FINE 5200: Managerial Finance
Essential Mathematical Concepts
08-sep-2021
Introduction and Context
This document is meant to reassure beginning Finance students that their math background is
sufficient, or else to identify concepts that demand additional study (before students are surprised by
strange terms). This review is not the best resource for learning these concepts from scratch; several
free texts with clear explanations are posted on Canvas. Should you need additional help, please ask.
A. Notation for sums and products
Ordered sequences (a.k.a. series) are common in many finance topics. Compact notation for adding
the terms in a series St or multiplying all the terms in the series are written like this:
Sums use  (Sigma), which sounds like “S”:
∑𝑇𝑡=1 𝑆𝑡 = 𝑆1 + 𝑆2 + ⋯ + 𝑆𝑇
Products use  (Pi), which sounds like “P”:
∏𝑇𝑡=1 𝑆𝑡 = 𝑆1 × 𝑆2 × ⋯ × 𝑆𝑇
B. Rules for the order of arithmetic operations (GEMDAS)
The order of operations (or operator precedence) is a set of rules for interpreting mathematical
expressions. A mathematical expression is a puzzle with numbers. Misread the order of operations
and your result may be intriguing, but like a jigsaw puzzle put together wrong, inaccurate. Your result
is more like Picasso than Pissarro.
Girl Sewing, Camille Pissarro (1895)
Girl Before a Mirror, Pablo Picasso (1932)
FINE 5200S –Essential Mathematical Concepts
Follow this order to accurately evaluate any mathematical expression you will see in this course. You
might have seen the acronym PEMDAS with the phrase “Please Excuse My Dear Aunt Sally” to help
remember it. An alternative mnemonic is GEMDAS with “Green Eyed Monsters Defy All Sanity”.
1. Grouping (or Parentheses) have the highest order of precedence. Follow GEMDAS in each of
the most interior parentheses or brackets, then move the next most interior group.
2. Exponents (powers and roots) are evaluated before multiplication and division.
3. Multiplication and Division is performed before addition and subtraction operations.
4. Addition and Subtraction occurs last, in the lowest order of precedence.
Here are a few problems from later in the course, with check figures at the end of the document.
Sample Problems (check figures are at the end of the document)
i.
The last four quarterly returns on Tesla (TSLA) shares are shown below, a small statistical sample:
𝑅1
𝑅2
𝑅3
𝑅4
198.4 13.90 19.01 5.024
Evaluate the sample mean, given by this expression:
Evaluate the sample variance, given by this expression:
Calculate the sample standard deviation, too:
ii.
𝑅̄ = ∑4𝑛=1
𝑅𝑛
𝜎𝑅2 = ∑4𝑛=1
𝜎𝑅 =
4
(𝑅𝑛 −𝑅̄ )2
3
√𝜎𝑅2
With the (strong) assumption that markets are perfectly fair, the return on equity required by
D
investors is given by the expression ROE = (1 − t )  [ ROA + ( ROA − k D )  ] where the variables are
E
defined as shown below:
ROE
Return on levered equity, as determined by market risk and leverage
ROA
Market risk-adjusted expected return on unlevered assets (without debt)
E
Market value of equity securities
D
Market value of debt securities
t
Marginal corporate tax rate
The market value of Tesla’s outstanding debt was USD 12.51b and the market value of Tesla’s
equity was USD 683.4b. Its bonds sold to yield 4.30%. Suppose its marginal tax rate is 10% and its
expected stock return is 15%. Calculate Tesla’s unlevered required return on its assets alone, ROA.
This figure is also known as the Weighted Average Cost of Capital.
FINE 5200S –Essential Mathematical Concepts
C. Exponents, the number e and logarithms
When a number R is multiplied n times over, the product is an exponential expression written as R n .
The inverse operation is exponential, too, but the exponent is the inverse fraction: [R n ]1/n = R.
Multiply two exponential expression that have the same root R, say R a and R b, and the product is R a+b.
By the same rule, R 2a  R 3b = R 2a+3b .
n
 1
A transcendental number that appears regularly in finance is labeled e and defined as e = lim 1 +  .
n
 n
The number e is ubiquitous in natural sciences, mathematical analysis and statistics. The inverse
operation to y = ex is known as the natural logarithm function, written as: x = ln (y) .
Sample Problems
iii.
Suppose R is an annual investment return. If (1 + R )10 = 6.19174, find R, which happens to be an
example of a Compound Annual Growth Rate (CAGR).
iv.
A bank account pays a nominal interest rate of 6% per annum, but you may choose any
compounding frequency you wish. If you deposit $1,000, how much will the account hold after
a year if you choose to compound quarterly (earning 6%/4 each quarter)? Compound weekly
(earn 6%/52 each week)? Compound continuously? Which compounding frequency is best for
you?
v.
If a paid-up whole-life insurance policy is growing at 3% per year, how long (in years) does it
take for the fund’s value to grow by 20%? To double in value?
D. Sums of Geometric Series
𝑡
It is convenient (and beautiful) to know how to simplify expressions like S(R) = ∑∞
𝑡=1 𝑹 , given a
constant R. To find a closed-form expression, divide each term by R. Now we have this expression:

S ( R )  t −1
=  R = 1 +  Rt = 1 + S ( R )
R
t =1
t =1
See how the sum on the right exactly equals S(R)! Isolating S(R) by elementary arithmetic yields a very
useful expression, known as the sum of a geometric series:

S ( R ) =  Rt =
t =1
R
1− R
Sample Problem
vi.
A never-ending series of equal cash flows is known as a perpetuity. You will soon recognize a
useful sum of an infinite geometric series as the present value of a perpetuity. It looks like this:
FINE 5200S –Essential Mathematical Concepts

t
 1  $1
PVperp ( $1) = ( $1)  
 =
r
t =1  1 + r 
Make the substitution R = 1
in the sum of a geometric series to prove the expression above.
1+ r
E. Essential Statistics for Random Variables
We consider random variables that take on values from a well-defined set of possible outcomes. The
set of outcomes associated with non-zero probability is known as the support of the random variable.
A probability is a number between zero and one, inclusive, such that the sum of probabilities over all
possible outcomes is 100% (one). The probability distribution function of a random variable is the list
of probabilities that correspond with each possible outcome of the random variable.
The expectation, or expected value, of a random variable X is its probability-weighted average value,
for a well-defined probability distribution over all the values the random variable can possibly take.
The expected value is also known as a random variable’s mean, written as E(X), and frequently
represented by the letter . (Mu makes the “em” sound). For a random variable X with a probability
distribution function p(X) defined over a discrete set of states of the world S, this expression describes
the expected value of X:
S
E(X) =
 p ( s )X ( s )
s =1
The variance of a random variable is a measure of its dispersion. Variance, frequently abbreviated as
2, is defined by these equivalent algebraic expressions:
 X2 = E [X − E(X)]2 =
S
 p (s)( X (s) − X )
2
s =1
It is often convenient to work with the square root of a variance, known as the standard deviation of
the random variable, which is measured in the same units as the random variable itself. A standard
deviation is frequently abbreviated as  X .
When we consider two random variables at the same time, it is useful to measure their covariance,
defined by these equivalent expressions:
 XY = E [ (X − E(X)) (Y − E(Y)) ] =
S
 p ( s ) ( X ( s ) − X ) (Y ( s ) − Y )
s =1
Please note that if X and Y are the same random variable, the covariance becomes a variance.
FINE 5200S –Essential Mathematical Concepts
Sample Problems
vii.
Return variance (or std dev) is the most common measure of portfolio risk. Suppose stock A
has expected return 10% and stock B has expected return 15%. Say  A is 8% and  B is 9%. Last,
suppose  AB is − 0.0024. What is the expected return of a portfolio with 50% of its value in A
and 50% in B? What is the portfolio’s standard deviation?
HINT: The return variance of a portfolio P formed with 50% invested in A and 50% in B, is
written as
 P2 = ( 50% )  A2 + ( 50% )  B2 + 2 ( 50% )  AB
2
2
2
F. Answers to Sample Problems
The answers to B.(i) are
Mean Return:
Variance of Returns:
Std Dev of Returns:
59.08% per quarter
8659.1
93.05% per quarter
For B.(ii), the ROA (or WACC) is 14.80%.
The CAGR for C.(iii) is 20%.
For C.(iv), a $100 deposit at 6% will grow to $1,061.36 with quarterly compounding, $1,061.80 with
weekly compounding, and $1,061.84 with continuous compounding. The difference between weekly
or daily compounding and continuous compounding is very small, but as a general principle, the owner
of the account earns most with continuous compounding.
In C.(v), a whole-life insurance policy growing at 3% / yr will grow by 20% after 6.168 years have passed
and will take 23.45 years to double in value.
1
R
1+ r = 1 = 1
=
For D.(vi), if R = 1
, then
. Recognizing the formula for the
1+ r
1− R 1+ r − 1
1 + r −1 r
1+ r 1+ r
$1
present value of a perpetuity as a geometric series, we have PV perp ( $1) =
.
r
In E.(vii), the expected return of the portfolio P is 12.5%. The variance of P equals 0.002425 and P’s
standard deviation of return is 4.924%. It is interesting to notice how much smaller the risk of the
50/50 portfolio is compared to A or B alone.