# Compound Interest ```Compound Interest
An Example

Suppose that you were going to invest
\$5000 in an IRA earning interest at an
annual rate of 5.5%
 How
would you determine the amount of
one year?
i1  50000.055  \$275
An Example

How much money would you have in your
IRA account?
F1  5000  i1  5000  5000(0.055)  50001  0.055  \$5275

How much interest would you get after two
years?
i2  52750.055  290.13
An Example

How much money would you have in your
IRA account after two years?
F2  5275  i2  5275  52750.055  52751  0.055  50001  0.0551  0.055

F10  50001  0.055  \$8540.72
10
Compound Interest

Notice that the interest in our account was
paid at regular intervals, in this case every
year, while our money remained in the
account. This is called compounding
annually or one time per year.
Compound Interest


Suppose that instead of collecting interest at the
end of each year, we decided to collect interest
at the end of each quarter, so our interest is paid
four times each year. What would happen to our
investment?
Since our account has an interest rate of 5.5%
annually, we need to adjust this rate so that we
get interest on a quarterly basis. The quarterly
rate is:
5.5 / 4  1.375%
Compound Interest

So for our IRA account of \$5000 at the end
of a year looks like:
41
0.055 

F1  50001 
  \$5280.72
4 

 After 10 years, we have:
0.055 

F10  50001 

4 

410
 \$8633.85
Compound Interest Formula

P dollars invested at an annual rate r,
compounded n times per year, has a
value of F dollars after t years.
 r
F  P  1  
 n

nt
Think of P as the present value, and F as
the future value of the deposit.
Yield

One may compare investments with
different interest rates and different
frequencies of compounding by looking at
the values of P dollars at the end of one
year, and then computing the annual rates
that would produce these amounts without
compounding.
Yield
Such a rate is called the effective annual
yield, annual percentage yield, or simply
yield.
 In our previous example when we
compounded quarterly, after one year we

0.055 

F1  50001 

4 

41
 \$5280.72
Yield

To find the effective annual yield, y, notice
that we gained \$280.72 on interest after a
year compounded quarterly. That interest
represents a gain of 5.61% on \$5000:
280.72
y
 0.0561
5000
Finding the Yield


To find the effective annual yield, find the
difference between our money after one year
and our initial investment and divided by the
initial investment.
Therefore, interest at an annual rate r,
compounded n times per year has yield y:
n
r

P 1    P
n
r
n


y
 1    1
P
 n
More on Yield

There may be times when we need to find the annual
rate that would produce a given yield at a specified
frequency of compounding. In other words, we need to
solve for r :
n
 r
y  1    1
 n
 r
y  1  1  
 n
 y  11/ n  1  r
n
 y  11/ n  1  r
n
1/ n
n  y  1  1  r


n
Compound Interest Formula


Notice that when we collected our interest more
times during each year, i.e. we compounded
more frequently, the amount of money in our
account was actually greater than if only
collected interest one time a year.
What would happen to our money if we
compounded a really large number of times?
Continuous Compounding

As n increases, approaches a constant value in the
Frequency Worksheet. Here’s why:
n=mr
nt
 r
 1
P1    P1  
 n
 m

mrt
 1 
 P 1  
 m 
m rt



As n gets really large, m also becomes really large, and:
m
1

1    2.71828182845905  e
 m
Continuous Compounding

The value of P dollars after t years, when
compounded continuously at an annual
rate r , is
F  Pe
 On
rt
the calculator use the button
(on TI-83: 2nd + LN )
 In Excel, use the function EXP(x )
Yield

The effective annual yield, y, for compounding
continuously at an annual interest rate of r is:
Pe  P
r
y
 e 1
P
r

To find the annual interest rate r if we know the
yield, y, we would have to solve for r in the
above equation. To do this you would use
logarithms:
r  ln  y  1
Present Values

If we are given the future value, F, the
annual interest rate r, the number of times
compounded per year n, and the length of
time invested t, we may solve the present
value P :
 r
P  F  1  
 n
P  F  e r t
 nt
Ratios

Sometimes we are not interested in the percentage that
an investment increases by. Rather, we would like to
know by what factor the investment increased or
decreased. Such factors are computed by find the ratio
of the future value to the present value. This ratio, R, for
continuous compounding is:
F Pert
R 
 e rt
P
P

This allows us to convert the interest rate for a given
period to a ratio of future to present value for the same
period.
Example Ratios


Suppose that in our IRA example, the annual
interest rate of 5.5% is compounded
continuously.
If we wanted to know the weekly rate our
investment would increase, we would simply
have 0.055/52 or 0.00105 or 0.105%. This
would mean that the ratio of the future value to
the present value between consecutive weeks
compounded continuously would be e0.055/52 or
1.00105
Example Ratios

Multiplying by this weekly ratio 52 times yields a
yearly ratio of (e0.055/52)52 = e(0.055/52)52 = e0.055. As we
would expect, this corresponds to the annual rate of
0.055.
The Project
• How can compound interest help us price a stock
option?
• Our annual risk-free rate of 4%, compounded
continuously, gives a weekly risk-free rate of rrf =
0.04/52  0.0007692. The weekly ratio
corresponding to this weekly rate is e0.04/52.
• We call Rrf = e0.04/52  1.0007695 the risk-free weekly
ratio for the Walt Disney option.
The Project
Compound interest can help us with option pricing
in a second way. Suppose that we know a
future value F for our 20 week option at the end
of the 20 weeks. We suppose that money will
earn at the risk-free annual interest rate or 4%
compounded continuously. This can be used to
find the present value, P, of the option.
P  F e
P  F e
 rt
 0.04 

20
52


Preliminary Reports






You will deliver your preliminary report on Monday March 28th, 2005.