1. Calculating probabilities
Calculating probabilities is essential in life insurance. This video fine-tunes your skills.
2. From one-year to multi-year survival probabilities
Consider a policyholder who is x-year-old at time 0. He survives u+t years with probability u+tpx.
3. From one-year to multi-year survival probabilities
He should first survive u years and reach age x+u, with probability upx.
4. From one-year to multi-year survival probabilities
Then, he should survive another t years with probability tpx+u.
5. From one-year to multi-year survival probabilities
Combining both viewpoints: the survival probability u+tpx must equal the product of upx and tpx+u.
6. The multiplication rule
The timeline illustrated that multi-year survival probabilities are calculated by multiplying smaller step survival probabilities. What if you break down the multi-year period into periods of one year?
7. The multiplication rule
With k an integer, the k-year survival probability of an x-year-old is the product of px, px+1 up to px+k-1. The latter is the probability that an (x+k-1)-year-old reaches age x+k.
8. Calculating survival probabilities in R
You calculate the 5-year survival probability of a 65-year-old. First extract px as 1 minus qx stored in the life table. You need the survival probability of a 65-year old until a 69-year-old. You then multiply these one-year survival probabilities to get the 5-year survival probability. Using the prod() function on the vector of relevant one-year survival probabilities gives you the result.
Alternatively you evaluate this probability as the ratio of l70 and l65. Up to rounding errors both calculations lead to the same result!
9. Cumulative product of survival probabilities in R
The cumprod() function is very convenient to evaluate multi-year survival probabilities kpx over a range of values for k. Applied to the entries p65, p66 until p69, cumprod() returns a vector of cumulative products. The first entry in the resulting vector is p65, second entry is p65 multiplied by p66, and the last entry is the product of p65 until p69. That's in fact kp65 where k runs from 1 to 5!
But when k starts from 0 the resulting vector should have 0p65 as its first entry. That's equal to 1 because the 65-year-old is alive at time 0. You combine the result of cumprod() with 1 using the c() function.
10. A deferred mortality probability
Now you want to know the probability that an x-year-old at time 0 will die at age x+k. This implies that (x) first survives k years, and then dies within the next year. This is a deferred mortality probability denoted with the symbol as presented here. To get this probability, you multiply the probability kpx that an x-year-old survives k years, and reaches age x+k, with the probability qx+k that an (x+k)-year-old dies in the next year.
11. A deferred mortality probability in R
You are now ready to calculate the probability that a 65-year-old will die at age 70. That is: he first survives 5 years, reaches age 70, and then dies within the next year. You'll see two different ways to calculate this probability in R. First, multiply the one-year survival probabilities at ages 65 to 69 with prod() and then multiply this survival probability with q70, the mortality rate at age 70. Second, take the ratio of the number of deaths at age 70 and the number of people alive at age 65. Both approaches give the same probability.
12. Deferred mortality probabilities in R
Let's calculate the k-year deferred mortality probabilities of a 65-year-old with k starting from 0. You need the vector kpx with k running from 0. You make sure that kpx starts from 1. You want this vector to have the same length as the vector with mortality rates running from q65 until the last qx available in the lifetable. As such, you can multiply kpx with the vector of mortality rates. The result is the vector with the deferred mortality probabilities.
13. Let's practice!
Calculating probabilities is at your fingertips. Let's go explore!