Engineering Economic Analysis

116 MORE INTERESTFORMULAS -! - -- ....-

r

• m = Interest rate per interest period

mn = Number of compounding subperiods innyears

Single Payment Interest Factors: Continuous Compounding

The single payment compound amount formula (Equation 3-3)

may be rewritten as

If we increasem,the number of compoundingsubperiodsper year, withoutlimit,mbecomes

very large and approaches infinity, andr/mbecomes very small and approaches zero.

This is the condition ofcontinuous compounding,that is, where the duration of the

interest period decreases from some finite durationD.tto an infinitely small durationdt,

and the number of interest periods per year becomes infinite. In this situation of continuous_

compounding:

F=P lim

(

1+..c.

)

mn

m~oo m

. (4-35)

An important limit in calculus is:

x~olim(1 +X)l/x= 2.71828 =e (4-36)

If we setx=r/m,thenmnmay be written as (1/x)(rn).Asmbecomesinfinite,xbecomesO.

Equation 4-35 becomes

F=P[lim(1x-+o +x)l/xyn

Equation4-36 tells us the quantity inside the bracketsequalse.So returning to Equation 3-3,

we find that

F=P(l+it becomes F=Pern (4-37)

and

P =F(1+i)-n becomes P=Fe-rn (4-38)

We see that for continuous compounding,

or, as shown earlier,

Effective interest rate per year (4.,34)