The expected yield to maturity on a defaultable obligation equals the nominal yield less expected default losses. However, in a mark-to-market world, one doesn't have the luxury of reporting one's performance on the basis of yield to maturity. Total return is calculated for an arbitrary holding period, and must reflect any mark-to-market gains or losses as at the close of the period-gains or losses that can be triggered by the bond's upgrade or downgrade. Thus to estimate expected total return, one must estimate not only expected default losses, but also the impact on capital price of expected ratings transitions. This paper begins with the observation that a bond which is blessed by more favourable transition characteristics is likely to produce a higher total return, and poses the question of how that benefit can be quantified. How much is it worth? To answer the question, I start by specifying a formal bond-pricing model reflective of ratings transitions. I survey various statistical methods and past research efforts to identify the ratings-transition matrix which best parametrises the model, and propose a novel test for selecting between competing matrices. Using this approach, I replicate several important studies of ratings transitions. I also use it to examine new published and unpublished data, testing for (and finding) ratings path-dependency, and otherwise exploring the effect of ratings changes on different bond sectors. I then turn to the question of whether it is possible to estimate bond-specific transition probabilities, and propose a way to do so. I combine these efforts into the specifications for a pricing model capable of answering the question: How much is it worth?

The expected yield to maturity on a defaultable obligation equals the nominal yield less expected default losses. However, in a mark-to-market world, one doesn't have the luxury of reporting one's performance on the basis of yield to maturity. Total return is calculated for an arbitrary holding period, and must reflect any mark-to-market gains or losses as at the close of the period-gains or losses that can be triggered by the bond's upgrade or downgrade. Thus to estimate expected total return, one must estimate not only expected default losses, but also the impact on capital price of expected ratings transitions. This paper begins with the observation that a bond which is blessed by more favourable transition characteristics is likely to produce a higher total return, and poses the question of how that benefit can be quantified. How much is it worth? To answer the question, I start by specifying a formal bond-pricing model reflective of ratings transitions. I survey various statistical methods and past research efforts to identify the ratings-transition matrix which best parametrises the model, and propose a novel test for selecting between competing matrices. Using this approach, I replicate several important studies of ratings transitions. I also use it to examine new published and unpublished data, testing for (and finding) ratings path-dependency, and otherwise exploring the effect of ratings changes on different bond sectors. I then turn to the question of whether it is possible to estimate bond-specific transition probabilities, and propose a way to do so. I combine these efforts into the specifications for a pricing model capable of answering the question: How much is it worth?

Coercivity is a characteristic property of the bilinear term in a weak form of a partial differential equation in both infinite space and the corresponding finite space utilized by a numerical scheme. This concept implies \textit{stability} and \textit{well-posedness} of the weak form in both the exact solution and the numerical solution. In fact, the loss of this property especially in finite dimension cases leads to instability of the numerical scheme. This phenomenon occurs in three major families of problems consisting of advection-diffusion equation with dominant advection term, elastic analysis of very thin beams, and associated plasticity and non-associated plasticity problems. There are two main paths to overcome the loss of coercivity, first manipulating and stabilizing a weak form to ensure that the discrete weak form is coercive, second using an automatically stable method to estimate the solution space such as the Discontinuous Petrov Galerkin (DPG) method in which the optimal test space is attained during the design of the method in such a way that the scheme keeps the coercivity inherently. In this dissertation, A stable numerical method for the aforementioned problems is proposed. A stabilized finite element method for the problem of migration risk problem which belongs to the family of the advection-diffusion problems is designed and thoroughly analyzed. Moreover, DPG method is exploited for a wide range of valuing option problems under the black-Scholes model including vanilla options, American options, Asian options, double knock barrier options where they all belong to family of advection-diffusion problem, and elastic analysis of Timoshenko beam theory. Besides, The problem of American option pricing, migration risk, and plasticity problems can be categorized as a free boundary value problem which has their extra complexity, and optimization theory and variational inequality are the main tools to study these families of the problems. Thus, an overview of the classic definition of variational inequalities and different tools and methods to study analytically and numerically this family of problems is provided and a novel adjoint sensitivity analysis of variational inequalities is proposed.

Master of Agribusiness / Department of Agricultural Economics / Allen M. Featherstone / Institutions of the Farm Credit System (FCS) focus on risk-based lending in accordance with regulatory direction. The rating of risk also assists retail staff in loan approval, risk-based pricing, and allowance decisions. FCS institutions have developed models to analyze financial and related customer information in determining qualitative and quantitative risk measures. The objective of this thesis is to examine empirical account data from 2006-2012 to review the probability of default (PD) rating methodology within the overall risk rating system implemented by a Farm Credit System association. This analysis provides insight into the effectiveness of this methodology in predicting the migration of accounts across the association’s currently-established PD ratings where negative migration may be an apparent precursor to actual loan default. The analysis indicates that average PD ratings hold relatively consistent over the years, though the distribution of the majority of PD ratings shifted to higher quality by two rating categories over the time period. Various regressions run in the analysis indicate that the debt to asset ratio is most consistently statistically significant in estimating future PD ratings. The current ratio appears to be superior to working capital to gross profit as a liquidity measure in predicting PD rating migration. Funded debt to EBITDA is more effective in predicting PD rating movement as a measure of earnings to debt than gross profit to total liabilities, although the change of these ratios over time appear to be weaker indicators of the change in PD rating potentially due to the variable nature of annual earnings of production agriculture operations due to commodity price volatility. The debt coverage ratio is important as it relates to future PD migration, though the same variability in commodity price volatility suggests the need implement multi-year averaging for calculation of earnings-based ratios. These ratios were important in predicting the PD rating of observations one year into the future for production agriculture operations. To further test the predictive ability of the PD ratings, similar regression analyses were completed comparing current year rating and ratios to future PD ratings beyond one year, specifically for three and five years. Results from these regression models indicate that current year PD rating and ratios are less effective in predicting future PD ratings beyond one year. Furthermore, because of the variation in regression results between the analyses completed for one, three and five years into the future, it is important to regularly capture ratio and rating information, at least annually.

為了研究總體因子與產業違約率之間的關聯性, 本文以信用投資組合觀點模型(CPV) 做為開端, 建立在具評等基礎下的違約損失模型, 並以投機等級違約率估計出移轉係數矩陣, 進而模擬各產業條件移轉矩陣, 藉以反應在各種不同總體情境下, 產業內各評等的移轉機率及違約機率。此外, 本文亦建立不分評等的簡化違約損失模型, 並將兩模型做一比較。最後, 我們以台灣537 家上市櫃公司做為投資組合樣本, 分別模擬出兩模型的條件違約損失分配。進一步計算風險指標,以此做為未來規劃資本計提的基礎。最後結果顯示, 投資組合違約情況確實受總體因子影響, 且發現若投資組合中評等越差公司之曝險越小, 將有助於降低組合資產風險。