Optimal Loan Interest Rate Contract Design

This paper analyzes optimal loan interest rate contracts under conditions of risky, symmetric information for one-period (static) and multi-period (dynamic) models. The optimal loan interest rate depends upon the volatility of, and co-variation among the market interest rate, borrower collateral, and borrower income, as well as the time horizon and the risk preferences of lenders and borrowers. For a risk-averse borrower with stochastic collateral, variable interest rate contracts are, in general, Pareto optimal. For plausible assumptions, the optimal loan interest rate for the multi-period model often exhibits muted responses to changes in market interest rates, making fixed rate loans a reasonable approximation for the optimal loan. Hence, in the absence of optimal contracts, long-term (short-term) borrowers tend to prefer fixed rate (variable) contracts.     

A Two-Factor Model for Low Interest Rate Regimes

This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

Implicit Inflation and Risk Premiums in the Brazilian Fixed Income Market

The breakeven inflation, the differential between nominal and real yields of bonds, is often used as a predictor of future inflation. The model presented here decomposes this interest rate differential into a risk premium and implicit inflation using a parametric formulation based on no-arbitrage conditions using nominal and indexed yield curves in Brazil, via an affine model of the Nelson–Siegel family. The measures of implicit inflation obtained from the model are shown to be unbiased estimators of future inflation for short horizons and carry some information for long horizons, and the model forecasts are superior to market surveys.     

Cohort and value-based multi-country longevity risk management

ABSTRACT

Multi-country risk management of longevity risk provides new opportunities to hedge mortality and interest rate risks in guaranteed lifetime income streams. This requires consideration of both interest rate and mortality risks in multiple countries. For this purpose, we develop value-based longevity indexes for multiple cohorts in two different countries that take into account the major sources of risks impacting life insurance portfolios, mortality and interest rates. To construct the indexes we propose a cohort-based affine model for multi-country mortality and use an arbitrage-free multi-country Nelson–Siegel model for the dynamics of interest rates. Index-based longevity hedging strategies have the advantages of efficiency, liquidity and lower cost but introduce basis risk. Graphical risk metrics are a way to effectively capture the relationship between an insurer's portfolio and hedging strategies. We illustrate the effectiveness of using a value-based index for longevity risk management between two countries using graphical basis risk metrics. To show the impact of both interest rate and mortality risk we use Australia and the UK as domestic and foreign countries, and, to show the impact of mortality only, we use the male populations of the Netherlands and France with common interest rates and basis risk arising only from differences in mortality risks.     

Price Stability in Economies with Habit Persistence

This paper shows that the divine‐coincidence does not hold in a sticky price model with external habit if a time‐varying tax rate on labor income is not implemented to fully eliminate the time‐varying distortions associated with external habit and monopoly power in goods market. The required labor income tax rate is inversely related to the risk‐free real interest rate and the markup in the goods market, but it is proportional to the degree of external habit. Under this circumstance, the optimal monetary policy commands a countercyclical interest rate, having a perfect negative correlation with tax rate in the sticky price model with external habit. If a time‐invariant tax is the only fiscal instrument, then the degree of external habit entails a gap between the private marginal rate of substitution between consumption and labor and the social marginal rate of substitution, generating an endogenous trade‐off between the stabilization of welfare‐relevant output gap and inflation. Under this circumstance, price stability is not the optimal policy. The monetary policy authority should optimally try to undo the time‐varying distortions associated with external habit and monopoly power in goods market by deviating from price stability.     

Real-world jump-diffusion term structure models

This paper considers interest rate term structure models in a market attracting both continuous and discrete types of uncertainty. The event-driven noise is modelled by a Poisson random measure. Using as numeraire the growth optimal portfolio, interest rate derivatives are priced under the real-world probability measure. In particular, the real-world dynamics of the forward rates are derived and, for specific volatility structures, finite-dimensional Markovian representations are obtained. Furthermore, allowing for a stochastic short rate in a non-Markovian setting, a class of tractable affine term structures is derived where an equivalent risk-neutral probability measure may not exist.     

利率市场化对我国商业银行影响的实证分析

对于商业银行而言,利率市场化的积极意义在于它将促进金融市场的深化发展和金融机构之间的公平竞争,为银行业快速发展和多元化经营创造良好的外部环境。但是从短期来看,利率市场化也将改变商业银行原有的利率决定机制和经营模式。利率市场化将大大提高利率波动的幅度和频率,并使利率的期限结构复杂化,对商业银行的经营能力和风险把控能力都将是重大考验。本文选取10家国内上市商业银行,对其非利息收入占比、利率敏感性缺口等指标进行实证检验,最后根据实证结果对商业银行应对利率市场化提出合理建议。     

Variable Rate Debt Instruments and Corporate Debt Policy

Sustained high rate of inflation has led to the creation of debt instruments with variable interest rate. The availability of these debt instruments presents management with the problem of the choice of the optimal debt portfolio. This paper deals with this problem assuming a given, and optimal, debt to equity ratio. Given expected monetary value maximization, an efficient frontier is derived in terms of the expected net income and probability of bankruptcy, where net income is defined as operating income minus debt repayment. This efficient frontier is shown to be also mean-variance efficient. It is also shown that in most cases the optimal debt portfolio includes more than one debt instrument. In other words, the firm will avoid the policy of minimizing the expected cost of its debt repayments or the policy of minimizing the costs of bankruptcy. The optimal solution itself is affected by market variables like the relative expected cost of different debt instruments and by firm specific variables like the variability of its operating income stream, and the covariance between the operating income and the debt repayments.     

公开市场操作与利率期限结构行为——基于混频数据信息的研究视角

在混频数据信息环境中,精准识别公开市场操作(央行政策利率)和国债收益率曲线(基准利率体系)之间的关联机制至关重要,其影响了货币政策期限结构传导的有效性。本文在混频Nelson-Siegel(N-S)利率期限结构模型框架下,引入央行政策利率,揭示公开市场操作与利率期限结构(水平、斜率、曲度)因子之间的作用机制。实证结果表明:混频数据信息条件下,引入的公开市场操作信息显著改进国债收益率曲线的拟合效果;斜率因子冲击对公开市场操作具有显著的正向影响,而利率期限结构因子对政策调控的反应不敏感。进一步研究表明,2015年以来,公开市场操作对斜率因子的影响逐渐扩大,政策利率向国债收益率曲线的传导效率得到显著提高,我国现代货币政策框架日益健全。     

Identifying volatility risk premia from fixed income Asian options

Fixed income options are frequently adopted by companies to hedge interest rate risk. Their payoff dependence on the cumulative short-term rate makes them particularly informative about interest rate volatility risk. Based on a joint dataset of bonds and Asian interest rate options, we study the interrelations between bond and volatility risk premia in a major emerging fixed income market. We propose a dynamic term structure model that generates an incomplete market compatible with a preliminary empirical analysis of the dataset. Approximation formulas for at-the-money Asian option prices avoid the use of computationally intensive Fourier transform methods, allowing for an efficient implementation of the model. The model generates a bond risk premium strongly correlated with a widely accepted emerging market benchmark index (EMBI-Global), and a negative volatility risk premium, consistent with the use of Asian options as insurance in this market.     

Path-dependent scenario trees for multistage stochastic programmes in finance

The formulation of dynamic stochastic programmes for financial applications generally requires the definition of a risk–reward objective function and a financial stochastic model to represent the uncertainty underlying the decision problem. The solution of the optimization problem and the quality of the resulting strategy will depend critically on the adopted financial model and its consistency with observed market dynamics. We present a recursive scenario approximation approach suitable for financial management problems, leading to a minimal yet sufficient representation of the randomness underlying the decision problem. The method relies on the definition of a benchmark probability space generated through Monte Carlo simulation and the implementation of a scenario reduction scheme. The procedure is tested on an interest rate vector process capturing market and credit risk dynamics in the fixed income market. The collected results show that a limited number of scenarios is sufficient to capture the exposure of the decision maker to interest rate and default risk.     

Unspanned stochastic volatility and fixed income derivatives pricing

We propose a parsimonious ‘unspanned stochastic volatility’ model of the term structure and study its implications for fixed-income option prices. The drift and quadratic variation of the short rate are affine in three state variables (the short rate, its long-term mean and variance) which follow a joint Markov (vector) process. Yet, bond prices are exponential affine functions of only two state variables, independent of the current interest rate volatility level. Because this result holds for an arbitrary volatility process, such a process can be calibrated to match fixed income derivative prices. Furthermore, this model can be ‘extended’ (by relaxing the time-homogeneity) to fit any arbitrary term structure. In its ‘HJM’ form, this model nests the analogous stochastic equity volatility model of Heston (1993) [Heston, S.L., 1993. A closed form solution for options with stochastic volatility. Review of Financial Studies 6, 327–343]. In particular, if the volatility process is specified to be affine, closed-form solutions for interest rate options obtain. We propose an efficient algorithm to compute these prices. An application using data on caps and floors shows that the model can capture very well the implied Black spot volatility surface, while simultaneously fitting the observed term structure.     

On cox processes and credit risky securities

A framework is presented for modeling defaultable securities and credit derivatives which allows for dependence between market risk factors and credit risk. The framework reduces the technical issues of modeling credit risk to the same issues faced when modeling the ordinary term structure of interest rates. It is shown how to generalize a model of Jarrow, Lando and Turnbull (1997) to allow for stochastic transition intensities between rating categories and into default. This generalization can handle contracts with payments explicitly linked to ratings. It is also shown how to obtain a term structure model for all different rating categories simultaneously and how to obtain an affine-like structure. An implementation is given in a simple one factor model in which the affine structure gives closed form solutions.     

Multi-factor Affine Term Structure Model with Single Regime Shift: Real Term Structure under Zero Interest Rate

In this paper, we extend the one-factor, single regime shift, affine term structure model with time-dependent regime-shift probability to a multi-factor model. We model the nominal interest rate and the expected inflation rate, and estimate the term structure of the real interest rate in the Japanese government bond market using inflation-indexed bond data under zero interest rates. Incorporating the economic structure that the Bank of Japan terminates the zero interest rate when the expected inflation rate gets out of deflationary regime, we estimate the yield curve of the real interest rate for less than 10 years, consistent with the expectation of the market participants in the Japanese government bond market, where inflation-indexed bonds are traded for only around 10 years.     

Optimal investment-consumption-insurance with random parameters

This paper discusses an optimal investment, consumption, and life insurance purchase problem for a wage earner in a complete market with Brownian information. Specifically, we assume that the parameters governing the market model and the wage earner, including the interest rate, appreciation rate, volatility, force of mortality, premium-insurance ratio, income and discount rate, are all random processes adapted to the Brownian motion filtration. Our modeling framework is very general, which allows these random parameters to be unbounded, non-Markovian functionals of the underlying Brownian motion. Suppose that the wage earner’s preference is described by a power utility. The wage earner’s problem is then to choose an optimal investment-consumption-insurance strategy so as to maximize the expected, discounted utilities from intertemporal consumption, legacy and terminal wealth over an uncertain lifetime horizon. We use a novel approach, which combines the Hamilton–Jacobi–Bellman equation and backward stochastic differential equation (BSDE) to solve this problem. In general, we give explicit expressions for the optimal investment-consumption-insurance strategy and the value function in terms of the solutions to two BSDEs. To illustrate our results, we provide closed-form solutions to the problem with stochastic income, stochastic mortality, and stochastic appreciation rate, respectively.     

China’s official rates and bond yields

Recent research shows that bond yields are influenced by monetary policy decisions. To learn how this works in a bond market that differs significantly from those in the US and Europe, we model Chinese bond yields using the one-year deposit interest rate as a state variable. We also include the spread between the one-year market interest rate and the one-year deposit interest rate as another factor. The model is developed in an affine framework and closed-form solutions are obtained. We then test the model empirically with a Markov Chain Monte Carlo simulation procedure. The results show that the new model that incorporates the official rate in China characterizes the changing shape of the yield curve well.     

结构性衍生产品的定价和风险管理技术

准确可靠的定价能力不仅是银行增加资金业务收益、提高风险管理水平的重要前提,而且也是增强银行核心竞争力的技术保障。结构性衍生产品的定价过程通常由三部分构成:一是准确理解和描述产品的结构,二是选择适当的定价模型、市场数据和校准方法,三是选择最优的数值算法,定价过程最终都借助计算机程序来实现。定价的本质就是准确度量风险的价值,实现风险的对冲。最好的风险管理技术应该采用动态 Delta、Vega、Gamma 对冲。中国银行业在结构性衍生产品交易领域起步较晚,先引进外部定价软件,在此基础上逐步自主开发是实现定价能力和风险管理的现实选择。     

Modeling Foreign Exchange Risk Premium in Armenia

This paper applies stochastic discount factor methodology to modeling the foreign exchange risk premium in Armenia. We use weekly data on foreign and domestic currency deposits, which coexist in the Armenian banking system. This coexistence implies elimination of the cross-country risks and transaction costs, leaving the pure foreign exchange risk. It is shown that there exists a systematic time-varying risk premium that increases with maturity. Using two-currency affine term structure and generalized autoregressive conditional heteroskedasticity (GARCH)-in-mean models, we find that the central bank's foreign exchange market interventions and ratio-of-deposit volumes significantly affect public expectations about foreign exchange fluctuations. We also find that the foreign exchange risk premium accounts for the largest part of the interest differential. When accounting for economic and institutional differences, our results can be extended to other countries.     

Are interest rate options important for the assessment of interest rate risk?

Fixed income options contain substantial information on the price of interest rate volatility risk. In this paper, we ask if those options will also provide information related to other moments of the objective distribution of interest rates. Based on dynamic term structure models within the class of affine models, we find that interest rate options are useful for the identification of interest rate quantiles. Two three-factor models are adopted and their adequacy to estimate Value at Risk of zero-coupon bonds is tested. We find significant difference on the quantitative assessment of risk when options are (or not) included in the estimation process of each of these dynamic models. Statistical backtests indicate that bond estimated risk is clearly more adequate when options are adopted, although not yet completely satisfactory.