Investments, tenth edition

  Continuous Compounding 

 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 

bod61671_ch05_117-167.indd   123

bod61671_ch05_117-167.indd   123

6/18/13   8:03 PM

6/18/13   8:03 PM

Final PDF to printer

---

124

P A R T   I I

  Portfolio Theory and Practice

two rates diverge as the compounding frequency continues to grow? Put differently, what 

is the limit of [1  1   T   3  APR] 

1/  T 

 , as  T  gets ever smaller? As  T  approaches zero, we effec-

tively approach  continuous compounding (CC),  and the relation of EAR to the annual 

percentage rate, denoted by  r  

 cc 

  for the continuously compounded case, is given by the 

exponential function

    

1 1 EAR 5 exp(r

cc

) 5 e

r

cc

 

 (5.9)   

 where   e  is approximately 2.71828. 

 To find  r  

 cc 

  from the effective annual rate, we solve Equation 5.9 for  r  

 cc 

  as follows:

ln(1 1 EAR) 5 r

cc

     

 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,  r  

 cc 

 , 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 Rates 

 While continuous compounding may at first seem to be a mathematical nuisance, work-

ing with such rates can actually simplify calculations of expected return and risk. For 

example, given a continuously compounded rate, the total return for any period  T,   r  

 cc 

 ( T ),  is 

simply exp( T   3   r  

 cc 

 ).  

6

   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 

6

 This follows from Equation 5.9. If    1 1 EAR 5 e

r

cc

,   then     (1 1 EAR)

T

5 e

r

cc

T

.