What is the duration of a 10 year zero coupon bond. Assume the bond is callable (ie, the issuer can buy it back) at face value at any time during its existence.
The key point in this question is that the bond is zero coupon, and can only be called at face value. Since the bond is zero coupon, its value will always be less than its par value at any time during its existence (as any interest rate will be a positive number). Therefore the issuer will never exercise the call. Thus the bond will have a duration equal to what an equivalent non-callable bond would have.
Since zero coupon bonds have a duration equal to their maturity, the bond's duration is 10 years.