Fixed Income- Effects of Interest Rate Level on Bonds with Embedded Options

CFA Level II Fixed Income
#1

I would like to know if my understanding is correct or no. Please do confirm the same.

FOR A CALLABLE BOND:

If the interest rate increases, the value of the call option decreases (Value of Call option is inversely related to level of interest rate), which implies the value of the callable bond increases. (Value of Callable Bond = Value of Option Free Bond - Value of Call Option).

FOR A PUTABLE BOND:

If the interest rate increases, the value of the put option increases (Value of Put option is directly related to level of interest rate), which implies the value of the increases bond increases. (Value of Putable Bond = Value of Option Free Bond + Value of Put Option).

#2

You’re ignoring the fact that as interest rates increase the value of the (option-free) bond decreases, and it decreases more than the the decrease in the value of the call or the increase in the value of the put.

#3

Yes sir, I did ignore that fact. So, would the final conclusion be, that increase in interest rates, increases the value of the Putable bond and decreases the value of the callable Bond?