Valuation Equation Research Articles

A mortgage valuation model: a tale of two equations

ABSTRACT In the Journal of Financial Economics, Gerardi, Willen, and Zhang (2023) examine the relationship between prepayment rates, race, and monetary policy. In this article, we identify an anomaly in GWZ‘s valuation equation that relates to the effect of prepayment rates on the ratio of market price to initial loan balance. We then show the mathematical steps to correct this anomaly.

Risk Measures in Simulation-Based Business Valuation: Classification of Risk Measures in Risk Axiom Systems and Application in Valuation Practice

Simulation-based company valuations are based on an analysis of the risks in the company to be valued. This means that risk analysis is decisively important in a simulation-based business valuation. The link between risk measures, risk conception and risk axiom systems has not yet been sufficiently elaborated for simulation-based business valuations. The aim of this study was to determine which understanding of risk underlies simulation-based business valuations and how this can be implemented via suitable risk measures in simulation-based business valuations. The contribution of this study is providing guidance for the methodologically correct selection of appropriate risk measures. This will help with avoiding valuation errors. To this end, the findings were combined from risk axiom systems with the valuation equations of simulation-based business valuations. Only position-invariant risk measures are suitable for simulation-based business valuations.

An Analysis on Business Risk of Jingdong and Meituan

The development of the Internet has brought huge market development and expansion opportunities for e-commerce companies, especially the largest e-commerce platform companies in the market today. Therefore, many investors attach great importance to the investment of these firms. This article mainly uses the CAPM model together with other valuation equations to provide valuation data, and then combines realistic factors to evaluate the e-commerce platforms, represented by two B2C self-operated platform e-commerce companies with different service contents, JD and Meituan. Results shows that compared to Jingdong, Meituan has a lower business risk. We will try to elaborate the difference in business risk in terms of these three causes related to target customers: products and services, internal company decisions and the impact of real factors, i.e. COVID-19. Through the research on the two e-commerce companies’ risk, this article provides implications for investors to have a lucid investment understanding of distinct types of e-commerce platform companies, to make their investment alternatives scientifically.

Terminal value calculation with discontinuous financing and debt categories

Empirical analyses indicate that active and passive debt management have limited power to explain the financing behavior of firms. Therefore, discontinuous financing, as a combination of active and passive debt management, might be a more realistic financing strategy. However, the properties of this financing strategy for the steady state have not yet been sufficiently analyzed. For this reason, we investigate analytically terminal value calculation with discontinuous financing and derive adjustment formulas for the period-specific levered cost of equity. Since a single adjustment of the entire debt at the beginning of every planning phase might still not be close to the real financing behavior of firms, we modify discontinuous financing by introducing debt categories, which are adjusted successively and include the maturity of debt. For this new financing strategy, we derive valuation equations and an adjustment formula for the constant cost of equity. Finally, we discuss the relevance and applicability of discontinuous financing with debt categories and its impact on the market value of a firm.

INFLATION, CENTRAL BANK AND SHORT-TERM INTEREST RATES: A NEW MODEL WITH CALIBRATION TO MARKET DATA

In this paper, we propose a new model for the joint evolution of the inflation rate, the Central Bank official interest rate and the short-term interest rate. Our model takes into account the fact that the Central Bank interest rate changes at random times, inflation is measured at fixed, regular times, while the short-term interest rate evolves essentially continuously. We derive the valuation equation for a contingent claim and show that it has a unique solution. The payoff may depend on all three economic factors of the model and the discount factor is allowed to include inflation. Our model is not an affine model. Although in some special cases the solution of the valuation equation might admit a closed form, in general it has to be solved numerically. This can be done efficiently by the algorithm that we provide. Taking as a benchmark the model of [H. W. Ho, H. H. Huang & Y. Yildirim (2014) Affine model of inflation-indexed derivatives and inflation risk premium, European Journal of Operational Research 235, 159–169], we show that our model performs better on European market data from 2008 to 2015. Our model uses many fewer parameters than the benchmark model: This is advantageous from the numerical point of view and suggests that our model describes the behavior of the economic factors more closely.

Unjust and Unreasonable: Misuse of the Dividend Growth Model in Public Utility Rate Setting

The dividend (Gordon) growth model (DGM) is a simple stock valuation equation in four variables: (1) price; (2) next-period cash flow; (3) a perpetual and constant rate of growth in the periodic cash flow; and (4) a perpetually-constant discount rate. The DGM can be expressed to equate the perpetually-constant discount rate to the cash flow yield on the current price plus the perpetual growth rate in the periodic cash flow. But that expression would provide a reliable estimate of a real-world discount rate only if both the cash flow yield and the growth rate used as inputs to the model were perpetually constant. In rate-regulation proceedings, however, utility expert witnesses use the DGM with current (not perpetual) dividend yields and analysts' 3-5 year (not perpetual) earnings growth rates. This practice contributes to approved equity cost of capital awards above plausible cost of capital estimates, shifting wealth from rate payers to utilities.

Additive logistic processes in option pricing

In option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an ell ^{p} vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide parsimonious and simple option pricing models capturing various important stylised facts at the minimum price of a single market observable input.

How to value a cultural festival? The case of Petronio Álvarez Pacific Music Festival in Colombia

ABSTRACT The article identifies factors related to the local context that influence the valuation of a local, traditional and internationally recognized music festival. Analyzing the case of Petronio Álvarez Pacific Music Festival held in Cali (Colombia), a valuation equation of the festival is estimated via an ordered probit model based upon microdata provided by a face-to-face survey (N = 1257). Results show that there are two key factors shaping the valuation of the festival: (i) previous experience of attending the festival, (ii) perceptions and expectations individuals have over the implications of the festival celebration (even if not participating), that is, the territory’s externalities (the creation of income and employment for the city) and personal benefits (cultural enrichment and enjoyment of a well-executed festival’s plan). Finally, our work provides policymakers with an instrument that reduces uncertainty about the characteristics of the festival, which can also be used as a managerial decision instrument based on empirical evidence about individuals’ perceptions.

Valuation with mixed financing strategies

In corporate valuation, it is common to assume either passive or active debt management. However, it is questionable whether these pure financing policies reflect the real financing policies of firms with a sufficient degree of accuracy. This shortcoming has led to the development of mixed financing strategies as combinations of pure financing strategies. Whereas hybrid financing is directly linked to the two-phase model, it is unclear how to apply discontinuous financing in such a setting. In this study, according to the two versions of hybrid financing, we analyze the implementation of discontinuous financing in a two-phase model. Thereby, we present a simpler and more intuitive derivation of the valuation equation for discontinuous financing to increase its acceptance and its use for corporate valuation practice. Moreover, we compare the different mixed financing strategies with each other theoretically, and we conduct simulations to elucidate the impact on market values and the sensitivities of input parameters. The study concludes that the presented mixed financing strategies can help in the attempt to reflect the real financing behavior of firms more accurately and, therefore, constitute a valuable alternative to pure financing strategies for valuation.

Additive Logistic Processes in Option Pricing

In option pricing it is customary to specify first a stochastic underlying model and then extract from it valuation equations. Still, it is possible to reverse this paradigm: starting from an arbitrage-free valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper we start from two simple arbitrage-free valuation formulae, one supported by a real-valued spot price and inspired by the log-sum-exponential function, and the other supported on positive spot prices and given by an l^p vector norm. We observe that such formulae lead respectively to Standard Logistic and Dagum (or log-skew logistic'') risk-neutral distributions for the underlying, and proceed to exhibit supporting stochastic processes of additive type for a market underlying having as time marginals two such distributions. By construction, the underlying price processes analyzed produce closed-forms for option prices which are even simpler of those of the Bachelier and Black-Scholes models. Additionally a single time-normalized moneyness curve fully characterizes the option value. Logistic additive processes provide parsimonious and simple option pricing models capturing various important stylized facts such as returns kurtosis, skewness and self-similarity, at the minimum price of a single market observable input.

Two New Strategies for Pricing Freight Options by Means of a Valuation PDE and by Functional Bounds

Freight derivative prices have been modeled assuming that the spot freight follows a particular stochastic process in order to manage them, like freight futures, forwards and options. However, an explicit formula for pricing freight options is not known, not even for simple spot freight processes. This is partly due to the fact that there is no valuation equation for pricing freight options. In this paper, we deal with this problem from two independent points of view. On the one hand, we provide a novel theoretical framework for pricing these Asian-style options. In this way, we build a partial differential equation whose solution is the freight option price obtained from stochastic delay differential equations. On the other hand, we prove lower and upper bounds for those freight options which enables us to estimate the option price. In this work, we consider that the spot freight rate follows a general stochastic diffusion process without restrictions in the drift and volatility functions. Finally, using recent data from the Baltic Exchange, we compare the described bounds with the freight option prices.

Robust XVA

AbstractWe introduce an arbitrage‐free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a risky counterparty, and hedges credit risk by taking a position in defaultable bonds. The investor does not know the exact return rate of her counterparty's bond, but she knows it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs leads to important differences with respect to classical credit risk valuation. The value of the super‐replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the closeout value throughout the life of the transaction. Our comparative statics analysis indicates that credit contagion has a nonlinear effect on the replication strategies and on the XVA.

Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion

Recently, the valuation of variable annuity products has become a hot topic in actuarial science. In this paper, we use the Fourier cosine series expansion (COS) method to value the guaranteed minimum death benefit (GMDB) products. We first express the value of GMDB by the discounted density function approach, then we use the COS method to approximate the valuation Equations. When the distribution of the time-until-death random variable is approximated by a combination of exponential distributions and the price of the fund is modeled by an exponential Lévy process, explicit equations for the cosine coefficients are given. Some numerical experiments are also made to illustrate the efficiency of our method.

The Value of Government Debt

The market value of government debt equals the present discounted value of primary surpluses. Applying present value decompositions from asset pricing to this valuation equation, I find that half of the variation in the market value of debt to GDP ratio corresponds to varying forecasts of future primary surpluses, and half to varying discount rates. Variation in expected growth rates is unimportant.

Nonlinear valuation under credit, funding, and margins: Existence, uniqueness, invariance, and disentanglement

Since the 2008 global financial crisis, the banking industry has been using valuation adjustments to account for default risk and funding costs. These adjustments are computed separately and added together by practitioners as if the valuation equations were linear. This assumption is too strong and does not allow to model market features such as different borrowing and lending rates and replacement default closeout. Hence we argue that the full valuation equations are nonlinear, and this paper is devoted to studying the nonlinear valuation equations introduced in Pallavicini et al (2011).We illustrate all the cash flows exchanged by the parties involved in a derivative contract, in presence of default risk, collateralisation with re-hypothecation and funding costs. Then we show how to obtain semi-linear PDEs or Forward Backward Stochastic Differential Equations (FBSDEs) from present-valuing said cash flows in an arbitrage-free setup, and we study the well-posedness of these PDEs and FBSDEs in a viscosity and classical sense.Moreover, from a financial perspective, we discuss cases where classical valuation adjustments (XVA) can be disentangled. We show how funding costs are offset by treasury valuation adjustments when one takes a whole-bank perspective in the valuation, while the same costs are not offset by such adjustments when taking a shareholder perspective. We show that although we use a risk-neutral valuation framework based on a locally risk-free bank account, our final valuation equations do not depend on the risk-free rate. Finally, we show how to consistently derive a netting set valuation from a portfolio level one.

Applications of the Replication Method for XVAs

Following in using zero-cost trade-able hedge instruments, we show that many forms of the valuation equations for PV and XVAs presented in the literature can be seen as particular cases of a more generic Replication framework. The convenient split between the effects of hedging our own default risk and funding obtained from the assumption of hedging with repo-able bonds allows for a systematic classification of the valuation equations according to their funding strategies. We describe strategies justifying for instance asymmetric funding in linear valuation by using three funding bonds. The Funding Invariance principle (Elouerkhaoui, 2016) i.e. the invariance against the choice of risk-free rate, is seen as a natural consequence of the Replication applied to the total risky derivative and is valid for all valuations derived from it. GVA, the adjustment when PV is discounted differently from the close-out amount, is crucial for the Funding Invariance to hold and allows a company to choose its PV discount rate with flexibility. Other applications of the Replication method include avoidance of double-counting and emergence of DVA2 and its counterpart CVA2.

The fiscal theory of the price level in a world of low interest rates

A central equation for the fiscal theory of the price level (FTPL) is the government budget constraint (or “government valuation equation”), which equates the real value of government debt to the present value of fiscal surpluses. In the past decade, the governments of most developed economies have paid very low interest rates, and there are many other periods in the past in which this has been the case. In this paper, we revisit the implications of the FTPL in a world where the rate of return on government debt may be below the growth rate of the economy, considering different sources for the low returns: dynamic inefficiency, the liquidity premium of government debt, or its favorable risk profile.This article is part of a Special Issue entitled ``Fiscal and Monetary Policies''.

Robust XVA

We introduce an arbitrage-free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a risky counterparty, and hedges credit risk by taking a position in defaultable bonds. The investor does not know the return rate of her counterparty's bond, but is confident that it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs leads to important differences with respect to classical credit risk valuation. The value of the super-replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the close-out value throughout the life of the transaction. Our comparative statics analysis indicates that credit contagion has a nonlinear effect on the replication strategies and on the XVA.

혼합잉여의 유형별 가치관련성에 관한 연구

This study investigates whether firm`s share prices correctly reflect two types of dirty surplus items in Korean capital market. The first of these is commonly referred to as “other comprehensive income,” which is a component of comprehensive income that is excluded from reported earnings, and therefore violates clean surplus accounting. The second accounting measure we consider is “hidden dirty surplus”, which arises when a firm issues or reacquires its own shares in a transaction that does not record the shares at fair market value. An Example this kind of transaction is a conversion of a bond into common stock at other than fair market value. Unlike other comprehensive income, which is readily observable from the comprehensive income statement under K-IFRS, hidden dirty surplus is unobservable. Therefore, it is very difficult for investor to estimate the valuation impact of equity transactions that give rise to hidden dirty surplus. As a result, hidden dirty surplus is less likely to be correctly priced than are other comprehensive income. Specifically, we compute other comprehensive income(OCI) as sum (1) the change in value of available-for-sale securities. (2) the foreign exchange gains and losses arising from translations of financial, (3) the effective portion of gains and losses on hedging instruments in a cash flow hedge, (4) the change in value of property, and (5) the actuarial gains and losses on defined benefit plans. Based on Landsman et al. (2011), we compute hidden dirty surplus(HDS) as the change in the book value of common equity, less OCI, less net income, plus dividends, less share price at middle of fiscal year times change in common shares outstanding. To achieve the afore-mentioned research objectives, we first assess whether investors properly value each of these two types of dirty surplus by estimating a residual income forecasting equation and an attendant valuation equation that includes both of these components, using financial statement and stock price data from 2011-2014. We next employ Carhart`s (1997) four-factor market model in order to see market-based financial performance measured by stock return incorporates these two components of dirty surplus. Using the four factor model, we analyze whether investors can obtain abnormal returns by holding long position in firms with relatively large OCI or HDS and short position in firms with relatively small OCI or HDS. The analysis yields following key findings. First, other comprehensive income are irrelevant for forecasting abnormal very comprehensive income in next period and setting stock prices. Second, hidden dirty surplus is also irrelevant for forecasting abnormal very comprehensive income in next period but investors appear to undervalue the HDS component of income, i. e., over-value equity. Third, hedge returns are insignificant when the sample firms are ranked based on the other comprehensive income, but it is positive and significant when the HDS is used to rank the sample firms. These results are suggest that investors fail to understand the lack of persistence of hidden dirty surplus, and therefore apparently over-valuing firms that have hidden dirty surplus.

CAPM with various utility functions: Theoretical developments and application to international data

This paper presents an extension of the Capital Assets Pricing Model (hereafter CAPM) where various utility functions are applied. Specifically, we propose an overall CAPM beta that accounts for the higher order moments and reflects the investor preferences and attitudes toward risk. We particularly develop CAPM betas for different classes of utility function: the negative exponential utility function, power utility function or “Constant Relative Risk Aversion (CRRA) Utilities” and hyperbolic utility function or “HARA Utilities” (hyperbolic absolute risk aversion). In order to validate our theoretical results, we analyze the impact of investors’ preferences on the valuation equation. Applying the International CAPM, the results indicate that our utilities-based betas differ largely from the traditional CAPM betas. Moreover, the results confirm the importance of higher order moments on the pricing equation. Finally, the results both empirically and theoretically post to the consistent effect of the risk av...