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Term structure estimation based on a generalized optimization framework
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

The current work is devoted to estimating the term structure of interest rates based on a generalized optimization framework. To x the ideas of the subject, we introduce representations of the term structure as they are used in nance: yield curve, discount curve and forward rate curve.

Yield curves are used in empirical research in nance and macroeconomic to support nancial decisions made by governments and/or private nancial institutions. When governments (or nancial corporations) need fundings, they issue to the public (i.e. the market) debt securities (bills, bonds, notes, etc ) which are sold at the discount rate at the settlement date and promise the face value of the security at the redemption date, known as maturity date. Bills, notes and bonds are usually sold with maximum maturity of 1 year, 10 years and 30 years respectively.

Let us assume that the government issues to the market zero-coupon bonds, which provide a single payment at maturity of each bond. To determine the price of the security at time of settlement, a single discount factor is used. Thus, the yield can be dened as the discount rate which makes the present value of the security issued (the zero-coupon bond) equal to its initial price. The yield curve describes the relationship between a particular yield and a bond's maturity. In general, given a certain number of bonds with dierent time to maturity, the yield curve will describe the one-to-one relationship between the bond yields and their corresponding time to maturity. For a realistic yield curve, it is important to use only bonds from the same class of issuer or securities having the same degree of liquidity when plotting the yields.

Discount factors, used to price bonds, are functions of the time to maturity. Given that yields are positive, these functions are assumed to be monotonically decreasing as the time to maturity increases. Thus, a discount curve is simply the graph of discount factors for dierent maturities associated with dierent securities.

Another useful curve uses the forward rate function which can be deduced from both the discount factor and the yield function. The forward rate is the rate of return for an investment that is agreed upon today but which starts at some time in the future and provides payment at some time in the future as well. When forward rates are used, the resulting curve is referred to as the forward rate curve. Thus, any of these curves, that is, the yield curve, the discount curve or the forward rate curve, can be used to represent what is known as the term structure of interest rate. The shapes that the term structure of interest rates can assume include upward sloping, downward sloping,  atness or humped, depending on the state of the economy. When the expectations of market participants are incorporated in the construction of these curves representing the term structure, their shapes capture and summarize the cost of credit and risks associated with every security traded.

However, constructing these curves and the choice of an appropriate representation of the term structure to use is not a straightforward task. This is due to the complexity of the market data, precisely, the scarcity of zero-coupon bonds which constitutes the backbone of the term structure. The market often provides coupons alongside market security prices for a small number of maturities. This implies that, for the entire maturity spectrum, yields can not be observed on the market. Based on available market data, yields must be estimated using traditional interpolation methods. To this end, polynomial splines as well as parsimonious functions are the methods mostly used by nancial institutions and in research in nance. However, it is observed in literature that these methods suer from the shape constraints which cause them to produce yield curves that are not realistic with respect to the market observations. Precisely, the yield curves produced by these methods are characterized by unrealistic t of the market data, either in the short end or in the long end of the term structure of interest rate.

To ll the gap, the current research models the yield curve using a generalized optimization framework. The method is not shape constrained, which implies that it can adapt to any shape the yield curve can take across the entire maturity spectrum. While estimating the yield curve using this method in comparison with traditional methods on the Swedish and US markets, it is shown that any other traditional method used is a special case of the generalized optimization framework. Moreover, it is shown that, for a certain market consistency, the method produces lower variances than any of the traditional methods tested. This implies that the method produces forward rate curve of higher quality compared to the existing traditional methods.

Interest rate derivatives are instruments whose prices depend or are derived from the price of other instruments. Derivatives instruments that are extensively used include the forward rate agreement (FRA) contracts where forward rate is used and the interest rate swap (IRS) where LIBOR rate is used as  oating rate. These instruments will only be used to build up the term structure of interest rates. Since the liquidity crisis in 2007, it is observed that discrepancies in basis spread between interest rates applied to dierent interest rate derivatives have grown so large that a single discount curve is no longer appropriate to use for pricing securities consistently. It has been suggested that the market needs new methods for multiple yield curves estimation to price securities consistently with the market. As a response, the generalized optimization framework is extended to a multiple yield curves estimation. We show that, unlike the cubic spline for instance, which is among the mostly used traditional method, the generalized framework can produce multiple yield curves and tenor premium curves that are altogether smooth and realistic with respect to the market observations.

U.S. Treasury market is, by size and importance, a leading market which is considered as benchmark for most xed-income securities that are traded worldwide. However, existing U.S. Treasury yield curves that are used in the market are of poor quality since they have been estimated by traditional interpolation methods which are shape constrained. This implies that the market prices they imply contain lots of noise and as such, are not safe to use. In this work, we use the generalized optimization framework to estimate high-quality forward rates for the U.S. Treasury yield curve. Using ecient frontiers, we show that the method can produce low pricing error with low variance as compared to the least squares methods that have been used to estimate U.S. Treasury yield curves.

We nally use the high-quality U.S. Treasury forward rate curve estimated by the generalized optimization framework as input to the essentially ane model to capture the randomness property in interest rates and the time-varying term premium. This premium is simply a compensation that is required for additional risks that investors are exposed to. To determine optimal investment in the U.S. Treasury market, a two-stage stochastic programming model without recourse is proposed, which model borrowing, shorting and proportional transaction cost. It is found that the proposed model can provide growth of wealth in the long run. Moreover, its Sharpe ratio is better than the market index and its Jensen's alpha is positive. This implies that the Stochastic Programming model proposed can produce portfolios that perform better than the market index.

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1539
Mathematics
##### Identifiers
ISBN: 978-91-7519-526-1 (print)OAI: oai:DiVA.org:liu-97410DiVA, id: diva2:647662
##### Supervisors
Available from: 2013-09-12 Created: 2013-09-12 Last updated: 2013-10-07Bibliographically approved
##### List of papers
1. High Quality Yield Curves from a Generalized Optimization Framework
Open this publication in new window or tab >>High Quality Yield Curves from a Generalized Optimization Framework
##### Abstract [en]

Traditional methods for estimating yield curves are special cases of a generalized optimization framework. For pricing out-of-sample in both the Swedish and U.S. interest rate swap (IRS) markets, it is shown that the framework dominates or is close to dominating the traditional methods in the comparison by first order stochastic dominance. When measuring the perceived variance for each traditional method, it is shown that, for the same level of market consistency, the framework produces lower variance. For these new yield curves, PCA of innovations in forward rates shows that the first three loadings (shift, twist and butterfly) do not explain movements in the short end, and that the subsequent loadings explain uncorrelated movements in the short end.

##### Keyword
Term structure estimation, Forward rate, Principal Components Analysis (PCA)
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-97405 (URN)
Available from: 2013-09-12 Created: 2013-09-12 Last updated: 2013-09-12Bibliographically approved
2. Multiple Yield Curves Estimation Using A Generalized Optimization Framework
Open this publication in new window or tab >>Multiple Yield Curves Estimation Using A Generalized Optimization Framework
##### Abstract [en]

After the credit crunch which started in 2007, significant basis spreads for exchanging floating payments of different tenors appeared. To deal with the problem, multiple yield curves estimation methods have been suggested. In this paper, a generalized optimization framework is extended to a multiple yield curve framework. As has been observed by practitioners, extending traditional cubic splines to multiple yield curves, though consistent with the market prices, does not provide smooth and realistic yield curves. When the parameters in the generalized optimization framework are selected to exactly match market prices, the yield curves are much more realistic, but small waves still remain due to noise in the input data. To avoid having a rough yield curve, we also study the least squares parameter setting in the generalized optimization framework. This method gives much smoother and more realistic yield curves with adjustments to market prices that are less than 0.2 basis points. When exact traditional methods are extended to estimate multiple yield curves, then even tiny pricing errors can cause a situation where the shape constraints prevent the method from finding realistic yield curves.

##### Keyword
Multiple yield curve estimation, Overnight Index Swap (OIS), Basis spread, Tenor Swap (TS)
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-97406 (URN)
Available from: 2013-09-12 Created: 2013-09-12 Last updated: 2013-09-12Bibliographically approved
3. Estimating U.S. Treasury Yield Curves By A Generalized Optimization Framework
Open this publication in new window or tab >>Estimating U.S. Treasury Yield Curves By A Generalized Optimization Framework
##### Abstract [en]

We show that traditional data sets for the U.S. Treasury yield curves contain large amounts of noise, in e.g. the Fama-Bliss discount file already the second factor loading for innovations in forward rates is a consequence of noise. We implement the quadratic and cubic McCulloch splines, Nelson-Siegel and Svensson models and compare these traditional models with a recently developed generalized optimization framework using daily CRSP data from 1961 to 2011. In out-of-sample tests, it is shown that the generalized optimization framework produces smaller pricing errors compared to the traditional methods. Factor loadings from the generalized optimization framework show that the short and long end of the forward rate curve move independently, where principal component 1-3 explain the long end, and subsequent principal components explain the short end. This is consistent with the behavior of the market where short rates are governed by central bank while long rates are dependent on e.g. the expectation of future inflation.

##### Keyword
structure estimation, U.S. Treasury, Principal component analysis, Forward rates
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-97407 (URN)
Available from: 2013-09-12 Created: 2013-09-12 Last updated: 2013-09-12Bibliographically approved
4. Optimal Investment in the Fixed-Income Market with Focus on the Term Premium
Open this publication in new window or tab >>Optimal Investment in the Fixed-Income Market with Focus on the Term Premium
##### Abstract [en]

A good estimation of expected returns is imperative when optimal investments are determined with Stochastic Programming model. However, existing Stochastic Programming models do not include a model for the time-varying term premium. In this paper Duffee’s essentially affine model is used to capture the randomness in interest rates and the time-varying term premium. To determine optimal investments, a two-stage Stochastic Programming model without recourse is proposed which models borrowing, shorting and proportional transaction costs. The proposed model is evaluated over the period 1961-2011 and the Sharpe ratio is better than the one’s that corresponds to the market index, and Jensen’s alpha is positive.

##### Identifiers
urn:nbn:se:liu:diva-97409 (URN)
Available from: 2013-09-12 Created: 2013-09-12 Last updated: 2013-09-12Bibliographically approved

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Ndengo Rugengamanzi, Marcel

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Cite
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