Spot interest rate

In a continuous time environment we do not assume discrete time intervals over which interest rates are applicable, rather a period of time in which a borrowing of funds would be repaid instantaneously. So we define the forward rate f(t, s) as the interest rate applicable for borrowing funds where the deal is struck at time f the actual loan is made at s (with s>t) and repayable almost instantly. In mathematics the period s — f is described as infinitesimally small. The spot interest rate is defined as the continuously compounded yield or interest rate r(f, T). In an environment of no arbitrage, the return generated by investing at the forward rate f(t, s) over the period s — t must be equal to that generated by investing initially at the spot rate r(f, T). So we may set... 

Forward rates may be calculated using the discount function or spot interest rates. If spot interest rates are known, then the bond price equation can be set as ... 

The stochastic process for default-free spot rates and default process are independent under martingale measure Q This last assumption implies that the default process is uncorrelated with default-free spot interest rates. 

The forward price is calculated only for the purpose of incorporating repo interest it should not he confused with a forward interest rate, which is the interest rate for a term starting in the future and which is calculated from a spot interest rate. Nor should it be taken to he an indication of what the market price of the hond might be at the time of trade termination, the price of which could differ greatly from the sell/buyback forward price. 

A zero-coupon bond is the simplest fixed-income security. It makes no coupon payments during its lifetime. Instead, it is a discount instrument, issued at a price that is below the face, or principal, amount. The rate earned on a zero-coupon bond is also referred to as the spot interest rate. The notation P t, T) denotes the price at time r of a discount bond that matures at time T, where T >t - The bond s term to maturity, T - t, is... 

This section describes the relationships among spot interest rates and the actual market yields on zero-coupon and coupon bonds. It explains how an implied spot-rate curve can be derived from the redemption yields and prices observed on coupon bonds, and discusses how this curve may be used to compare bond yields. Note that, in contrast with the common practice, spot rates here refer only to rates derived from coupon-bond prices and are distinguished from zero-coupon rates, which denote rates actually observed on zero-coupon bonds trading in the market. 

Equation (16.7) differs from the conventional redemption yield formula in that every cash flow is discounted, not by a single rate, but by the zero-coupon rate corresponding to the maturity period of the cash flow. To apply this equation, the zero-coupon-rate term structure must be known. These rates, however, are not always readily observable. Treasury prices, on the other hand, are and can be used to derive implied spot interest rates. (Although in the market the terms are used interchangeably, from this point on, zero coupon will be used only of observable rates and... 

Forward rates can be calculated using the no-arbitrage argument. We use this basic premise to introduce the concept of OAS. Consider Table 12.1, which shows the spot interest rates for two interest periods. We can determine that the one-period interest rate starting one period from now is 7.009 percent. This is the implied one-period forward rate. 

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The Spot and Forward Rate Relationship In the discussion to date, we have assumed discrete time intervals and interest rates in discrete time. Here, we consider the relationship between spot and forward rates in continuous time. For this, we assume the mathematical convenience of a continuously compounded interest rate. 

In this section, we describe the relationship between the price of a zero-coupon bond and spot and forward rates. We assume a risk-free zero-coupon bond of nominal value 1, priced at time t and maturing at time T. We also assume a money market bank account of initial value P t, T) invested at time t. The money market account is denoted M. The price of the bond at time t is denoted P t, T) and if today is time 0 (so that t > 0), then the bmid price today is unknown and a random factor (similar to a future interest rate). The bond price can be related to the spot rate or forward rate that is in force at time t. 

What is the importance of this result for our understanding of the term structure of interest rates First, we see (again, but this time in continuous time) that spot rates, forward rates and the discount function are all closely related, and... 

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

In the previous chapter, and indeed in previous analysis, we have defined the forward rate as the interest rate applicable to a loan made at a future point in time and repayable instantaneously. We assume that the dynamics of the forward rate follow a Wiener process. The spot rate is the rate for borrowing undertaken now and maturing at T, and we know from previous analysis that it is the geometric average of the forward rates from 0 to T that is... 

Model inputs Arbitrage models use the term structure of spot rate as an input, and this data is straightforward to obtain. Equilibrium models require a measure of the investor s market risk premium, which is rather more problematic. Practitioners analyse historical data on interest rate movements, which is considered less desirable. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

The starting point is that we set discount curves in all the main currencies, which are the relevant OIS curve. We can extract the discount factors for each currency from these curves, which we call DfccY for general discount factor and Dfojs for the relevant discount factor for the OIS in that currency. If we assume that EX rates are not correlated to interest rates (a big assumption, but necessary in this analysis), this implies that forward FX rates - which are a deposit product, as forward EX rates are simply spot FX rate adjusted for the deposit interest rate in each currency - are not a function of the discounting level in each currency. This further implies that the ratio of forward discount factors is constant. 

The assumption of forward FX rates being uncorrelated to funding rates is perhaps the biggest issue for discussion. Certainly one would be right to state that FX spot rates do have positive correlation with changes in interest rates. The impact is greater where one of the currencies is a core currency such as USD, EUR, GBP and possibly CHF, which are held as reserve deposits by other... 

In the spot-rate scenario where the expected future rate is high, the interest rate r(T) will exert very little influence, while it exerts more weight at lower levels. Therefore, the forward rate will be lower than the expected spot rate, and this is described below, where... 

In chapter 2 of the companion volume to this book in the boxed-set library, Corporate Bonds and Structured Financial Products, we introduced the concept of the yield curve, and reviewed some preliminary issues concerning both the shape of the curve and to what extent the curve could be used to infer the shape and level of the yield curve in the future. We do not know what interest rates will be in the future, but given a set of zero-coupon (spot) rates today we can estimate the future level of forward rates using a yield curve model. In many cases however we do not have a zero-coupon curve to begin with, so it then becomes necessary to derive the spot yield curve from the yields of coupon bonds, which one can observe readily in the market. If a market only trades short-dated debt instruments, then it will be possible to construct a short-dated spot curve. 

It is important for a zero-coupon yield curve to be constructed as accurately as possible. This because the curve is used in the valuation of a wide range of instruments, not only conventional cash market coupon bonds, which we can value using the appropriate spot rate for each cash flow, but other interest-rate products such as swaps. 

These models are two more general families of models incorporating Vasicek model and CIR model, respectively. The first one is used more often as it can be calibrated to the observable term structure of interest rates and the volatility term structure of spot or forward rates. However, its implied volatility structures may be unrealistic. Hence, it may be wise to use a constant coefficient P(t) = P and a constant volatility parameter a(t) = a and then calibrate the model using only the term structure of market interest rates. It is still theoretically possible that the short rate r may go negative. The risk-neutral probability for the occurrence of such an event is... 

Now let s turn to the interest rates that should be used for discounting. In Chapter 3, we emphasized two things. First, every cash flow should be discounted at its own discount rate using a spot rate. So, if we discounted a cash flow of 1 using the spot rate for period t, the present value would be... 

A. Biihler and H. Zimmerman, A Statistical Analysis of the Term Structure of Interest Rates in Switzerland and Germany, Journal of Fixed Income 6, no. 3 (December 1996), pp. 55-67. Germany (1988-96)—Spot ZC Switzerland IM-lOY 3 71/18/4 75/16/3... 

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... 

The first two chapters of this section discuss bond pricing and yields, moving on to an explanation of such traditional interest rate risk measures as modified duration and convexity. Chapter 3 looks at spot and forward rates, the derivation of such rates from market yields, and the yield curve. Yield-curve analysis and the modeling of the term structure of interest rates are among the most heavily researched areas of financial economics. The treatment here has been kept as concise as possible. The References section at the end of the book directs interested readers to accessible and readable resources that provide more detail. 

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