Riskless Interest Rate Research Articles

Arbitrage Pricing of Russian Options and Perpetual Lookback Options

Let $X = \{X_t,t \geq 0\}$ be the price process for a stock, with $X_0 = x > 0$. Given a constant $s \geq x$, let $S_t = \max\{s,\sup_{0\leq u \leq t} X_u\}$. Following the terminology of Shepp and Shiryaev, we consider a "Russian option," which pays $S_\tau$ dollars to its owner at whatever stopping time $\tau \in \lbrack 0,\infty)$ the owner may select. As in the option pricing theory of Black and Scholes, we assume a frictionless market model in which the stock price process $X$ is a geometric Brownian motion and investors can either borrow or lend at a known riskless interest rate $r > 0$. The stock pays dividends continuously at the rate $\delta X_t$, where $\delta \geq 0$. Building on the optimal stopping analysis of Shepp and Shiryaev, we use arbitrage arguments to derive a rational economic value for the Russian option. That value is finite when the dividend payout rate $\delta$ is strictly positive, but is infinite when $\delta = 0$. Finally, the analysis is extended to perpetual lookback options. The problems discussed here are rather exotic, involving infinite horizons, discretionary times of exercise and path-dependent payouts. They are also perfectly concrete, which allows an explicit, constructive treatment. Thus, although no new theory is developed, the paper may serve as a useful tutorial on option pricing concepts.

Consols In the Cir Model

A consol is a default‐free financial instrument paying a constant stream of one unit of money. A synonym is a perpetuity. the valuation of a consol presents a particular difficulty: the time horizon of this instrument is infinity, and hence the usual technique of replacing the physical probability measure by a new probability measure represents serious problems with regard to absolute continuity of the two measures. We will work out explicit formulas when the instantaneous riskless interest rate follows a square‐root process under the risk‐free measure. Several mathematical properties will be investigated. Yor and Geman and Yor have considered the problem of pricing consols and carry out a more fundamental analysis (see References). This paper is self‐contained and emphasizes properties or techniques not covered by those authors.

Sovereign Debt: Optimal Contract, Underinvestment, and Forgiveness

ABSTRACTIn this paper we develop a time consistent rational expectations model which analyzes the equilibrium loan contract between a borrowing country and a foreign bank. The loan contract specifies both the amount of the loan and the promised interest payments, and rationally reflects the investment decisions of the country and the possibilities of renegotiation and repudiation of the debt. An important feature of the model is that at the initial negotiation of the loan there is uncertainty about whether the country will renegotiate for partial forgiveness in the future, and whether it will eventually repudiate the debt, even having successfully renegotiated. Moreover, the probabilities of renegotiation and repudiation, and the amount of possible forgiveness are endogenously determined. In the model the repudiation decision is directly related to the underinvestment problem; the objective of the renegotiation is precisely to alleviate this problem. The model is used to analyze the effects of four variables on both the optimal contract and the country's welfare: the degree of penalties that a bank can impose on a defaulting country, the uncertainty of production, the productivity of investments and the riskless interest rate. The analysis has policy implications as well as testable predictions.

The money and bond markets in France: Segmentation vs. integration

When rates of return on bonds are computed over extremely short holding periods, the ex post cross-sectional relationship between realized return and risk is linear. It is therefore possible, at any time, to extrapolate the cross-sectional relationship to a zero risk level, and thus to determine the implied instantaneous riskless rate of interest. We apply this technique to French bond price data. Using a rather unique data set in which prices are sampled daily, we are able to compare the overnight rate implied in bond price data to the actual overnight money market rate. We conclude that the two rates are significantly different, which is evidence of segmentation between the two markets. The institutional set-up prevailing in France during the sample period explains the segmentation result.

Dynamic firm behavior within an uncertain environment

In this paper we study the impact of an uncertain environment on the optimal dynamic investment policies of a value maximizing firm. We present a model in which risk-averse behavior of the shareholders is incorporated. We derive the policies that can be optimal for the firm and present solutions under different scenarios. After incorporating a dynamic version of the Capital Asset Pricing Model we can derive a new formula for the shareholders' time preference rate, which consists of the riskless interest rate and a risk premium.

Habit Formation: A Resolution of the Equity Premium Puzzle

The equity premium puzzle, identified by Mehra and Prescott, states that, for plausible values of the risk aversion coefficient, the difference of the expected rate of return on the stock market and the riskless rate of interest is too large, given the observed small variance of the growth rate in per capita consumption. The puzzle is resolved in the context of an economy with rational expectations once the time separability of von Neumann-Morgenstern preferences is relaxed to allow for adjacent complementarity in consumption, a property known as habit persistence. Essentially habit persistence drives a wedge between the relative risk aversion of the representative agent and the intertemporal elasticity of substitution in consumption.

Great and Almost-Great Magnitudes in Economics

Which ratios and key magnitudes should an educated economist know? An inevitably personal selection is offered below. The list is presented in the form of a quiz to liven up the exercise of pondering them, and therefore a question mark follows each title. 1) The ratio of savings (personal and total) to income? 2) Labor's share of output? 3) The capital-output ratio? 4) Ratio of the value of output of agricultural land to its market price? 5) The “real” riskless rate of interest in various eras? 6) The rate of return on common stocks? 7) The price-earnings ratio for common stocks? 8) The velocity of money? 9) The shares of the labor force in the farming, manufacturing, and service sectors, now and in earlier times? For those of you whose eyes share with mine the sinful propensity of straying to the “answer” before you have formulated your own, the ratios are partially concealed in paragraphs of explanation.

The Consumption-Based Capital Asset Pricing Model

The paper provides conditions on the primitives of a continuous-time economy under which there exist equilibria obeying the Consumption-Based Capital Asset Pricing Model (CCAPM). The paper also extends the equilibrium characterization of interest rates of Cox, Ingersoll, and Ross (1985) to multi-agent economies. We do not use a Markovian state assumption. THIS WORK PROVIDES sufficient conditions on agents' primitives for the validity of the Consumption-Based Capital Asset Pricing Model (CCAPM) of Breeden (1979). As a necessary condition, Breeden showed that in a continuous-time equilibrium satisfying certain regularity conditions, one can characterize returns on securities as follows. The expected rate of return on any security in excess of the riskless interest rate (the security's expected excess rate of return) is a multiple common to all securities of the instantaneous covariance of this excess return with aggregate consumption increments. This common multiple is the Arrow-Pratt measure of risk aversion of a representative agent. (Rubinstein (1976) published a discrete-time precursor of this result.) The exis- tence of equilibria satisfying Breeden's regularity conditions had been an open issue. We also show that the validity of the CCAPM does not depend on Breeden's assumption of Markov state information, and present a general asset pricing model extending the results of Cox, Ingersoll, and Ross (1985) as well as the discrete-time results of Rubinstein (1976) and Lucas (1978) to a multi-agent environment. Since the CCAPM was first proposed, much effort has been directed at finding sufficient conditions on the model primitives: the given assets, the agents' preferences, the agents' consumption endowments, and (in a production econ- omy) the feasible production sets. Conditions sufficient for the existence of continuous-time equilibria were shown in Duffie (1986), but the equilibria demonstrated were not shown to satisfy the additional regularity required for the CCAPM. In particular, Breeden assumed that all agents choose pointwise interior consumption rates, in order to characterize asset prices via the first order conditions of the Bellman equation. Interiority was also assumed by Huang (1987) in demonstrating a representative agent characterization of equilibrium, an approach exploited here. The use of dynamic programming and the Bellman equation, aside from the difficulty it imposes in verifying the existence of interior 1 Financial support from the National Science Foundation is gratefully acknowledged. We thank

Real interest rates and the cost of capital: A comparison of the United States and Japan

Previous studies have reported a lower cost of capital in Japan than in the United States and attributed the difference to lower real interest rates in Japan. This paper argues that the approach employed in previous studies is biased in favor of the hypothesis that Japanese capital costs are lower. The paper employs an alternative approach. It presents rational expectations tests of the hypothesis that real riskless interest rates are equal across the two countries. The hypothesis that real riskless rates are equal is rejected by data from the period 1976–1985. Although real riskless rates are not equal, the difference favored the U.S. as often as Japan, and by equal orders of magnitude during the decade 1976–1985. The results therefore do not support the nation that Japan has consistently benefitted from lower costs of capital over the decade. Forward parity and ex ante purchasing power parity are also rejected by the data, though these rejections need not imply unexploited financial market opportunities and therefore cannot be said to have caused the differences in real interest rates.

An analysis of intertemporal and cross-sectional determinants of earnings response coefficients

Stock pride change associated with a given unexpected earnings change (the earnings response coefficient) exhibits cross-sectional and temporal variation. We predict and document evidence that the earnings response coefficient is a function of riskless interest rates and the riskiness, growth and/or persistence of earnings. The earnings response coefficient also varies cross-sectionally with the holding period return interval. Collectively, our results explain the previously reported differential earnings response coefficient with respect to size. Moreover, by including the factors noted above, the empirical specification of the earnings/returns relation is significantly improved.

Arbitrage and Diversification in a General Equilibrium Asset Economy

This paper presents a theory of equilibrium asset pricing that generalizes the recent work of Connor. The model extends Connor's results to more general sets of asset returns and consumer preferences, introduces production, and provides a framework for analyzing exact and approximate equilibrium asset pricing. The other major contribution of the paper is the introduction of geometric arguments that exploit the properties of induced prefer- ences over assets. This method of analyzing asset pricing provides an intuitively appealing way of analyzing equilibrium asset pricing theories. unifies several well-known asset-pricing models. This paper extends Connor's finite-asset model in a number of directions. These extensions are nontrivial, because they allow the theory to encompass asset-pricing models that were excluded by Connor's assumptions. The second major contribution of the paper is to introduce a method of argument that exploits the geometric properties of induced preferences over assets, and characterizes asset prices using these proper- ties. This technique provides an intuitively appealing way of analyzing asset pricing theories. (For a simple geometric exposition of the main ideas in this paper, see Milne (1987).) 0.2. To place this paper in the context of the literature, it is important to appreciate the development of asset-pricing models over the last two decades. The first-and perhaps the most famous-asset-pricing model is the Capital Asset Pricing Model (CAPM) developed by Sharpe (1964) and Lintner (1965). The model demonstrated a striking result: assuming (i) consumers characterize portfolios in terms of the means and variance of rates of return, (ii) there exists a risk-free asset, and (iii) asset equilibrium; then any asset's expected rate of return could be written as a linear function of the riskless interest rate, and the expected rate of return on the endowment of risky assets (the so-called market portfolio). The original CAPM result was derived from a one period model, but Merton (1973) showed that in a continuous time model where returns could be char- acterized by a diffusion process, the same mean-variance argument could be applied to obtain the CAPM formula. It was well-known that if consumers had von Neumann-Morgenstern prefer- ences, then either quadratic utility and/or multivariate normality of returns were

Normality, Solvency, and Portfolio Choice

This paper examines whether investors with power utility functions choose mean-vari? ance- (MV) efficient portfolios when returns are approximately normally distributed and there is borrowing or lending at a riskless interest rate. The results show that the unlevered portfolios of power utility investors plot very closely to the MV-efficient frontier. How? ever, there are marked differences in the mix of risky assets, regardless of whether the portfolios are highly concentrated or widely diversified. Such differences allow power investors to remain solvent even when they lever their optimal portfolios to a greater ex? tent than less MV investors who risk bankruptcy. It is concluded that the investment policies of power utility and MV investors with similar risk aversion measures are not as similar as is commonly believed. This is particularly true for high power inves? tors, unless explicit solvency constraints are imposed on the MV problem, and for low power investors when quadratic utility approximations are made to the power utility func? tions. These differences in the investment policies of power utility and MV investors lead us to question the widely-accepted assertion that the assumptions of homogeneous beliefs, normality, a riskless asset, and risk-averse investors imply the simple MV CAPM where all investors, including power utility investors, hold combinations ofthe market portfolio and the riskless asset.

Calculating the market value of riskless cash flows

This paper uses arbitrage arguments to calculate the market value of riskless after-tax cash flows. The market value equals the present value of riskless after-tax cash flows discounted at the after-corporate-tax riskless interest rate. The market value equals the adjusted present value of riskless after-tax cash flows only when the incremental debt used in the adjusted present value calculations equals the market value of the remaining after-tax cash flows. Also, the analysis provides valuation formulas when interest and tax rates are certain but not uniform and when interest rates are uncertain.

Option Bounds in Discrete Time: Extensions and the Pricing of the American Put

In a recent paper (Perrakis and Ryan 1984), upper and lower bounds on European option prices were derived in discrete time. These bounds were based on an extension of the discrete-time derivation of Black-Scholes pricing, first presented in the work of Black and Scholes (1973). The bounds are functions of the stock and exercise prices, the riskless rate of interest, the time to maturity, and the entire distribution of stock returns. In other words, they need more information than Black-Scholes for their computation, and the derived results are weaker.' On the other hand, they are significantly more general in their assumptions, insofar as they do not rely on the continuous-time riskless hedge of the original Black-Scholes derivations or on the assumption of a binomial distribution of stock returns used in Cox, Ross, and Rubinstein (1979) and Rendleman and Bartter (1979). For this reason they are likely to be particularly useful in valuing options on assets, for which the Black-Scholes methodology is clearly This paper generalizes and tightens the discrete-time bounds for European options and develops similar bounds for American puts. The stock return has a distribution with a finite range and is subject to the restriction that the stock be beta. The bounds are derived from arbitrage portfolios using the stock, the option, and the riskless asset. They converge to the previous ones when the range coincides with the nonnegative real line, and they both become equal to the binomial option model under a two-state stock return. Upper and lower bounds for the American put are derived recursively by the same method, with arbitrage portfolios involving the stock, the riskless asset, and American and European puts. An example is provided with a trinomial return distribution. * Part of this research was done while I was a visiting professor (1983-84), Ecole Superieure de Commerce et d'Administration des Entreprises, Reims, France. I wish to thank Peter Ryan, Albert Madansky, and a referee of this Journal for helpful advice and comment, while retaining all responsibility for any remaining errors or omissions. 1. Note, however, that this is also true of other BlackScholes generalizations, such as Merton (1976).

Dividend yields are equity risk premiums

I n the tenth anniversary issue of this Journal, Michael Rozeff [1984] showedthat the equity risk premium can be proxied by the prospective dividend yield. This is an interesting and important result, due to the inherent unobservability of equity yields (hence equity risk premiums) compared to the easy availability of current dividend yields. My purpose here is to consider the conditions under which the approximation holds. Specifically, I will show that Rozeff's result holds -only for (1) long time periods, as opposed to individual years,' (2) the aggregate stock market, as opposed to an individual firm's equity, and (3) periods of little or no inflation. Rozeff bases his result on the Golden Rule of Accumulation (wherein the growth rate of physical output equals the real rate of interest in an equilibrium in which consumption per capita is maximized) in the context of the Gordon growth model. He defines the equity risk premium as the required return on equity minus the riskless rate of interest. He then converts real growth and the real riskless interest rate into nominal values by adding the expected inflation rate to each, essentially a Fisher effect adjustment applied to both variables. When he subtracts the riskless rate from both sides of the Gordon model, he derives his Equation (4)Z:

Ordering Uncertain Options under Inflation: A Note

ABSTRACTStochastic dominance rules (SD) have been extended to the case where investors are allowed to borrow and lend at the riskless interest rate. Stochastic dominance rules with a riskless asset (SDR) are much more effective than SD rules. However, it seems that this benefit is eliminated by an uncertain inflation, since riskless assets become risky once uncertain inflation is considered.We prove in this paper that SDR criteria are valid also in the face of uncertain (and independent) inflation. Moreover, while the mean‐variance (MV) efficient set increases with uncertain inflation, the stochastic dominance efficient sets decrease.

Stochastic Dominance and the Investment Horizon With Riskless Assets

This paper analyses the relationship between the efficient sets of investment portfolios and the investment holding period. Investors are allowed to hold risky assets as well as the riskless asset. The main result is that dominance in each period implies dominance in the multiperiod case. This finding holds with respect to first, second and third degree stochastic dominance. The riskless interest rate may vary from one period to another without changing the results of this paper.

Risk and the determinants of aggregate investment

In his survey of investment studies, Jorgenson (1971) summarized that integration of uncertainty into the theory and econometrics of investment was the most important open question in the study of investment. This paper follows Jorgenson's suggestion and tries to take a step in this direction by the incorporation of the theory of corporate finance and especially a measure of risk in the theory of investment. 1 1 The relationship between financial risk and investment was developed by Modigliani and Miller (1958). Their work however was mainly in a cross-section rather than a time series context. A systematic link between financial variables and investment was developed by Grunfeld who viewed the ‘market value of the firm as a sensitive indicator upon which investment decisions are based’. Grunfeld's model was extended and tested in a macro-economic context by Griliches and Wallace, who introduced two financial variables – an index of stock prices and rate of interest. However, they disregarded the ‘risk variable’ as a relevant determinant of investment. This variable is also missing from the Jorgenson's neoclassical model of investment. One possible explanation for the omission of a risk variable is the empirical orientation of these models and the absence of a reasonable time series measure of market risk in the academic literature. The inclusion of the risk variable in a model of investment seems to us a relevant direction for research. We are doing this by deriving the partial effect of risk on investment and testing it using empirical data. In Section I we analyse the partial effect of risk on investment decisions and we show the similarity between the effect of an increase in the riskless rate of interest and the effect of an increase in market risk. The analysis in this section is mainly a review and extension of the classical model and in particular the formal model of Modigliani and Miller (1958, 1963). In Section II we present an extension of the Griliches–Wallace model to include the effect of market risk. We also discuss a dynamic version of the model in which firms can revise their planned investment in response to more recent observations. In Section III we discuss the construction of a time series measure of risk as well as the definition of other variables which are used in an empirical test of the model. In Section IV we present and discuss the empirical results of the estimation of the model using quarterly' data for the period 1948–70. Section V summarizes the study and main conclusions.

State Preference and the Riskless Interest Rate: A Markov Model of Capital Markets

Journal Article State Preference and the Riskless Interest Rate: A Markov Model of Capital Markets Get access Avraham Beja Avraham Beja New York University and Tel-Aviv University Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 46, Issue 3, July 1979, Pages 435–446, https://doi.org/10.2307/2297012 Published: 01 July 1979 Article history Received: 01 September 1976 Accepted: 01 January 1978 Published: 01 July 1979

Borrowing, Short-Sales, Consumer Default, and the Creation of New Assets

There appears to be some confusion in the literature on asset-pricing models about the role of default-risk in short-selling and borrowing by consumers. For example, in the Sharpe-Lintner ([16] [9]) model it is usual to assume that all consumers are able to borrow or lend without restriction, at the riskless rate of interest. This assumption has been recognized to involve an inconsistency in the theory, because with personal borrowing there is always a positive probability of default with the S-L assumptions, so that the consumer's bond (as seen by any lender) is not a perfect substitute for the safe asset.