MATH 403A, fall 98, CHAPTER 3 Section 3.6

Assumptions for fractional Ages: Interpolation Rules

Basic rules:

• Uniform distribution of death (UUD) (linear interpolation).

• Constant force of mortality (exponential interpolation).

• Balducci’s assumtion (harmonic or hyperbolic interpolation).

In words  UUD  Constant Force Balducci
Linear  Exponential Harmonic
s(x + t)  Interpolated value  (1 − t)s(t)+  s(x)e^{−μt}  \frac{1}{s(x+t)} =
of the survival function ts(x + t)  where
for non integral ages  μ = − ln p_{x}  \frac{1−t}{s(x)} + \frac{t}{s(x+t)}

Probability of somebody

tq_{x} already age x dying tq_{x}

in the next t years, 0 ≤ t ≤ 1.

Probability of somebody  e^{−μt} =
tp_{x} already age x living (p_{x})^{t}

through the next t years, 0 ≤ t ≤ 1.

Probability of somebody

1−tq_{x+t} already age x + t dying  (1 − t)q_{x}

in the remainder of the current year, 0 ≤ t ≤ 1.

μ_{x+t}  Force of mortality  \frac{q_{x}}{1−tq_{x}}  μ
tp_{x}μ_{x+t}  p.d.f. of T  q_{x}

Homework

1. When x is an integer, we can gain insight by looking at the random variable S defined by T = K +S. That is, S is the fractional part of year lived in the year of death. Show that under the uniform distribution assumption, K and S are independent random variables. What is the distribution of S in this case?

2. What about under the constant force of mortality assumption?

3. Under the assumtion of uniform distribution of deaths, show that (a) {∘ \atop e}_{x} = e_{x} + 1/2.

(b) Var(T) = Var(K) + 1/12.

4. You are given that mortality follows Baldicci’s law within each year of life. You are also given that q_{x} = .06 and q_{x+1} = .09. Find the probability that a life aged x + 2/3 dies within the next 2/3 of a year.

5. Same problem but assuming constant force of mortality in each year of age.

6. Textbook problems: 3.27–3.30