Deferred mortality probabilities

In this exercise, you will help Cynthia to better understand the concept of a $k$-year deferred mortality probability for an 18-year-old. This is the probability that one first survives $k$ years, reaches age $18+k$ and then dies in the next year:

$$ \begin{aligned} {}_{k|}q_{18} &= {}_kp_{18} \cdot q_{18+k}. \end{aligned} $$ These probabilities with $k = 0, 1, 2, \ldots$ determine a discrete probability distribution. They run over all possible ages at death for the 18-year-old and express the corresponding probability to die at each of these ages.

The mortality rates $q_x$ and the one-year survival probabilities $p_x$ have been preloaded as qx and px.

This is a part of the course

“Life Insurance Products Valuation in R”

View Course

Exercise instructions

Define kpx as the survival probabilities ${}_kp_{18}$ of an 18-year-old for $k = 0, 1, 2, \ldots$
Assign the deferred mortality probabilities ${}_{k|}q_{18}$ to the variable kqx by multiplying kpx with the mortality rates qx from 18 + 1 until length(px).
Compute the sum() of kqx to verify that it equals one.
Visualize the kqx against 0:(length(kqx) - 1).

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Compute the survival probabilities of (18)
kpx <- c(___, ___(px[(___):(length(px) - 1)]))

# Compute the deferred mortality probabilities of (18)
kqx <- ___ * qx[(___):___]

# Print the sum of kqx
___

# Plot the deferred mortality probabilities of (18)
plot(___, ___, 
    pch = 20, 
    xlab = "k", 
    ylab = expression(paste(""['k|'], "q"[18])),
    main = "Deferred mortality probabilities of (18)")

Edit and Run Code