Term structure estimation based on a generalized optimization framework

Abstract

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.