Fixed income risk

Fixed Income Risk Modeling for Portfolio Managers... 

The implications of this new model class are in contrast to most term structure models discussed in the literature, which assume that the bond markets are complete and fixed income derivatives are redundant securities. Collin-Dufresne and Goldstein [ 18] and Heiddari and Wu [36] show in an empirical work, using data of swap rates and caps/floors that there is evidence for one additional state variable that drives the volatility of the forward rates. Following from that empirical findings, they conclude that the bond market do not span all risks driving the term structure. This framework is rather similar to the affine models of equity derivatives, where the volatility of the underlying asset price dynamics is driven by a subordinated stochastic volatility process (see e.g. Heston [38], Stein and Stein [71] and Schobel and Zhu [69]). 

Note that the impact of this correlation effect is not in contradiction to the results found by Bakshi, Cao and Chen [5], Nandi [62] and Schobel and Zhu [69] for equity options. They found higher option prices given positive correlations and vice verca. On the other hand, we have a risk-neutral bond price process, where the source of uncertainty is negatively assigned (see e.g. (7.2)). Thus, assuming a USV bond model with negative correlated Brownian motions is the fixed income market analog of a stochastic volatility equity market model, with positive correlated sources of uncertainty. ... 

Cheyette, O., 1992. Term structure dynamics and mortgage valuation. J. Fixed Income 1 (4), 28 1. Das, S., 1997. A Direct Discrete-Time Approach to Poisson-Gaussian Bond Option Pricing in the Heath-Jarrow-Morton Model Working Paper. Harvard Business School, Boston, MA, pp. 1 14. Das, S., Foresi, S., 1996. Exact solutions for bond and option prices with systematic jump risk. Rev. Deriv. Res. 1, 1-24. 

Duffie, D., Kan, R., 1996. A yield-factor model of interest rates. Math. Financ. 6 (4), 379-406. Fitton, P., McNatt, J., 1997. The four faces of an interest rate model. In Fabozzi, F. (Ed.), Advances in Fixed Income Valuation Modelling and Risk Management. FJF Associates, New Hope, PA. 

Lacey, N., Nawalkha, S., 1993. Convexity, risk, and returns. J. Fixed Income 3 (3), 129-145. Phoa, W., 1998. Advanced Fixed Income Analytics. FJF Associates, New York (chapter 4). 

Cohler, G., Feldman, M., Lancaster, B., 1997. Price of risk ctmstant (PORC) going beyond OAS. J. Fixed Income 6(4). 

RISKS ASSOCIATED WITH INVESTING IN FIXED INCOME SECURITIES... 

Risk can thought of as the possibility of unpleasant surprise. Fixed-income securities expose the investor to one or more of the following types of risk (1) interest rate risk (2) credit risk (3) call and prepayment risk (4) exchange rate risk (5) liquidity risk and (6) inflation or purchasing power risk. 

A fundamental property is that an upward change in a bond s price results in a downward move in the yield and vice versa. This result makes sense because the bond s price is the present value of the expected future cash flows. As the required yield decreases, the present value of the bond s cash flows will increase. The price/yield relationship for an option-free bond is depicted in Exhibit 1.9. This inverse relationship embodies the major risk faced by investors in fixed-income securities—interest rate risk. Interest rate risk is the possibility that the value of a bond or bond portfolio will decline due to an adverse movement in interest rates. 

Once we estimate the cash flows for a fixed-income security, the next step is to determine the appropriate interest rate for discounting each cash flow. Before proceeding, we pause here to note that we will once again use the terms interest rate, discount rate, and required yield interchangeably throughout the chapter. The interest rate used to discount a particular security s cash flows will depend on three basic factors (1) the level of benchmark interest rates (2) the risks that the market perceives the securityholder is exposed to and (3) the compensation the market expects to receive for these risks. 

Thomas S. Y. Ho, Key Rate Durations Measures of Interest Rate Risk, Journal of Fixed Income (September 1992), pp. 29-44. 

This chapter explores interest rate options—a vitat part of the European fixed income securities market. The first section tooks at exchange-traded options, where 20 bittion worth of bond options and over 250 billion of options on short-term rates change hands every day. Next, we ll look at the flexible OTC markets for interest rate options, including caps, collars, swaptions, and structured products. Finally, having explained the products themselves, we ll move on to explore how they can be used to hedge interest rate risk. 

This discussion covers the main factors affecting bond returns in the European fixed income market, namely, the random fluctuations of interest rates and bond yield spreads, the risk of an obligor defaulting on its debt, or issuer-specific risk, and currency risk. There are also other, more subtle sources of risk. Some bonds such as mortgage-backed and asset-backed securities are exposed to prepayment risk, but such instruments still represent a small fraction of the total outstanding European debt. Bonds with embedded options are exposed to volatility risk. However, it is not apparent that this risk is significant outside derivatives markets. 

In today s asset management industry, organizations operations often extend beyond the European fixed income market. Controlling risk firm wide therefore calls for a detailed understanding of what the levels of risk in each market are and how markets interact with each other. The purpose of this last section is to provide a few elements of comparison between the euro and US dollar fixed income markets. 

We compare in Exhibit 23.16 the volatilities of a few selected euro and US dollar factors. The common denominator is that euro volatilities are less than their US dollar counterparts. This is true for all factors if we ignore the volatility bursts sometimes observed over a few months for some factors (for instance the Industrial A factor in Exhibit 23.16). The average level of systematic risk observed amongst euro-denomi-nated fixed income instruments is more generally low compared to other markets. Exhibit 23.16 shows one case where euro volatilities seem to be catching up with US levels. A more systematic analysis of how euro volatilities have recently evolved since 2002 would show that this is an exception. On average, euro volatilities have remained low with respect to US ones. Note that this is consistent first with the predictions of the swap factor model, euro spread levels and swap volatility being low compared to other markets. 

For more details, see Lionel Martellini, Stephane Priaulet, and Philippe Priaulet, Fixed-Income Securities Valuation, Risk Management and Portfolio Strategies (Hoboken, NJ John Wiley C Sons, Inc., 2003). 

L. Martellini and P. Priaulet, Fixed-Income Securities Dynamic Methods for Interest Rate Risk Pricing and Hedging (New York John Wiley 8c Sons, 2000). France (1995-98)—Spot ZC IM-lOY 3 66.64/20.52/6.96... 

Tracking error calculations have a relatively long history in the equity markets of measuring the relative risk of a portfolio against an index. The popularity of this methodology in equities has led many fixed-income managers to adopt the same approach, but we believe it is not as appropriate for the fixed-income markets. 

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