Interest Rate Formulas
If you put a sum of money in the bank and let the
interest accumulate, the amount of money you will have some time in the
future is given by the formula
where
P is the initial investment,
r
is the interest rate per period,
t is the number of periods, and
A is the amount of money in the bank after the
t periods.
If you know any three of the four numbers
A,
P,
r,
t
you can solve for the fourth by using the following formulas

  
  
 
 ln(A/P)
ln(1+r)  =  log(A/P)
log(1+r)  . 

 

Note that you can use log instead of ln in the calculation
for
t; you will get the same number even though the numerator and
denominator of the fraction will be different. If you want to know how inflation affects the value
of your money, the formula
will determine this. Here,
P is the present
value of your money and
F is the future value of your money. Now,
r is the inflation rate per period and
t is the number of
periods. In other words,
P dollars in today's dollars is equivalent
to
F dollars after
t periods in the future. If you want to
know how much money was worth in the past, use
where
F is the current value of your money
and
P is the equivalent in past dollars. If you are taking out a loan, use the following formula
The amount you borrow is
L, the interest rate
per period is
r, the number of periods is
t, and
P
is the payment per period. If you know what you can make in payments, at
what rate you will get, then you can determine how much you can borrow
with the formula
which we get from the previous formula by solving
for
L. If you change your payments to a new payment
P^{ ¢ },
you can determine the new number of periods by the formula
For annuities, the formulas are similar as those
for loans. If you need a certain amount of money in the future and want
to put away some money every month (or in some other frequency), the formula
is
where
P is how much you put in the bank each
period,
r is the interest rate per period,
t is the number
of periods, and
A is how much money you need in the future. If you
know how much you wish to invest each period and want to find out how much
money you have in the future, we solve this formula for
A to get
Keep in mind that inflation will eat away at your
money, so if you need $10,000 now, you will need more 20 years from now. If you have an amount of money invested and you receive
regular payments from this investment, the appropriate formula is the same
as the loan formula,
where
P is your income per period,
A
is the amount of money you started with in the bank,
r is the interest
rate per period, and
t is the number of payments you receive. If
you know the amount to invest and you want to determine the amount of money
you started with, use
If you have a perpetual annuity, one that pays you
without a fixed ending date, then your payment is given by the equation
This is the same as if you have an amount
A
of money in the bank collecting interest at a rate of
r, and you
withdraw the interest each period. If you have a desired income
P,
you can
find out how much money A you need in the bank by using the formula A = P/r. To help you use these formulas, here are a few examples.
Example 1.
If you invest $1,000 in the bank at 6%, compounded
yearly, after 7 years you have
If the interest rate is compounded monthly instead
of yearly, then in 7 years you have
1000(1+.06/12)^{84} = $1520.37. 

If you hoped to have $1,750 in the bank 7 years later,
you need a yearly interest rate of
r = (1750/1000)^{1/7}1 = .083, 

or 8.3%. On the other hand, if your interest rate
was 7.5% compounded monthly, the amount of time it will take for your $1,000
to increase to $1,750 is
t =  ln(1750/1000)
ln(1+.075/12)  = 89.8, 

and this number is in months since our interest rate,
which is .075/12, is the yearly rate of 7.5% converted into a monthly rate.
Therefore, it takes 89.8/12 = 7.48 years, or almost 7 and a half years,
for this to happen.
Example 2.
You need $75,000 now to send your kid through
college. However, he or she won't go to college for 12 more years, and
the college estimates its tuition and other expenses will rise at the rate
of 9% per year. Thus, in 12 years you will need
75000(1+.09)^{12} = $210,949.86 

for college expenses.
For another example, suppose inflation in the
last fifty years has averaged 3%. If you can buy a house today for $100,000,
what would be the equivalent amount in 1935 dollars? This value is
100000(1+.03)^{50} = $22,810.70, 

so, at this rate of inflation, a house costing $22,811
fifty years ago would cost $100,000 today.
Example 3.
You need $50,000 to buy a used Jaguar, but you
only have $5,000 for a down payment. If the bank will loan you the rest
at 11% for a five year loan, then your monthly payment will be
 45000(.11/12)
(1(1+.11/12)^{60})  = $978.41 

Note that we had to convert the interest rate to
a monthly rate. Also, five years has 60 months, so you will make 60 payments
of $978.41. If you are able to pay $1,050 each month, you will pay off
the loan in

ln  æ
ç
è   1050
105045000(.11/12)   ö
÷
ø  
ln(1+.11/12)  = 54.7, 

or in just under 55 monthly payments. At $978.41
for 60 months you pay a total of $978.41·60 = $58,704.60, while
at $1,050 for 55 months you pay a total of $1,050·55 = $57,750.00.
So, you save about $1,000 over the life of the loan.
Example 4.
To get back to the college expenses problem, you
need to have approximately $211,000 in 12 years for your kid's college
expenses. If you can invest money each month at an 8% yield, each month
you need to deposit
 211000(.08/12)
((1+.08/12)^{144}1)  = $877.31. 

Again, we had to convert the interest rate to a monthly
rate and use the number of months in 12 years.
Example 5.
You have saved for your retirement, and today
you have $400,000 in your retirement plan. If your plan is earning 4% interest
per year, and you want to receive monthly payments for the next 20 years,
you will receive
 400000(.04/12)
(1(1+.04/12)^{240})  = $2,423.92 

each month. If you only want to receive money for
15 years you will receive
 400000(.04/12)
(1(1+.04/12)^{180})  = $2,958.75 

per month. However, if you wish to receive a perpetual
annuity, you will receive
400000(.04/12) = $1,333.33 

each month.
File translated from T_{E}X by T_{T}H,
version 1.95.
On 10 Dec 1998, 18:41.
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