Proof of the perpetuity equation

AbstractA perpetuity is a perpetual annuity. Although there have been a number of different derivations, which we discuss in detail, we present what appears to be the first mathematical proof of the perpetuity equation based on the fundamental properties of the real numbers (Result (2.2.1) of Dieudonne (1960) [14])

Should I endorse a third party? Authorization strategies for brand manufacturers in a refurbishing market

Original equipment manufacturers (OEM) may have little or no control over third-party (3P) refurbishing firms. With the rapid growth of the refurbished market for electronic products, we study whether it is beneficial for an OEM to cooperate with a 3P via authorization schemes that boost an OEM’s brand reputations, increase their sales, and strengthen consumer acceptance of authorized 3P’s refurbished products. We examine the conditions under which both the OEM and the 3P benefit from the authorization strategy, studying the trade-off between the indirect benefit of authorizing a 3P to increase market share and the downside of cannibalizing new-product sales. To estimate our model’s behavioral parameters, we conduct an extensive experiment on MTurk to capture consumer preferences and cannibalization effects. The experimental study examines the price-perceived quality relationship along with brand value, seller identity (OEM, 3P), and product condition; its results show that the discount and seller identity play a large role in consumer choice and that cannibalization is generally linear in price. We subsequently construct a revenue maximizing model that incorporates this linear cannibalization effect, along with the authorization fees. We show that refurbished products offered by authorized 3Ps have higher demand than those that are not authorized and that it is beneficial for 3Ps to participate in these schemes despite the authorization fees. We conclude that authorization can be a win–win strategy for OEMs and 3Ps, especially when low-end consumer demand and average reduction of refurbishing costs are relatively high, and the level of cannibalization is relatively low. To achieve win–win solutions, it is important for OEMs and 3Ps to consider brand recognition, consumer behavior related to refurbished products, and remanufacturable supply.</p

Dynamic portfolio optimization with Credit Default Swaps

This dissertation studies Merton\u27s optimal portfolio problem applied to an investor who trades in a Credit Default Swap (CDS). This work began under the guidance of Dr Capponi following his previous analysis Capponi and Figueroa [2011] with corporate bonds. We study the a portfolio optimization problem of a speculative investor allocating the wealth to an equity, a money market account and to a Credit Default Swap. This work is differentiated from Capponi and Figueroa [2011] since they analyze a portfolio of an equity, money market account and a zero-coupon corporate bond. The primary difference is our consideration of a CDS instead of a defaultable bond. Note that there is a significant difference in the reward structure of these two assets. The CDS has a continuous payment while the bond does not. Also, we consider the correlation between the CDS and the equity. Initially, we assume a market existing in a single regime and market coefficients are assumed to be constant. Under these assumptions, the dynamics of the CDS price are determined in terms of a stochastic differential equation. The utility maximization problem is solved using a Hamilton-Jacobi-Bellman equation and verification theorems are shown for its solution. Explicit optimal strategies are obtained for an investor with a logarithmic utility function. With data from the last 150 years, we see that the corporate bond market suffers repeated clustered defaults. Since we are considering CDS that are written on corporate bonds, we incorporate this market behavior by modeling the economy to exist in multiple regimes. The market coefficients are assumed to depend on the regime in place. This regime is modeled by a nite state continuous Markov process. The CDS price dynamics is determined in terms of a Markov modulated stochastic differential equation and the Utility maximization problem is solved in the same way as above. Explicit optimal strategies are obtained for a logarithmic investor in a market that exists in 3 regimes. In the last chapter, we extend the work done in Capponi and Figueroa [2011] to the situation where the assets are correlated with one another. Over the last 10 years, it has been observed that the credit markets and the equity markets are correlated. To incorporate this behavior, we model the default process as a doubly stochastic Poisson process where the default rate is a function of the stock price. The CDS price dynamics are determined as a function of the stock price and the utility maximization problem is solved in a similar fashion. Explicit strategies are obtained for a logarithmic investor. We see that the optimal investment in the CDS and stock are dependent on each other

Comparison study and empirical analysis on prices of collateralized debt obligations

This thesis attempts two things. Basically, I have used the three-factor model from [1] and performed analysis on a different data set. I have also modified this model and performed analysis on the same data set. The objective therefore is two-fold, the first being a comparison study of the two models, the second to analyze the information contained in the index spreads of collateralized debt obligations. My model performs better on three of the five data sets used. The analysis performed also gives us an insight into economic implications of the prices. On an average, 60% of the spread is due to firm-specific risk, 30% is due to industry-wide risk and 10% is due to economy-wide risk

Models for Point-of-Sale (POS) Market Entry

Point of Sale (PoS) or Buy-Now-Pay-Later (BNPL) type financing has observed a strong rise in the last decade, and accounts for about 10% of the unsecured lending market in the USA. This market is largely covered by FinTechs, and traditional financial institutions are beginning to demonstrate interest in entering this market. In this short communication, we identify why the POS market is attractive and what drives its growth, examine the market participants, and illustrate the various ways in which financial institutions can enter this market

Global Sourcing under Tariffs: Time Series Analysis of Product-Level Evidence

Global sourcing is a complex process to acquire products and services from international sources, and therefore is subject to various disruptions. This paper focuses on the potential disruptions arising from the large-scale tariffs during 2018-2019. Drawing on tariff implications from analytic models of global supply network design, we specifically examine the patterns of global sourcing and how tariffs could disrupt global supply chains by investigating time series of monthly sourced amounts. We draw 222 manufacturing firms from FactSet Shipping database, with 3,348,595 unique observations covering time period between January 2014 and December 2019. By aggregating sourcing amounts on each firm, and applying multivariate time series clustering algorithm, we identify seven unique clusters for these firms. We further examine the disruptive effects of these tariffs by using intervention analysis of time series for each cluster. We find that firms in each cluster increase their sourcing amounts before or during tariff time periods. Our results further show that while some firms have a significant disruptive effect, other firms still maintain pre-tariff sourcing behaviors. An additional analysis reveals that firm size, growth potential, and firm profitability are associated with firms’ ability to deal with disruptions. Overall, our results have important implications for global supply chain management

Revenue management in a refurbishing duopoly with cannibalization

Refurbishing has promising economic potential, yet firms remain wary of its potential cannibalization effect on new product sales. Firms need to make strategic choices in competitive settings, involving varying brand strengths and collection constraints. The current study provides an empirical characterization of consumer behavior in such complex settings, which in turn informs an analytical model of optimal pricing policies. The results show that cannibalization (or switching) in a competitive environment has a linear relationship with price discounts, unlike the inverted U-shaped relationship observed in monopolistic settings. The experimental results also highlight the relevance of cross-cannibalization, especially for weaker firms (with low brand value) that compete with stronger firms to sell refurbished products. Our subsequent analytical results reveal how both competitors should adjust their refurbished product prices when internal and cross-cannibalization coefficients change. Weaker firms should focus on surpassing stronger competitors in terms of collection and refurbishing efforts, which implies a predominant focus on operations. Finally, we show that refurbished product prices are a function of, and profits are concave in relation to, the number of core products collected by firms

Value of Intermediate Imaging in Adaptive Robust Radiotherapy Planning to Manage Radioresistance

In radiotherapy, uncertainties in tumor radioresistance and its progression can degrade the efficacy of deterministic treatments. While a robust methodology can overcome this, it often produces overly conservative or suboptimal decisions, especially when there are changes in time. We aim to develop an adaptive radiotherapy planning framework that can reduce over-conservatism yet remain robust to the uncertainties in radioresistance. Specifically, intermediate imaging is used to update the uncertainty at each stage and curb over-conservatism. While additional imaging reduces uncertainty, it accrues costs such as extra radiation to organs, which deters continuous imaging. We probe this trade-off in uncertainty and cost of observation by computing and comparing results from two-stage, three-stage, and four-stage robust models. The three robust models are also compared to two currently practiced deterministic methods, one that does not account for radioresistance and one that assumes a constant radioresistance. All five models are evaluated on a clinical prostate case. The three robust models improve control of the tumor compared to the deterministic model ignoring radioresistance, at comparable radiation dose to critical organs. The robust models also reduce tumor overdose and organ dose compared to the deterministic model assuming a constant radioresistance. Increasing the number of intermediate imaging leads to further improvements, especially on tumor dose criteria under best-case and nominal scenarios. Under the worst-case, intermediate images provide no additional benefit as robust optimization inherently protects against the worst-case. The proposed method is generic and can include additional sources of uncertainties that reduce the effect of radiation

Revenue management in a refurbishing duopoly with cannibalization

Refurbishing has promising economic potential, yet firms remain wary of its potential cannibalization effect on new product sales. Firms need to make strategic choices in competitive settings, involving varying brand strengths and collection constraints. The current study provides an empirical characterization of consumer behavior in such complex settings, which in turn informs an analytical model of optimal pricing policies. The results show that cannibalization (or switching) in a competitive environment has a linear relationship with price discounts, unlike the inverted U-shaped relationship observed in monopolistic settings. The experimental results also highlight the relevance of cross-cannibalization, especially for weaker firms (with low brand value) that compete with stronger firms to sell refurbished products. Our subsequent analytical results reveal how both competitors should adjust their refurbished product prices when internal and cross-cannibalization coefficients change. Weaker firms should focus on surpassing stronger competitors in terms of collection and refurbishing efforts, which implies a predominant focus on operations. Finally, we show that refurbished product prices are a function of, and profits are concave in relation to, the number of core products collected by firms

Valuing Switching Options with the Moving-Boundary Method

Switching options can be deployed in various complex switching problems such as tolling agreements and the offshoring–backshoring problem. Closed form solutions to valuing switching options are not only hard, but also computationally intensive when solving numerically. We develop a new computational method to value switching options based on the moving boundary method. We show how the free boundary problem arising from switching options can be converted into a sequence of fixed boundary problems. We formulate the problem, and solve the optimal switching problem in two regimes over a finite time horizon. We establish the theoretical guarantees for this method (maximum principles, uniqueness and convergence). We demonstrate this with a numerical example and show the sensitivity of the solution with respect to problem parameters