Dependent and Independent Decrement Rates in Multiple Decrement Models

Understand Dependent and Independent Decrement Rates

Key rules to remember

Total force of decrement

μ_{x+t} = Σ_j μ^j_{x+t}

The total force is the sum of the forces for each cause.

Survival probability

t p_x = exp(−∫₀ᵗ μ_{x+s} ds)

This is the probability of remaining in the group, not leaving by any cause.

Dependent (absolute) rate

t q^j_x = ∫₀ᵗ s p_x μ^j_{x+s} ds

s p_x is the survival probability in the multiple decrement table, so all causes are included.

Independent (net) rate

q'^j_x = 1 − exp(−∫₀¹ μ^j_{x+t} dt)

The rate in the associated single decrement table for cause j.

Product of single survival probabilities

p_x = Π_j (1 − q'^j_x)

Holds because the total force is the sum of the individual forces.

Total of dependent rates

Σ_j q^j_x = 1 − p_x

Use this as a check.

UDD in the multiple decrement table

s p_x μ^j_{x+s} = q^j_x for 0 ≤ s ≤ 1, so the t-year probability t q^j_x = t × q^j_x

On the left, t q^j_x is the probability of leaving by cause j within time t (the t is the prefix). On the right, t multiplies the one-year rate q^j_x. Under UDD in the multiple decrement table the t-year probability is linear in t, for 0 ≤ t ≤ 1.

Independent rate from dependent rates (UDD in the multiple decrement table)

q'^j_x = 1 − (1 − Σ_i q^i_x)^(q^j_x ÷ Σ_i q^i_x)

Equivalently 1 − p_x^(q^j_x ÷ (1 − p_x)). Use when all dependent rates are given. Why it works: under UDD in the multiple decrement table, s p_x μ^j_{x+s} = q^j_x and s p_x μ_{x+s} = 1 − p_x. So μ^j_{x+s} ÷ μ_{x+s} = q^j_x ÷ (1 − p_x), a constant. Then ∫₀¹ μ^j_{x+s} ds = (q^j_x ÷ (1 − p_x)) × (−ln p_x), which gives the formula.

Dependent rate from independent rates, two causes (UDD in each single decrement table)

q^1_x = q'^1_x × (1 − ½ q'^2_x)

Exact when each cause is uniform in its own single decrement table. For more than two causes, q^j_x = ∫₀¹ Π_{i≠j} (1 − t q'^i_x) × q'^j_x dt, with UDD in each single table. With two causes this integral gives the formula above.

How to solve Dependent and Independent Decrement Rates questions

1. 1Write down which rates you are given (dependent or independent) and which you need.
2. 2Name the causes and the age and period involved. Write the notation clearly, such as q^d_x, q^w_x and q'^d_x.
3. 3Find the assumption stated. UDD in the multiple decrement table gives the exponent formula. UDD in each single decrement table gives the formula with (1 − ½ q').
4. 4If you know the force of decrement for each cause, integrate it to get the independent rate: q'^j_x = 1 − exp(−∫μ^j dt).
5. 5Apply the correct conversion formula. For two causes under UDD in the single tables, q^1_x = q'^1_x(1 − ½q'^2_x).
6. 6Check using Σ q^j_x = 1 − p_x and p_x = Π(1 − q'^j_x), or the order q^j_x ≤ q'^j_x.
7. 7State the result to the accuracy needed, often 4 to 6 decimal places, and give the assumption you used.

Quickest way: Two-cause conversion under UDD in the single decrement tables

Common mistakes in Dependent and Independent Decrement Rates

• Treating dependent rates as if they were independent and multiplying (1 − q^j) together.

  The two sets of rates use similar symbols and both are probabilities.

  Fix: Only independent rates multiply: p_x = Π(1 − q'^j_x). Dependent rates add: Σ q^j_x = 1 − p_x.

• Using the (1 − ½ q') formula when the question states UDD in the multiple decrement table.

  Students memorise one formula and ignore which table the assumption applies to.

  Fix: Underline the assumption in the question. Multiple decrement table UDD gives the exponent formula. Single table UDD gives the (1 − ½ q') formula.

• Getting q' smaller than q.

  Students forget that other causes remove lives before cause j can act in the dependent table.

  Fix: Remember q'^j ≥ q^j. If your answer breaks this, recheck the working.

• Forgetting that the total force is the sum of the forces, and integrating one cause only to find p_x.

  Focus on one cause hides the other causes.

  Fix: Use μ = Σ μ^j for survival in the multiple decrement table. Use μ^j alone only for the single decrement table of cause j.

• Using the two-cause formula for three or more causes without adjusting.

  The simple (1 − ½ q'^2) factor is learned as a general rule.

  Fix: With more than two causes, use the exponent formula, or work from p_x = Π(1 − q'^j) and the ratio of forces. State the assumption clearly.

Worked examples