The question is a probability problem that involves calculating the percentage of policyholders who will renew at least one policy next year based on the given probabilities and the distribution of policy types among policyholders.

Let's define the events:

• $A$ = event that a policyholder has an auto policy
• $H$ = event that a policyholder has a homeowner policy

From the company records:

• $P(A \cap H) = 0.20$ (20% of policyholders have both an auto and a homeowner policy)
• $P(A) = 0.70$ (70% of policyholders have an auto policy)
• $P(H) = 0.50$ (50% of policyholders have a homeowner policy)

To find the probabilities of the complementary events:

• $P(A^c \cap H^c) = P(A) - P(A \cap H) = 0.70 - 0.20 = 0.50$ (50% have only an auto policy)
• $P(A^c \cap H) = P(H) - P(A \cap H) = 0.50 - 0.20 = 0.30$ (30% have only a homeowner policy)

The company estimates the renewal rates:

• 40% of those with only an auto policy will renew
• 70% of those with only a homeowner policy will renew
• 80% of those with both policies will renew at least one

The calculation for the proportion of policyholders that will renew at least one policy is: $0.40 \times P(A^c \cap H^c) + 0.70 \times P(A^c \cap H) + 0.80 \times P(A \cap H)$ $= 0.40 \times 0.50 + 0.70 \times 0.30 + 0.80 \times 0.20$ $= 0.20 + 0.21 + 0.16$ $= 0.57$

Thus, the percentage of policyholders that will renew at least one policy next year is 57%, which corresponds to option D.