CFAI QM Time Value Formula Flashcards
For N = 1, the expression for the future value of amount PV is
FV1 = PV(1 + r)
Future value after N periods:
FVN = PV(1 + r)N
With more than one compounding period per year, the future value formula can
be expressed as
The expression for the future value of a sum in N years with continuous compounding is
The effective annual rate is calculated as follows:
With continuous compounding, we can solve for the effective annual rate as follows:
The Future Value of a N time period Ordinary Annuity:
Given a future cash flow that is to be received in N periods and an interest rate per
period of r, we can use the formula for future value to solve directly for the present
value as follows:
In general, with more than one compounding period in a year, we can express the formula for present value as
Because the annuity payment (A) is a constant in this equation, it can be factored out as a common term. Thus the sum of the interest factors has a shortcut expression:
To derive a formula for the present value of a perpetuity, we can modify Equation 10 to account for an infinite series of cash flows
As long as interest rates are positive, the sum of present value factors converges:
Solving Equation 2 for r and replacing the interest rate r with the growth rate g produces the following expression for determining growth rates:
The Number of Annual Compounding Periods Needed for an Investment to Reach a Specific Value
N = [ln(FV/PV)]/ln(1 + r)
To quickly approximate the number of periods, practitioners sometimes use an ad hoc rule called the Rule of 72
Divide 72 by the stated interest rate to get the approximate number of years it would take to double an investment at the interest rate.
