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Increasing liability sensitivity refers to enhancing a financial institution's exposure to interest rate risk, typically by focusing on issues related to how liabilities (debts) react to changes in interest rates. This can be done effectively through non-core purchased liabilities, which usually have shorter maturities and can be more responsive to interest rate changes than core deposits.

Borrowing more through non-core purchased liabilities can lead to increased liability sensitivity because these liabilities are often more volatile and may adjust more quickly when there are fluctuations in prevailing interest rates. Therefore, an increase in reliance on these types of liabilities can effectively magnify the institution's responsiveness to interest rate changes.

In contrast, lengthening the maturities of loans or issuing long-term subordinated debt would typically lead to a decrease in liability sensitivity, as both strategies result in the institution being locked into longer-term borrowing costs that are less likely to change in response to interest rate adjustments. Thus, while the banking institution may still need to manage its liabilities effectively, the focus here is specifically on increasing sensitivity, which is more aligned with option C.