Worked Example: Converting an Annual Disability Intensity

A hypothetical disability model has an active-to-disabled transition intensity of \(0.24\) per year. Convert it to a monthly intensity and a daily intensity, using 12 months and 365 days per year. Then use the small-interval approximation to estimate the transition probability over three months, conditional on being active at the start.

Method. Convert the rate by dividing by the number of new time units in a year. For the short-interval estimate, multiply the annual intensity by the interval expressed in years. The latter is a first-order approximation, not an exact finite-interval probability.

Monthly intensity. There are 12 months in a year, so \(0.24/12=0.02\) per month.

Daily intensity. There are 365 days in the stated year, so \(0.24/365=0.000657534\ldots\) per day. To seven decimal places, this is \(0.0006575\) per day.

Three-month approximation. Three months is \(3/12=0.25\) years. The small-interval approximation is \(0.24(0.25)=0.06\), or 6%. Equivalently, using monthly units gives \(0.02(3)=0.06\).

Check and common mistake. Both time scales give the same rate-time product: \(0.24(0.25)=0.02(3)\). Multiplying \(0.24\) by 12 to obtain a monthly intensity would give \(2.88\) per month, which incorrectly makes the numerical rate larger when the unit interval becomes shorter. The correct monthly rate is \(0.02\) per month.

Why the Numerical Value Changes

The conversion rule follows directly from the definition of a local rate. Let \(t\) measure time in the original unit, and let \(u=kt\) measure time in a new unit, where \(k\) is the number of new units corresponding to one old unit. For example, if \(t\) is in years and \(u\) is in months, then \(k=12\).

A small interval \(\Delta u\) in the new scale corresponds to \(\Delta t=\Delta u/k\) in the old scale. The transition probability over the interval is the same event whichever unit is used to describe it. Dividing that probability by the new interval, rather than the old interval, gives the new intensity:

\mu_{\text{per new unit}} = \lim_{\Delta u\to 0} \frac{P(\text{transition during }\Delta u\mid\text{state at start})}{\Delta u} = \frac{1}{k}\, \lim_{\Delta t\to 0} \frac{P(\text{transition during }\Delta t\mid\text{state at start})}{\Delta t} = \frac{\mu_{\text{per old unit}}}{k}.

The equality uses \(\Delta t=\Delta u/k\). The conditional transition probability in the numerator refers to the same time interval in either description; only the denominator’s numerical measure changes. This is why moving from years to months divides the numerical intensity by 12.

A useful consequence is that the intensity multiplied by the length of a short interval is unchanged by a consistent unit conversion. This product is dimensionless. It is the quantity used in the first-order small-interval approximation, but it is not, by itself, a claim that an intensity equals a finite-interval probability.

Unit-rescaling rule: If one unit on a new time scale represents \(1/k\) of an old time unit, so that the new time coordinate is \(u=kt\), then the numerical intensity per new unit is the old intensity divided by \(k\). For a matching interval, \(\mu_{\text{old}}\Delta t=\mu_{\text{new}}\Delta u\).

Changing the Time Coordinate

A unit conversion is one example of rescaling a time coordinate. If the new coordinate is \(u=kt\), then each unit of \(u\) represents \(1/k\) units of \(t\). The formula applies to the local intensity at the corresponding time. It changes the numerical rate because the coordinate’s unit has changed, not because the modeled transition has become more or less likely.

This distinction is useful when comparing model specifications. For a fixed transition and a fixed instant, a rate per year and a rate per month should give the same local description when paired with intervals expressed in their respective units. If they do not, check the conversion factor and the time units before changing the actuarial assumption.

Worked Example: Converting Two Long-Term Care Intensities to Quarters

A hypothetical long-term care model reports an intensity of \(0.30\) per year for a transition from independent living to needing care, and \(0.08\) per year for a different transition from needing care to a higher-care state. Convert both to intensities per quarter. Then compare the rate-time products over two years using each time scale.

Method. Four quarters make one year, so divide each annual intensity by 4. For the check, express two years either as 2 years or as 8 quarters. The calculation checks units; it does not calculate a transition probability.

First transition. The quarterly intensity is \(0.30/4=0.075\) per quarter. Over two years, the annual-scale product is \(0.30(2)=0.60\). The same duration is 8 quarters, giving \(0.075(8)=0.60\).

Second transition. The quarterly intensity is \(0.08/4=0.02\) per quarter. The annual-scale product over two years is \(0.08(2)=0.16\). On the quarterly scale it is \(0.02(8)=0.16\).

Check and common mistake. Both conversions preserve the rate-time product. Multiplying the annual rates by 4 would produce \(1.20\) and \(0.32\) per quarter, respectively, and would make the products over 8 quarters four times too large. The correct quarterly intensities are \(0.075\) and \(0.02\) per quarter.

Arbitrary Linear Rescalings and a Practical Check

The rescaling formula is not restricted to familiar calendar units. A system might use a coordinate that counts several units for each year, provided that the relationship is stated clearly. If the new coordinate is six times the old one, then the numerical intensity per new coordinate unit is one-sixth of the old numerical intensity. The rate-time product can then be used as a quick consistency check.

Worked Example: Checking a Rescaled Model Clock

A hypothetical model uses \(t\) in years and specifies a transition intensity of \(0.18\) per year. A reporting system instead uses the coordinate \(u=6t\). Convert the intensity to units of \(u\). Then check the conversion for an interval of \(0.2\) units of \(u\), without treating the rate-time product as an exact transition probability.

Method. Since \(u=6t\), there are six units of \(u\) per year. Divide the annual intensity by 6. Convert the specified interval back to years and compare the two rate-time products.

Converted intensity. The intensity per unit of \(u\) is \(0.18/6=0.03\) per \(u\)-unit.

Convert the interval. From \(u=6t\), an interval of \(0.2\) units of \(u\) corresponds to \(\Delta t=0.2/6=1/30\) year.

Check the products. On the original scale, \(0.18(1/30)=0.006\). On the rescaled clock, \(0.03(0.2)=0.006\). The products agree, as required.

Check and common mistake. The new rate must be smaller because each unit of \(u\) is only one-sixth of a year. Using \(0.18(6)=1.08\) per \(u\)-unit would reverse the conversion and would fail the product check: \(1.08(0.2)=0.216\), not \(0.006\).

Common Unit Traps

Reporting a number without its unit. Write \(0.02\) per month, not just \(0.02\). The unit determines how the rate is interpreted.
Multiplying in the wrong direction. Converting a rate per year to a rate per month divides by 12. Converting a rate per month to a rate per year multiplies by 12.
Changing the time unit but not the interval. If a rate is converted to a monthly basis, a duration stated in years must also be expressed in months when forming a rate-time product.
Confusing a changed numerical rate with a changed assumption. Equivalent annual and monthly descriptions have different numbers but represent the same local rate when converted consistently.
Using an unstated calendar convention. If a problem specifies 365 days per year, use that conversion. Do not replace it with another year-length convention without instruction.

For exam calculations aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus, a reliable first step is to write the time unit beside every intensity and every interval. Convert one or the other before combining them, then check that equivalent descriptions give the same rate-time product. This catches many unit errors before they affect a multi-state calculation.

Check Your Understanding

For each problem, retain the stated time convention and show units through the conversion.

A hypothetical transition intensity is \(0.36\) per year. Convert it to an intensity per month and per day, using 12 months and 365 days per year. (Answer: \(0.03\) per month and approximately \(0.0009863\) per day.)
A model specifies an intensity of \(0.015\) per week. Using 52 weeks per year, convert it to an annual intensity. (Answer: \(0.78\) per year.)
A transition intensity is \(0.045\) per quarter. Convert it to an annual intensity using 4 quarters per year. (Answer: \(0.18\) per year.)
A hypothetical intensity is \(0.21\) per year, but a calculation uses months. Find the monthly intensity and the rate-time product for a six-month interval. (Answer: \(0.0175\) per month; product \(0.105\).)
A model uses \(u=24t\), where \(t\) is measured in years, and has intensity \(0.48\) per year. Find the intensity per unit of \(u\), then check the rate-time product for an interval of \(0.3\) units of \(u\). (Answer: \(0.02\) per \(u\)-unit; both scales give a product of \(0.006\).)

What You'll Learn