Zero coupon methodology


Zero coupon yield curve

The yield curve currently used in Ghana is a primary market auction yield curve which is produced by the BoG. The Ghanaian bond market needs a secondary market benchmark zero-coupon yield curve for pricing corporate bonds and other securities. The market also needs forward yield curve for pricing forward contracts and other derivatives. There are important reasons why the Ghanaian bond market urgently needs these benchmark yield curves.

First, these curves could enhance the development of the entire financial market of Ghana. As stated earlier, they would be used for pricing corporate bonds and other financial instruments and derivatives. Presently, the corporate bond market of Ghana is not active; and there are no active markets for derivatives and asset-backed securities. As at the end of , for instance, only seven companies had their bonds trading on the GFIM. According to IOSCO , one of the key impediments to the development of the corporate bond market is the nonexistence of benchmark yield curve.

Financial market participants in Ghana need to know the level of, and the movements in the secondary market bond yields for the purpose of trade decision making.

The purpose of this article is to model the secondary market zero-coupon and forward yield curves for the GoG bonds using various methods of yield curve modeling. The article seeks to compare the piecewise cubic hermite method with other methods such as the piecewise cubic spline method with not-a-knot end conditions , the penalized smoothing spline method, and the Nelson—Siegel—Svensson NSS method.


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  • Zero-Coupon and Forward Yield Curves for Government of Ghana Bonds;
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  • What is Bootstrapping Yield Curve?.

We therefore use the terms penalized smoothing spline and VRP interchangeably in the article. Virtually, the article is comparing the spline-based methods of yield curve modeling with the parametric methods, using data from an illiquid African bond market Ghana. The remaining of the article is organized as follows. Section 2 reviews relevant literature; Section 3 discusses the methodology and data; Section 4 discusses the results; and Section 5 provides conclusion, recommendations and theoretical implications. There are two main categories of yield curve fitting methods. These are the parametric methods and the spline-based methods.

Parametric methods involve the specification of a single-piece function defined over the entire maturity range.

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Model parameters are determined through minimization of squared deviation of theoretical prices from observed prices. According to James and Webber , even though these parametric methods capture the overall shape of the yield curve fairly well, they are recommended when good accuracy is not a requirement. Considering the illiquidity of the Ghanaian bond market and data constraints, we prefer a method that would enhance the accuracy of the yield curve as much as possible.

A spline is a piecewise polynomial function, consisting of several individual polynomial segments that are joined together at knot points. Instead of using a single functional form over the entire maturity range, the spline-based methods employ the use of piecewise polynomials to fit the yield curve over the maturity range. To ensure continuity and smoothness, the splines join at the knot points and must be differentiable at the knot points.

There is a wide range of spline-based methods of fitting the yield curve, varying in complexity. Choudhry thinks that although the spline approach can lead to unrealistic shapes for the forward curve due to its divergence at the long end , it is an accessible method and one that gives reasonable accuracy for the zero-coupon yield curve.

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We therefore recommend a spline-based method for modeling the zero-coupon yield curve for the Ghanaian bond market. Our specific choice of spline-based method is mentioned elsewhere later in this section. The Bank for International Settlements BIS has recommended that central banks adopt methods for estimating zero-coupon yields.

After a meeting held in concerning the estimation of zero-coupon yield, many central banks have been reporting their zero-coupon yield estimates, as well as the methods of estimation, to the BIS Table 1 shows the methods used by some central banks to estimate zero-coupon yields. Most of these central banks use either a variation of the NSS model or a form of spline-based methods. Many African central banks including the BoG are yet to adopt a method of estimating the zero-coupon yields.

Yet still, others also use indicative yields e. Only very few African countries currently produce yield curves based on secondary market trades e. Nevertheless, there are some researches going on to propose yield curve modeling methods for the African central banks and bond markets. In , the then Bond Exchange of South Africa adopted a method of estimating zero-coupon yield curves. These curves served the purpose of providing benchmarking and valuation tools for the South African bond market.

Subsequently, the Johannesburg Stock Exchange JSE considered these curves to be outdated; and a new set of curves were generated using a different methodology referred to as Monotone Preserving Interpolation. The method is based on the Monotone Convex method by Hagan and West , , who emphasize the importance of shape preservation in yield curve modeling. The JSE zero-coupon yield curves now comprise three different daily yield curves: one for the nominal bond market, one for the nominal swaps market, and one for the inflation-linked bond market.

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Yield estimation in South Africa is very active because the South African bond market produces the volume of trade and data needed for the daily curves, as far as benchmarking is concerned. However, Ghana cannot boast of such volume of trade and data availability. While the input data for South African daily yield curves are from liquid bond and swap trades, Ghana can only rely on limited data from its illiquid but developing bond market.

We therefore would not strictly adopt the Monotone Preserving method used by South Africa; but adopt a variation which could preserve shape.

Intermediate Accounting 6A: Zero Coupon Bonds.

In Kenya, according to Muthoni, Onyango, and Ongati , there is currently no agreed-upon method used to construct yield curves. The existing practice is that financial companies use in-house methods to construct yield curves for pricing and other decision-making purposes. This is because some market participants think the yield curve produced by Nairobi Securities Exchange NSE has some limitations.

However, because all variations of linear interpolation result in curves which are not differentiable, we do not recommend the use of any form of linear interpolation for modeling yield curve for Ghana. In Nigeria, the Central Bank of Nigeria is at the forefront of providing the necessary prerequisites to develop the Nigerian bond market. One good step in this regard is the initiative to commission a project to fit the Nigerian government yield curve Sholarin, Sholarin seeks to use bootstrapping and piecewise cubic spline method to model the Nigerian zero-coupon yield curve for the Central Bank of Nigeria.

However, we do not recommend this piecewise cubic spline method for Ghana, due to a reason mentioned shortly under this section. Both papers seek to estimate real interest rates using yield curve.

Bootstrapping Yield Curve

Logubayom, Nasiru, and Luguterah also seek to forecast the weekly bill rates in the primary market. Ida and Albert investigate the relationship between the primary market bill rates, inflation rates and exchange rates in Ghana. Iyke analyzes the comovements of the BoG monetary policy rates and the bill rates in Ghana. The yields used in all these works are primary market yields, as the only form of yield curve presently in Ghana is the primary market yield curve. It is therefore not surprising that in a BoG working paper, Dzigbede and Ofori recommend that a research should be focused on building a framework to model the yield curve for the GoG debt securities.

Due to illiquidity and data constraints in the Ghanaian bond market, resulting in wide gaps in-between data points, we need to use an interpolation method which can preserve shape and is differentiable as well.


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  4. The method must be smoother than linear interpolation, but less smooth than piecewise cubic spline interpolation. This method is the piecewise cubic hermite interpolation. It is also a spline-based method based on cubic polynomials; but the estimation process is quite different from the traditional piecewise cubic spline method. The purpose of this work is to propose a framework for modeling secondary market zero-coupon and forward yield curves for GoG bonds.

    Specifically, this work compares the use of the Hermite method with other methods such as the Cubic Spline method, the penalized smoothing spline method, and the NSS method. The article seeks to make the following contributions:. This would be the first work, to the best of our knowledge, to model both the zero-coupon and forward yield curves for the Ghanaian bond market.

    Spot Rates, Forward Rates, and Bootstrapping

    The article compares the Hermite method with other spline-based methods Cubic Spline and penalized smoothing spline for modeling secondary market daily yield curves for an illiquid African bond market. This work empirically shows that for illiquid and inactive secondary bond markets, the Cubic Hermite method could work better than the Cubic Spline method with not-a-knot end conditions. The article also in general compares the spline-based methods with parametric methods of yield curve modeling based on data from an illiquid African bond market.

    This work serves as a step toward the development of a database of secondary market daily yield curves for GoG bonds. This would provide a source where daily yield curves could be accessed by researchers, financial analysts, policy makers, investors, and other market participants. The shape of the yield curve provides useful information in the bond market. There are some main theories that seek to explain the shape of the yield curve. These are collectively known as the expectations theories. The first of these theories is the pure expectations theory or the unbiased expectations theory.

    This theory asserts that the forward yields are unbiased predictors of future spot yields zero-coupon yields. In other words, forward yields are what investors expect spot yields or zero-coupon yields to be in future. The broadest interpretation of this theory is that bonds of any maturity are perfect substitutes for one another Chartered Financial Analyst [CFA] Institute, Thus, instead of investing in a 2-year bond at once, one could choose to first invest in a 1-year bond; and at maturity, reinvest the proceeds in another 1-year bond making a total of 2-year horizon, and yielding the same total returns as if it is an investment in a 2-year bond.

    Thus, an upward-sloping yield curve means investors expect the future short-term yields to increase; and a downward-sloping yield curve means investors expect future short-term yields to decrease. A flat yield curve therefore means investors expect short-term yields to remain constant in future. The pure expectations theory does not make any reference to risk premium as a factor affecting the shape of the yield curve. The second of the expectations theories is the local expectations theory. The local expectations theory does not explicitly assert that bonds of any maturity are perfect substitutes for one another; but rather, it asserts that the expected return for every bond over short time periods is the risk-free rate.

    While the pure expectations theory requires no risk premium along the entire maturity spectrum of the yield curve, the local expectations theory only requires existence of risk free at the very short end of the yield curve no risk-free requirement is made for the longer ends of the yield curve.


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    • Construct and analyze zero curves;
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    Therefore, unlike the pure expectations theory, the local expectations theory is applicable to both risk-free and risky bonds CFA Institute, It is the liquidity preference theory or the liquidity premium theory. This theory views bonds of different maturities as substitutes but not perfect substitutes. It is therefore sometimes referred to as the biased expectations theory. It is based on the premise that investors prefer liquid short-term bonds to long-term bonds because the former are free of inflation and interest rate risks.

    Investors would prefer to pay premium to buy short-term assets rather than to buy long-term assets. Thus, investors would have to be paid liquidity risk premium for holding long-term bonds in lieu of short-term bonds. Because of this term premium, per the theory, long-term bond yields tend to be higher than short-term yields.

    The liquidity preference theory therefore predicts upward-sloping yield curves. The theory also implies that the forward yields do not only reflect expectations about future spot zero-coupon yields but they also reflect expectations about risk premiums for holding long-term bonds. Accordingly, the theory implies that forward yields reflect higher yields demanded by investors for buying long-term bonds. The third theory is the market segmentation theory or the segmented market theory.

    Method for calculating a CNO Zero Coupon Yield Curve - CNO France - The french Bond Association

    This theory asserts that markets for different maturities of bonds are completely segmented. This implies that bonds of different maturities are not substitutes for one another. According to the theory, the shape of the yield curve is not a reflection of expected future spot rates; and neither does it reflect liquidity risk premiums. Instead, the theory asserts that the shape of the yield curve is a reflection of the demand and supply activities of the segmented market participants with respect to the specific maturities of interest.

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