CASO DE ESTUDIO: CASA AVELINO DUARTE
6. CASO DE ESTUDIO: CASA DAVID VIEIRA DE CASTRO
This discussion emphasizes also the crucial difference between a rate of return on an investment, and an interest rate, which are related, but different, concepts. The rate of return is indeed the difference between payoff and initial investment, divided by the latter.
In the example, 5% = ($105− $100)/$100 is the rate of return on the investment. The rate of interest corresponds instead to the (annualized) rate of return on the investment within the compounding period, but it differs from it otherwise. For instance, if the rate of interest is 5% and it accrues semi-annually, then within a six-month period the rate of return on the investment is 2.5%, that is, from $100 we have in six months $102.5. If we annualize this semi-annual return we obtain 5%, which corresponds to the rate of interest.
Note, however, the rate of interest and the rate of return differ for a one-year horizon. In one year, the original investment will pay $105.0625, as we obtained earlier, and thus the rate of return is 5.0625%>5%. When the horizon is longer, the discrepancy between the annualized interest rate figure and the annualized rate of return on the investment is larger.
2.2.1 Discount Factors, Interest Rates, and Compounding Frequencies The examples above illustrate that discount factors and interest rates are intimately related, once we make explicit the compounding frequency. Given an interest rate and its com-pounding frequency, we can define a discount factor. Vice versa, given a discount factor, we can define an interest rate together with its compounding frequency. In this section we make the relation explicit.
Two compounding frequencies are particularly important: semi-annual compounding and continuous compounding. The semi-annual compounding frequency is the standard benchmark, as it matches the frequency of coupon payments of U.S. Treasury notes and bonds. The continuous compounding, defined below, is also important, mainly for its analytical convenience. As we shall see, formulas and derivations are much simpler under the assumption that the interest on an investment accrues infinitely frequently. This is of course an abstraction, but a useful one.
INTEREST RATES 35
2.2.1.1 Semi-annual Compounding Let’s begin with an example:
EXAMPLE 2.3
Let t = August 10, 2006, and let T = August 10, 2007 (one year later). Consider an investment of $100 at t at the semi-annually compounded interest r = 5%, for one year. As mentioned earlier, this terminology means that after six months the investment grows to $102.5 = $100× (1 + 5%/2), which is then reinvested at the same rate for another six months, yielding at T the payoff:
Payoff at T = $105.0625 = ($100)× (1 + r/2) × (1 + r/2) = ($100) × (1 + r/2)2 Given that the initial investment is $100, there are no cash flows to the investor during the period, and the payoff at T is risk free, the relation between money at t ($100) and money at T (= $105.0625 = payoff at T ) establishes a discount factor between the two dates, given by
Z(t, T ) = $100
payoff at T = 1 (1 + r/2)2
This example underlies the following more general statement:
Fact 2.4 Let r2(t, T ) denote the (annualized) semi-annually compounded interest rate between t and T . Then r2(t, T ) defines a discount factor as
Z(t, T ) = 1
1 + r2(t,T )2 2×(T −t) (2.2)
The logic of this fact lies in the example above. The semi-annually compounded interest rate r2(t, T ) defines a payoff at maturity T given by
Payoff at T = Investment at t ×
1 + r2(t, T ) 2
2×(T −t) .
Since the payoff at T is known at t, the relation between investment today at t and the payoff at T defines the time value of money, and Z(t, T ) given in Equation 2.2 defines the rate of exchange between money at T and money at t.
Similarly, given a discount factor Z(t, T ), we can obtain the semiannually compounded interest rate. The following example illustrates the point.
EXAMPLE 2.4
On March 1, 2001 (time t) the Treasury issued a 52-week Treasury bill, with maturity date T = February 28, 2002. The price of the Treasury bill was $95.713. As we have learned, this price defines a discount factor between the two dates of Z(t, T ) = 0.95713. At the same time, it also defines a semi-annually compounded interest rate equal to r2(t, T ) = 4.43%. In fact, $95.713 × (1 + 4.43%/2)2 = $100. The
36 BASICS OF FIXED INCOME SECURITIES
semi-annually compounded interest rate can be computed from Z(t, T ) = 0.95713 by solving for r2(t, T ) in Equation 2.2:
Fact 2.5 Let Z(t, T ) be the discount factor between dates t and T . Then the semi-annually compounded interest rate r2(t, T ) can be computed from the formula
r2(t, T ) = 2 ×
1
Z(t, T )2×(T −t)1 − 1
(2.4)
2.2.1.2 More Frequent Compounding Market participants’ time value of money – the discount factor Z(0, T ) – can be exploited to determine the interest rates with any compounding frequency, as well as the relation that must exist between any two interest rates which differ in compounding frequency. More precisely, if we let n denote the number of compounding periods per year (e.g., n = 2 corresponds to semi-annual compounding), we obtain the following:
Fact 2.6 Let the discount factor Z(t, T ) be given, and let rn(t, T ) denote the (annualized) n-times compounded interest rate. Then rn(t, T ) is defined by the equation
Z(t, T ) = 1
For instance, a $100 investment at the monthly compounded interest rate r12(0, 1) = 5%
yields by definition
Thus, the monthly compounded interest rate r12(0, 1) = 5% corresponds to the discount factor Z(0, 1) = $100/$105.1162 = 0.95133, and vice versa.
2.2.1.3 Continuous Compounding. The continuously compounded interest rate is obtained by increasing the compounding frequency n to infinity. For all practical purposes, however, daily compounding – the standard for bank accounts – closely matches the continuous compounding, as we see in the next example.
INTEREST RATES 37
Table 2.1 Interest Rate and Compounding Frequency
Compounding Frequency n rn(t, t + 1)
Annual 1 5.000%
Semi-annual 2 4.939%
Monthly 12 4.889%
Bi-monthly 24 4.883%
Weekely 52 4.881%
Bi-weekly 104 4.880%
Daily 365 4.879%
Bi-daily 730 4.879%
Hourly 8760 4.879%
Continuous ∞ 4.879%
EXAMPLE 2.5
Consider the earlier example in which at t we invest $100 to receive $105 one year later. Recall that the annually compounded interest rate is r1(t, t + 1) = 5%, the semi-annually compounded interest rate is r2(t, t + 1) = 4.939%, and the monthly compounded interest rate is r12(t, t + 1) = 4.889%. Table 2.1 reports the n−times compounded interest rate also for more frequent compounding. As it can be seen, if we keep increasing n, the n− times compounded interest rate rn(t, t + 1) keeps decreasing, but at an increasingly lower rate. Eventually, it converges to a number, namely, 4.879%. This is the continously compounded interest rate. Note that in this example, there is no difference between the daily compounded interest rate (n = 252) and the one obtained with higher frequency (n > 252). That is, we can mentally think of continuous compounding as the daily compounding frequency.
Mathematically, we can express the limit of rn(t, T ) in Equation 2.6 as n increases to infinity in terms of the exponential function:
Fact 2.7 The continuously compounded interest rate r(t, T ), obtained from rn(t, T ) for n that increases to infinity, is given by the formula
Z(t, T ) = e−r(t,T )(T −t) (2.7) Solving for r(t, T ), we obtain
r(t, T ) = −ln (Z(t, T ))
T− t (2.8)
where “ln(.)” denotes the natural logarithm.
Returning to Example 2.5, we can verify Equation 2.8 by taking the natural logarithm of Z(t, T ) = $100/$105 = .952381 and thus obtaining
r(t, T ) = −ln(Z(t, t + 1))
1 = 4.879%
38 BASICS OF FIXED INCOME SECURITIES
2.2.2 The Relation between Discounts Factors and Interest Rates
The previous formulas shows that given a discount factor between t and T , Z(t, T ), we can define interest rates of any compounding frequency by using Equations 2.2, 2.5, or 2.7.
This fact implies that we can move from one compounding frequency to another by using the equalities implicit in these equations. For instance, for given interest rate rn(t, T ) with n compounding frequency, we can determine the continuously compounded interest rate r(t, T ) by solving the equation
e−r(t,T )(T −t)= Z(t, T ) = 1
1 + rn(t,T )n n×(T −t) (2.9)
Because of its analytical convenience, in this text we mostly use the continuously com-pounded interest rate in the description of discount factors, and for other quantities. Trans-lating such number into another compounding frequency is immediate from Equation 2.9, which, more explicitly, implies
r(t, T ) = n × ln
1 +rn(t, T ) n
(2.10) rn(t, T ) = n ×
er(t,T )n − 1
(2.11) To conclude, then, this section shows that the time value of money can be expressed equivalently through a discount factor, or in terms of an interest rate with its appropriate compounding frequency. At times, it will be convenient to focus on discount factors and at other times on interest rates, depending on the exercise. We should always keep in mind that the two quantities are equivalent.
2.3 THE TERM STRUCTURE OF INTEREST RATES
In the previous sections we noted that the primitive of our analysis is the discount factor, from which we define interest rates of various compounding frequencies. Interest rates, though, have a big advantage over discount factors when we analyze the time value of money: their units can be made uniform across maturities by annualizing them. The following example illustrates the point.
EXAMPLE 2.6
On June 5, 2008, the Treasury issued 13-week, 26-week and 52-week bills at prices
$99.5399, $99.0142, and $97.8716, respectively. Denoting t = June 5, 2008, and T1, T2, and T3the three maturity dates, the implied discount factors are Z(t, T1) = 0.995399, Z(t, T2) = 0.990142, and Z(t, T3) = 0.978716. The discount factor of longer maturities is lower than the one of shorter maturities, as Fact 2.1 would imply. The question is then: How much lower is Z(t, T3), say, compared to Z(t, T2) or Z(t, T1)? Translating the discount factors into annualized interest rates provides a better sense of the relative value of money across maturities. In this case, the continuously compounded interest rates are
r(t, T1) = −ln(0.995399)
0.25 = 1.8444%;
THE TERM STRUCTURE OF INTEREST RATES 39
r(t, T2) = −ln(0.990142)
0.5 = 1.9814%;
r(t, T3) = −ln(0.978716)
1 = 2.1514%.
The time value of money rises with maturity: The compensation that the Treasury has to provide investors to make them part with money today to receive money in the future, i.e., hold Treasury securities, increases the longer the investment period.
The term structure of interest rates is defined as follows:
Definition 2.3 The term structure of interest rates, or spot rate curve, or yield curve,