Investments, tenth edition

Compounding 

Period

EAR 5 [1 1 r

f

 (T )]

1/T

2

 1 5 .058

APR 5 r

f

 (T)*(1/T) 5 .058

T

r

f

 (T)

APR 5 [(1 1 EAR)^T 2 1]/T

r

f

 (T)

EAR 5 (1 1 APR*T)^(1/T) 2 1

1 year

1.0000

.0580

.05800

.0580

.05800

6 months

0.5000

.0286

.05718

.0290

.05884

1 quarter

0.2500

.0142

.05678

.0145

.05927

1 month

0.0833

.0047

.05651

.0048

.05957

1 week

0.0192

.0011

.05641

.0011

.05968

1 day

0.0027

.0002

.05638

.0002

.05971

Continuous

r

cc

 5 ln(1 1 EAR) 5 .05638

EAR 5 exp(r

cc 

) 2 1 5 .05971

 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. 

                    

e

X

c e l

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  C H A P T E R  

5

  Risk, Return, and the Historical Record 

125

period compounding. As another example, look again at Equation 5.1. There, the relation-

ship between the real rate  rr,  the nominal rate  rn,  and the inflation rate  i,   rr  <  rn   2   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,  

7

   that is,  rr  

 cc 

   5   rn  

 cc 

   2   i  

 cc 

 .

  

 

 

   

  

7

    1 1 r (real) 5

1 1 r (nominal)

1 1 inflation

1 ln31 1 r(real)4 5 ln a

1 1 r (nominal)

1 1 inflation

b 5 ln31 1 r(nominal)4 2 ln(1 1 inflation)

1 r

cc

(real) 5 r

cc

(nominal) 2 i

cc

  

 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%; ( b ) an annually, continuously compounded rate ( r  

 cc 

  ) of 12%. Which alternative 

should you choose? 

 CONCEPT CHECK 

5.2 

  Financial time series often begin in July 1926, the starting date of a widely used accu-

rate return database from the Center for Research in Security Prices at the University of 

Chicago. 

  Table 5.2  summarizes the history of short-term interest rates in the U.S., the inflation 

rate, and the resultant real rate. You can find the entire post-1926 history of the monthly 

rates of these series on the text’s Web site,   www.mhhe.com/bkm   (link to the student mate-

rial for Chapter 5). The real rate is computed from the monthly T-bill rate and the percent 

change in the CPI.  

 The first set of columns of  Table 5.2  lists average annual rates for the various series. 

The average interest rate over the more recent half of our history (1969–2012), 5.35%, was 

noticeably higher than in the earlier half, 1.79%. The reason is inflation, the main driver 

of T-bill rates, which also had a noticeably higher average value in the recent half of the 

sample, 4.36%, than in the earlier period, 1.74%. Nevertheless, nominal interest rates over 

the recent period were still high enough to leave a higher average real rate, 0.95%, com-

pared with a paltry 10 basis points (.10%) for the earlier half. 

    5.3 

Bills and Inflation, 1926–2012 

 Table 5.2 

 Statistics for 

T-bill rates, 

inflation rates, 

and real rates, 

1926–2012