Vacancies and Rent Dispersion in a Stochastic Search Model with Generalized Tenant Demand

I model imperfect information, derive a downward sloping market demand curve, and explain vacancies in a partial equilibrium model of a rental housing market. Tenants can be completely described by an exogenous demand curve, perhaps arising from differences in income, preferred location, or tastes, and view vacant units based on a stochastic arrival of rental information. Free entry of these landlords induces excess rental housing capacity (equilibrium vacancies). I determine the equilibrium distribution of rents for vacant units, show that this rent distribution may be discontinuous, and explore the equilibrium vacancy rate to changes in exogenous parameters. The resulting characterization of equilibrium distributions of rents may be amenable to econometric modeling exploring the relationship between market rents and vacancies.     

Admissible uncertainty in the intertemporal asset pricing model

We embed the Sharpe-Lintner, two-parameter asset pricing theory in an intertemporal general equilibrium model. The investment opportunity set changes stochastically over time; in general the short-term and long-term interest rates and the distribution of the rate of return of the market portfolio are non-stationary. This non-stationarity, which is admissible in the Sharpe-Lintner model, has two implications: First, it may bias econometric methods which fail to explicitly take into account the non-stationarity. Second, the sequential application of the Sharpe-Lintner model in the discounting of stochastic cash flows becomes computationally complex and of little practical use.     

What Drives the Owner‐Occupied and Rental Housing Markets? Evidence from an Estimated DSGE Model

Most dynamic stochastic general equilibrium (DSGE) models with a housing market do not explicitly include a rental market and assume a tight mapping between house prices and rents over the business cycle. However, rents are much smoother than house prices in the data. We match this feature of the data by adding both an owner‐occupied housing market and a rental market in a standard DSGE model. The intertemporal preference shock accounts for more than half of the variation in house prices and contributes to residential investment fluctuations through the liquidity constraint, and nominal rigidity in rental contracts captures the variation in the price‐rent ratio.     

Misspecified Recovery

Asset prices contain information about the probability distribution of future states and the stochastic discounting of those states as used by investors. To better understand the challenge in distinguishing investors' beliefs from risk‐adjusted discounting, we use Perron–Frobenius Theory to isolate a positive martingale component of the stochastic discount factor process. This component recovers a probability measure that absorbs long‐term risk adjustments. When the martingale is not degenerate, surmising that this recovered probability captures investors' beliefs distorts inference about risk‐return tradeoffs. Stochastic discount factors in many structural models of asset prices have empirically relevant martingale components.     

Asymmetric Properties of Office Rent Adjustment

In this paper we use an error correction model for understanding the changes in real office rents for a panel of 15 U.S. MSA’s over the period 1990-2007. We find that office rents in all cities react positively to a rise in office employment and lagged rent changes, while lagged deviations from equilibrium rent levels exhibit a slow and partial adjustment over time. Given the non-negativity constraint of vacancy rates we extend the basic model by examining whether rents react to positive changes in employment conditional on the vacancy rate level. Our results show that office rents react significantly stronger to increases in employment when vacancy rates are below the long-term average. We also repeat the analysis for clusters of cities based on similarities in rent and employment dynamics using multi dimensional scaling. The cluster results confirm the overall conclusions and show that our results are not solely valid for the full panel of cities.     

A Comparison of Federal Government Office Rents with Market Rents

This study examines Federal Government office leases using data from Texas and Oklahoma during the 1981–1991 time period. The lease indifference model presented here indicates that landlords may be willing to accept lower rents from government tenants due to reduced tenant risk, but that such discounts may be offset by other premiums implicit in the lease contract. The data collected for this study reveal that rents paid by the government are significantly higher than average market rents during this time period. A time-series, cross-sectional regression analysis of the spread between market rents and office rents to government tenants in nine metropolitan markets suggests that the difference is affected in part by expense pass-throughs, lease period, amount of space leased, and local market conditions.     

Mossin's Theorem Given Random Initial Wealth

The purpose of this article is to reexamine Mossin's Theorem under random initial wealth. Conditions necessary and sufficient for Mossin's Theorem depend on the stochastic dependence between risks. The correlation coefficient, however, is not an adequate measure of stochastic dependence in the general expected‐utility model, and so other notions of dependence are used to investigate Mossin's Theorem. The inadequacy of the correlation coefficient is illustrated with two counterexamples. Then, using notions of positive and negative dependence measures, we provide necessary and sufficient conditions for a generalized Mossin Theorem to hold. In addition, a generalized Mossin Theorem is interpreted using the notion of a mean preserving spread made popular by Rothschild and Stiglitz (1970) . Given a fair premium and dependent stochastic conditions, we show that an individual can obtain a final wealth distribution with less weight in its tails by selecting less than or more than full insurance.     

A bivariate model for evaluating equity-linked policies with surrender option

This article proposes a bivariate lattice model for evaluating equity-linked policies embedding a surrender option when the underlying equity dynamics is described by a geometric Brownian motion with stochastic interest rate. The main advantage of the model stays in that the original processes for the reference fund and the interest rate are directly discretized by means of lattice approximations, without resorting to any additional transformation. Then, the arising lattices are combined in order to establish a bivariate tree where equity-linked policy premiums are computed by discounting the policy payoff over the lattice branches, and allowing early exercise at each premium payment date to model the surrender decision.     

Long-term factorization in Heath–Jarrow–Morton models

The long-term factorization decomposes the stochastic discount factor (SDF) into discounting at the rate of return on the long bond and a martingale that defines a long-term forward measure. We establish sufficient conditions for existence of the long-term factorization in HJM models. A condition on the forward rate volatility ensures existence of the long bond volatility. This yields existence of the long bond and convergence of \(T\)-forward measures to the long forward measure. It contrasts with the familiar risk-neutral factorization that decomposes the SDF into discounting at the short rate and a martingale defining the risk-neutral measure.     

Value at risk methodology of international index portfolio under soft conditions (fuzzy-stochastic approach)

The approach to modelling uncertainty of the international index portfolio by the value at risk (VAR) methodology under soft conditions by fuzzy-stochastic methodology is described in the paper. The generalised term uncertainty is understood to have two aspects: risk modelled by probability (stochastic methodology) and vagueness sometimes called impreciseness, ambiguity, softness is modelled by fuzzy methodology. Thus, hybrid model is called fuzzy-stochastic model. Input data for a stochastic model are unique distribution functions and crisp (real) data. Input data for fuzzy model are fuzzy numbers and crisp (real) data. Input data for hybrid model are fuzzy probability distribution functions, unique distribution functions, and crisp (real) data. Softly defined VAR model is constructed as hybrid model because it is supposed that the input data are difficult to determine as crisp numbers or as some unique distribution functions. Risk is modelled by stochastic methodology on the VAR basis and vagueness is modelled through the fuzzy numbers. The analytical delta normal VAR methodology for international index portfolio under soft conditions is described including illustrative example. It is shown, that methodology described could be considered to be generalised sensitivity analysis.     

Authors’ Reply: Corporate Hedging in the Insurance Industry: The Use of Financial Derivatives by U.S. Insurers - Discussion by L. Lee Colquitt; Arlette C. Wilson; Gray G. Venter; Morton Lane; Joan Lamm-Tennant,January 1997

Abstract

This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ultimate ruin. Explicit results are obtained when dividends are paid out to the stockholders according to a constant barrier strategy.     

Martingales in life insurance

Abstract

This paper presents a general probabilistic model, including stochastic discounting, for life insurance contracts, either a single policy or a portfolio of policies. In § 4 we define prospective reserves and annual losses in terms of our model and we show that these are generalisations of the corresponding concepts in conventional life insurance mathematics. Our main results are in § 5 where we use the martingale property of the loss process to derive upper bounds for the probability of ruin for the portfolio. These results are illustrated by two numerical examples in § 6.     

Probability distribution of returns in the Heston model with stochastic volatility*

Abstract

We study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker‐Planck equation exactly and, after integrating out the variance, find an analytic formula for the time‐dependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow‐Jones index for time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log‐returns with a time‐dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow‐Jones data for 1982–2001 follow the scaling function for seven orders of magnitude.     

Local Office Rent Dynamics

This study applies rent adjustment models for ten major European office markets. We capture long-run equilibrium relationships of demand and supply variables and their short-term corrections in a two equation error correction model. We test whether the local nature of office markets makes a model based on national economics inaccurate if local and national markets do not move in tandem. For this we employ a unique dataset, which includes both disaggregated and national variables to model changes in real prime rents for a group of premier and second tier office market cities across Europe for the period 1990–2006. We explicitly compare results that are derived from models that include different levels of geographic aggregation. Results of the two stage error correction model indicate that international office rents adjust to short-run changes in office related economic activity, lagged rent changes, and to the deviation of rents from their long-run values. At the same time our results offer no proof that error correction mechanism models for office rents improve significantly by specifying economic growth figures beyond the national aggregated level for the cities included in our analysis.     

Credit ratings and convertible bond prices: a simulation-based valuation

This study presents a simulation-based model of convertible bond prices under the assumption of stochastic interest rates. The model is developed such that the convertible bond price explicitly depends on the credit rating at the time of issuance. Key ideas explored in this study include terminating the simulated sample path immediately when the issuer defaults on the bond at time t, which is the same as the investor and the issuer optimally exercising their options and discounting the resulting cash flows at a risk-free rate. In turn, the defaulted group of sample paths belongs to the bottom xth percentile of the realized stock prices at each time, which is exogenously given by the cumulative or marginal default probability of a firm that has the same rating as the issuer. Upon calibrating the model, we can see that the moneyness of convertible bonds is strongly responsible for influencing the convertible bond price when the rating changes. Furthermore, the effects of stochastic interest rates are shown to be possibly significant when the interest rate risk’s market price is not zero.     

Optimal regulation of network expansion

We model the regulation of irreversible capacity expansion by a firm with private information about capacity costs, where investments are financed from the firm's cash flows and demand is stochastic. The optimal mechanism is implemented by a revenue tax that increases with the price cap. If the asymmetric information has large support, then the optimal mechanism consists of a laissez‐faire regime for low‐cost firms. That is, the firm's price cap corresponds to that of an unregulated monopolist, and it is not taxed. This “maximal distortion at the top” is necessary to provide information rents, as direct subsidies are not feasible.     

Residential Search and Mobility in a Housing Market Equilibrium Model

In this paper, we propose an equilibrium model for the housing market which provides an explanation for observed housing consumption of households over their lifetimes. The moving behavior of households is described as a stochastic dynamic process in which households moving decisions depend on information which is obtained over time. Households move when the offer exceeds an endogenously determined threshold. On the basis of the households moving behavior, the steady-state distribution of households over the housing stock is obtained. On the supply side of the market, landlords are looking for households to occupy their vacant dwellings. Their strategy is to set rents in a mixed strategy in order to profit from imperfect information. After formulating search behavior of households as well as the behavior of landlords, the market equilibrium is derived. We explore the sensitivity of the equilibrium to changes in the structural parameters.     

CAPITAL BUDGETING DECISIONS WHEN CASH FLOWS AND PROJECT LIVES ARE STOCHASTIC AND DEPENDENT

The purpose of this study is twofold: (1) to develop an operational economic state and simulation capital budgeting procedure for allowing cash flows and project lives to be dependent and (2) to provide empirical evidence of the impact of stochastic project lives on mean-variance and mean-semivariance capital budgeting decisions. The required number of input estimates for the proposed model is small. For individual projects, incorrectly assuming deterministic project lives when project lives are stochastic often results in large overestimates of expected net present values and large underestimates of the variance of the net present value. Similar results occur for the mean-variance and mean-semivariance portfolio models. The primary managerial implication of this study is that the inclusion of stochastic project lives in capital budgeting decisions is critical to obtain appropriate risk-return estimates.     

An equilibrium model of wealth distribution

I present an explicitly solved equilibrium model for the distribution of wealth and income in an incomplete-markets economy. I first propose a self-insurance model with an inter-temporally dependent preference [Uzawa, H. 1968. Time preference, the consumption function, and optimal asset holdings. In: Wolfe, J.N. (Ed.), Value, Capital, and Growth: Papers in Honour of Sir John Hicks. Edinburgh University Press, Edinburgh, pp. 485-504]. I then derive an analytical consumption rule which captures stochastic precautionary saving motive and generates stationary wealth accumulation. Finally, I provide a complete characterization for the equilibrium cross-sectional distribution of wealth and income in closed form by developing a recursive formulation for the moments of the distribution of wealth and income. Using this recursive formulation, I show that income persistence and the degree of wealth mean reversion are the main determinants of wealth-income correlation and relative dispersions of wealth to income, such as skewness and kurtosis ratios between wealth and income.     

Historical simulation approach to the estimation of stochastic discount factor models

We propose an approach to the estimation of the parameters of stochastic discount factor (SDF) models which is based on the idea that the next period joint distribution of the variables in a SDF and asset returns can be well approximated by their joint historical distribution. The estimates of the SDF parameters may therefore be found as the values of the parameters at which the mean of the historical distribution of the product of the SDF with an asset return equals one. Each time period, the estimates are updated using the most recent periods of data and hence can change over time. This method can be viewed as an alternative to the approaches that specify a particular functional form relating the SDF parameters to proxies for the state of the world.