Zero Coupon Bond

In Fixed Income markets, instruments such as bonds and notes make periodic payments of interest to their owners, until what is known as the maturity date.

Payments are received at predetermined intervals which are defined by the instruments type. For example, US Government securities such as Treasury bonds will pay their owners twice a year - or semiannually - while Corporate bonds will pay only once a year.

A zero coupon bond makes no payments of interest during it's lifetime. Instead, the owner of such an instrument receives a single payment at the maturity date of the instrument.

Typically a zero coupon bond is priced at a deep discount to it's redemption or par value. The difference between par and the purchase price is effectively the interest received by the ower, for acquiring the instrument.

A bill is identical to a zero coupon with the exception of the maturity or lifetime of the instrument. Zero coupon bonds typically are issued for thirty years, while bills are issued for much shorter periods - one year or less.

A zero coupon bond is also known as a strip for a curious historical reason. In modern times few bonds are physically issued - instead ownership is recorded on a register.

Before the development of adequate computer power, bonds were physically issued and retained by the owner. Each bond consisted of a series of coupons which were clipped, or removed by the owner and presented for payment.

A zero coupon bond, then, is nothing more than a bond with the coupons removed, or striped. This, as previously explained, entitles the owner to a single payment at maturity of the instrument.
A zero-coupon bond has negative amortization: no interest payments are made (the principal accumulates) until maturity.

The principal at year t is:

Specified interest rate or yield
Principal value at a given point
The tax rate on a such a bond is imputed:
Tax rate
Time of maturity
Principal value at a given point
Thus one pays Pt-1r at time t.

A zero-coupon bond is a special kind of bond. Normally, the holder of a bond has two rights: the right to be periodically paid interest and the right to be paid back the amount of money borrowed, the principal. These interest payments are called coupons. The historical reason for this is that on paper bonds, these coupons where physical pieces of paper that would have to be cut off and turned in to collect the interest

A zero-coupon bond is different in that it does not pay a periodic interest. The interest is in fact all paid at the maturity of the bond, together with the principal. For a one-year bond with a time to maturity of 1 year, an interest percentage of 5% and "principal" of 100 euro, we hence receive 105 euros at the end. One important thing to remember here is that it is common that zero-coupon bonds have a "round" final payoff; hence, it is more probable that we would have a payoff of 100 euros, and need to pay 95.24 for this bond.

For longer-dated bonds, the initial payment I depends as follows on the final payout:


with M the value of the bond at maturity, r the interest rate percentage, and t the time to maturity. As an example, consider a bond with a time to maturity of 15 years, an interest rate of 7% (or 0.07), and a value at maturity of 1000 euro. For this, we would need to pay 362 euro initially.

Zero-coupon bonds are used for three main reasons: as an aid in doing calculations, for short-dated bonds, and for very long-dated bonds. Below, these three uses will be briefly discussed.

Conceptually, zero-coupon bonds are the fundamental building block of bond mathematics. Any future payment can be seen as a zero-coupon bond and hence be discounted to its present value. In particular, a "normal" bond consists of a strip of small zero-coupon bonds (the coupon payments) and one big zero-coupon bond (the principal). Note that it is entirely possible for the present value of the principal to be smaller than that of a coupon, especially if the bond has a very long time to maturity and/or the interest rate is high.

Short-dated government bonds (for instance T-bills) are often zero-coupon bonds. Having just one simple payment to track makes things considerably easier. Furthermore, time deposits in which the interest is automatically reinvested in the deposit are also a form of zero-coupon bond

Very long-dated bonds also may be zero-coupon bonds. The reason for this is that in a normal bond, the present value of the coupons would be high compared to the present value of the principal. This would mean that the effective duration of the bond isn't very long at all, as the investor already gets most of his investment out in a decade or so. To avoid this, extremely long-dated zero-coupon bonds are used; these have only one payment, at the end, avoiding this problem.

In terms of risk, short-dated zero-coupon government bonds are about the most riskless in existence. Longer dated zero-coupon bonds can be very risky, though. The reason is that the effective duration is so long. If a company were to issue a normal bond and it were to default on it, the investor would still have received the accrued interest up to that point, which could be substantial, especially given the fact that companies often have to pay (a lot) more interest than a government due to the fact they may default. However, for a zero-coupon bond, this isn't the case - it is either all or nothing.

Zero-coupon bonds, in summary, are bonds in which the accrued interest and principal are all paid back together at maturity. This single payment structure makes them easy to do math on. Hence, they are an important theoretical concept in finance. The single payback also means that the time to maturity is very well-defined, and this means they can be used to get exactly the bond duration an investor wants

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