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Chapter 8: Problem 69

Use the future value formulas for simple and compound interest in one year to derive the formula for effective annual yield.

Short Answer

Expert verified
The formula for effective annual yield is \( EY = (1 + r/n)^n - 1 \).

Step by step solution

01

Set up the future value formula for simple interest

Begin with the formula for future value with simple interest, which is \( FV = PV + PV \cdot r \), where \( FV \) is the future value, \( PV \) is the present value, and \( r \) is the interest rate.
02

Set up the future value formula for compound interest

Next, set up the equation for future value with compound interest, which is \( FV = PV(1 + r/n)^(nt) \), where \( n \) is the number of compounding periods in a year and \( t \) is the number of years. Since we are considering one year, this simplifies to \( FV = PV(1 + r/n)^n \).
03

Set both future values equal to each other

In order to derive the formula for effective annual yield, set the two future values equal to each other and solve for \( r \). This gives \( PV + PV \cdot r = PV(1 + r/n)^n \). Simplifying further, we get \( 1 + r = (1 + r/n)^n \).
04

Derive the formula for effective annual yield

Rearranging the previous step's equation, the effective annual yield \( EY \) can be found since \( EY = r \), the simple interest rate. Isolating \( r \) gives \( EY = (1 + r/n)^n - 1 \).

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