Insurance Policies from Mortality Rates

[nattyann]

1. 
   Julio’s mortality for 1 ≤ t ≤ 4 is assumed to be governed by the law tpx =.2(4 − t). Harold’s mortality for 1 ≤ t ≤ 5 is governed by tpx = .3(4 − t). Ifi = .07, find the price at time 0 of an insurance policy which will pay 100, 000at the end of the year in which the first of Julio or Harold dies
   

 

[tkhunny]

Harold lives a whole year after he dies? What have you done, so far?

Note: "end of the year" is good. It's a nice, discrete problem. We don't really need the continuous functions. We can use them if desired.

Hint: Year 1

p(Julio is living) = 0.2 * (4-1) = 0.2 * 3 = 0.6
p(Harold is living) = 0.3 * (4-1) = 0.3 * 3 = 0.9

p(both alive) = 0.6 * 0.9 = 0.54 Pay Nothing and move on to year 2.
p(Julio Alive and Harold Dead) = 0.6 * (1 - 0.9) = 0.6 * 0.1 = 0.06 Pay 100,000 at the end of the year and terminate the policy.
p(Harold Alive and Julio Dead) = (1 - 0.6) * 0.9 = 0.4 * 0.9 = 0.36 Pay 100,000 at the end of the year and terminate the policy.
p(Both Dead) = (1 - 0.6) * (1 - 0.9) = 0.4 * 0.1 = 0.04 Pay 100,000 at the end of the year and terminate the policy.
Checking: 0.54 + 0.06 + 0.36 + 0.04 = 1.00 -- Looks like we got everything.

Price at Year 0, then: (0.54 * 0 + 0.06 * 100000 + 0.36 * 100000 + 0.04 * 100000)/1.07 = (0.54 * 0 + 0.46 * 100000)/1.07 = ... You do the arithmetic.

Year 2
Careful. We care ONLY about both alive at the end of year 1. We already paid, otherwise.

Let's see what you get.

 

[nattyann]

Harold lives a whole year after he dies? What have you done, so far?

Note: "end of the year" is good. It's a nice, discrete problem. We don't really need the continuous functions. We can use them if desired.

Hint: Year 1

p(Julio is living) = 0.2 * (4-1) = 0.2 * 3 = 0.6
p(Harold is living) = 0.3 * (4-1) = 0.3 * 3 = 0.9

p(both alive) = 0.6 * 0.9 = 0.54 Pay Nothing and move on to year 2.
p(Julio Alive and Harold Dead) = 0.6 * (1 - 0.9) = 0.6 * 0.1 = 0.06 Pay 100,000 at the end of the year and terminate the policy.
p(Harold Alive and Julio Dead) = (1 - 0.6) * 0.9 = 0.4 * 0.9 = 0.36 Pay 100,000 at the end of the year and terminate the policy.
p(Both Dead) = (1 - 0.6) * (1 - 0.9) = 0.4 * 0.1 = 0.04 Pay 100,000 at the end of the year and terminate the policy.
Checking: 0.54 + 0.06 + 0.36 + 0.04 = 1.00 -- Looks like we got everything.

Price at Year 0, then: (0.54 * 0 + 0.06 * 100000 + 0.36 * 100000 + 0.04 * 100000)/1.07 = (0.54 * 0 + 0.46 * 100000)/1.07 = ... You do the arithmetic.

Year 2
Careful. We care ONLY about both alive at the end of year 1. We already paid, otherwise.

Let's see what you get.

I've been told by my professor that it should actually be .3*(5-t) for Harold's mortality. However would this not give me a probability greater then 1 for the first year?

 

[tkhunny]

a probability greater then 1

You are thinking correctly.

t = 1 ==> 0.3*4 = 1.2 -- That's no good. You cannot have a 120% chance of Death.

On the other hand, that makes the problem a lot easier. Simply throw it out. It is not feasible.