# Putable Bonds and Positive Convexity

I don’t have the specific page number in the text - but I have in my notes that the text stated that putable bonds have positive convexity everywhere. For interest rate increases, I understand why Putable Bonds would have positive convexity, as their price would decrease, at decreasing rates, because the value of the put option goes up as interest rates increase (making the putable bond more valuable). I am struggling with the scenario where interest rates decrease though - with a decrease in rates, I would think that the value of the put option bascially becomes worthless and the bond essentially becomes a straight bond, but the premium you paid or decrease in coupons you are receiving (for the put option) would cause the putable bond to increase, but at decreasing rates (you are penalized for the premium paid or decreased coupon payments). Isn’t this negative convexity and not positive convexity?

No. Putable bonds will always have pos conv.

Why are you thinking that the premium and coupon change with interest rates? The premium is already paid in the beginning and the coupon is set.

Putable bonds will act like non putable bonds for int rate decreases (reg pos conv) and will act even more suck up (greater pos conv) than non putable bonds for int rate increases.

put options are desinged to reduce the downside risk of the underlying bonds, or effectively creaitng positive convexity. recall the second order approx for cange in bond price: - D * delta i + C * (delta i)^2. a higher C means more downside protection because a positive convexity bond has a duration that falls as yields increase and increase as yields decrease. decreasing duration when yields are increasing is desirable as the sensitivity of the bond to increasing rates lessens, meaning less and less negative price movements for the bond. the opposite is true for increasing yields

puttable bonds are less risky than straight bonds and the put provides downside protection, so we are increasing the convexity relationship of the bond. it is this positive C that results from the put option which is downside protection.

the premium paid for the bond doesn’t affect the value of the put option or the value of the bond, it’s a sunk cost paid upfront