The question basically tests three things:
Parallelism, verb tense, brevity

(A) had expected it to and its business will improve
Problems:
Brevity: Unnecessarily uses IT to refer to profits. Its not required.
Parallelism: The sentence requires "that.. and that.."(X and Y) parallelism.

(B) had expected and that its business would improve
Correct Option

(C) expected it would and that it will improve its business
Problems:
Brevity: it would is unnecessary and wordy.
Verb Tense:
(1)We use past perfect tense when we want to indicate that one action occurred before another action in the past. Here, analysts expected profits to be at certain level(event 1) before the the company announced its profits(event 2). Event 1 happened before event 2 and thus when we are talking about event 1 in a sentence which also mentions event 2, we must use past perfect tense to refer event 1 to indicate that event 1 happened before event 2. So we need analysts had expected.
(2) We use will usually when we are certain about some event to happen in the future. Since the sentence has nothing that could indicate that the business will improve, it is better to use "its business would improve"

(D) expected them to and its business would improve
Problems:
Brevity: them to is unnecessary and wordy.
Parallelism: The sentence requires "that.. and that.."(X and Y) parallelism.
Verb Tense: We use past perfect tense when we want to indicate that one action occurred before another action in the past. Here, analysts expected profits to be at certain level(event 1) before the the company announced its profits(event 2). Event 1 happened before event 2 and thus when we are talking about event 1 in a sentence which also mentions event 2, we must use past perfect tense to refer event 1 to indicate that event 1 happened before event 2. So we need analysts had expected.

(E) expected and that it will have improved its business
Problems:
Verb Tense:
(1)We use past perfect tense when we want to indicate that one action occurred before another action in the past. Here, analysts expected profits to be at certain level(event 1) before the the company announced its profits(event 2). Event 1 happened before event 2 and thus when we are talking about event 1 in a sentence which also mentions event 2, we must use past perfect tense to refer event 1 to indicate that event 1 happened before event 2. So we need analysts had expected.
(2) Incorrectly uses the future perfect (will have improved) that implies the action will be completed rather than ongoing.

Hope that helps..

Great Explanation once again. Thanks Hardik!!
No queries this time.