# Callable Bond

A callable bond is a bond that can be redeemed by the issuer before its maturity date at a predetermined call price. It gives the issuer the flexibility of calling away the bond when the interest rates drop by issuing a new bond at lower coupon rate. It behaves like a conventional fixed-rate bond with an embedded call option.

A callable bond may have a call protection i.e. a provision that stops the issuer from paying off the bond during an initial period, say 5 years. The call price, the price at which the issuer may pay off the bond, may be higher than the face value of the bond and may decline as the bond nears its maturity date. The excess of call price over the face value is called call premium.

A callable bond has lower interest rate risk but higher reinvestment risk. Due to lower duration, it is less sensitive to interest rate movements. However, the possibility of redemption before maturity exposes it to a situation in which the bond-holder might have to reinvest the redemption proceeds at lower rate thereby resulting in significant reinvestment risk. In event of a decrease in interest rates, the issuer may recall the bond at the call price which forms an upper limit on the bond price. An investor may be interested in holding a callable bond if it expects the interest rates to increase.

Yield on a callable bond is called yield to call. The yield to call varies with time, highest at the start of call period and approaches yield to maturity as the bond nears its maturity date.

## Price

A callable bond can be valued by discounting its coupon payments and call price using the following formula:

$$ P=\frac{c}{m}\times F\times\frac{1-\left(1+\frac{r}{m}\right)^{-n\times m}}{\frac{r}{m}}+\frac{C}{\left(1+\frac{r}{m}\right)^{n\times m}} $$

Where **P** is the callable bond price, **c** is the coupon rate, **m** is the number of coupon payments per year, **F** is the face value of the bond, **r** is the market interest rate applicable to the bond of equivalent risk, **n** is the number of years till the call date and **C** is the call price.

Since call price depends on the call date, the price varies.

## Example

A $1,000 par value bond carrying 6% semi-annual coupon was issued on 1 January 2016 and it matures on 31 December 2025. The bond is call protected till 31 December 2020, afterwards it can be called by giving a 30-day notice. The call price exceeds the face value by 5%, 4%, 3%, 2% and 1% at the start of each year in the call period. Your economist estimates that the interest rates in by the start of 2022 will be low enough for the issue to call the bonds.

Imagine its 1 January 2018. Let’s value the bond based on your economist’s estimation of most likely call date if relevant market interest rate is 6.5% per annum.

The call premium by start of 2022 will be 3%, so the applicable call price is $1,030.

The call date is 4 years from now (i.e. from 1 January 2018 to 1 January 2022).

Let’s plug the values in the callable bond price equation and we find that the bond value is $1,005.86:

$$ P=\frac{6\%}{2}\times$1,000\times\frac{1-\left(1+\frac{6.5\%}{2}\right)^{-4\times2}}{\frac{6.5\%}{2}}\\+\frac{$1,030}{\left(1+\frac{6.5\%}{2}\right)^{4\times2}}=$1,005.86 $$