# Zero Coupon Bond

Zero coupon bond (also called deep-discount bond or pure discount bond) is a bond that pays no coupon payments over its life. It is issued at a significant discount to its face value and its total return results from the difference between its face value and the issue price. The value of a zero-coupon bond equals the present value of its face value which increases as the bond reaches maturity. A zero-coupon bond is also called a zero.

Zero-coupon bond differs from a traditional bond in the following ways:

- A regular bond pays periodic coupon payments while a zero-coupon bond doesn't pay any periodic cash flows;
- The return on a traditional bond results from both interest payments and price movestment while a zero-coupon bond's total return results from difference between issue price and matuirty value;
- A zero-coupon bond has zero reinvestment risk while a traditional bond has significant reinvestment risk;
- A zero-cooupon bond has higher interest rate risk than a traditional bond.

Sometimes normal coupon-paying bonds are broken down into their principal and coupon components such that each payment is a coupon bond in itself. Such zero coupon components are called strips.

## Formula

### Value of a Zero-Coupon Bond

Becuase a zero coupon bond has only one cash flow which occurs at the time of maturity of the bond, its price/value equals the present value of that cash flow discounted at the required rate of return.

The following formula can be used to work out value of a zero-coupon bond

$$ Value\ of\ Zero-Coupon\ Bond= P =\frac{FV}{{(1+\frac{Yield}{m})}^{n\times m}} $$

Where **FV** stands for face value fo the bond, **yield** stands for the annual required rate of return given the risk inherent in the bond, **m** is the number of compounding periods per year and **n** is the total number of years till maturity.

### Yield on a Zero-Coupon Bond

Given the current price (or issue price) of a zero-coupon bond (denoted as **P**), its face value (also called maturity value) of **FV** and years till maturity **n**, we can find out its annualized yield to maturity using the following equation:

$$ Yield\ on\ Zero-Coupon\ Bond=\left(\frac{FV}{P}\right)^\frac{1}{n}-1 $$

## Example

On 1 January 2013, Andrews invested $50,000 in 100 $1,000 par value zero coupon bonds issued by Stonehenge Travel Plc. The bonds were issued at a yield of 7.18%. The forecasted yield on the bonds as at 31 December 2013 is 6.8%. Find the value of the zero coupon bond as at 31 December 2013 and Andrews expected income for the financial year 2013 from the bonds.

Value of the zero coupon bond as at 31 December 2013 = $1,000 ÷ (1 + 6.8%)^{9} = $553.17

Value of Andrews total holding = 100 × $553.17 = $55,317

Expected accrued income = value at the end of a period − value at the start of a period = $55,317 − $50,000 = $5,317

This gain of $5,317 is made up of the unwinding of discount (the increase in present value as it nears maturity) plus 'capital gain' portion that results from positive movement in market yield on the bond.

The value of zero coupon bond will continue to increase till it reach $100,000 at the time of its maturity. Total dollar amount of interest earned by Andrews would be $50,000 ($100,000 minus $50,000).