Explanation

For a callable bond priced at par and callable at par:

• One-sided down-duration: Measures price sensitivity when yields increase (prices decrease)
• One-sided up-duration: Measures price sensitivity when yields decrease (prices increase)

When yields decrease (prices increase), the call option becomes more valuable to the issuer. The bond's price appreciation is limited because the issuer is likely to call the bond when it reaches the call price. This creates negative convexity on the upside.

When yields increase (prices decrease), the call option becomes less valuable, and the bond behaves more like an option-free bond, allowing for normal price depreciation.

Therefore, one-sided up-duration is lower than one-sided down-duration because the call option restricts price appreciation when rates fall.