PROYECTO LUMÍNICO MEDIANTE EXPERIMENTACIÓN

later date T . We denote the discount factor between these two dates by Z(t, T ).

In the above example, the two dates are t = August 10, 2006 and T = February 08, 2007. The discount factor is Z (t, T ) = 0.97477.

In short, the discount factor Z(t, T ) records the time value of money between t and T . Since it is a value (what is the value today of $1 in the future), it is essentially a price, describing how much money somebody is willing to pay today in order to have $1 in the future. In this sense, the notion of a discount factor is un-ambiguous. In contrast, as we shall see below, the related notion of an interest rate is not un-ambiguous, as it depends on compounding frequency, for instance. Exactly because discount factors unambiguously represent a price – an exchange rate between money today versus money tomorrow – they are at the heart of fixed income securities analysis. In the following sections we describe their characteristics in more detail.

2.1.1 Discount Factors across Maturities

Definition 2.1 and Example 2.1 highlight that the discount factor at some date t (e.g., August 10, 2006) depends on its maturity T (e.g., February 8, 2007). If we vary the maturity T , making it longer or shorter, the discount factor varies as well. In fact, for the same reason that investors value $1 today more than $1 in six months, they also value $1 in three months more than $1 in six months. This can be seen, once again, from the prices of U.S. Treasury securities.

EXAMPLE 2.2

On August 10, 2006 the U.S. government also issued 91-day bills with a maturity date of November 9, 2006. The price was $98.739 for $100 of face value. Thus, denoting again t = August 10, 2006, now T₁ = November 9, 2006, and T₂ = February 8, 2007, we find that the discount factor Z(t, T1) = 0.98739, which is higher than Z(t, T2) = 0.97477.

1These data are obtained from the Web site http://www.treasurydirect.gov/RI/OFBills, accessed on August 22, 2006.

DISCOUNT FACTORS 31

This example highlights an important property of discount factors. Because it is always the case that market participants prefer $1 today to $1 in the future, the following is true:

Fact 2.1 At any given time t, the discount factor is lower, the longer the maturity T . That is, given two dates T1and T2with T1< T2, it is always the case that

Z(t, T1) ≥ Z(t, T2) (2.1)

The opposite relation Z(t, T1) < Z(t, T2) would in fact imply a somewhat curious behavior on the part of investors. For instance, in the example above in which T₁ = November 9, 2006 and T2 = February 8, 2007, if Z(t, T1) was lower than Z(t, T2) = 0.97477, it would imply that investors would be willing to give up $97.477 today in order to receive $100 in six months, but not in order to receive the same amount three months earlier. In other words, it implies that investors prefer to have $100 dollar in six months rather than in three months, violating the principle that agents prefer to have a sum of money earlier rather than later. Moreover, a violation of Relation 2.1 also generates an arbitrage opportunity, which we would not expect to last for long in well functioning financial markets (see Exercise 1). In Chapter 5 we elaborate on this topic, showing also that a violation of Relation 2.1 amounts to the assumption that future nominal interest rates be negative.

2.1.2 Discount Factors over Time

A second important characteristic of discount factors is that they are not constant over time, even while keeping constant the time-to-maturity T−t, that is, the interval of time between the two dates t and T in the discount factor Z(t, T ). As time goes by, the time value of money changes. For instance, the U.S. Treasury issued a 182-day bill on t1= August 26, 2004, with maturity T₂ = February 24, 2005, for a price of $99.115. This price implies a discount factor on that date equal to Z(t₁, T₁) = 0.99115. This value is much higher than the discount factor with the same time to maturity (six months) two years later, on August 10, 2006, which we found equal to 0.97477.

Figure 2.1 plot three discounts factors over time, from January 1953 to June 2008.²The top solid line is the 3-months discount factor, the middle dotted line is the 1-year discount factor, and bottom dashed line is the 3-year discount factor. First, note that indeed on each date in the sample, the discount factor with shorter time to maturity is always higher than the discount factor with longer time to maturity. Second, the variation of discount factors over time is rather substantial. For instance, the 3-year discount factor is as low as 0.6267 in August 1981, and as high as 0.95 in June 1954 and in June 2003.

Why do discount factors vary over time? Although this is a topic of a later chapter, it is useful to provide here the most obvious, and intuitive, reason. Figure 2.2 plots the time series of expected inflation from 1953 to 2008.³ Comparing the discount factor series

2Data excerpted from CRSP ( Fama Bliss discount bonds) ©2009 Center for Research in Security Prices (CRSP), The University of Chicago Booth School of Business. We discuss methodologies to estimate discount factors from bond data in Section 2.4.2 and in the Appendix.

3The expected inflation series is computed as the predicted annual inflation rate resulting from a rolling regression of inflation on its 12 lags. We present more details in Chapter 7.

32 BASICS OF FIXED INCOME SECURITIES

Figure 2.1 Discount Factors

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 60

65 70 75 80 85 90 95 100

Discount Factor (%)

3 months 1 year 3 years

Source: Center for Research in Security Prices (CRSP)

plotted in Figure 2.1 with the expected annual inflation series in Figure 2.2 it appears that expected inflation is an important determinant of discount factors. The intuition is also quite straightforward: Inflation is exactly what determines the time value of money, as it determines how much goods money can buy. The higher the expected inflation, the less appealing it is to receive money in the future compared to today, as this money will be able to buy a lesser amount of goods.

Although expected inflation is the most obvious culprit in explaining the variation over time of discount factors, it is not the only one. In Chapter 7 we look at various explanations that economists put forward to account for the behavior of discount factors and interest rates. These explanations are related to the behavior of the U.S. economy, its budget deficit, and the actions of the Federal Reserve, as well as investors’ appetite for risk (or lack thereof). These macro economic conditions affect the relative supply and demand of Treasury securities and thus their prices.

2.2 INTEREST RATES

Grasping the concept of a rate of interest is both easier and more complicated than absorbing the concept of a discount factor. It is easier because the idea of interest is closer to our everyday notion of a return on an investment, or the cost of a loan. For instance, if we invest

$100 for one year at the rate of interest of 5%, we receive in one year $105, that is, the original capital invested plus the interest on the investment. The same investment strategy could be described in terms of a discount factor as well: The discount factor here is the

INTEREST RATES 33

Figure 2.2 Expected Inflation

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

−2

Data Source: Bureau of Labor Statistics.

exchange rate between having $105 in one year or $100 today, that is 0.9524 = $100/$105.

This latter number, which is equivalent to the 5% rate of interest, perhaps less intuitively describes the return on an investment.

The concept of an interest rate, however, is also more complicated, because it depends on the compounding frequency of the interest paid on the initial investment. The compounding frequency is defined as follows:

Definition 2.2 The compounding frequency of interest accruals refers to the annual