# Putable Bonds

Putable bonds are bonds which entitle the bondholder to return the bond to the issuer on specified dates before its maturity date. Putable bonds have an embedded put option. They allow investors to sell the bonds back to the issuer and reinvest the proceeds in new bonds paying higher yields.

Putable bonds differ from a conventional bond (also called plain-vanilla bond) in that they allow the bondholder to sell them back to the issuer at a pre-determined price before maturity date while conventional bonds pay principal back only at their maturity date. The putable bonds contain an inherent price floor i.e. a price below which the market price of the bond won’t fall in an event of increase in market interest rate. However, this price floor is not free because putable bonds are issued at a price higher than that of an otherwise identical conventional bond. This also means that the yield on a putable bond is lower than the yield on a conventional bond of same maturity, coupon rate and payment frequency. However, the value of a putable bond converges to the value of a conventional bond as the bonds near their maturity date.

Putable bonds are exactly opposite to callable bonds which are bonds that allow the bond issuer to purchase the bonds before their maturity date.

## Pricing of a Putable Bond

Since a putable bond is essentially a combination of a conventional bond and put option, it can be priced using the following equation:

$$P_P=P_C+v_p$$

Where PP is the price of a putable bond, PC is the price of a conventional bonds which equals the present value of its coupon payments and redemption value and vp is the value of the put option.

Just like a conventional bond, value of a putable bond equals the future value of the bond cash flows. However, in case of a putable bond, an assessment is made at each put date whether the bond will be sold back to the bond issuer or not. For this purpose, the present value (at put date) of the bond cash flows that occur after that put date is calculated using the relevant future interest rates and compared with the floor price of the bond i.e. the put price. If the put price exceeds the present value of remaining cash flows, it follows that the bond will be sold by the bond-holder. The present value at time 0 of the putable bond equals the present value of coupon payments that occur before the put date and the put price calculated using relevant spot interest rates. This is illustrated using the following example

## Example: Putable Bond Price

Let’s consider a $100 par bond paying 4.5% annual coupon and maturing in 2 years. The bond-holder has an option to put the bond to the issuer after at the end of each year. Current spot rate and forward rate is 3%, 1-year forward rate one year from now is 5% and the 2-year spot rate is 4%. Find out if the bondholder will sell the bond back or not and determine the bond price. We need to find the present value of the bond cash flows at the end of Year 1 using the 1-year forward rate one year from now: $${\rm PV}_1=\frac{4.5\%\times100+100}{1+5\%}=99.52$$ Because the put price i.e.$100 is higher than the expected market price of $99.52, the bond issuer will sell the bond after Year 1. Hence, the value of the putable bond at time 0 should equal the put price of$100 plus the 1-year coupon of $4.5 discounted using the 1-year spot rate. $${\rm PV}_0=\frac{4.5\%\times100+100}{1+3\%}=101.45$$ If the bond had no put provision, i.e. had it been a conventional bond, its value must have been$100.994 (determined by discounting the relevant cash flows using relevant spot rates). The value of a putable bond at $101.45 is higher than value of an identical conventional bond by$0.46 and this is the value of the put option.