Annual and Fractional-Year Associated Probabilities Under UDD

Why Fractional-Year Probabilities Need Care

An associated single-decrement table gives an annual probability for one decrement cause under a specified basis. Many actuarial calculations, however, require a probability over only part of a year: for example, the probability of a decrement before a midyear valuation date. Under the uniform distribution of decrements (UDD) assumption, the associated survival pattern within the year is linear. This makes the calculation direct, but the starting point of the interval matters.

In particular, the probability of decrement over the first \(t\) years of the year is not generally the same as the probability over a \(t\)-year interval that starts later. The latter is conditional on survival to that later point. This tutorial focuses on fractional-year calculations using annual associated probabilities already specified by an associated table or obtained from multiple-decrement data.

The prime identifies an associated single-decrement probability, not the observed multiple-decrement probability. The distinction matters: the associated table describes a single-cause basis, whereas the multiple-decrement data describe a population exposed to competing causes. The UDD formulas below apply within an associated table. They do not, by themselves, convert observed multiple-decrement probabilities into associated annual probabilities.

From the Start of the Year

Write \(q'=q_x^{\prime(j)}\) temporarily. If the interval begins at age \(x\) and lasts \(t\) years, UDD gives the survival and decrement probabilities by linear interpolation between the start and end of the year:

At \(t=1\), these reduce to the annual probabilities: \({}_1p_x^{\prime(j)}=1-q_x^{\prime(j)}\) and \({}_1q_x^{\prime(j)}=q_x^{\prime(j)}\). At \(t=0\), survival is certain and the decrement probability is zero. These endpoint checks are useful for detecting an incorrect interpolation.

Intervals Beginning Partway Through the Year

Suppose the interval starts at age \(x+s\), where \(0\le s<1\), and continues for \(t\) years, with \(0\le t\le1-s\). The person must first survive in the associated table from age \(x\) to age \(x+s\). Therefore, the probability over the later interval is conditional on survival to its beginning.

Under UDD, survival from age \(x\) to age \(x+s\) is \(1-sq_x^{\prime(j)}\), while survival from age \(x\) to age \(x+s+t\) is \(1-(s+t)q_x^{\prime(j)}\). Taking the ratio gives conditional survival over the interval. The conditional decrement probability is its complement:

These formulas require \(s+t\le1\), and the denominator is positive whenever the associated table has survivors at age \(x+s\). They are conditional probabilities: the denominator adjusts for the fact that the calculation starts among those who have already survived to \(x+s\). If an interval crosses an integer age, apply the appropriate conditional survival calculation separately within each year and multiply the survival probabilities across the pieces.

Keeping Separate Associated Tables Separate

If an associated table is provided for each cause, apply the UDD fractional-year formulas to each table using that cause's own annual associated probability. These are probabilities in separate single-decrement tables. Their fractional decrement probabilities are not automatically competing probabilities for one shared population, and generally should not be added as if they were mutually exclusive outcomes.

Common Errors and Practical Checks

The main exam trap is confusing an annual associated probability with a fractional-year probability, or confusing a probability from the beginning of the year with one conditional on reaching a later point. A reliable calculation identifies the annual basis, the interval start \(s\), and the interval length \(t\) before substituting values.

• Using an annual probability for part of a year. From the start of the year, UDD gives \(tq_x^{\prime(j)}\), not \(q_x^{\prime(j)}\), as the fractional decrement probability.
• Ignoring conditional exposure. For an interval beginning at \(x+s\), divide by \(1-sq_x^{\prime(j)}\). The expression \(tq_x^{\prime(j)}\) alone is not the conditional interval probability unless \(s=0\).
• Using the wrong elapsed time. The survival numerator uses \(s+t\), the total time elapsed from age \(x\) to the interval end. The interval length itself remains \(t\).
• Combining associated causes as if they were observed competing outcomes. Each cause-specific associated table has its own single-decrement basis. Use multiple-decrement relations when the question asks for competing-cause probabilities.

As a final check, verify that the interval ends within the year, the conditional denominator represents survival to the interval start, and the calculated survival and decrement probabilities sum to one. When the interval starts at age \(x\), the formula should reduce to the simple linear interpolation. When \(t=1-s\), it should give the conditional probability from age \(x+s\) through the end of the year.

Fractional-year associated probabilities are useful whenever cash flows, valuation dates, or benefit events occur between integer ages. Correct conditioning ensures that a probability over the remaining part of a year is measured among those still exposed at the interval's beginning. These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.