Assuming that we issue the sqrt(3) command to estimate the square root of three, we would get the following error message:

>>>sqrt(3)
Traceback (most recent call last):
  File "<pyshell#17>", line 1, in <module>
    sqrt(3)
NameError: name 'sqrt' is not defined

The last line of the error message tells us that the sqrt() function is not defined. Later in the chapter, we learn that the sqrt() function is included in a module called math and that we have to load (import) the module before we can call the functions contained in it. A module is a package that contains a set of functions around a specific subject.

>>>abcde
Traceback (most recent call last):
  File "<pyshell#0>", line 1, in <module>
    abcde
NameError: name 'abcde' is...

A perpetuity describes the situations where equivalent periodic cash flows happen in the future and last forever. For example, we receive $5 at the end of each year forever. A real-world example is the UK government bond, called consol, that pays fixed coupons. To estimate the present value of a perpetuity, we use the following formula if the first cash flow occurs at the end of the first period:

Here, PV is the present value, C is a perpetual periodic cash flow that happens at a fixed interval, and R is the periodic discount rate. Here C and R should be consistent. For example, if C is annual (monthly) cash flow, then R must be an annual (monthly) discount rate. This is true for other frequencies too. Assume that a constant annual cash flow is $10, with the first cash flow at the end of the first year, and that the annual discount rate is 10 percent. Compare the following two ways to name the C and R variables:

>>>x=10       # bad way for variable names...

After assigning values to a few variables, we could use the dir() function to show their existence. In the following example, variables n, pv, and r are shown among other names. At the moment, just ignore the first five objects in the following code, which start and end with two underscores:

>>>pv=100
>>>r=0.1
>>>n=5
>>>dir()
['__builtins__', '__doc__', '__loader__', '__name__', '__package__', 'n', 'pv', 'r']

>>>rate=0.075
>>>rate
0.075

For basic math operations in Python, we use the conventional mathematical operators +, -, *, and /. These operators represent plus, minus, multiplication, and division operations respectively. All these operators are embedded in the following line of code:

>>>3.09+2.1*5.2-3/0.56
8.652857142857144

Although we use integer division less frequently in finance, a user might type the division sign twice (//) accidentally to get a weird result. The integer division is done with double slash //, which would return an integer value that is the largest integer than the final output. The result of 7 divided by 3 is 2.33, and 2 will be the largest integer smaller than 2.33. This example is shown in the following code:

>>>7/3
2.3333333333333335

For Python 2.x versions, 7/3 could be 2 instead of 2.333. Thus, we have to be careful. In order to avoid an integer division, we could use 7/2 or 7/2., that is...

For our FV=PV(1+R)^n, we use a power function. The floor function would give the largest integer smaller than the current value. The remainder is the value that remains after an integer division. Given a positive discount rate, the present value of a future cash flow is always smaller than its corresponding future value.

The following formula specifies the relationship between a present value and its future value:

In this formula, PV is the present value, FV is the future value, R is the cost of capital (discount rate) per period, and n is the number of periods. Assume that we would receive $100 payment in two years and that the annual discount rate is 10 percent. What is the equivalent value today that we are willing to accept?

>>>100/(1+0.1)**2
82.64462809917354

Here, ** is used to perform a power function. The % operator is used to calculate the remainder. Refer to the following example for the implementation of these operators:

>>...

The default precision for Python has 16 decimal places as shown in the following example. This is good enough for most finance-related problems or research:

>>>7/3
2.3333333333333335

We could use the round() function to change the precision as follows:

>>>payment1=3/7
>>>payment1
0.42857142857142855
>>>payment2=round(y,5)
>>>payment2
0.42857

Assume that the units for both payment1 and payment2 are in millions. The difference could be huge after we apply the round() function with just two decimal places! If we use one dollar as our unit, the exact payment is $428,571. However, if we use millions instead and apply two decimal places, we end up with 430,000, which is shown in the following example. The difference is $1,429:

>>>payment1*10**6
428571.4285714285
>>>payment2=round(payment1,2)
>>>payment2
0.43
>>>payment2*10**6
430000.0

To understand each math function, we apply the help() function, such as help(round), as shown in the following example:

>>>help(round)
Help on built-in function round in module builtins:
round(...)
    round(number[, ndigits]) -> number
Round a number to a given precision in decimal  
digits (default 0 digits).This returns an int when 
called with one argument, otherwise the same type as 
the number. ndigits may be negative.

>>>dir()
['__builtins__', '__doc__', '__loader__', '__name__', '__package__', 'x']...

When learning finance with real-world data, we deal with many issues such as downloading data from Yahoo! finance, choosing an optimal portfolio, estimating volatility for individual stocks or for a portfolio, and constructing an efficient frontier. For each subject (topic), experts develop a specific module (package). To use them, we have to import them. For example, we can use import math to import all basic math functions. In the following codes, we calculate the square root of a value:

>>>import math
>>>math.sqrt(3)
1.732050807568772

To find out all functions contained in the math module, we call the dir() function again as follows:

>>>import math
>>>dir(math)
['__doc__', '__loader__', '__name__', '__package__', 'acos', 'acosh', 'asin', 'asinh', 'atan', 'atan2', 'atanh', 'ceil', 'copysign', &apos...

There are several functions we are going to use quite frequently. In this section, we will discuss them briefly. The functions are print(), type(), upper(), strip(), and last expression _. We will also learn how to merge two string variables. The true power function pow() discussed earlier belongs to this category as well.

>>>import math            
>>>print('pi=',math.pi)
pi= 3.141592653589793

>>>pv=100.23
>>>type(pv)
<class 'float'>
>>>n=10
>>>type(n)
<class 'int'>
>>>