Zero Coupon Bond  Definition
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ZeroCoupon 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 zerocoupon bond, as the name implies, does not pay a coupon (interest). So, why would people buy a zerocoupon 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 bonds face value will be payable. At that date, the bond will pay its full, face value. For this reason, zerocoupon bonds are often referred to as an accrual bond. Many zerocoupon 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 zerocoupon bonds.
A Little More on What is a ZeroCoupon 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 semiannually 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 zerocoupon 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 investors return. The amount of interest or coupon received includes the principle paid for the bond and interest (which is compounded annually or semiannually) at a stated yield. Unlike the regular, couponpaying bonds, a zerocoupon 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 zerocoupon 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/financialdictionary/bonds/zerocouponbond859
 https://www.investopedia.com/terms/z/zerocouponbond.asp
 https://en.wikipedia.org/wiki/Zerocoupon_bond
 http://www.businessdictionary.com/definition/zerocouponbond.html
Academic Research on Zero Coupon Bonds
 Fundamental solutions forzerocoupon bondpricing models, Pooe, C. A., Mahomed, F. M., & Soh, C. W. (2004). Nonlinear Dynamics,36(1), 6976. This paper explains the transformation approach and the explanation given was that this approach is used to reduce the onefactor bondpricing 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 twozero coupon bondpricing 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 Danishzerocoupon bondyields, Engsted, T., & Tanggaard, C. (1994). Applied Financial Economics,4(4), 265278. According to this research work, the result obtained from the coinintegration analysis of the term structure of the interest rate which adopts newlyconstructed yields on main discount bond gotten from the Danish bond market (from 19761991). The systems of interest rates of different maturity were analysed in this research and they were seen as a vector autoregressive system. The coinintegration 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 bondarbitrage: An illustration of the regulatory dialectic at work, Finnerty, J. D. (1985). Financial Management, 1317. 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.
 Zerocoupon bondprices in the Vasicek and CIR models: Their computation as groupinvariant solutions, Sinkala, W., Leach, P. G. L., & O'hara, J. G. (2008).Mathematical Methods in the Applied Sciences,31(6), 665678. This paper computes the prices of the zerocoupon bonds gotten from the CoxIngersollRoss and Vasicek interest rate models as a groupinvariant 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 tozerocoupon bondvolatility, 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 BEKKGARCH model and the DCCGARCH model) in evaluating the volatility of a portfolio of zerocoupon 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 thezerocoupon bondand its fair premium under a structural credit risk model with jumps, Dong, Y., Wang, G., & Wu, R. (2011). Journal of Applied Probability,48(2), 404419. 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 zerocoupon bond were investigated for the proposed model. The fair premium and the price of the bond are connected with the Laplace transform of the firms expected present market value and the default time. The result obtained from this paper was that the closedform expressions for them when the jumps have a hyperexponential distribution and using the closedform expression, numerical answers were obtained for the credit spread, default probability and the bonds fair premium.
 An inverse problem arose in thezerocoupon bondpricing, Deng, Z. C., Yu, J. N., & Yang, L. (2010). Nonlinear Analysis: Real World Applications,11(3), 12781288. This paper defines the zerocoupon 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 zerocoupon bond that must be known i9s that it contains an important quality (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 zerocoupon bond. The research work analyses the opposite problem of analyzing the market risk price from the current market price of the zerocoupon bond. The result gotten from this paper is very important and may be applied to various derivatives pricing problems.
 Pricing American interest rate option onzerocoupon bondnumerically, ShuJin, L., & ShengHong, L. (2006). Applied Mathematics and Computation,175(1), 834850. This research work explains the finite volume method which is a method in which an American put option on the zerocoupon bond numerically in the presence of a single model factor of the shortterm rate. As regards the price of the zerocoupon 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 zerocoupon bond and the American put option are provided and the optimal interest rate was also given.
 Zerocoupon yields and the crosssection of bond prices, Pancost, N. A. (2018). Available at SSRN 2157271. This paper estimates a dynamic termstructure model with a timevarying risk premia on a panel of Treasury coupon bond, without depending on an interposed zerocoupon 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 zerocoupon yields. This paper also shows that price risk estimated over from vector autoregression as an important factor that does not propose the return of the actual Treasury bonds.
 A Computational Scheme for a Problem in theZerocoupon BondPricing, Chernogorova, T., & Valkov, R. (2010, November). InAIP Conference Proceedings(Vol. 1301, No. 1, pp. 370378). 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 zerobond pricing method. This paper shows that the system matrix of the discretization scheme is an Mmatrix; hence, the discretization is termed as monotone. This result provides a nonnegativity of the price with respect to the time if and only if the former distribution is nonnegative. The diverse numerical experiment indicates higher accuracy with the comparison of known differences in the scheme.
 Approximating thezerocoupon bondprice in a general onefactor model with constant coefficients, Stehlikova, B. (2014). arXiv preprint arXiv:1408.5673. This research paper admonishes a general onefactor 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 CoxIngersollRoss and Dothan model.
 Direct Estimating Price of a DefaultableZeroCoupon BondUsing 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 zerocoupon bond was considered. This conception is the only method that allows the obtainment of an explicit equation for the price of the risky zerocoupon 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.
 TaxExemptZeroCoupon BondPricing, Williamson, S. H. (1982). National Tax Journal,35(4), 497500. This paper explains the exemption of tax in the zerocoupon bond in the pricing of the coupon bonds. This paper explains the correlation between the exemption of tax and the zerocoupon bond pricing.