### Zero-Coupon Bonds Definition

A bond is a debt instrument issued by the government or by a company. It promises to make routine payments (coupon payments) to the holder.

A zero-coupon bond, as the name implies, does not pay a coupon (interest). So, why would people buy a zero-coupon bond? Basically, the bond is sold at a significant discount from its face value. The trading value goes up as the bond approaches its priority date. The priority date is the date on which the bond’s face value will be payable. At that date, the bond will pay its full, face value. For this reason, zero-coupon bonds are often referred to as an accrual bond.

Many zero-coupon bonds are sold under those conditions. In other cases, a regular corporate bond is stripped of its interest payment by the company issuing the bond. The bonds are reclaimed and repackaged as zero-coupon bonds.

### A Little More on What is a Zero-Coupon Bond

Bonds are methods for companies to raise capital. Bond purchasers are lenders to the company. The company is the debtor. Most bonds pay interest (a coupon) annually or semi-annually throughout the duration of the bond.

When the bond reaches a maturity date (the end of the bond period), the holder is entitled to receive the face value (a stated amount) of the bond.

Many bonds (like the zero-coupon bond) do not pay regular interest payments. These types of bonds are sold at the time of issuance for an amount less than face value. This is known as selling the bond at a discount. For example, a $10 bond may be sold for $9. At the end of 1 year, the bond will pay the holder $10. Effectively, the bondholder earned $1 on a $9 investment for one year. In the above example, the $1 is the “investor’s return”.

The amount of interest or coupon received includes the principle paid for the bond and interest (which is compounded annually or semi-annually) at a stated yield. Unlike the regular, coupon-paying bonds, a zero-coupon bond has an imputed interest rate (rather than an established interest rate). To illustrate, if a bond with a face value of $1,000 matures in 20 years with a 5.5% annual yield, can be purchased at $3,378. This represents $1,000 in value in 20 years if the money compounds annually for 20 years.

The bondholder must pay federal (an potentially state) income taxes on the bond coupon payments. On a zero-coupon bond, the bondholder has imputed income. This is also known as “phantom income”, as the bondholder does not actually receive any funds.

### Calculating the Price of a Bond

Below is the formula for calculating the present value of a zero coupon bond:

Price = M / (1 + r)^n

where M = the date of maturity

r = Interest Rate

n = # of Years until Maturity

If an investor wishes to make a 4% return on a bond with $10,000 par value due to mature in 2 years, he will be willing to pay:

$10,000 / (1 + 0.04)^2 = $9,245.

So, the bond is being sold at 92% of its face value.

### References for Zero Coupon Bonds

- https://investinganswers.com/financial-dictionary/bonds/zero-coupon-bond-859
- https://www.investopedia.com/terms/z/zero-couponbond.asp
- https://en.wikipedia.org/wiki/Zero-coupon_bond
- http://www.businessdictionary.com/definition/zero-coupon-bond.html

### Academic Research on Zero Coupon Bonds

- · Fundamental solutions for
**zero**–**coupon bond**pricing models, Pooe, C. A., Mahomed, F. M., & Soh, C. W. (2004).*Nonlinear Dynamics*,*36*(1), 69-76. This paper explains the transformation approach and the explanation given was that this approach is used to reduce the one-factor bond-pricing equation into the heat equation in which the fundamental solution is known. According to this paper, these transformations are subsequently applied in constructing the fundamental solution for a two-zero coupon bond-pricing equation. After this process, the pricing model of the equation was then obtained from the closed form analytical solutions of the Cauchy initial values. - · A cointegration analysis of Danish
**zero**–**coupon bond**yields, Engsted, T., & Tanggaard, C. (1994).*Applied Financial Economics*,*4*(4), 265-278. According to this research work, the result obtained from the coin-integration analysis of the term structure of the interest rate which adopts newly-constructed yields on main discount bond gotten from the Danish bond market (from 1976-1991). The systems of interest rates of different maturity were analysed in this research and they were seen as a vector autoregressive system. The coin-integration implications of the expected hypothesis were tested in this paper. The result obtained generally supports the hypothesis that claims the Danish nominal terms structure is controlled by one common stochastic trend and that the interest rate increase is randomly found to be stationary. - · Zero coupon bond arbitrage: An illustration of the regulatory dialectic at work, Finnerty, J. D. (1985).
*Financial Management*, 13-17. According to this research work, an example of how structural frictions in the world capital market can develop profit arbitrage opportunities was provided and explained. This process explains how a company gets an arbitrage profit by simultaneously making use of zero coupon Eurobonds and also by purchasing a cash matching portfolio of stripped United States Treasury bond. This paper also describes and explains the market irregularities that were created by this opportunity. - · Zero‐coupon bond prices in the Vasicek and CIR models: Their computation as group‐invariant solutions, Sinkala, W., Leach, P. G. L., & O’hara, J. G. (2008).
*Mathematical Methods in the Applied Sciences*,*31*(6), 665-678. This paper computes the prices of the zero-coupon bonds gotten from the Cox-Ingersoll-Ross and Vasicek interest rate models as a group-invariant solution. Before any other process, the first method adopted by this paper is a determination of the symmetries in the valuation of the partial differential equation that is similar and compatible to the terminal condition and then the desired solution among the invariant solutions that were as a result of the symmetries gotten from this process was also provided. The second process was to point to other liable studies that show these models are using the symmetries given by the valuation of the partial differential equation. - · Comparison of multivariate GARCH models with application to
**zero**–**coupon bond**volatility, Su, W., & Huang, Y. (2010). The main aim of this paper is to study the main difference in the formulation of the multivariate GARCH models and to apply two of the popular formulations (the BEKK-GARCH model and the DCC-GARCH model) in evaluating the volatility of a portfolio of zero-coupon bonds. This paper defines the Multivariate GRACH model as one of the most important tools for explaining and forecasting the volatility of the time series when volatility fluctuates over time. This characteristic demonstrates its availability in modelling the movement of the multivariate time series with a difference in the conditional covariance matrix. - · Pricing the
**zero**–**coupon bond**and its fair premium under a structural credit risk model with jumps, Dong, Y., Wang, G., & Wu, R. (2011).*Journal of Applied Probability*,*48*(2), 404-419. This research thesis considered the structural form credit risk model with jumps. According to this paper, the price, credit spread and the fair premium of the zero-coupon bond were investigated for the proposed model. The fair premium and the price of the bond are connected with the Laplace transform of the firm’s expected present market value and the default time. The result obtained from this paper was that the closed-form expressions for them when the jumps have a hyper-exponential distribution and using the closed-form expression, numerical answers were obtained for the credit spread, default probability and the bond’s fair premium. - · An inverse problem arose in the zero-coupon bond pricing, Deng, Z. C., Yu, J. N., & Yang, L. (2010).
*Nonlinear Analysis: Real World Applications*,*11*(3), 1278-1288. This paper defines the zero-coupon bond as a special bond without coupon which is mostly purchased at a certain price today while at maturity, the bond is redeemed for a fixed price. Some of the main feature of the zero-coupon bond that must be known i9s that it contains an important quality λ (t) which is mostly regarded as the market price of risk and cannot be directly observed but it should be noted that it has a very important influence on the zero-coupon bond. The research work analyses the opposite problem of analyzing the market risk price from the current market price of the zero-coupon bond. The result gotten from this paper is very important and may be applied to various derivatives pricing problems. - · Pricing American interest rate option on
**zero**–**coupon bond**numerically, ShuJin, L., & ShengHong, L. (2006).*Applied Mathematics and Computation*,*175*(1), 834-850. This research work explains the finite volume method which is a method in which an American put option on the zero-coupon bond numerically in the presence of a single model factor of the short-term rate. As regards the price of the zero-coupon bond, an integral delegation of the early exercise rate is gotten which can both be used to find the exercise rate and be used as an error indicator. According to the numerical result obtained in this paper, the price of the zero-coupon bond and the American put option are provided and the optimal interest rate was also given. - · Zero-coupon yields and the cross-section of bond prices, Pancost, N. A. (2018).
*Available at SSRN 2157271*. This paper estimates a dynamic term-structure model with a time-varying risk premia on a panel of Treasury coupon bond, without depending on an interposed zero-coupon yield curve on a selection of maturities. This model according to this research allows the incorporation of prices and the relaxed return of coupon bond and incorporates them into the testing and estimation of the model. Specification test was also carried out using the infeasible zero-coupon yields. This paper also shows that price risk estimated over from vector auto-regression as an important factor that does not propose the return of the actual Treasury bonds. - · A Computational Scheme for a Problem in the
**Zero**‐**coupon Bond**Pricing, Chernogorova, T., & Valkov, R. (2010, November). In*AIP Conference Proceedings*(Vol. 1301, No. 1, pp. 370-378). AIP. According to the research of this paper, a finite volume difference scheme for a degenerate parabolic equation with dynamical boundary was derived based on the conditions of the zero-bond pricing method. This paper shows that the system matrix of the discretization scheme is an M-matrix; hence, the discretization is termed as monotone. This result provides a non-negativity of the price with respect to the time if and only if the former distribution is non-negative. The diverse numerical experiment indicates higher accuracy with the comparison of known differences in the scheme. - · Approximating the
**zero**–**coupon bond**price in a general one-factor model with constant coefficients, Stehlikova, B. (2014).*arXiv preprint arXiv:1408.5673*. This research paper admonishes a general one-factor short rate model where the instantaneous interest rate is propelled by univariate diffusion with time irrespective of the volatility and drift. This paper constructs a recursive formula for the coefficient of the Taylor expansion of the price of the bond and its logarithm and its time to maturity. Numerical examples were provided and compared with the known and exact values as in the case of Cox-Ingersoll-Ross and Dothan model. - · Direct Estimating Price of a Defaultable
**Zero**–**Coupon Bond**Using Conception of Continuous Coupon Bond, Voloshyn, I. (2014). This research thesis adopts the use of conception of a continuous coupon bond having a continuous accrual of coupons on an easily fixed rate for pricing a risky zero-coupon bond was considered. This conception is the only method that allows the obtainment of an explicit equation for the price of the risky zero-coupon bond from the fixed bond. In other to apply this conception, a simple condition for inversion was introduced. This condition for inversion is possible only if a recovery rate is among the present value of the leftover cash flow. - · Tax-Exempt
**Zero**–**Coupon Bond**Pricing, Williamson, S. H. (1982).*National Tax Journal*,*35*(4), 497-500. This paper explains the exemption of tax in the zero-coupon bond in the pricing of the coupon bonds. This paper explains the correlation between the exemption of tax and the zero-coupon bond pricing.