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continuous compounding
Compounding of interest an
infinite number of times per year
at intervals of microseconds.
CHAPTER 4 Time Value of Money 157
Comparing the calculator, table, and spreadsheet values, we can see that the
calculator and spreadsheet values agree generally with the values in Table 4.7 but
are more precise because the table factors have been rounded.
Continuous Compounding
In the extreme case, interest can be compounded continuously. Continuous
compounding involves compounding over every microsecond—the smallest time
period imaginable. In this case, m in Equation 4.18 would approach infinity.
Through the use of calculus, we know that as m approaches infinity, the equation
becomes
FV n (continuous compounding)PV (e in ) (4.19)
where e is the exponential function 10, which has a value of 2.7183. The future
value interest factor for continuous compounding is therefore
FVIF i,n (continuous compounding)e in (4.20)
EXAMPLE To find the value at the end of 2 years (n2) of Fred Moreno’s $100 deposit
(PV $100) in an account paying 8% annual interest (i0.08) compounded
continuously, we can substitute into Equation 4.19:
FV 2 (continuous compounding)$100e 0.082
$1002.7183 0.16
$1001.1735$117.35
Calculator Use To find this value using the calculator, you need first to find the
value of e 0.16 by punching in 0.16 and then pressing 2nd and then e x to get 1.1735.
10. Most calculators have the exponential function, typically noted by e x , built into them. The use of this key is especially
helpful in calculating future value when interest is compounded continuously.