Compound Interest

Basics

• Compound interest is more long-term
• Capital is invested again at the end

Formula

Value Acquired

• $A_n$: Value acquired after n periods
• $C_0$: Initial Capital
• r: Rate of interest per period
• n: Number of periods

Total Interest

• $A_n$: Value acquired after n periods
• $C_0$: Initial Capital
• r: Rate of interest per period
• n: Number of periods

Recap

Capitalization

Capitalization is when you capitalize or actualize an investment to a future date.

Example: You invest $1000 at 5% interest for 3 years. The interest is compounded annually. What is the value of the investment at the end of 3 years?

Unique Investment

If a single investment is made, the formula for the future value of the investment is:

$A_n = C_0 \times (1 + r)^n$

Where:

• $A_n$: Value acquired after n periods
• $C_0$: Initial Capital
• r: Rate of interest per period
• n: Number of periods

Multiple Investments (Annuities)

If multiple annuities are made at the end of each period, the formula for the future value of the investment is:

$A_n = a \times \frac{(1+\text{r})^{n}-1}{\text{r}}$

Where:

• $A_n$: Value acquired after n periods
• a: Annuity invested at the end of each period
• r: Rate of interest per period
• n: Number of periods

Discounting

Discounting is when you reduce the value of an investment to a present date (NPV) => $V_0$.

Example: You are offered an investment that will pay you $10,000 in 5 years. If you can earn 6% on your money, what is the most you should pay for this investment?

Unique Investment

If a single investment is made, the formula for the present value of the investment is:

$C_0 = C_n \times (1+\text{r})^{-\text{n}} \\ = \frac{C_n}{(1 + r)^n}$

Where:

• $C_0$: Initial Capital
• $C_n$: Value acquired after n periods
• r: Rate of interest per period
• n: Number of periods

Multiple Investments (Annuities)

If multiple annuities are made at the end of each period, the formula for the present value of the investment is:

$C_0 = a \times \frac{1 - (1 + r)^{-n}}{\text{r}}$

Where:

• $C_0$: Initial Capital
• a: Annuity invested at the end of each period
• r: Rate of interest per period
• n: Number of periods

Interest Rates

Proportional Rates

Proportional rates are used to convert rates from one period to another. For example, converting an annual rate to a monthly rate.

Examples

To convert an annual rate to monthly rate, we just divide the annual rate by 12 because there is 12 months in one year..

• Annual to Monthly: $r = \frac{r_{\text{annual}}}{12}$
• Annual to Quarterly: $r = \frac{r_{\text{annual}}}{4}$
• Annual to Semi-Annual: $r = \frac{r_{\text{annual}}}{2}$

Formula

$T_n = \frac{T_a}{\text{number of n in one year}}$

Where:

• $T_n$: The rate for n periods (monthly, weekly, etc...)
• $T_a$: Annual rate
• n: Number of periods in one year

Equivalent Rates

Equivalent rates are used to compare different rates. For equivalent, we use the formula:

$(1+T_n)^{n} = 1+T_a \\ T_n = (1 + T_a)^{\frac{1}{n}} - 1$

Where:

• $T_n$: The rate for n periods (monthly, weekly, etc...)
• $T_a$: Annual rate
• n: Number of periods in one year

Examples

• Annual to monthly: $T_n = (1 + T_a)^{\frac{1}{12}} - 1$
• Annual to quarterly: $T_n = (1 + T_a)^{\frac{1}{4}} - 1$
• Annual to semi-annual: $T_n = (1 + T_a)^{\frac{1}{2}} - 1$