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Finance

CFA Level 1

Volume 1 - Quantitative Methods

Future Value Formula

$$FV_N = PV(1 + r)^N$$

The future value after N time periods ($FV_N$) is equal to the present value ($PV$) multiplied by one plus the interest rate ($r$) raised to the $N_{th}$ power.

  • Note that both the interest rate $r$ and the time period $N$ must be in the same units (i.e. if $N$ is stated in months, then $r$ should be the monthly interest rate, unannualized).

  • Another way of saying this is that the future value after $N$ periods ($FV_N$) is the present value ($PV$) scaled by a factor of $(1 + r)^N$.

Compounding

The future value formula (above) is used when interest is compounded annually. When interest is compounded more frequently, one uses:

$$FV_N = PV \left( 1 + \frac{r_s}{m} \right)^{mN}$$

This can be used for any discrete division of a time interval. When interest is compounded continuously, the formula becomes:

$$FV_N = PVe^{r_sN}$$

  • $r_s$ = stated annual interest rate/Quoted interest rate (when interest is compounded more frequently than annually, financial institutions often state the annual interest rate as opposed to breaking it down into more precise units).

  • $m$ = Number of compounding periods per year (i.e. 12 if compounded monthly).

  • $N$ = Years ($N$ can always be interpreted as years in the above equation).

  • $FV_N$ = Future value after $N$ years.

  • $PV$ = Present value.

Annual Percentage Yield (APY)

As one can see from the above formulas, the effective annual rate (synonymous to annual percentage yield) is often slightly different than the stated annual interest rate, which can make a large difference over time.

This leads to the need for a "standardized" rate, a rate that allows interest rates that are compounded at different frequencies to be compared. Enter APY. The annual percentage yield gives one the ability to aptly compare stated annual interest rates with different compounding periods.

If compounding is discrete (most often the case):

$$APY (EAR) = \left( 1 + \frac{r_s}{m} \right)^m - 1$$

If compounding is continuous:

$$APY (EAR) = e^{r_s} - 1$$

Resources

Market Data Processing