# Bond Discount and Bond Premium

When a bond’s coupon rate is lower than the market interest rate, the bond sells at a price lower than its face value. The difference is called bond discount. Similarly, when a bond’s coupon rate is higher than the market interest rate, the bond sells at a price higher than the face value and the difference is called bond premium.

As part of the bond issuance process, the issuer sets a coupon rate keeping in view the current market interest rate and its assessment of the credit risk of the bond. However, market interest rates are volatile, and the credit risk assessment might be different from the default premium investors require in the market.

## Bond discount

If the market requires a rate higher than the coupon rate, the demand for the company’s bond is low and hence it must be issued at a price lower than the face value.

Let’s say your company wants to raise $50 million by issued $1,000 par value bonds maturing in 10 years and paying 6% semi-annual coupon rate. If the market yield on the bond turns out to be 6.4%, the bond price at which your company will be able to sell each bond works out to $970.79:

$$ P=\frac{c}{m}\times F\times\frac{1-{(1+\frac{y}{m})}^{-n\times m}}{\frac{y}{m}}+\frac{F}{{(1+\frac{y}{m})}^{n\times m}}\\=\frac{6\%}{2}\times$1,000\times\frac{1-{(1+\frac{6.4\%}{2})}^{-10\times2}}{\frac{6.4\%}{2}}+\frac{$1,000}{{(1+\frac{6.4\%}{2})}^{10\times2}}\\=$970.79\ $$

Where c is the annual coupon rate, m is the coupon payments per year, F is the face value of the bond, y is the annual yield and n is the total years till maturity.

You can work out the price using Excel PV function: PV(6.2%/2,10*2,-6%/2*1000,-1000).

Your company will be able to raise $48.54 million (=$50 million/$1,000×$970.79). Because the face value of bonds is $50 million, you will be required to pay $50 million at maturity date. The difference of $1.46 million represent the bond discount.

The total amount of bond discount is directly proportional to the difference between the coupon rate and bond yield (i.e. market interest rate) and the time to maturity. You will be required to amortize the bond discount over the life of the bond. This will result in your interest expense to be higher than the interest payment. Your will effective interest rate will be higher than the coupon rate.

## Bond premium

If the bond’s coupon rate is set higher than the expected rate of return, the demand for bond will be higher and it can be sold at a price higher than the par value. The difference represents the bond premium.

Continuing with the example above, if the annual coupon rate is 7% instead of 6% and the market interest rate is 6.4%, your bond will sell at $1,043.82 raising a total amount of $52.19 million:

$$ P=\frac{c}{m}\times F\times\frac{1-{(1+\frac{y}{m})}^{-n\times m}}{\frac{y}{m}}+\frac{F}{{(1+\frac{y}{m})}^{n\times m}}\\=\frac{7\%}{2}\times$1,000\times\frac{1-{(1+\frac{6.4\%}{2})}^{-10\times2}}{\frac{6.4\%}{2}}+\frac{$1,000}{{(1+\frac{6.4\%}{2})}^{10\times2}}\\=$1,043.82\ $$

The amount by which the bond proceeds exceed the face value of the bond is the bond premium. It equals $2.19 million.

The bond premium causes the interest expense to be lower than the interest payment such that the effective rate of interest is lower than the coupon rate.

Bond discount and bond premium i.e. the difference between market price of the bond and face value decreases as the bond nears its maturity as illustrated by the following chart: