Human Survival at Older Ages and the Implications for Longevity Bond Pricing

Abstract

Governments are concerned about the future of pension plans, for which increasing longevity is judged to be an important risk to their future viability. We focus on human survival at age 65, the starting age point for many pension products. Using a simple model, we link basic measures of life expectancy to the shape of the human survival function and consider its various forms. The model is then used as the basis for investigating actual survival in England and Wales. We find that life expectancy is increasing at a faster rate than at any time in history, with no evidence of this trend slowing or any upper age limit. With interest growing in the use of longevity bonds as a way to transfer longevity risks from pension providers to the capital markets, we seek to understand how longevity drift affects pension liabilities based on mortality rates at the point of annuitization, versus what actually happens as a cohort ages. The main findings are that longevity bonds are an effective hedge against longevity risk; however, it is not only the oldest old that are driving risk, but also more 65-year-olds reaching less extreme ages such as 80. In addition, we find that the possibility of future inflation and interest rates could be as an important a risk to annuities as longevity itself.     

Securitization of Longevity Risk in Reverse Mortgages

Abstract

The reverse mortgage market has been expanding rapidly in developed economies in recent years. The onset of demographic transition places a rapidly rising number of households in an age window in which reverse mortgages have potential appeal. Increasing prices for residential real estate over the last decade have further stimulated interest.

Reverse mortgages involve various risks from the provider-s perspective that may hinder the further development of these financial products. This paper addresses one method of transferring and financing the risks associated with these products through the form of securitization. Securitization is becoming a popular and attractive alternative form of risk transfer of insurance liabilities. Here we demonstrate how to construct a securitization structure for reverse mortgages similar to the one applied in traditional insurance products.

Specifically, we investigate the merits of developing survivor bonds and survivor swaps for reverse mortgage products. In the case of survivor bonds, for example, we are able to compute premiums, both analytically and numerically through simulations, and to examine how the longevity risk may be transferred to the financial investors. Our numerical calculations provide an indication of the economic benefits derived from developing survivor bonds to securitize the “longevity risk component” of reverse mortgage products. Moreover, some sensitivity analysis of these economic benefits indicates that these survivor bonds provide for a promising tool for investment diversification.     

Catastrophe Risk Bonds

Abstract

This article examines the pricing of catastrophe risk bonds. Catastrophe risk cannot be hedged by traditional securities. Therefore, the pricing of catastrophe risk bonds requires an incomplete markets setting, and this creates special difficulties in the pricing methodology. The authors briefly discuss the theory of equilibrium pricing and its relationship to the standard arbitrage-free valuation framework. Equilibrium pricing theory is used to develop a pricing method based on a model of the term structure of interest rates and a probability structure for the catastrophe risk. This pricing methodology can be used to assess the default spread on catastrophe risk bonds relative to traditional defaultable securities.     

Hedging Longevity Risk When Interest Rates are Uncertain

Abstract

This paper proposes an asset liability management strategy to hedge the aggregate risk of annuity providers under the assumption that both the interest rate and mortality rate are stochastic. We assume that annuity providers can invest in longevity bonds, long-term coupon bonds, and shortterm zero-coupon bonds to immunize themselves from the risks of the annuity for the equity holders subject to a required profit. We demonstrate that the optimal allocation strategy can lead to the lowest risk under different yield curves and mortality rate assumptions. The longevity bond can also be regarded as an effective hedging vehicle that significantly reduces the aggregate risk of the annuity providers.     

Collateral affects return risk: evidence from the euro bond market

Covered bonds and senior bonds are prominent securities in the euro bond market. Senior bonds are unsecured, while covered bonds are secured—backed by collateral. Our results show that the presence of collateral reduces the total risk in individual bonds by more than 70%. Compared to diversified portfolios of senior bonds, diversified portfolios of covered bonds have a significantly lower level of systematic risk. However, the fraction of systematic risk to total risk is higher for covered bonds. By decomposing the variance of bond returns, we find that around 33% of the risk in senior bonds is systematic, versus 53% in covered bonds. Both types of bonds contain instrument-specific risk.

     

Tracking bond indices in an integrated market and credit risk environment

Abstract

The management of credit risky assets requires simulation models that integrate the disparate sources of credit and market risk, and suitable optimization models for scenario analysis. In this paper we integrate Monte Carlo simulation models for credit risk with scenario optimization, and develop a methodology for tracking broadly defined corporate bond indices. Testing of the models shows that the integration of the multiple risk factors improves significantly the performance of tracking models. Good tracking performance can be achieved by optimizing strategic asset allocation among broad classes of corporate bonds. However, extra value is generated with a tactical model that optimizes bond picking decisions as well. It is also shown that adding small corporate bond holdings in portfolios that track government bond indices improves the risk/return characteristics of the portfolios. The empirical results to substantiate the findings of this study are obtained by backtesting the model over a recent 30 month period.     

Asymptotics of bond yields and volatilities for extended CIR models under the real-world measure

ABSTRACT

The Cox–Ingersoll–Ross CIR short rate model is a mean-reverting model of the short rate which, for suitably chosen parameters, permits closed-form valuation formulae of zero-coupon bonds and options on zero-coupon bonds. This article supplies proofs of the formulae for the expected present value of payoffs under the real-world probability measure, known as actuarial valuation. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended CIR models when suitable conditions are imposed on the model parameters.     

Nonlinearities in the Brazilian Yield Curve

ABSTRACT

The article indicates the yield curve can be modeled using a continuous estimator as smooth transition regression, instead of traditional switch models, because bonds are traded continuously in the financial market. The results indicate that nonlinearity in the yield curve explains the pitfalls of monetary policy. The positive correlation between inflation and spread is consistent with a rise on uncertainty due to inflation risk or seems to indicate Brazilian Central Bank’s monetary policy credibility in the sample period. Therefore, if dependence on international capital exists, the Brazilian economic policy makers must monitor the movements in yield and analyze its feedback frequently in order to guide their plans and decisions.     

A continuous-time model for reinvestment risk in bond markets

When comparing standard bond market models with practice we observe that, whereas the literature places no restrictions on the time to maturity of traded bonds, this is actually the case in practice. Hence, standard models ignore the reinvestment risk present in practice when considering contacts with longer time to maturity than the longest bond traded in the market. In this paper we propose a model including this reinvestment risk. We place a restriction on the bonds traded in the market by limiting the time to maturity of traded bonds. At fixed times, new bonds are issued in the market, thus extending the time of maturity of traded bonds. The initial prices of the new bonds issued in the market depend on the information generated by the market and a stochastic variable independent thereof describing the reinvestment risk. In order to quantify and control the reinvestment risk we apply the criterion of risk-minimization.     

Bond Systematic Risk and the Option Pricing Model

In this paper we examine the behavior of the systematic risk of corporate bonds. A model that assumes β is constant is compared with a model that allows systematic risk to vary in a manner consistent with the Black-Scholes-Merton Options Pricing Model. This procedure captures some fundamental properties of the movement of bond β and provides a starting point for improved models of the process generating bond returns.     

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.     

The Risk Microstructure of Corporate Bonds: A Case Study from the German Corporate Bond Market

This article presents joint econometric analysis of interest rate risk, issuer‐specific risk (credit risk) and bond‐specific risk (liquidity risk) in a reduced‐form framework. We estimate issuer‐specific and bond‐specific risk from corporate bond data in the German market. We find that bond‐specific risk plays a crucial role in the pricing of corporate bonds. We observe substantial differences between different bonds with respect to the relative influence of issuer‐specific vs. bond‐specific spread on the level and the volatility of the total spread. Issuer‐specific risk exhibits strong autocorrelation and a strong impact of weekday effects, the level of the risk‐free term structure and the debt to value ratio. Moreover, we can observe some impact of the stock market volatility, the respective stock's return and the distance to default. For the bond‐specific risk we find strong autocorrelation, some impact of the stock market index, the stock market volatility, weekday effects and monthly effects as well as a very weak impact of the risk‐free term structure and the specific stock's return. Altogether, the determinants of the spread components vary strongly between different bonds/issuers.     

A mean/variance approach to long-term fixed-income portfolio allocation

Abstract

Long-term investments in bonds offer known returns, but with risks corresponding to defaults of the underwriters. The excess return for a risky bond is measured by the spread between the expected yield and the risk-free rate. Similarly, the risk can be expressed in the form of a default spread, measuring the difference between the yield when no default occurs and the expected yield. For zero-coupon bonds and for actual market data, the default spread is proportional to the probability of default per year. The analysis of market data shows that the yield spread scales as the square root of the default spread. This relation expresses the risk premium over the risk-free rate that the bond market offers, similarly to the risk premium for equities. With these measures for risk and return, an optimal bond allocation scheme can be built following a mean/variance utility function. Straightforward computations allow us to obtain the optimal portfolio, depending on a pre-set risk-aversion level. As for equities, the optimal portfolio is a linear combination of one risk-free bond and a risky portfolio. Using the scaling law for the default spread allows us to obtain simple expressions for the value, yield and risk of the optimal portfolio.     

Computing the survival probability in the Madan–Unal credit risk model: application to the CDS market

We obtain a quasi-analytical approximation of the survival probability in the credit risk model proposed in [Madan, D.B. and Unal, H., Pricing the risk of default. Rev. Deriv. Res., 1998, 2(2), 121–160]. Such a formula, which extensive numerical simulations reveal to be accurate and computationally fast, can also be employed for pricing credit default swaps (CDSs). Specifically, we derive a quasi-analytical approximate expression for CDS par spreads, and we use it to estimate the parameters of the model. The results obtained show a rather satisfactory agreement between theoretical and real market data.     

Dynamical analysis of corporate bonds based on the yield spread term-quality surface

Our aim of this research is to propose a model which estimates implied relative credit reliability from the yield spread of defaultable bonds and evaluates their spread risk. We introduce “yield spread term-quality surface” (YSTQS) which is defined on the space of duration and credit reliability of the issuers, and express their yield spread. First, we review the general pricing theorem of defaultable bonds with unpredictable recovery in the no-arbitrage context based on the external hazard rates. Second, we show that the dynamics of state variables determine the shape of the YSTQS, and they drive the YSTQS if the loss-adjusted hazard rates are described by a function of them. Finally, we show an empirical analysis of our model with daily yield spread, duration, and the credit ratings of corporate bonds.

Attracting investors for public health programmes with Social Impact Bonds

ABSTRACT

Social impact bonds (SIBs) have emerged as an innovative financing mechanism. This paper explores how health SIBs align with investors’ expectations and the conditions required to attract investors. At present, health SIBs are unlikely to provide sufficient financial returns given their financial risk to attract mainstream investors, so may be better suited to investors who are prepared to accept lower financial returns blended with particular health impact returns.     

Learning minimum variance discrete hedging directly from the market

Option hedging is a critical risk management problem in finance. In the Black–Scholes model, it has been recognized that computing a hedging position from the sensitivity of the calibrated model option value function is inadequate in minimizing variance of the option hedge risk, as it fails to capture the model parameter dependence on the underlying price (see e.g. Coleman et al., J. Risk, 2001, 5(6), 63–89; Hull and White, J. Bank. Finance, 2017, 82, 180–190). In this paper, we demonstrate that this issue can exist generally when determining hedging position from the sensitivity of the option function, either calibrated from a parametric model from current option prices or estimated nonparametricaly from historical option prices. Consequently, the sensitivity of the estimated model option function typically does not minimize variance of the hedge risk, even instantaneously. We propose a data-driven approach to directly learn a hedging function from the market data by minimizing variance of the local hedge risk. Using the S&P 500 index daily option data for more than a decade ending in August 2015, we show that the proposed method outperforms the parametric minimum variance hedging method proposed in Hull and White [J. Bank. Finance, 2017, 82, 180–190], as well as minimum variance hedging corrective techniques based on stochastic volatility or local volatility models. Furthermore, we show that the proposed approach achieves significant gain over the implied BS delta hedging for weekly and monthly hedging.     

Teaching children and youths about risk and risk analysis: what are the goals and the risk analytical foundation?

Abstract

Work has recently been initiated to teach children and youths about risk and risk analysis, highlighting conceptual issues, risk assessment, risk communication, as well as risk management (decision-making). Using this work as a point of departure, this article provides a discussion of the foundation of this type of teaching: what do we wish to obtain, in short- and long-term perspectives, and what is the scientific risk analysis basis? Key pillars for the development of a scientific platform for such teaching is presented and discussed.     

Corporate Hedging in the Insurance Industry

Abstract

In this paper we investigate the extent to which insurance companies utilize financial derivatives contracts in the management of risks The data set we employ allows us to observe the universe of individual insurer transactions for a class of contracts, namely, those normally thought of as off-balance-sheet (OBS) We provide information on the number of insurers using various types of derivatives contracts and the volume of transactions in terms of notional amounts and the number of counterparties. Life insurers are most active in interest rate and foreign exchange derivatives, while property/casualty insurers tend to be active in trading equity option and foreign exchange contracts Using a multivariate probtt analysis, we explore the factors that potentially influence the existence of OBS activities. We also investigate questions relating to whether certain subsets of OBS transactions (for example, exchange traded) are related to such things as interest rate risk measures, organizational form and other characteristics that may discriminate between desired risk/return profiles across a cross-section of insurers. We find evidence consistent with the use of derivatives by insurers to hedge risks posed by guaranteed investment contracts (GICs), coilater-alized mortgage obligations (CMOs), and other sources of financial risk