The problem of estimating the volatility of zero coupon bond interest rate

Financial literature and financial industry use often zero coupon yield curves as input for testing hypotheses, pricing assets or managing risk. They assume this provided data as accurate. We analyse implications of the methodology and of the sample selection criteria used to estimate the zero coupon bond yield term structure on the resulting volatility of spot rates with different maturities. We obtain the volatility term structure using historical volatilities and Egarch volatilities. As input for these volatilities we consider our own spot rates estimation from GovPX bond data and three popular interest rates data sets: from the Federal Reserve Board, from the US Department of the Treasury (H15), and from Bloomberg. We find strong evidence that the resulting zero coupon bond yield volatility estimates as well as the correlation coefficients among spot and forward rates depend significantly on the data set. We observe relevant differences in economic terms when volatilities are used to price derivatives.

$80,000,000 State of Illinois general obligation college savings bonds : series of October 2002 (zero coupon bonds)

"Due: August 1, as shown below."

$101,855,028.75 State of Illinois general obligation college savings bonds : series of October 2000 (zero coupon bonds)

"New issue."

$80,000,000 State of Illinois general obligation college savings bonds : series of October 2000 (zero coupon bonds).

"Due: August 1, as shown below."

Machine learning Vasicek model calibration with Gaussian processes

In this article, we calibrate the Vasicek interest rate model under the risk neutral measure by learning the model parameters using Gaussian processes for machine learning regression. The calibration is done by maximizing the likelihood of zero coupon bond log prices, using mean and covariance functions computed analytically, as well as likelihood derivatives with respect to the parameters. The maximization method used is the conjugate gradients. The only prices needed for calibration are zero coupon bond prices and the parameters are directly obtained in the arbitrage free risk neutral measure.

Machine learning Gaussian short rate

Dissertação para obtenção do Grau de Doutor em Estatística e Gestão do Risco

Osakeindeksiobligaation fair value Case: SHB Paras Kolmesta 5010

Tutkielmassa selvitetään monimutkaisen indeksiobligaation arvon määritystä obligaation juoksuaikana. Tutkittava indeksiobligaatio sijoittaa osakkeisiin, korkoon ja raaka aineisiin. Kyseisessä indeksi-obligaatiossa sijoitetaan korioptioihin ja ne ovat lisäksi niiltä osin kvantto optioita, kun positio on ollut tarpeen suojata valuutta kurssimuutoksia vastaan. Tämän lisäksi indeksiobligaatio sijoittaa nolla kuponkilainaan. Sijoittajalle on haastavaa ymmärtää oikein ja läpinäkyvästi monimutkaisen osakeindeksiobligaation arvonmääritystä sen juoksuaikana. Tässä tutkielmassa avataan monimutkaisen osakeindeksiobligaation arvonmäärityksen perusteet. Tutkielmassa huomataan, että sijoittajalla voi helposti olla vaara mielikuva siitä, miten arvo määrittyy. Tämä johtuu siitä, että arvonmääritys eroaa huomattavasti siitä mikä käy ilmi velkakirjan ehdoissa. Tutkielman keskeisin anti on se, että kyseinen osakeindeksiobligaatio antaa heikkoa tuottoa kesken juoksuajan, mikäli joko osakkeet, raaka-aineet tai korot kehittyvät negatiivisesti. Lisaksi raaka aineoptioiden hinnan määritys eroaa siitä, mitä sijoittaja olettaa sen olevan velkakirjan ehtoja lukiessaan. Raaka-aineiden hinnat määritellään forward hinnoista

Credit risk & forward price models

Doutoramento em Gestão

Three essays on the term structure of interest rates

A PhD Dissertation, presented as part of the requirements for the Degree of Doctor of Philosophy from the NOVA - School of Business and Economics

Valuation of defaultable bonds and debt restructuring

In this paper we develop a contingent valuation model for zero-coupon bonds with default. In order to emphasize the role of maturity time and place of the lender’s claim in the hierarchy of debt of a firm, we consider a firm that issues two bonds with different maturities and different seniorage. The model allows us to analyze the implications of both debt renegotiation and capital structure of a firm on the prices of bonds. We obtain that renegotiation brings about a significant change in the bond prices and that the effect is dispersed through different channels: increasing the value of the firm, reallocating payments, and avoiding costly liquidation. Moreover, the presence of two creditors leads to qualitatively different implications for pricing, while emphasizing the importance of bond covenants and renegotiation of the entire debt.

Data uncertainty in real estate forecasting

The rapid expansion of the TMT sector in the late 1990s and more recent growing regulatory and corporate focus on business continuity and security have raised the profile of data centres. Data centres offer a unique blend of occupational, physical and technological characteristics compared to conventional real estate assets. Limited trading and heterogeneity of data centres also causes higher levels of appraisal uncertainty. In practice, the application of conventional discounted cash flow approaches requires information about a wide range of inputs that is difficult to derive from limited market signals or estimate analytically. This paper outlines an approach that uses pricing signals from similar traded cash flows is proposed. Based upon ‘the law of one price’, the method draws upon the premise that two identical future cash flows must have the same value now. Given the difficulties of estimating exit values, an alternative is that the expected cash flows of data centre are analysed over the life cycle of the building, with corporate bond yields used to provide a proxy for the appropriate discount rates for lease income. Since liabilities are quite diverse, a number of proxies are suggested as discount and capitalisation rates including indexed-linked, fixed interest and zero-coupon bonds. Although there are rarely assets that have identical cash flows and some approximation is necessary, the level of appraiser subjectivity is dramatically reduced.

The appraisal of data centres: deconstructing the cash flow

This paper analyses the appraisal of a specialized form of real estate - data centres - that has a unique blend of locational, physical and technological characteristics that differentiate it from conventional real estate assets. Market immaturity, limited trading and a lack of pricing signals enhance levels of appraisal uncertainty and disagreement relative to conventional real estate assets. Given the problems of applying standard discounted cash flow, an approach to appraisal is proposed that uses pricing signals from traded cash flows that are similar to the cash flows generated from data centres. Based upon ‘the law of one price’, it is assumed that two assets that are expected to generate identical cash flows in the future must have the same value now. It is suggested that the expected cash flow of assets should be analysed over the life cycle of the building. Corporate bond yields are used to provide a proxy for the appropriate discount rates for lease income. Since liabilities are quite diverse, a number of proxies are suggested as discount and capitalisation rates including indexed-linked, fixed interest and zero-coupon bonds.

Structural macro factors and the affine term structure of interest rates

This work consists of three essays investigating the ability of structural macroeconomic models to price zero coupon U.S. government bonds. 1. A small scale 3 factor DSGE model implying constant term premium is able to provide reasonable a fit for the term structure only at the expense of the persistence parameters of the structural shocks. The test of the structural model against one that has constant but unrestricted prices of risk parameters shows that the exogenous prices of risk-model is only weakly preferred. We provide an MLE based variance-covariance matrix of the Metropolis Proposal Density that improves convergence speeds in MCMC chains. 2. Affine in observable macro-variables, prices of risk specification is excessively flexible and provides term-structure fit without significantly altering the structural parameters. The exogenous component of the SDF is separating the macro part of the model from the term structure and the good term structure fit has as a driving force an extremely volatile SDF and an implied average short rate that is inexplicable. We conclude that the no arbitrage restrictions do not suffice to temper the SDF, thus there is need for more restrictions. We introduce a penalty-function methodology that proves useful in showing that affine prices of risk specifications are able to reconcile stable macro-dynamics with good term structure fit and a plausible SDF. 3. The level factor is reproduced most importantly by the preference shock to which it is strongly and positively related but technology and monetary shocks, with negative loadings, are also contributing to its replication. The slope factor is only related to the monetary policy shocks and it is poorly explained. We find that there are gains in in- and out-of-sample forecast of consumption and inflation if term structure information is used in a time varying hybrid prices of risk setting. In-sample yield forecast are better in models with non-stationary shocks for the period 1982-1988. After this period, time varying market price of risk models provide better in-sample forecasts. For the period 2005-2008, out of sample forecast of consumption and inflation are better if term structure information is incorporated in the DSGE model but yields are better forecasted by a pure macro DSGE model.

La hipótesis de expectativas en la curva cero cupón: un análisis por cointegración

El cumplimiento de la hipótesis de expectativas (HE) ha sido contrastado en varios países por medio de diferentes métodos. En Colombia no se ha estudiado de manera conjunta las relaciones de largo plazo entre las tasas cero cupón. El presente trabajo busca contrastar el cumplimiento de la hipótesis de expectativas estimando un modelo multivariado con corrección de errores, de acuerdo a la metodología propuesta por Hall, Anderson y Granger (1992). La significancia de la prima por liquidez en las relaciones de largo plazo y las pruebas estadísticas del modelo favorecen el contraste de la HE. Sin embargo, la existencia de dos relaciones de cointegración y el rechazo de las relaciones teóricas esperadas indican el incumplimiento de la HE en Colombia.

Estructura a plazo, hipótesis de expectativas y paridad descubierta de intereses en Colombia

El desarrollo del mercado financiero en Colombia, ha hecho que la integr ación con los merados financieros internacionales sea cada vez más evidente. Es por esto que el estudio del grado de relación de nuestras tasas de interés con las tasas de interés de las tasas internacionales, cobra relevancia. Este estudio, busca evidencia sobre el cumplimiento de la hipótesis de paridad descubierta de intereses y la hipótesis de expectativas racionales, a través del uso de las curvas cero cupón de Colombia y Estados Unidos, siguiendo la metodología derivada del estudio de Bekaert (2002). Se encuentra que el cumplimiento de ambas hipótesis, simultáneamente, no es un hecho común entre los diferenc iales de tasas de Colombia y Estados Unidos. Además, en algunos casos, se encuentra que las hipótesis no se cumplen entre las tasas de interés de la misma nacionalidad.