Mathematics of interest rates
[edit] Simplified Calculation
Formulae are presented in greater detail at
time value of money.
In the formulae below,
i or
r are the interest rate, expressed as a true percentage (i.e. 10% = 10/100 = 0.10).
FV and
PV represent the future and present value of a sum.
n represents the number of periods.
These are the most basic formulae:
The above calculates the future value of
FV of an investment's present value of
PV accruing at a fixed interest rate of
i for
n periods.
The above calculates what present value of
PV would be needed to produce a certain future value of
FV if interest of
i accrues for
n periods.
or
The above two formulae are the same and calculate the compound interest rate achieved if an initial investment of
PV returns a value of
FV after
n accrual periods.
The above formula calculates the number of periods required to get
FV given the
PV and the interest rate
i. The log function can be in any base, e.g. natural log (ln)
[edit] Compound
Formula for calculating compound interest:
Where,
- P = principal amount (initial investment)
- r = annual nominal interest rate (as a decimal)
- n = number of times the interest is compounded per year
- t = number of years
- A = amount after time t
Example usage: An amount of $1,500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years.
A. Using the formula above, with P = 1500, r = 4.3/100 = 0.043, n = 4, and t = 6:
So, the balance after 6 years is approximately $1,938.84.
[edit] Translating different compounding periods
Each time unpaid interest is compounded and added to the principal, the resulting principal is grossed up to equal P(1+i%).
A) You are told the interest rate is 8% per year, compounded quarterly. What is the equivalent effective annual rate?
The 8% is a nominal rate. It implies an effective quarterly interest rate of 8%/4 = 2%. Start with $100. At the end of one year it will have accumulated to:
$100 (1+ .02) (1+ .02) (1+ .02) (1+ .02) = $108.24
We know that $100 invested at 8.24% will give you $108.24 at year end. So the equivalent rate is 8.24%. Using a financial calculator or a
tableis simpler still. Using the Future Value of a currency function, input
- PV = 100
- n = 4
- i = .02
- solve for FV = 108.24
B) You know the equivalent annual interest rate is 4%, but it will be compounded quarterly. You need to find the interest rate that will be applied each quarter.
$100 (1+ .009853) (1+ .009853) (1+ .009853) (1+ .009853) = $104
The mathematics to find the 0.9853% is discussed at
Time value of money, but using a financial calculator or
table is easier. Input
- PV = 100
- n = 4
- FV = 104
- solve for interest = 0.9853%
C) You sold your house for a 60% profit. What was the annual return? You owned the house for 4 years, paid $100,000 originally, and sold it for $160,000.
$100,000 (1+ .1247) (1+ .1247) (1+ .1247) (1+ .1247) = $160,000
Find the 12.47% annual rate the same way as B.) above, using a financial calculator or
table. Input
- PV = 100,000
- n = 4
- FV = 160,000
- solve for interest = 12.47%
[edit] Example question:
In January 1970 the
S&P 500 index stood at 92.06 and in January 2006 the index stood at 1248.29. What has been the annual rate of return achieved? (ignoring dividends).
[edit] Answer:
[edit] Doubling
The number of time periods it takes for an investment to double in value is
where
is the interest rate as a fraction.
Let
p be the interest rate as a percentage ( i.e., 100
i ). Then the product of
p and the doubling time
t is fairly constant:
interest doubling time product
[edit] Periodic compounding
The amount function for compound interest is an exponential function in terms of time.
- n = Number of compounding periods per year (note that the total number of compounding periods is
)
As
n increases, the rate approaches an upper limit of
er. This rate is called
continuous compounding, see below.
Since the principal
A(
0) is simply a coefficient, it is often dropped for simplicity, and the resulting
accumulation function is used in
interest theory instead. Accumulation functions for
simple and compound interest are listed below:
Note:
A(
t) is the amount function and
a(
t) is the accumulation function.