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Classic

Compound Interest

Basics

Compound Interest: Capital is invested again at the end of each period to produce new interest

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

Formula

Value Acquired

An=C0×(1+r)n \text{A}_\text{n} = \text{C}_0\times(1+\text{r})^n

  • AnA_n: Value acquired after n periods
  • C0C_0: Initial Capital
  • r: Rate of interest per period
  • n: Number of periods

Total Interest

Total Interest=C0×(1+r)nC0 \text{Total Interest} = \text{C}_0\times(1+\text{r})^n - C_0

Total Interest=AnC0 \text{Total Interest} = \text{A}_\text{n} - C_0

  • AnA_n: Value acquired after n periods
  • C0C_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?

Capitalization Example Image Capitalization Example Image

Unique Investment

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

An=C0×(1+r)nA_n = C_0 \times (1 + r)^n

Where:

  • AnA_n: Value acquired after n periods
  • C0C_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:

An=a×(1+r)n1rA_n = a \times \frac{(1+\text{r})^{n}-1}{\text{r}}

Where:

  • AnA_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) => V0V_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?

Discounting Example Image Discounting Example Image

Unique Investment

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

C0=Cn×(1+r)n=Cn(1+r)nC_0 = C_n \times (1+\text{r})^{-\text{n}} \\ = \frac{C_n}{(1 + r)^n}

Where:

  • C0C_0: Initial Capital
  • CnC_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:

C0=a×1(1+r)nrC_0 = a \times \frac{1 - (1 + r)^{-n}}{\text{r}}

Where:

  • C0C_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=rannual12r = \frac{r_{\text{annual}}}{12}
  • Annual to Quarterly: r=rannual4r = \frac{r_{\text{annual}}}{4}
  • Annual to Semi-Annual: r=rannual2r = \frac{r_{\text{annual}}}{2}

Formula

Tn=Tanumber of n in one yearT_n = \frac{T_a}{\text{number of n in one year}}

Where:

  • TnT_n: The rate for n periods (monthly, weekly, etc...)
  • TaT_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+Tn)n=1+TaTn=(1+Ta)1n1(1+T_n)^{n} = 1+T_a \\ T_n = (1 + T_a)^{\frac{1}{n}} - 1

Where:

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

Examples

  • Annual to monthly: Tn=(1+Ta)1121T_n = (1 + T_a)^{\frac{1}{12}} - 1
  • Annual to quarterly: Tn=(1+Ta)141T_n = (1 + T_a)^{\frac{1}{4}} - 1
  • Annual to semi-annual: Tn=(1+Ta)121T_n = (1 + T_a)^{\frac{1}{2}} - 1

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