Time value of money: present value and future value

The time value of money (TVM) is the idea that a peso available today is worth more than a peso you will receive in the future. Why? Because today's peso can be invested and earn interest. This concept underpins almost the entire CFA curriculum: valuing bonds, stocks, investment projects and much more.

Amounts in this article are in Mexican pesos (MXN), but the math works the same in any currency.

Future value (FV)

If you invest an amount today at an interest rate $r$ per period, for $N$ periods, with compound interest (interest also earns interest), at the end you will have:

$$FV = PV \times (1 + r)^N$$

Where:

• $PV$ = present value (the amount you invest today)
• $r$ = interest rate per period
• $N$ = number of periods

Present value (PV)

Solving the previous formula for $PV$ tells us what an amount we will receive in the future is worth today:

$$PV = \frac{FV}{(1 + r)^N}$$

This process is called discounting, and $r$ is called the discount rate.

Compounding more than once a year

Many rates are quoted as an annual rate, but interest is calculated more often (monthly, quarterly, etc.). If the stated annual rate is $r_s$ and it is compounded $m$ times per year:

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

Here is how the future value of $1,000 at a 12% annual rate changes over one year:

The more frequent the compounding, the higher the future value, although each increase is smaller. The limit is continuous compounding: $FV = PV \times e^{r_s N}$.

Effective annual rate (EAR)

To compare rates with different compounding frequencies, we use the effective annual rate: what the investment actually earns in one year.

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

For example, a 12% annual rate compounded monthly is equivalent to an EAR of $(1.01)^{12} - 1 = 12.68\%$.

Summary

• A peso today is worth more than a peso tomorrow because it can be invested.
• Future value: $FV = PV(1 + r)^N$. Present value: $PV = FV / (1 + r)^N$.
• With compounding $m$ times per year, use $r_s / m$ per period and $mN$ periods.
• The effective annual rate lets you compare rates with different compounding frequencies.