Discount function

The two latter difficulties can be resolved by relaxing the assumption of stable preferences and full information about future mental states. In fact, if we allow actors future orientation (that is, their discount function) to fluctuate over time, actors may move in and out of the high consumption mode (Skog 1997). However, according to George Ainslie (1992), the basic assumption of dynamic consistency in Becker and Murphy s theory is at variance with the empirical evidence. Ainslie claims that people... 

This function, which includes the cognitive element of willpower, represents Ainslie s complete discount function. It discounts the future less than the original hyperbolic function (Skog 1997) and is also less deeply bowed. 

However, even a less radical theory of discounting than the hyperbolic theory can solve the difficulties of Becker and Murphy s addiction theory. It is sufficient to assume that people s rate of discounting typically fluctuate unsystematically over time and that people are therefore not always equally farsighted. The theory of fluctuating discount functions postulate that people discount the future recursively and that they base their judgments on realistic expectations about their own future mental states. Hence, this theory can be seen as a straightforward extension of classical rational choice theory. Still, these consu-... 

Hence, the picoeconomic theory of addiction and the modified version of the rational choice theory of addiction obtained by allowing for fluctuations in discount functions do in fact have a fairly large common core. 

Manufacturers of prepolymers will often provide the weight of curative per 100 units of prepolymer. This is based on the midpoint NCO for the grade and the normal index value for the curative. The calculations can readily be carried out using a calculator or spreadsheet. From a practical point of view, a set of shop scales with a discount function has been used to provide the weight required. 

Equation (3.22) describes the bond price as a function of the spot rate only, as opposed to the multiple processes that apply for aU the forward rates from t to T. As the bond has a nominal value of 1, the value given by Equation (3.22) is the discount factor for that term the range of zero-coupon bond prices would give us the discount function. 

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... 

At time 1, the discount function is specified by two possible functions / P(O) and P 0) which correspond respectively to the upside and the downside outcomes. Therefore, at time n, the binomial process is given by the discount function ( ) which can move upwards to a function pf ( ) and downwards to a function. ) for = 0 to n. 

The two functions specify the deviations of the discount functions from the implied forward functions. To satisfy arbitrage-free conditions, they define an implied binomial probability tt that is independent of time T, while the initial discoxmt function P(T) is given by ... 

From an elementary understanding of the markets, we know that there is a relationship between a set of discount factors, and the discount function, the par yield curve, the zero-coupon yield curve and the forward yield curve. If we know one of these functions, we may readily compute the other three. In practice, although the zero-coupon yield curve is directly observable from the yields of zero-coupon... 

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 ... 

A discount factor is a value for a discrete point in time, whereas markets often prefer to think of a continuous value of discount factors that applies a specific discount factor to any time t. This is known as the discount function, which is the continuous set of discrete discount factors and is indicated by df, = S(t,). 

The discount function relates the current cash bond yield curve with the spot yield curve and the implied forward rate yield curve. From Equation (5.3) we can set ... 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

One of the main criticisms of cubic and polynomial functions is that they produce forward rate curves that exhibit unrealistic properties at the long end, usually a steep fall or rise in the curve. A method proposed by Vasicek and Fong (1982) avoids this feature, and produces smoother forward curves. Their approach characterises the discount function as exponential in shape, which is why splines, being polynomials, do not provide a good fit to the discount function, as they have a different curvature to exponential functions. Vasicek and Fong instead propose a transform to the argument T of the discount function v(T). This transform is given by... 

The technique proposed by McCulloch (1975) used aregression cubic spline to approximate the discount function, and he suggested that the number of node points that are used be roughly equal to the square root of the number of bonds in the sample, with equal spacing so that an equal number of bonds mature between adjacent nodes. A number of writers have suggested that this approach produces accurate results in practice. The discount function is constrained to set v(0) = 1. Given these parameters the discount function chosen is the one that minimises the function (5.14). As this is a discount function and not a yield curve, Equatimi (5.14) can be solved using the least squares method. 

The term structure method described by McCulloch (1971) involved fitting a discount function, rather than a spot curve, using the market prices of a sample... 

Using any of the methods described in Chapter 5 or the discount function approach summarised above, we can construct curves for both the nominal and the real implied forward rates. These two curves can then be used to infer market expectations of future inflation rates. The term stmcture of forward inflation rates is obtained from both these curves by applying the Fisher identity ... 

We use Bloomberg Generic closing prices (BGN prices) for bonds. The inputs of the model are market gross prices of all instruments in the basket. The model we use falls into the category of discount function fitting models. ... 

The set of discount factors for every period from one day to thirty years and longer is termed the discount function. Since the following discussion is in terms of PV, discount factors may be used to value any financial instrument that generates future cash flows. For example, the present value... 

Expression (3.24) states that the market value of a risk-free bond on any date is determined by the discount function on that date. 

In practice, the term structure of coupon bonds is not complete, so the coefficients in (3.33) cannot be identified. To address this problem, McCulloch (1971, 1975) prescribes a spline estimation method that assumes zero-coupon bond prices vary smoothly with term to maturity. This approach defines price as a discount function of maturity, P N), which is a given by (3.34). 

The function/) (A0 is usually specified by setting the discount function as a polynomial. In certain texts, including McCulloch, this is done by applying a spline function, which is discussed in the next chapter. (For further information, see the References section, particularly Suits et al (1978).)... 

There has been a good deal of research on the empirical estimation of the term structure, the object of which is to construct a zero-coupon or spot curve (or, equivalently, a forward-rate curve or discount function) that represents both a reasonably accurate fit to market prices and a smooth function—that is, one with a continuous first derivative. Though every approach must make some trade offbetween these two criteria, both are equally important in deriving a curve that makes economic sense. 

These considerations introduce what is known in statistics as error or noise into market prices. To handle this, smoothing techniques are used in the derivation of the discount function. 

FIGURE 5.2 Discount Function Derived from U.S. Treasury Prices on December P.r P003 ... 

FIGURE 5.2 is the graph of the discount function derived by bootstrapping from the U.S. Treasury prices as of December 23, 2003- FIGURE 5.3 shows the zero-coupon yield and forward-rate curves corresponding to this discount function. Compare these to the yield curve in FIGURE 5.1... 

FIGURE 5.3 Zero-Coupon (Spot) Yield and Forward-Rate Curves Corresponding to the Discount Function ... 

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