It is evident from Table 5.1 (and Equation 5.8) that the difference between APR and EAR grows with the frequency of compounding. This raises the question: How far will these two rates diverge as the compounding frequency continues to grow? Put differ- ently, what is the limit of [1 + T × APR]1/ T, as T gets ever smaller? As T approaches zero, we effectively approach continuous compounding (CC), and the relation of EAR to the
Table 5.1
Annual percentage rates (APR) and effective annual rates (EAR). In the first set of columns, we hold the equivalent annual rate (EAR) fixed at 5.8% and find APR for each holding period. In the second set of columns, we hold APR fixed at 5.8% and solve for EAR.
Compounding Period
EAR = [1 + rf (T ) ] 1/T – 1 = 0.058 APR = rf (T ) × (1 / T ) = 0.058 T rf (T ) APR = [(1 + EAR )T – 1] / T rf (T ) EAR = (1 + APR × T )(1/T ) – 1
1 year 1.0000 0.0580 0.05800 0.0580 0.05800
6 months 0.5000 0.0286 0.05718 0.0290 0.05884
1 quarter 0.2500 0.0142 0.05678 0.0145 0.05927
1 month 0.0833 0.0047 0.05651 0.0048 0.05957
1 week 0.0192 0.0011 0.05641 0.0011 0.05968
1 day 0.0027 0.0002 0.05638 0.0002 0.05971
Continuous rcc = ln(1 + EAR) = 0.05638 EAR = exp(rcc ) − 1 = 0.05971
e X c e l
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annual percentage rate, denoted by rcc for the continuously compounded case, is given by the exponential function
1 + EAR = exp (rcc) = e r cc (5.9) where e is approximately 2.71828.
To find rcc from the effective annual rate, we solve Equation 5.9 for rcc as follows:
ln(1 + EAR) = rcc
where ln(•) is the natural logarithm function, the inverse of exp(•). Both the exponen- tial and logarithmic functions are available in Excel and are called EXP(•) and LN(•), respectively.
The continuously compounded annual percentage rate, rcc, that provides an EAR of 5.8%
is 5.638% (see Table 5.1). This is virtually the same as the APR for daily compounding.
But for less frequent compounding, for example, semiannually, the APR necessary to provide the same EAR is noticeably higher, 5.718%. With less frequent compounding, a higher APR is necessary to provide an equivalent effective return.
Example 5.5
Continuously Compounded RatesWhile continuous compounding may at first seem to be a mathematical nuisance, work- ing with such rates can sometimes simplify calculations of expected return and risk. For example, given a continuously compounded rate, the total return for any period T, rcc(T ), is simply exp (T × rcc ).4 In other words, the total return scales up in direct proportion to the time period, T. This is far simpler than working with the exponents that arise using discrete period compounding. As another example, look again at Equation 5.1. There, the relationship between the real interest rate rreal, the nominal rate rnom, and the inflation rate i, rreal ≈ rnom − i, was only an approximation, as demonstrated by Equation 5.3. But if we express all rates as continuously compounded, then Equation 5.1 is exact,5 that is, rcc(real) = rcc (nominal) − icc .
4This follows from Equation 5.9. If 1 + EAR = e r cc , then (1 + EAR)T = e r cc T.
51 + r (real) = 1 + _____________1 + inflationr (nominal)
⇒ ln [ 1 + r (real) ] = ln
( 1 + r (nominal) _____________
1 + inflation
) = ln [ 1 + r (nominal) ] − ln(1 + inflation)
⇒ r (real) = r (nominal) − i
A bank offers two alternative interest schedules for a savings account of $100,000 locked in for 3 years:
(a) a monthly rate of 1%; and (b) an annually, continuously compounded rate, rcc, of 12%. Which alternative should you choose?
Concept Check 5.2
5.3 Bills and Inflation, 1926–2015
Financial time series often begin in July 1926, the starting date of a widely used return database from the Center for Research in Security Prices at the University of Chicago.
Table 5.2
Statistics for T-bill rates, inflation rates, and real rates, 1926–2015
Sources: Annual rates of return from rolling over 1-month T-bills: Kenneth French; annual inflation rates: Bureau of Labor Statistics.
Average Annual Rates Standard Deviation T-Bills Inflation Real T-Bill T-Bills Inflation Real T-Bill
All months 3.46 3.00 0.56 3.12 4.07 3.81
First half 1.04 1.68 −0.29 1.29 5.95 6.27
Recent half 4.45 3.53 0.90 3.11 2.89 2.13
–15 –10 –5 0
Percentage Points
5 10 15 20
T-Bills Inflation
1926 1936 1946 1956 1966 1976 1986 1996 2006 2016
Figure 5.2 Interest and inflation rates, 1926–2015
Table 5.2 summarizes the history of returns on 1-month U.S. Treasury bills, the inflation rate, and the resultant real rate. You can find the entire post-1926 history of the monthly rates of these series in Connect (link to the material for Chapter 5).
The first set of columns of Table 5.2 lists average annual rates for three periods. The average interest rate over the more recent portion of our history, 1952–2015 (essentially the post-war period), 4.45%, was noticeably higher than in the earlier portion, 1.04%. The rea- son is inflation, the main driver of T-bill rates, which also had a noticeably higher average value, 3.53%, in the later portion of the sample than in the earlier period, 1.68%. Neverthe- less, nominal interest rates in the recent period were still high enough to leave a higher aver- age real rate, .90%, compared with a negative 29 basis points (–.29%) for the earlier period.
Figure 5.2 shows why we divide the sample period at 1952. After that year, inflation is far less volatile, and, probably as a result, the nominal interest rate tracks the inflation rate with far greater precision, resulting in a far more stable real interest rate. This shows up as the dramatic reduction in the standard deviation of the real rate documented in the last column of Table 5.2. Whereas the standard deviation is 6.27% in the early part of the
sample, it is only 2.13% in the later portion. The lower standard deviation of the real rate in the post-1952 period reflects a similar decline in the standard deviation of the inflation rate. We conclude that the Fisher relation appears to work far better when inflation is itself more predictable and investors can more accurately gauge the nominal interest rate they require to provide an acceptable real rate of return.