In a question should we check for impairment for held to maturity or available for sale securities?

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Yes.

But in the CFAI ebook, Q 13, the Main company owns held to maturity securities in Cathay Corp. The Cost is $40k and Mv is $37k. Clearly there is an impairment of 3k. However, they’ve recorded 40k as the carrying value in the balance sheet and have not considered the impairment. I was wondering why is that.

The fact that the market value has changed does not, in and of itself, represent an impairment. The market value could go back up to $40,000 tomorrow.

If Cathay Corporation defaulted on the bonds, that would most likely represent an impairment.