MATH 403A, fall 98, CHAPTER 5 Actuarial Present Values (Net Single Premiums)

In these situations we have a payment b_{t} at the time t if the person survives up to that time.

The Actuarial Present Value = Net Single Premium (NSP) is

(1)  continuous NSP = ∫_{0}^{∞} b_{t} v^{t} _{t}p_{x}dt
(2)  discrete NSP = ∑_{k=0}^{∞} b_{k} v^{k} _{k}p_{x}dt

(3)

Temporary Life Annuities:

ä_{x:\overline{n}|} = ∑_{k=0}^{n−1} v^{k} _{k}p_{x} a_{x:\overline{n}|} = ∑_{k=1}^{n} v^{k} _{k}p_{x} ā_{x:\overline{n}|} = ∫_{0}^{n} v^{t} _{t}p_{x} dt

Whole Life Annuities:

ä_{x} = ∑_{k=0}^{∞} v^{k} _{k}p_{x} a_{x} = ∑_{k=1}^{∞} v^{k} _{k}p_{x} ā_{x} = ∫_{0}^{∞} v^{t} _{t}p_{x} dt

Deferred Life Annuities:

n|ä_{x} = ∑_{k=n}^{∞} v^{k} _{k}p_{x}  _{n|}a_{x} = ∑_{k=n+1}^{∞} v^{k} _{k}p_{x}  _{n|}ā_{x} = ∫_{n}^{∞} v^{t} _{t}p_{x} dt

n–year Certain Life Annuities:

ä\overline{x:\overline{n}|} = ∑_{k=0}^{n−1} v^{k} + ∑_{k=n}^{∞} v^{k} _{k}p_{x} = ä_{\overline{n}|} + _{n|}ä_{x}

a\overline{x:\overline{n}|} = ∑_{k=1}^{n} v^{k} + ∑_{k=n+1}^{∞} v^{k} _{k}p_{x} = a_{\overline{n}|} + _{n|}a_{x}

ā\overline{x:\overline{n}|} = ∫_{0}^{n} v^{t} dt + ∫_{n}^{∞} v^{t} _{t}p_{x} dt = ā_{\overline{n}|} + _{n|}ā_{x}

Shortcut: If you have constant force of mortality μ and constant force of interest δ, then the net single premium for a whole life annuity of 1 per year is

ā_{x} = \frac{1}{μ + δ}

Relationship between a and A

1 = δ ā_{x} + Ā_{x}
1 = δ ā_{x:\overline{n}|} + Ā_{x:\overline{n}|}

other notation:

nE_{x} = A_{x:\overline{n}|^{1}}  s̄_{x:\overline{n}|} = \frac{ā_{x:\overline{n}|}}{_{n}E_{x}}

For discrete annuities we have similar formulas

1 = d ä_{x} + A_{x}
1 = d ä_{x:\overline{n}|} + A_{x:\overline{n}|}
A_{x} = v ä_{x} − a_{x}
A_{x^{1}:\overline{n}|} = v ä_{x:\overline{n}|} − a_{x:\overline{n}|}
A_{x:\overline{n}|} = v ä_{x:\overline{n}|} − a_{x:\overline{n−1}|}

Annuities payid m–thly

A_{x:\overline{n}|}^{(m)} = \frac{i}{i^{(m)}} A_{x^{1}:\overline{n}|}^{(m)} + A_{x:\overline{n}|^{1}}^{(m)}

ä_{x:\overline{n}|}^{(m)} = \frac{1 − A_{x:\overline{n}|}^{(m)}}{d^{(m)}}

Apportionable and complete annuities

Apportionable Complete

Annuity–Due Annuity–Immediate

Actuarial Present Value of 1/m per m–th with ä_{x:\overline{n}|}^{\{m\}} = (δ/d^{(m)}) ā_{x:\overline{n}|} å_{x:\overline{n}|}^{(m)} = (δ/i^{(m)}) ā_{x:\overline{n}|} adjustment payment at death

Adjustment payment  ā_{\overline{1/m−t}|}/(m ā_{\overline{1/m}|})  s̄_{\overline{t}|}/(m s̄_{\overline{1/m}|})

at Death (pmt in the opposite (pmt in the same direction) direction) t = T − ⌊mT ⌋/m.

1. You are given that the forces of mortality and interest are constant μ and δ respectively. Find the net single premium for a continuous, deferred and temporary life annuity that pays at a rate of 1 per year between times 10 and 20 (in years) if the annuitant (currently aged 35) is alive.

2. Y is the present value random variable for a special discrete 4– year annuity immediate issued to (95) with the benefit at the end of year k equal to k^{2} (k = 1, 2, 3, 4). Mortality follows deMoivre’s Law with w = 100. You are given that v = .8. Find the expected value for Y.

3. You are given that A_{x^{1}:\overline{n}|} = .20, _{n}E_{x} = .3, i = .04. Find ä_{x:\overline{n}|}.

4. You are given that Ā_{40} = .35 and i = .07. You are also given that death are uniformly distributed within each year of age. Find ä_{40}.

5. You are given that ä_{x:\overline{n}|} = 11, A_{x:\overline{n}|} = .34, A_{x} = .25. You are also given that deaths are uniformly distributed within each year of age. Find ā_{x}.

6. Y is the present value random variable for a continuous life annuity of 1 per year issued to (35). You are given that μ = .06 and δ = .05. Find the variance of Y.

7. Y is the present value random variable for a continuous life annuity of 1 per year issued to (x). You are given Ā_{x} = 10, ^{2}ā_{x} = 6.25, i = .06. Find Var[Y].

8. Y is the present value random variable for a special discrete 3– year increasing life annuity–immediate issued to (x). You are given that q_{x+k} = .1(k + 1) for k = 0, 1, 2, 3. The payments are b_{k+1} = 50(k + 1) for k = 0, 1, 2. You are also given that v = .8. find Var[Y].

9. Y is the present value random variable for a continuous whole life annuity of 1 per year issued to (35). Mortality follows deMoivre’s Law with w = 110 and i = .0625. Find P [Y ≥ E(Y)].

10. Express the p.d.f. of Y, the present value random variable for a continuous 5–year deffered 10 annuity of 100 issued to a life aged 35, in terms of the p.d.f. of the remaining future life random variable T.

11. You are given ä_{x:\overline{n}|} = 10.00, _{n}E_{x} = .14, i = .08. You need to find ä^{(}12)_{x:\overline{n}|}. Find the difference between the actuarial present value of the monthly annuity under the assumption of UDD and under the assumption that the commutation function D_{x} is linear btween integral ages. (D_{x} = l_{x}v^{x}).

12. A fast food restaurant is giving away “a hamburger, fries, and a large orange drink every day for the rest of your life” to the lucky winner of a contest. If the winner is 35 years old, find the difference between the actuarial present value of the prize using the assumption that there are 365 days per year (with the prize paid at the beginning of each day) vs. the assumption that the prize is payable continuously. You are given the following assumptions: A_{35} = .22 (based on a mortality table that accounts for the given food intake), d = .07, deaths are uniformly distributed within each year of age, A year’s worth of prize food costs $1186.25.

13. You are given q_{x+1} = 2q_{x} = .02, d = .1, a_{x+2} = 9. Find ä_{x}.

14. You are given ä_{x} = 10.25, ä_{x+1} = 10, q_{x} = .02. Find i.

15. You are given ā_{x} = 10.55, ā_{x+1} = 10.35, δ = .06. Death are uniformly distributed within each year of age. Find q_{x}.

16. ä_{x}^{(12)} = 4.75, i^{(12)} = .12, _{1/12}q_{x} = .005. Find ä_{x+1/12}^{(12)}.

17. True of false: å_{x}^{(m)} = v^{1/m} ä_{x}^{{m}}.

18. Betty pays her life insurance premium of $3,000 at the beginning of each calendar quarter. The policy provides a refund of the unearned premium upon death in such a way that hte premiums form an apportionable annuity–due. You are given that the effective annual rate of interest is 12%. If Betty dies on February 1 (assuming twelve 30–day month per year) find the amount of the refund upon death.