Market Risk (Equities, FX, Commodities)

Interest rates influence the calculation of premiums, reserves and benefits in the long-term. Theoretically, calculation of such actuarial values is based on the assumption of constant interest rates although interest rates constantly move over time. To obtain a more realistic assessment in valuation, it is beneficial if stochastic interest rates are used. Accurate calculation of reserves ensures that the insurance company can pay claims. A Reserve is a sum of money held by a financial institution such as a life office or a pension fund to cover for the difference between present value of future liabilities including expenses and present value of future premiums. A term insurance contract is an insurance policy that pays the sum assured to the beneficiaries if the policyholder dies within the duration of the policy. The purpose of this study was to compare the reserves that would be needed for a term insurance policy using the Vasicek and the Cox-Ingersol-Ross models. The models were chosen and are widely used because they are tractable and their ease of implementation. The stages of this research activity started by estimating the parameters for the models using Maximum Likelihood Estimation Method. Interest rates were then simulated for the models under study. Next, the reserve value of term insurance policy was determined using the simulated interest rates for the models using the Prospective Method for four randomly generated people of different ages. At the final stage, the results of the reserve values for the models were interpreted and compared. Kenya’s Life Table 2001–2003 was used as the reference in determination of mortality assumptions in reserve calculation. The goodness of fit of the models were done using Likelihood Ratio Test and CIR model was a better fit for the data. Interest rates were highly volatile, a feature replicated better by CIR model. Reserve values were also high for CIR model. Reserve values were higher for male insureds due to a higher mortality rate for men than women while the benefit reserves for the younger age were lower as compared to the older ages.

CHAPTER 4 - The Role of Money in Trade-Balance Stability: Synthesis of the Elasticity and Absorption Approaches

The optimal strategies for a long-term static investor are studied. Given a portfolio of a stock and a bond, we derive the optimal allocation of the capitals to maximize the expected long-term growth rate of a utility function of the wealth. When the bond has a constant interest rate, three models for the underlying stock price processes are studied: Heston model, 3/2 model, and jump diffusion model. We also study the optimal strategies for a portfolio in which the stock price process follows a Black-Scholes model and the bond process has a Vasicek interest rate that is correlated to the stock price.

An evaluator of warrant models is essential for enhancing the current procedures used in the country's stock market and economy. This would provide additional insight for investors to make informed investment decisions and help to better understand the dynamics of warrant pricing in Malaysia. It has been established that the conventional process for valuing equity warrants using call option pricing models, like the Black-Scholes model, has numerous flaws, including the presumption of constant volatility and interest rates. As a matter of fact, existing alternative models were more concerned with providing techniques for pricing than with conducting empirical tests of warrant pricing models. Consequently, it is crucial to have a mathematical model that takes into account stochastic volatility and stochastic interest rates when pricing and analyzing equity warrants. Recent research by Roslan, Ibrahim, Jameel, and Ibrahim (2020) and Roslan, Ibrahim, Karim (2020) examined the pricing of hybrid equity warrants in the presence of stochastic factors. There was no comparative analysis conducted in these studies to ascertain the relative efficacy of the two methods used. This study's main goal is to empirically investigate how well equity warrant pricing models perform in the Malaysian market. This evaluation will take into account the impact of stochastic interest rates, using the Cox-Ingersoll-Ross (CIR) stochastic interest rate model, and stochastic volatility, using the Heston stochastic volatility model. To perform the analysis, the study will make use of a number of mathematical techniques, such as partial differential equations, Fourier transform approaches, and the change of measure method. The research methodology is divided into three main phases. It is anticipated that the research's findings will provide new perspectives that go beyond the domain of warrant pricing, creating chances for more widespread applications in fields like foreign exchange, insurance, and mortgage contracts.

ABSTRACTOption‐pricing models that assume a constant interest rate may misprice futures options if the interest rate fluctuates significantly or if the price of the underlying asset is correlated with the interest rate. The futures option‐pricing model of Ramaswamy and Sundaresan allows for a stochastic interest rate and correlation of the underlying asset's price with the interest rate. Using a data set of daily closing prices for Comex gold futures options, this paper tests the Ramaswamy and Sundaresan model against a constant interest rate model. Results indicate that the stochastic interest rate model is a superior predictor of market prices.

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.

Chapter 23 - Market Risk, Interest Rates and Bank Intermediary Role in Developing Economies

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount have found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract for the underlying asset follows a geometric Brownian motion with constant interest rate, volatility and nonnegative dividend yield. In this paper, option price is derived for the case of the underlying asset that follows a geometric Brownian motion with time-dependent drift and volatility, which is more important for real life applications. The option price formulae are derived for the case of a drift that includes nonnegative or negative dividend. The latter yields a solution type that is new to the literature. A negative dividend corresponds to a negative foreign interest rate for foreign exchange options, or storage costs for commodity options. It may also appear in pricing options with transaction costs or real options, where the drift is larger than the interest rate.

Around the time the Bretton Woods exchange rate system was breaking down in global markets, interest rates in many local markets became much less predictable, leaving businesses and investors exposed to far more interest rate risk than they were familiar with. The financial community began looking for ways to hedge and speculate this risk. To address this need to hedge interest rate risk, the futures exchanges in Chicago and London (and other financial centers) began offering interest rate contracts in addition to their historically traded commodity futures. In 1981, IBM and the World Bank entered into the first interest rate swap, soon followed by a foreign exchange swap. As the Black-Scholes model and other options pricing models became generally accepted, options on interest rates became widely traded in deep, liquid markets. Decades later, the interest rate derivatives markets, both exchange traded and over the counter (OTC), have grown into a huge global marketplace (notional amounts in billions of U.S. dollars equivalent). (See Exhibit 10.1.) These so-called plain vanilla interest rate derivatives totaled almost U.S. $355 trillion at the end of 2006, and there are many other interest rate derivatives that the Bank for International Settlements (BIS) does not report statistics on. At first glance, the BIS total appears to be roughly 10 times global gross domestic product. However, some OTC derivatives are double counted (e.g., each side of the same risk may be on the books of two banks; in practice this is difficult to net). A significant percentage of the total (exchange traded plus OTC) is a hedge against other interest rate risk but does not get netted in the statistics just cited. Most important, the numbers shown reflect notional amounts (i.e., the amounts used to as a benchmark for determining the dollar value of interest rate payments). The market value of these instruments is much smaller. Whole books have been written on each type of interest rate derivative, but the next section provides a sampling of the common types of interest rate derivatives you are likely to see on a broker-dealer fixed income trading floor.

We present a pricing kernel that summarizes well the main features of the dynamics of interest rates and risk in postwar U.S. data and use it to uncover how the pricing kernel has moved with the short rate in this data. Our findings imply that standard monetary models miss an essential link between the central bank instrument and the economic activity that monetary policy is intended to affect, and thus we call for a new approach to monetary policy analysis. We sketch a new approach using an economic model based on our pricing kernel. The model incorporates the key relationships between policy and risk movements in an unconventional way: the central bank's policy changes are viewed as primarily intended to compensate for exogenous business cycle fluctuations in risk that threaten to push inflation off target. This model, while an improvement on standard models, is considered just a starting point for their revision. It leads to critical questions that researchers need to answer as they continue to revise their approach to monetary policy analysis. Copyright © 2009 by the National Bureau of Economic Research.

Performance of inflation targeting based on constant interest rate projections

Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims

The progress or decline of a company is influenced by the productivity of the workers. The pension program is one of the programs that can guarantee the welfare of employees. This study aims to implement the Cox Ingersoll Ross (CIR) model in approximating interest rates and using them to determine normal pension contributions. In order to be able to describe the actual situation, namely the interest rate which is the basis for determining pension contributions is fluctuating, the CIR model was chosen. The method used in actuarial calculations in this study is the Projected Unit Credit method. Based on the calculations, the Mean Absolute Percentage Error (MAPE) value is 6.45%. This suggests that the pattern of forecasted interest rates and market rates move in much the same way. In addition, the results show that the calculation of normal contributions using the CIR model tends to be larger than using a constant interest rate. The forecast results can help reduce risks in the future.Keywords: Cox Ingersoll Ross (CIR), Normal Cost, Pension, Projected Unit Credit (PUC). AbstrakKemajuan atau kemunduran dari perusahaan dipengaruhi oleh produktivitas para pekerja. Program pensiun merupakan salah satu program yang dapat menjamin kesejahteraan pegawai. Penelitian ini bertujuan mengimplementasikan model Cox Ingersoll Ross (CIR) dalam mengaproksimasi tingkat suku bunga dan menggunakannya dalam menentukan iuran normal pensiun. Supaya dapat menggambarkan keadaan yang sebenarnya, yaitu tingkat suku bunga yang menjadi dasar penentuan iuran pensiun bersifat fluktuatif maka dipilih model CIR. Metode yang digunakan dalam perhitungan aktuaria adalah metode Projected Unit Credit (PUC). Berdasarkan hasil perhitungan, diperoleh nilai Mean Absolut Percentage Error (MAPE) 6,45%. Hal ini menunjukkan bahwa pola perkiraan suku bunga dan suku bunga pasar bergerak dengan cara yang hampir sama. Berdasarkan metode PUC, perhitungan iuran normal dengan model CIR cenderung lebih besar dibandingkan dengan menggunakan tingkat bunga konstan. Hasil perkiraan tersebut dapat membantu mengurangi risiko di masa yang akan datang.

Are Real Estate Investment Trusts (REITs) an Inflation Hedge?

In this paper we assess the partial hedging problems by formulating hedging strategies that minimize conditional value-at-risk (CVaR) of the portfolio loss under stochastic interest rate environment. The combination of stochastic interest and CVaR hedging method makes the valuing approach more complex than the existing model with constant interest rate. We take up two issues in searching the optimal CVaR hedging strategy: given the initial capital constraint we minimize the CVaR of the portfolio loss; by prescribing a bound on the risk, we also minimize the hedging cost. As an illustration of this hedging technique we derive hedging strategies for a European call option with the Black Scholes setting under HJM framework; explicit formulas are presented. We also investigate CVaR hedging problems by using the real financial data.