Future Value Calculations

How to calculate the future value of some cash.

• A dollar now is worth more than a dollar tomorrow.
• Can be denoted in basis points as well.
• Reverse of Present Value Calculations

Definitions §

• $k$: periods per year
• $\tau$: duration (in years)
• $n$: number of compounding periods.
  • $n=\tau \cdot k$
• $F_0$: Present value = Principal = Amount lent/borrowed at time $t_0$
  • $F_{\tau }$: Future Value = Amount at time $t_0+\tau$
• $r$: annual interest
• $R$: Interest Rate

Also:

• Banker’s Rule is a common way to simplify math calculations.
• $\text{Discount Rate}=\frac{1}{1+\tau \cdot r}$.

def. Simple Interest ⇒ Only principal is invested at the end of each year

$$F_{\tau }=(1+\tau \cdot r)F_0$$

def. Compound Interest Divide into $k$ periods per year, and compound for $n$ periods:

$$F_{n\text{periods}}={\left(1+\frac{r}{k}\right)}^n{F}_0$$

Divide into $k$ periods per year, and compound for $\tau$ years (equivalent formula):

$$F_{\tau \text{ years}}={\left(1+\frac{r}{k}\right)}^{\tau k}{F}_0$$

⇒ Conceptual Demonstration of Compounding Interest:

def. Continuous Compounding Compound for $\tau$ years (equivalent formula):

$$F(x)=e^{\tau \cdot r}\cdot F_0$$

⇒ You can get this formula from compound interest and setting $k\to \infty$

def. Fractional Compounding

$$F_x={\left(1+\frac{r}{k}\right)}^x{F}_0\text{ where }x\in {\mathcal{R}}^{+}$$

⇒ You can do things like ”$5.35\text{ months}$ of daily compounding.”

Note

For a fixed $\tau$, $F_k(\tau )$ is monotonic for $k$. ⇒ Proof in slides. (Using binomial expansion)