Callable and putable bonds

Question

I am trying to synthesize everything I have learned about callable and putable bonds. Is the following information accurate?

A callable bond bought at a discount will have yield rate greater than coupon rate. For a fixed redemption value, it will be redeemed as late as possible. Compared to an equivalent bond without a call, it will be priced the same.
A callable bond bought at a premium will have yield rate lower than coupon rate. For a fixed redemption value, it will be redeemed as early as possible. Compared to an equivalent bond without a call, it will be priced lower. 
A putable bond bought at a discount will have yield rate higher than coupon rate. For a fixed redemption value, it will be redeemed as early as possible. Compared to an equivalent bond without a put, it will be priced higher.
A putable bond bought at a premium will have yield rate lower than coupon rate. For a fixed redemption value, it will be redeemed as late as possible. Compared to an equivalent bond without a put, it will be priced the same.

D Stanley · Answer

I see a few questionable things:

Compared to an equivalent bond without a call, it will be priced the same.

Adding a call option reduces the value of the bond since it provides optionality to the issuer (adds risk to the buyer). A callable bond would have to be sold at a deep discount (have very little chance of being called) to be priced the same as the equivalent non-callable version.

Similar for the putable bond - it adds optionality to the purchaser and thus increases the price.

it will be redeedmed as early/late as possible.

Not sure what this means. Options don't have to be exercised, but they could be exercised as soon as the strike of a put/call is above/below the  market price.

Plus you can generalize one thing even further:

any bond bought at a discount will have a yield greater than the coupon rate (since you get the coupons plus the gain between the redemption value and the purchase price)

Callable and putable bonds

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