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
- : Value acquired after
n
periods - : Initial Capital
r
: Rate of interest per periodn
: Number of periods
Total Interest
- : Value acquired after
n
periods - : Initial Capital
r
: Rate of interest per periodn
: 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:
Where:
- : Value acquired after
n
periods - : Initial Capital
r
: Rate of interest per periodn
: 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:
Where:
- : Value acquired after
n
periods a
: Annuity invested at the end of each periodr
: Rate of interest per periodn
: Number of periods
Discounting
Discounting is when you reduce the value of an investment to a present date (NPV) => .
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:
Where:
- : Initial Capital
- : Value acquired after
n
periods r
: Rate of interest per periodn
: 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:
Where:
- : Initial Capital
a
: Annuity invested at the end of each periodr
: Rate of interest per periodn
: 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:
- Annual to Quarterly:
- Annual to Semi-Annual:
Formula
Where:
- : The rate for
n
periods (monthly, weekly, etc...) - : 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:
Where:
- : The rate for
n
periods (monthly, weekly, etc...) - : Annual rate
n
: Number of periods in one year
Examples
- Annual to monthly:
- Annual to quarterly:
- Annual to semi-annual:
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