Valuation Equation Research Articles

Government Maturity Structure Twists

This paper examines the impact of government debt maturity restructuring on ination and the real economy using a New Keynesian model that features a time-varying maturity structure of nominal debt and allows for changes in the monetary/scal policy mix. The irrelevance of open market operations changing the duration of government liabilities (holding the total market value of debt constant) is violated when the slope of the yield curve is nonzero in a scally-led policy regime. When the yield curve is downward-sloping, shortening the maturity structure increases the government discount rate, which generates scal ination and an expansion in output. The opposite results obtain when the yield curve is upwardsloping. Conditional maturity restructuring policies depending on the slope of the yield curve can smooth macroeconomic uctuations and oer substantial welfare benets. In short, this paper highlights the importance of bond risk premia, in conjunction with the government debt valuation equation, as a transmission channel for open market operations.

The Implied Growth Rate in the Valuation of JSE Listed Companies

Valuation is the heart of finance and investment decision making. Shareholders, analysts and company officers all need to be able to measure the worth of a company if they are to make investment decisions or alternatively create value within the company. Although several methods exist to value companies, the discounted cash flow (DCF) model is (arguably) both the most theoretically sophisticated as well as the most widely used approach. Its theoretical elegance lies in the fact that it looks to future cash flows which are anticipated from the assets, and discounts these, using simple time value of money principles, to produce a net present value of the business enterprise. Its accuracy depends on the ability of analysts to correctly forecast future cash flows, and in particular, to estimate the growth rate of these in perpetuity. We reverse-engineer the DCF valuation equation to calculate the market’s implied terminal growth rate (ITGR). To do so, we use historical financial statements and market capitalisations of the top 160 Johannesburg Stock Exchange (JSE) listed companies between 1980 and 2015. By constructing four non-overlapping five year periods, we are able to calculate the ITGR by equating free cash flows to the market value of equity for each company. Because the results vary widely, we use medians to estimate an ITGR of between 10% pa and 16% pa. We triangulate our results using other approaches to the problem, and conclude that the values typically shown in textbooks and used by practitioners are too low. We also report that higher values of the ITGR should be employed for more established companies.

Invariance, Existence and Uniqueness of Solutions of Nonlinear Valuation PDEs and FBSDEs Inclusive of Credit Risk, Collateral and Funding Costs

We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011). These equations take the form of semi-linear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions allowing us to extend valuation to credit risk and default closeout, including collateral margining with possible re-hypothecation, and treasury funding costs, we show how such cash flows, when present-valued in an arbitrage free setting, lead to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence and uniqueness of such solutions in a viscosity and classical sense, discussing the role of the hedging strategy. We show an invariance theorem stating that even though we start from a risk-neutral valuation approach based on a locally risk-free bank account growing at a risk-free rate, our final valuation equations do not depend on the risk free rate. Indeed, our final semi-linear PDE or FBSDEs and their classical or viscosity solutions depend only on contractual, market or treasury rates and we do not need to proxy the risk free rate with a real market rate, since it acts as an instrumental variable. The equations derivations, their numerical solutions, the related XVA valuation adjustments with their overlap, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works.

The value relevance and information content of cash and stock dividends in China

Due to their unique institutional features, the Chinese stock markets provide an interesting experimental setting in which to examine how cash and stock dividends convey value-relevant information about future performance and future cash dividends. Applying market valuation equations and prediction models to a large sample of Chinese firms from 2003 to 2011 produces several interesting results. When cash dividends are not paid, then stock dividends are positively associated with firm value, and convey information about future cash dividends and future earnings. When cash dividends are paid, however, it is these that convey information about future cash dividends. Following the share tradability reform and convergence with International Financial Reporting Standards, the value relevance and information content of cash dividends are predicted and found to be stronger, while the value relevance of stock dividends weakens over the same time period. Our evidence suggests that, in a weak information environment, where management have limited control over cash dividend distributions, stock dividends play an important role as information conduits.

Option-Implied Term Structures

The illiquidity of long-maturity options has made it difficult to study the term structures of option spanning portfolios. This paper proposes a new estimation and inference framework for these option-implied term structures that addresses long-maturity illiquidity. By building a sieve estimator around the risk-neutral valuation equation, the framework theoretically justifies (fat-tailed) extrapolations beyond truncated strikes and between observed maturities while remaining nonparametric. New confidence intervals quantify the term structure estimation error. The framework is applied to estimating the term structure of the variance risk premium and finds that a short-run component dominates market excess return predictability.

A Fast Monte Carlo Algorithm for Estimating Value at Risk and Expected Shortfall

Risk management today focuses heavily on estimating the location and conditional expectation of the left tail of the probability distribution for returns or portfolio value. The Holy Grail in derivatives pricing is a closed-form valuation equation such as in the Black–Scholes model, which takes a small number of input parameters and produces the exact arbitrage-free properties of the target portfolio, including value-at-risk (VaR) and expected shortfall (ES). But closed-form solutions are rare and largely limited to highly idealized markets. Lattice-based approximation techniques are available for more general settings, but they also have serious constraints. When all else fails, there is Monte Carlo simulation. Simulation always works, in principle, but the amount of calculation required in practice can be tremendous, which provides a strong incentive to find ways to speed up the process. Antithetic variates, control variates, and importance sampling are all helpful. In this article, the authors propose a new technique for estimating VaR and ES that is simple but remarkably powerful. Their first step is to determine which underlying risk factor is the most important. Next, for each simulated value of this primary factor, they simulate values for the remaining factors, requiring that every path generated exceed the VaR threshold. By not computing numerous paths that do not end up in the tail, the procedure can achieve the same accuracy as standard Monte Carlo simulation but several orders of magnitude faster.

Pricing Composite and Quanto Derivatives under Stochastic Correlation and Stochastic Volatility

There is growing trade in options whose payoffs are a function of multiple stochastic variables. One common type is an option on an asset denominated in one currency with a payoff in a different currency. Two closely related structures of this kind are quantos and composite options. Pricing formulas for European payoff contracts are available under Black–Scholes assumptions, but they require the assumption of constant volatility and correlations, neither of which is supported empirically. Romo develops a model of time-varying volatilities and correlations under a multiasset Wishart process, which is solved using Fourier and Laplace transform methods to produce semi-closed-form valuation equations. One structure especially exposed to stochastic correlation risk is a forward-starting quanto, which is easily priced in this framework. <b>TOPICS:</b>Options, risk management

Barrier Caps and Floors under the LIBOR Market Model with Double Exponential Jumps

The LIBOR market model (LMM) has become the standard model for many kinds of interest rate derivatives, such as cap contracts. It assumes that the distribution of the one-period interest rate at each future repricing date is lognormal, so that every caplet can be easily priced using the Black model. Volatilities at different dates are tied together by assumptions that restrict the number of parameters to a manageable set. But interest rates are strongly affected by things like monetary policy decisions, which can lead to sharp, nondiffusive jumps when policy changes. The effect may be seen in the leptokurtosis of the empirical distributions of interest rates and in volatility smiles for caplets. The low probability of extreme rate moves in the standard LMM is especially problematical for barrier options. In this article, the authors introduce a new jump-diffusion process for the LLM that can capture the effects of jumps on interest rate barrier contracts. The authors allow exponential jumps, both up and down, so the model is called the LMM with a double-jump process, or LMMDJ. Explicit valuation equations are provided for the four classes of barrier contracts and are shown to be quite accurate in Monte Carlo simulations. <b>TOPICS:</b>Derivatives, quantitative methods

The Value Relevance and Information Content of Cash and Stock Dividends in China

The Chinese stock markets provide a unique experimental setting in which to examine how cash and stock dividends convey value relevant information about future performance and future cash dividends. Utilising market valuation equations based on Ohlson (1995), as well as prediction models, and applying them to a large sample of Chinese firms from 2003-2011, we find the following. First, stock dividends are positively associated with firm value, and convey information about future cash dividends and future earnings in the absence of cash dividends. Second, when cash dividends are paid, it is these that convey information about future cash dividends. Third, the value relevance and information content of cash dividends is stronger following the split share reforms and the adoption of IFRS, while the value relevance of stock dividends weakens over the same time period. Our evidence suggests that, in China, where the information environment is not as well-developed as in Western stock exchanges, stock dividends play an important role where cash dividends are not informative and this may help to explain their prevalence.

A Supply Approach to Valuation

A new methodology for equity valuation arises from the perspective of managers' supply of capital assets. Under q-theory, managers optimally adjust the supply of assets to changes in their market value. The first-order condition of investment then provides a valuation equation that infers asset prices from managers' costs of supplying the assets. This equation fits well the Tobin's q levels across many testing assets, including portfolios formed on q. With current investment-to-capital as the only input, the supply approach does not require cash flow forecasts or discount rate estimates, both of which are notoriously difficult to obtain in practice. The Author 2013. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com., Oxford University Press.

Formal Identification of Sentiment Effects in Asset Markets

It is generally accepted that excessive exuberance or gloom in investor sentiment contributes to booms and crashes in asset prices but, because of its complex interaction with other aspects of the valuation process, these effects are not easy to identify with statistical confidence and this limits the scope for crafting an adequate and early policy response. To fill this gap, we develop an ex ante valuation approach that assigns different measures of sentiment to separate roles in the valuation equation. One measure of sentiment is assigned to capture risk aversion effects, while a broader-based investor sentiment index is assigned to capture changes in the perceived prospects for long-term earnings growth. The ratio of equity valuation when sentiment variables are included in the valuation exercise to the valuation when they are excluded is an obvious indicator of sentiment effects but this is difficult to assess statistically. We show that the ratio of average squared ‘implied’ long-run earnings growth ‘with’ and ‘without’ sentiment produces a sentiment indicator that can be assessed with statistical significance. Out-of-sample testing using the Dow 30 index shows that sentiment effects can often be confidently identified as widespread, significant and large. We find that the growth ratio is more prescient as an early warning indicator of mis-valuations. Being able to draw attention to such statistically verifiable arbitrage opportunities in a timely fashion offers macro-prudential policy makers a more targeted policy response than making alterations to poorly focused policy instruments, such as interest rates.

Valuation Equations for Stochastic Volatility Models

We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of the state space. We allow for various types of model behavior: the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and the asset-price process may be a strict local martingale. Our main result is a necessary and sufficient condition on the uniqueness of classical solutions to the valuation equation: the value function is the unique nonnegative classical solution to the valuation equation among functions with at most linear growth if and only if the asset price is a martingale.

Singular risk-neutral valuation equations

Many risk-neutral pricing problems proposed in the finance literature do not admit closed-form expressions and have to be dealt with by solving the corresponding partial integro-differential equation. Often, these PIDEs have singular diffusion matrices and coefficients that are not Lipschitz-continuous up to the boundary. In addition, in general, boundary conditions are not specified. In this paper, we prove existence and uniqueness of (continuous) viscosity solutions for linear PIDEs with all the above features, under a Lyapunov-type condition. Our results apply to European and Asian option pricing, in jump-diffusion stochastic volatility and path-dependent volatility models. We verify our Lyapunov-type condition in several examples, including the arithmetic Asian option in the Heston model.

Executive remuneration, and the principles of the Financial Stability Forum of 2009

PurposeAs a consequence of the global financial crisis and widespread disquiet over executive bonuses and other remuneration, in April 2009 the Financial Stability Forum enunciated principles for sound compensation as part of an effort to ensure the effective governance of compensation. The core problem this article seeks to address is the measurement of the contribution of corporate executives to the intrinsic value of the firm as part of an initial step in the process of implementing the Financial Stability Forum's principles. Unless the contribution of corporate executives can be measured in a manner that satisfies the requirements of sound research methodology, rigorous epistemology, and statutory requirements, it is doubtful whether these principles can be operationalized. Thus, the purpose of this paper is to show how the contribution of corporate executives can be estimated from audited financial statements. From the core problem and purpose of this article, its title is drawn.Design/methodology/approachRelevant sections of the report of the Financial Stability Forum 2009, the UK Corporate Governance Code of 2010, and the UK Companies Act of 2006, in conjunction with important reviews such as the Turner Review of 2009 and the Walker Review of 2009 were studied. Data inputs from audited financial statements were applied to appropriate well‐established non‐controversial valuation equations that are based on “first‐principles” of corporate finance, and the contribution of corporate executives to the intrinsic value of the firm was estimated in order to illustrate the validity of this approach.FindingsThe paper shows that contribution to the intrinsic value of the firm made by corporate executives can be measured in a non‐controversial and transparent way, and once done, can form the basis for quantifying executive remuneration on the basis of valued‐added. No attempt is made to address the fractional share of value‐added that should be placed in a bonus‐pool.Originality/valueFrom an extensive survey of publicly available literature, there is no evidence to suggest that the measurement of the contribution of corporate executives to the intrinsic value of the firm, as part of an initial step in the process of implementing the principles of the Financial Stability Forum 2009, has yet been published.

IT assets, organization capital and market power: Contributions to business value

How much IT capital contributes to the economic value of firms is a relevant but complex issue, since the contribution may come from different sources that are difficult to separate. In this paper, we model the determinants of the economic value of multi-asset firms with market power when the adjustment costs of investing in IT capital turn into organization capital, increasing the future cash flow of the firm. The resulting valuation equation, new in the literature, has four simultaneous sources of economic value: (i) purchase costs of the assets, (ii) adjustment costs, (iii) organization capital, and (iv) rents from market power. The model is tested with a unique data base from Spanish banks in a time period when these banks invested heavily in IT capital. We find that 54% of the economic value of the representative bank corresponds to the purchase cost of material and immaterial assets, including IT capital. The remaining 46% corresponds to the contributions of: adjustment costs (17%), organization capital (7%) and rents from market power (22%).

Valuation of Two-Factor Interest Rate Contingent Claims Using Green's Theorem

Over the years a number of two-factor interest rate models have been proposed that have formed the basis for the valuation of interest rate contingent claims. This valuation equation often takes the form of a partial differential equation that is solved using the finite difference approach. In the case of two-factor models this has resulted in solving two second-order partial derivatives leading to boundary errors, as well as numerous first-order derivatives. In this article we demonstrate that using Green's theorem, second-order derivatives can be reduced to first-order derivatives that can be easily discretized; consequently, two-factor partial differential equations are easier to discretize than one-factor partial differential equations. We illustrate our approach by applying it to value contingent claims based on the two-factor CIR model. We provide numerical examples that illustrate that our approach shows excellent agreement with analytical prices and the popular Crank–Nicolson method.

Underwriting in Property-Casualty Insurance Markets: Regulation, Risk and Volatility

We offer a novel explanation of underwriting volatility in property-liability insurance markets in terms of private uncertainty over public regulatory policy. Underwriting involving random losses to policyholders is one source of risk to the equity value of insurance firms. Solvency regulations, however, pose a second source of risk to equity value when the implementation of such regulations randomly affects the return to underwriting but exhibits imperfect correlation with market conditions over time. Using a differential game in a standard no-arbitrage environment to model interaction between these two sources of risk, we derive the valuation equation for property-liability underwriting inclusive of the respective best-reply underwriting strategy of a representative insurance firm and the implementation strategy of a representative regulator. Owing to the conflicting effects of these strategies on firm equity, regulations adopted to reduce the solvency risk of insurers can, paradoxically, increase underwriting volatility in an otherwise efficient insurance market. Under certain parametric conditions, we also show that, even in the presence of complete financial markets, a subgame perfect Nash equilibrium of this game can exhibit a limit cycle in the value of underwriting which mirrors empirical evidence on the presence of cycles in property-liability insurance markets.

A direct test of hyperbolic discounting using market asset data

This paper introduces a framework that generalizes exponential discounting in a net present value model by including a quasi-hyperbolic discount parameter in the asset valuation equation. Using observed market asset data, a statistically significant quasi-hyperbolic parameter is obtained, thus rejecting exponential discounting.

The Valuation Differences Between Stock Option and Restricted Stock Grants for US Firms

Abstract: In this study, we document a significant shift over the past several years from stock option‐based compensation to restricted stock‐based compensation. Additionally, we evaluate whether stock option grants and restricted stock grants result in similar valuation consequences for firms. We estimate cross‐sectional valuation equations that include the value of stock option and restricted stock grants summed over the current and past two years, residual income, and book value of equity, after controlling for endogeneity. Consistent with prior research, our findings indicate that the market on average values stock option grants positively. However, in contrast to stock option grants, restricted stock grants are valued negatively. This result is consistent with restricted stock grants lacking the positive incentive effects of stock options and being viewed as a liability or expense to the firm.

A Test Of The M&amp;M Capital Structure Theories

&lt;p class="MsoNormal" style="text-align: justify; margin: 0in 0.5in 0pt; mso-pagination: none;"&gt;&lt;span style="font-size: 10pt;"&gt;&lt;span style="font-family: Times New Roman;"&gt;Modigliani and Miller (1958, 1963) predict two very specific relationships between firm value and the amount of debt in the firm&amp;rsquo;s capital structure, depending on whether or not the firm pays corporate income taxes.&lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp; &lt;/span&gt;Although there has been some testing of the implications of these equations, I have been unable to find anyone who has tested them in the exact forms specified by MM.&lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp; &lt;/span&gt;In this paper, the MM equations are tested exactly as specified.&lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp; &lt;/span&gt;The results of these tests indicate that neither the MM tax nor the no-tax valuation equations are accurate predictors of firm value.&lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp; &lt;/span&gt;Specifically, the value of the unlevered firm accounts for much less of firm value than predicted and the sign of the coefficient of the interest tax shield variable is negative, instead of positive as MM predict.&lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp; &lt;/span&gt;These results are robust to the inclusion of control variables for firm growth, earnings variability, and the existence of other tax shields.&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;