Financial Economics and

Investments, Spring 25
Topic 1

Introduction to Financial Economics

 Contents
1. The Economic Way of Thinking
2. Loanable Funds Market
3. Asset Classes and Financial Instruments
4. Statistics for Financial Economics
 The Economic Way of Thinking
Economics is the study of
• how agents choose to allocate scarce resources, and
• how those choices affects society.

Economic agents: decision makers.

• In economics, agents are
– self interest
– rational
• using all available information to achieve your goals.
• Rational investors weigh the benefits and costs of each action and try to make the
best decision possible.
 The Economic Way of Thinking
• Optimization: choosing the best feasible option with given
constraints (information, knowledge, budget, experience,
training, etc).
• Equilibrium: a situation in which everyone is simultaneously
optimizing, so nobody would benefit personally by changing
his or her own behavior, given the choices of others.
– Next: one example of equilibrium
– “good equilibria” vs. “bad equilibria”
Assume all you care about is to arrive your destination sooner and you don't need to exit any time
soon.
1. Will you change lane if you are one of the red cars?
2. Will you change lane if you are one of the blue cars?
3. Is it an equilibrium?
 Optimization
1. What is the benefit of obtaining a graduate school education?
2. What is the cost of obtaining a graduate school education?
 Optimization
• Benefit of something is the gain or pleasure that it brings and
is determined by preferences - by what a person likes and
dislikes and the intensity of those feelings.
– Preference and Utility

• Cost is what you must give up to get something.

 Optimization
Assume you are under a scholarship that all your tuition and school
related expenses are paid by the university. Then your cost of
spending one year getting a master degree is ZERO.
• Is it a true statement?
 Optimization
Trade-offs arise when some benefits must be given up in order to
gain others.
o An economic agent faces a trade-off, or exchange, when the agent needs
to give up at least one thing to get something else.
o All optimization problems involve trade-offs.
 Optimization
If an option is the best choice, you will be made
• better off, or at least not worse off, as you move toward it
• worse off, or at least not better off, as you move away from it

Two Optimization Techniques

• Marginal analysis
• Net benefit analysis
• Two optimization techniques yield identical answers. You can use
whichever technique you find easier for the particular problem.
 Marginal analysis
• Marginal benefit (MB)is the additional benefit generated by moving from
one feasible alternative to the next feasible alternative.

• Marginal cost (MC) is the additional cost generated by moving from one
feasible alternative to the next feasible alternative.

• Making decisions using marginal analysis: should I take one more step?
– Yes if MB ≥ MC
– No if MB ≤ MC
– Yes or No if MB = MC
• Economists usually assume “Yes” if MB = MC
 Net benefit analysis
• Net benefit = total benefit - total cost

• Net benefit analysis:

1. Calculate the net benefit of each alternative.
2. Pick the alternative with the highest net benefit.
 Contents
1. The Economic Way of Thinking
2. Loanable Funds Market
3. Asset Classes and Financial Instruments
4. Statistics for Financial Economics
 Loanable Funds Market
Three major types of financial capital markets:
• Stock markets
• Bond markets
– Corporate bonds (Hypothetical)
– Government bonds Loanable funds
market
• Loan markets
– Home loans, car loans, etc.
Others: the market for mutual fund shares, etc.
• For simplification, disregard the secondary market.
• A primary market is a financial market in which new issues of a security,
such as a bond or a stock, are sold to initial buyers by the corporation or
government agency borrowing the funds. A secondary market is a financial
market in which securities that have been previously issued can be resold.
 Loanable Funds Market
• How do borrowers and lenders interact with each other in a
perfectly competitive loanable funds market?
• How is the price of borrowing and lending determined in a
perfectly competitive loanable funds market?
Who demand loanable funds?
• Borrowers? Lenders?

Agents Loanable funds

demanders?
Firms

Households

Foreign agents

Governments

Banks
Since the early 1980s, except for 1991, the United States
has been a net borrower from the rest of the world.
Who demand loanable funds?
• Borrowers? Lenders?

Agents Loanable funds

demanders?
Firms

Households

Foreign agents

Governments

Banks
• https://fiscaldata.treasury.gov/americas-finance-guide/national-debt/
 Treasury
• Treasury bills: less than 12 months
– Zero-coupon bond
– Typically issued at a discount from the par amount (also called face value)
• Treasury notes: 2, 3, 5, 7, or 10 years
– Usually receive interest payments semi-annually until the maturity date of the note
• Treasury bonds: 20 or 30 years
– Usually receive interest payments semi-annually until the maturity date of the bond
• https://www.investopedia.com/ask/answers/difference-between-bills-notes-and-bonds/
Bonds are priced at a discount, at par, or a premium.
• Below par is a term describing a bond whose market price is trading below its face value.
– A discount bond is a bond that is issued for less than its par, or trades for less than its par in the
secondary market.
• Above par is a term used to describe the price of a bond when it is trading above its face
value.
– A premium bond is one for which the market price of the bond is higher than the face value.

• Coupon rate
𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 =
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
• Current yield
𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 =
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
• Yield to maturity (YTM) ((to be covered in future classes if time permits)
– The yield to maturity (YTM) is the expected annual rate of return earned on a bond, assuming the
debt security is held until maturity.
https://treasurydirect.gov/
 Treasury
Fixed income market: fixed-income or debt securities promise either
a fixed stream of income or a stream of income determined by a
specified formula.
• The money market
– U.S. Treasury bills (T-bills) are the most marketable of all money market
instruments.
• The bond market
– Treasury Notes and Treasury Bonds
– The bond market is composed of longer term borrowing or debt
instruments than those that trade in the money market.
Who demand loanable funds?
• Borrowers? Lenders?

Agents Loanable funds

demanders?
Firms

Households

Foreign agents

Governments

Banks
 Banks
A financial institution is a firm that operates on both sides of the
markets for financial capital.
• It is a borrower in one market and a lender in another.

Key financial institutions are

• Government-sponsored mortgage lenders
• Pension funds
• Insurance companies
• Asset management companies
• Commercial banks
YouTube Video: What Do Banks Do?
 Banks
Commercial banks borrow from savers in one market and lend to
borrowers in another market.
• Banks play the role of “middlemen”.
• Commercial banks borrow from savers at a lower rate and then lend to
borrowers at a higher rate to make profits.
• Commercial banks also provide services including
– risk sharing
– Information
– liquidity
 Banks
Risk sharing
• By allowing investors to spread their money over many different
assets, investors can reduce their risk while maintaining a high
expected return on their investment.
Information
• The prices of financial securities represent the beliefs of other
investors and financial intermediaries about the future revenue
stream from holding those securities.
• This aggregation of information makes funds flow to the right
firms.
 Banks
Liquidity
• The financial system allows savers to quickly convert their
investments into cash.

• Liquid assets: the assets can be sold and instantly converted

into cash with comparatively little risk of loss.
– Examples of liquid assets include cash, reserves held at the central
bank, and short-term government securities.
– https://www.investopedia.com/terms/l/liquidity.asp
Asset Liquidity (How Easy Selling/Liquidating Is)
Explained in One Minute
 Banks
Commercial banks and financial intermediaries are the
organizations that connect savers and borrowers.
• The loanable funds market improves the allocation of resources in an
economy by enabling savers to lend their excess money to borrowers.
Who demand loanable funds?
• Borrowers? Lenders?

Agents Loanable funds

demanders?
Firms Major item
Households

Foreign countries

Governments

Banks
 Loanable Funds Market
Who demand loanable funds: borrowers, or debtors.
• Economic agents who need to borrow.
o Such as businesses, home buyers, college students, etc.
o Business investment is the main item that makes up the demand for
loanable funds.
• 𝐷𝐷𝐿𝐿𝐿𝐿 is also influenced by the actions of banks.
In a perfectly competitive loanable funds market
• Real interest rate (𝑅𝑅) : the price of loanable funds (or credit) borrowing and
lending
• Quantity of loanable funds (or credit) demand: the amount of loanable
funds (or credit) that borrowers are willing to and able to borrow at a given
real interest rate in a certain period.

After observing the real interest rate (𝑅𝑅), you

need to make a decision on how much to borrow
for your student loan.
• If 𝑅𝑅 is higher, would you be inclined to take out
more or fewer student loans?
• If 𝑅𝑅 is lower, would you be inclined to take out
more or fewer student loans?
Loanable funds (or credit) demand curve
• A curve that plots the quantity of loanable funds (or credit) demanded at
different real interest rates.
The 𝐷𝐷𝐿𝐿𝐿𝐿 curve shifts with changes in:

Perceived business opportunities for firms

The 𝐷𝐷𝐿𝐿𝐿𝐿 curve shifts with changes in:

Household preferences and expectations

The 𝐷𝐷𝐿𝐿𝐿𝐿 curve shifts with changes in:

Bank actions
The 𝐷𝐷𝐿𝐿𝐿𝐿 curve shifts with changes in:

Government policies
The 𝐷𝐷𝐿𝐿𝐿𝐿 curve shifts with changes in:
• Perceived business opportunities for firms
• Household preferences or expectations
• Bank actions
• Government policies
• And more
Who supply loanable funds?
• Borrowers? Lenders?

Agents Loanable funds

suppliers?
Households

Firms

Foreign countries

Governments

Banks
 Loanable Funds Market
Who supply loanable funds: savers, or lenders.
• Economic agents who save and lend credit.
• 𝑆𝑆𝐿𝐿𝐿𝐿 is also influenced by the actions of banks.

In a perfectly competitive loanable funds market

• Real interest rate (𝑅𝑅): the price of loanable funds (or credit) borrowing
and lending
• Quantity of loanable funds (or credit) supplied: the amount of loanable
funds (or credit) that savers/lenders are willing to and able to save/lend
at a given real interest rate in a certain period.

The law of loanable funds supply?

Why do households save?
• Retirement
• Future large purchases
o car, house, vacation, wedding, etc.
• Start a business
• A “rainy day” (insurance)
 Loanable Funds Market
Why do households save?
• Take-home income = consumption + saving
o Consumption: consume today
o Saving: consume in the future
• Future consumption

• If the real interest rate were to increase, would you be inclined

to save more or less?
 Loanable Funds Market
Question: if the real interest rate were to increase, would you be inclined to
save more or less?

• Effect 1 (substitution): A higher real interest rate implies a higher return to

savings. You would save more!
• Effect 2 (income): A higher real interest rate implies that you can save less
to generate a retirement nest egg of a certain size. You would therefore
save less! link

• In aggregate, effect 2 is weaker than effect 1, meaning that higher real

interest rate increases the amount of money that households save.
Loanable funds (or credit) supply curve
• A curve that plots the quantity of loanable funds (or credit) supplied at
different real interest rates.
The 𝑆𝑆𝐿𝐿𝐿𝐿 curve shifts with changes in:
• Saving motives of households, firms and other savers
• Bank actions
• Government policies
• And more
The 𝑆𝑆𝐿𝐿𝐿𝐿 curve shifts with changes in:

Government policies
Banking Explained – Money and Credit
 Contents
1. The Economic Way of Thinking
2. Loanable Funds Market
3. Asset Classes and Financial Instruments
4. Statistics for Financial Economics
 Financial Assets
• In economics, an investment refers to productive real
physical capital, such as machinery and the construction assets
of buildings.
• In finance, an investment (aka financial investment) is financial
the current commitment of money or other resources assets
(such as stocks and bonds) in the expectation of reaping do not
future benefits. contribute
directly to
– You are forgoing either current leisure or the income in the the
expectation that your future career will be sufficiently productive
enhanced to justify this commitment of time and effort. capacity of
the economy
 Financial Assets
• What ultimately determines the material wealth of a society?
• Financial assets simply define the allocation of income or wealth
among investors.
– When investors buy securities issued by companies, the firms use the
money so raised to pay for real assets, such as equipment, or technology.
So investors’ returns ultimately come from the income produced by the
real assets that were financed by the issuance of those securities.
• We will focus almost exclusively on financial assets. But keep in
mind that the successes or failures of these financial assets
ultimately depend on the performance of the underlying real
assets.
 Financial Assets
Three broad types of financial assets:
• Fixed income (or debt securities)
– promise either a fixed stream of income or a stream of income
determined by a specified formula.
• Equity
– represents an ownership share in the corporation.
• Derivatives
– such as options and futures contracts provide payoffs that are determined
by the prices of other assets such as bond or stock prices.
– commodity and derivative markets allow firms to adjust their exposure to
various business risks.
 Financial Assets: Players
Brokers and dealers play distinct roles, although sometimes a
single entity can act as both.
• Brokers act on behalf of clients to find and execute trades.
They connect buyers and seller and earn commissions or fees
for their services.
• Dealers provide the prices and liquidity by buying and selling
securities from their own inventories. They provide liquidity to
the markets and earn profits from the spread.
 Financial Assets: Players
• The ask price is the price you (investor) would have to pay to
buy a security (e.g., T-bill) from a securities dealer.
• The bid price is the slightly lower price you (investor) would
receive if you wanted to sell a security (e.g., T-bill) to a dealer.
• The bid–ask spread is the difference in these prices, which is
the dealer’s source of profit.
 Financial Assets: Fixed Income
Fixed Income Market: more than three times the size of global equity markets.

1. The money market

– The money market consists of very short-term, highly marketable debt securities.
– Many of these securities trade in large denominations and so are out of the reach of
individual investors. (Money market funds, however, are easily accessible to small
investors.)
2. The Bond Market
– The bond market is composed of longer term borrowing or debt instruments than those
that trade in the money market.
– This market includes Treasury notes and bonds, corporate bonds, municipal bonds,
mortgage securities, and federal agency debt (these instruments are sometimes said to
comprise the fixed-income capital market).
 Financial Assets: Fixed Income
Money market instruments:
1. Treasury Bills
2. Certificates of Deposit (CD)
3. Commercial Paper
4. Bankers’ Acceptances (BA)
5. Eurodollars
6. Repos and Reverses
7. Federal Funds
8. Brokers’ Calls (to be covered in future classes)
9. Money Market Funds
 Financial Assets: Fixed Income
• Funds in the bank’s reserve account are called federal funds.
– Banks with excess funds lend to those with a shortage.
• The Federal Funds Rate (FFR) is the interest rate that banks borrow and
lend their excess reserves to each other overnight.
• The FFR is the central interest rate in the U.S. financial market and central
to the conduct of monetary policy in the U.S. as it influences a wide range
of market interest rates.
– It influences other interest rates such as the prime rate, which is the rate banks
charge their customers with higher credit ratings.
– Additionally, it indirectly influences longer-term interest rates such as
mortgages, loans, and savings, all of which are very important to consumer
wealth and confidence.
 Financial Assets: Fixed Income
Bond Market Instruments:
1. Treasury Notes and Bonds
2. Inflation-Protected Treasury Bonds
3. Federal Agency Debt
4. International Bonds
5. Municipal Bonds
6. Corporate Bonds
7. Mortgage and Asset-Backed Securities
 Financial Assets: Fixed Income
Some prominent issuers of Federal Agency Debt are
• The Federal Home Loan Banks (FHLB)
• The Federal National Mortgage Association (FNMA, or Fannie
Mae)
• The Government National Mortgage Association (GNMA, or
Ginnie Mae)
• The Federal Home Loan Mortgage Corporation (FHLMC, or
Freddie Mac)
 Financial Assets: Fixed Income
A mortgage-backed security (MBS) is either an ownership claim in a pool of mortgages or an
obligation that is secured by such a pool.
• https://www.investopedia.com/terms/s/securitization.asp
 Financial Assets: Equity
Common stocks, also known equities, represent ownership shares in a corporation.
• Each share of common stock entitles its owner to one vote on any matters of
corporate governance.
– Corporations occasionally issue two classes of common stock, one bearing the right to
vote, the other not
• Two most important characteristics
– Residual claim means that stockholders are the last in line of all those who have a claim
on the assets and income of the corporation.
• In a liquidation of the firm’s assets, the shareholders have a claim to what is left after all other
claimants such as the tax authorities, employees, suppliers, bondholders, and other creditors have
been paid.
• Preferred stocks are senior to common stock but subordinate to bonds in terms of claim.
– Limited liability: corporate shareholders may at worst have worthless stock. They are
not personally liable for the firm’s obligations.
Financial Assets: Equity
 Financial Assets: Equity
• The dividend yield is only part of the return on a stock investment
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 =
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

• Earnings per share (EPS) is the monetary value of earnings per outstanding share of common
stock for a company during a defined period of time.
– It is a key measure of corporate profitability
𝑁𝑁𝑁𝑁𝑁𝑁 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝐸𝐸𝐸𝐸𝐸𝐸 =
𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊ℎ𝑡𝑡𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

• Price–earnings ratio (P/E) is the ratio of the current stock price to last year’s earnings per share.
𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑃𝑃/𝐸𝐸 =
𝐸𝐸𝐸𝐸𝐸𝐸 𝑜𝑜𝑜𝑜 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
 Financial Institutions
• Security issuers (firms)
• Ultimate owner of the security (individual investors)
• Governments
• Financial institutions: stand between the security issuers and the ultimate owner of the
security
– About half of all stock is held by large financial institutions such as pension funds, mutual funds,
insurance companies, and banks.
– Underwriters: corporations do not market their own securities to the public. Instead, they hire
agents, called investment bankers, to represent them to the investing public
• Investment bankers advise the issuing corporation on the prices it can charge for the securities issued,
appropriate interest rates, and so forth. Ultimately, the investment banking firm handles the marketing of the
security in the primary market, where new issues of securities are offered to the public.
• Chapter 3.1 How Firms Issue Securities
– Specialization and economies of scales
 Financial Institutions
Financial intermediaries act as middlemen between lenders and borrowers. They
include banks, investment companies, insurance companies, and credit unions.
– Financial intermediaries can issue their own securities to raise funds to purchase the
securities of other corporations.
• First, by pooling the resources of many small investors, they are able to lend
considerable sums to large borrowers.
• Second, by lending to many borrowers, intermediaries achieve significant
diversification, so they can accept loans that individually might be too risky.
• Third, intermediaries build expertise through the volume of business they do
and can use economies of scale and scope to assess and monitor risk.
• Investment companies also can design portfolios specifically for large investors
with particular goals.
 Financial Assets: Derivative
The values of derivative assets, also called contingent claims, derive from the
values of other assets.

Derivative Instruments:
1. Futures: A futures contract calls for delivery of an asset (or, in some
cases, its cash value) at a specified delivery or maturity date for an
agreed-upon price, called the futures price, also to be paid at contract
maturity.
– https://www.investopedia.com/terms/f/futurescontract.asp
– The long position is held by the trader who commits to purchasing the asset.
– The short position commits to delivering the asset.
2. Options: call and put
 Financial Assets: Derivative
• Each contract calls for delivery of 5,000 bushels of corn.
 Financial Assets: Derivative
• Options: call and put
– A call option gives its holder the right to purchase an asset for a
specified price, called the exercise price or strike price, on or before
a specified expiration date.
– A put option gives its holder the right to sell an asset for a specified
exercise price on or before a specified expiration date.
• Call options must be purchased; futures contracts are entered
into without cost.
• The purchase price of an option is called the premium
Financial Assets: Derivative
 Contents
1. The Economic Way of Thinking
2. Loanable Funds Market
3. Asset Classes and Financial Instruments
4. Statistics for Financial Economics
 Variance and Standard Deviation

• The population variance is the sum N

of the squared differences between ∑ (x − μ) i
2

each observation and the population σ =

2 i=1
mean divided by the population size N
• The sample variance is the sum of
the squared differences between n
each observation and the sample
mean divided by the sample size ∑ (x − x)
i
2

minus 1 s =
2 i=1
n -1
 Variance and Standard Deviation
n

Unbiased sample variance ∑ (x − x)

i
2

𝒔𝒔𝟐𝟐𝒏𝒏−𝟏𝟏 s =
2 i=1
n -1

n
Biased sample variance ∑ (x − x)
i
2

𝒔𝒔𝟐𝟐𝒏𝒏 s =
2 i=1
n -1
 Variance and Standard Deviation
Sample variance, 𝑠𝑠 2 , can be computed as follows:
(∑𝑛𝑛 𝑥𝑥 )2
∑𝑛𝑛 𝑥𝑥 2
− 𝑖𝑖=1 𝑖𝑖
𝑖𝑖=1 𝑖𝑖
• 𝑠𝑠 2 = 𝑛𝑛
𝑛𝑛−1

∑𝑛𝑛 𝑥𝑥 2
−𝑛𝑛 𝑥𝑥̅ 2
• 𝑠𝑠 2 = 𝑖𝑖=1 𝑖𝑖
𝑛𝑛−1
 Variance and Standard Deviation
Given two small publicly traded companies,
Alpha and Beta, whose possible returns (in
PMF percentages) are discrete (for simplicity)
Alpha’s 𝑃𝑃 𝑥𝑥 = 0% = 25% and presented in the table on the right:
1. Calculate the population variance of
return 𝑃𝑃 𝑥𝑥 = 5% = 25% Alpha's returns.
𝑋𝑋 𝑃𝑃 𝑥𝑥 = 10% = 25% 2. Calculate the population variance of
𝑃𝑃 𝑥𝑥 = 15% = 25% Beta's returns.
3. Interpret the variances.
Beta’s 𝑃𝑃 𝑦𝑦 = −10% = 20% 4. Assuming the returns of Alpha and
return 𝑃𝑃 𝑦𝑦 = 0% = 20% Beta are independent, determine the
𝑌𝑌 𝑃𝑃 𝑦𝑦 = 10% = 30% mean and variance of your portfolio's
rate of return with 40% allocated to
𝑃𝑃 𝑦𝑦 = 15% = 30% Alpha and 60% allocated to Beta.
• The population standard N

deviation, σ, is the (positive) ∑ (x − μ)

i
2

square root of the population σ= i=1

N
variance
n
• The sample standard deviation, s,
is the (positive) square root of the
∑ i
(x − x) 2

S= i=1
sample variance n -1

• Variance measures the average squared "deviation" from the mean and
has the unit of the squared unit of xi
• By taking , we get back to the "standard" original unit of xi
• A measure of the “average” scatter around the mean
 Variance and Standard Deviation
Which one has smaller standard deviation?
• A
A

B
 Variance and Standard Deviation
• Stock A
– Average return last year = 50%
– Standard deviation = 5%
• Stock B:
– Average return last year = 100%
– Standard deviation = 5%

• Ignore the market risk and the correlations between individual stocks and the
market.
– Is the statement “stock A and B are equally risky” true or false?
 Coefficient of Variation
The coefficient of variation (CV), is a measure of relative dispersion that
expresses the standard deviation as a percentage of the mean (provided the
mean is positive and NOT close to zero)
• When the means of two objects are different, it is better to compare them
using CV rather than σ2 or s2
• Always in percentage (%)

Population coefficient of Sample coefficient of

variation variation
σ   s
CV =   ⋅ 100% CV =   ⋅ 100%
μ  x 
 Coefficient of Variation Misuse
• The coefficient of variation should typically only be used for
data measured on a ratio scale. That is, the data should be
continuous and have a meaningful zero.
• If the mean value is near zero, the coefficient of variation is
sensitive to small changes in the mean.
Are SAT math scores a good indicator of college success?
• The graph below gives the SAT math scores from a test given before admission to college and the
GPAs at college graduation from a random sample of 11 students at one small university.

Covariance and correlation are numerical measures of the linear relationship between two variables as intuitively
indicated in a scatter plot
• A positive value indicates a direct or increasing linear relationship
• A negative value indicates a decreasing linear relationship
 Covariance
• The population covariance
N

∑ (x − µ i x )(y i − µ y )
Cov (x , y) = σ xy = i=1
N

• The sample covariance

n

∑ (x − x)(y − y)
i i
Cov (x , y) = s xy = i=1
n −1

– No causal effect is implied

 Covariance
Year 2021 Year 2022 Year 2023
S&P 500 𝑋𝑋 $3800 $5000 $4100
Alpha stock 𝑌𝑌 $20 $30 $22

You would like to understand whether there is a linear relationship

between the annual average performance of the S&P 500 and the
performance of Alpha Inc stock from 2021 to 2023. Consider the
summarized prices in the table above.
1. Find the population covariance.
2. Assess the strength of the linear relationship using the population
covariance calculated.
 Covariance
For any constants 𝑎𝑎1 , 𝑎𝑎2 , 𝑏𝑏1 , 𝑏𝑏2 , variables 𝑋𝑋, 𝑌𝑌 have rules:
• 𝐶𝐶𝐶𝐶𝐶𝐶 𝑋𝑋, 𝑎𝑎1 = 0
• 𝐶𝐶𝐶𝐶𝐶𝐶 𝑋𝑋, 𝑋𝑋 = 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋)
• 𝐶𝐶𝑜𝑜𝑜𝑜 𝑋𝑋, 𝑌𝑌 = 𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌, 𝑋𝑋
• 𝐶𝐶𝑜𝑜𝑜𝑜 𝑎𝑎1 + 𝑏𝑏1 𝑋𝑋, 𝑎𝑎2 + 𝑏𝑏2 𝑌𝑌 = 𝑏𝑏1 𝑏𝑏2 𝐶𝐶𝐶𝐶𝐶𝐶 𝑋𝑋, 𝑌𝑌

– From this property, the covariance depends on units of measurement. Its unit is the
product of the units of X and Y
• i.e., not invariant to the scaling of X and Y
– Value of covariance varies if a variable such as height is measured in feet or inches
– It measures the direction, but not strength, of the linear relationship between X and Y
Positive Covariance Negative Covariance

Zero Covariance Zero Covariance (quadratic)

Source: Dr. Ping Yu

 Correlation
The correlation coefficient gives a standardized measure of the
linear relationship between two variables.
• It is generally more useful than covariance
– free of units
– provides both the direction and strength of a linear relationship
• Also called Pearson’s product-moment correlation coefficient
or Pearson’s r, was developed by Karl Pearson from a related
idea introduced by Francis Galton in the 1880s
 Correlation
• A population correlation is
𝐶𝐶𝐶𝐶𝐶𝐶(𝑥𝑥, 𝑦𝑦)
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑥𝑥, 𝑦𝑦 = 𝜌𝜌𝑥𝑥𝑥𝑥 =
𝜎𝜎𝑥𝑥 𝜎𝜎𝑦𝑦
• A sample correlation is
𝐶𝐶𝐶𝐶𝐶𝐶(𝑥𝑥, 𝑦𝑦)
𝑟𝑟𝑥𝑥𝑥𝑥 =
𝑠𝑠𝑥𝑥 𝑠𝑠𝑦𝑦
• A useful rule: a linear relationship exists if
2
|𝑟𝑟𝑥𝑥𝑥𝑥 | ≥
𝑛𝑛
• Both 𝜌𝜌𝑥𝑥𝑥𝑥 and 𝑟𝑟𝑥𝑥𝑥𝑥 ∈ [−1,1]
• Because 𝜎𝜎𝑥𝑥 and 𝜎𝜎𝑦𝑦 are positive, 𝜎𝜎𝑥𝑥𝑥𝑥 and 𝜌𝜌𝑥𝑥𝑥𝑥 always have the same
sign, and 𝜌𝜌𝑥𝑥𝑥𝑥 = 0 if and only if (iff) 𝜎𝜎𝑥𝑥𝑥𝑥 = 0 .
• This is also true for 𝑟𝑟𝑥𝑥𝑥𝑥 .
Figure on the left: Retail sales by quarter
• When 𝑟𝑟 = 0, there is no linear
relationship between x and y, but not
necessarily a lack of relationship
How Ice Cream Kills! Correlation vs. Causation
 Expectation and Variance
• The expected value (expectation, or mean), 𝐸𝐸[𝑋𝑋], of a discrete random variable 𝑋𝑋 is
defined as
𝐸𝐸 𝑋𝑋 = 𝜇𝜇𝑋𝑋 = � 𝑥𝑥𝑥𝑥(𝑥𝑥)
𝑥𝑥
– where the notation indicates that the summation extends over all possible values of 𝑥𝑥
– weighted average
• The expectation of the squared deviations about the mean, 𝑋𝑋 − 𝜇𝜇𝑋𝑋 2 , is called the
variance, given by
𝜎𝜎𝑋𝑋2 = 𝐸𝐸 𝑋𝑋 − 𝜇𝜇𝑋𝑋 2 = �{ 𝑥𝑥 − 𝜇𝜇𝑋𝑋 2 𝑃𝑃(𝑥𝑥)}
𝑥𝑥
– Alternative form 𝜎𝜎𝑋𝑋2 = 𝐸𝐸 𝑋𝑋 2
− 𝜇𝜇𝑋𝑋2 2
= ∑𝑥𝑥 𝑥𝑥 𝑃𝑃 𝑥𝑥 − 𝜇𝜇𝑋𝑋2
• The standard deviation is the positive square root of the variance
 Expectation and Variance
• Let 𝑋𝑋 be a discrete random variable with probability distribution 𝑃𝑃(𝑥𝑥), and let g(𝑥𝑥) be some function
of 𝑋𝑋. Then the expected value, E[𝑔𝑔 𝑥𝑥 ], of that function is defined as follows:
𝐸𝐸 𝑔𝑔 𝑋𝑋 = � 𝑔𝑔 𝑥𝑥 𝑃𝑃(𝑥𝑥) LOTUS: law of the
𝑥𝑥 unconscious statistician
– In general, 𝐸𝐸 𝑔𝑔 𝑋𝑋 ≠ 𝑔𝑔(𝐸𝐸[𝑋𝑋]) unless g(𝑥𝑥) is linear
• How about g 𝑥𝑥 = 𝑋𝑋 2 ?
• Let 𝑋𝑋 be a discrete random variable with mean 𝜇𝜇𝑋𝑋 and variance 𝜎𝜎𝑋𝑋2 , and let 𝑎𝑎 and 𝑏𝑏 be any constant
fixed numbers. Define the random variable 𝑌𝑌 as 𝑎𝑎 + 𝑏𝑏𝑏𝑏. Then, the mean and variance of 𝑌𝑌 are

𝜇𝜇𝑌𝑌 = 𝐸𝐸 𝑎𝑎 + 𝑏𝑏𝑏𝑏 = 𝑎𝑎 + 𝑏𝑏𝜇𝜇𝑋𝑋

𝜎𝜎𝑌𝑌2 = 𝑉𝑉𝑉𝑉𝑉𝑉 𝑎𝑎 + 𝑏𝑏𝑏𝑏 = 𝑏𝑏 2 𝜎𝜎𝑋𝑋2
𝜎𝜎𝑌𝑌 = 𝑏𝑏 𝜎𝜎𝑋𝑋
• Linearity of expectation
o Let 𝑋𝑋 and 𝑌𝑌 be two discrete random variables (independent or not) and 𝑍𝑍 = 𝑋𝑋 + 𝑌𝑌
o 𝐸𝐸 𝑍𝑍 = 𝐸𝐸 𝑋𝑋 + 𝐸𝐸[𝑌𝑌]
It does NOT mean
𝑃𝑃 𝑍𝑍 = 𝑃𝑃 𝑋𝑋 + 𝑃𝑃(𝑌𝑌)
 Discrete RV Continuous RV
Can take any value in an interval
Can take on no more than a
There’re uncountably many real
countable number of values
numbers in the interval

Can assign probabilities to a Can assign probabilities only

specific value to a range of values
 Discrete RV Continuous RV

Expectation Expectation
∞
𝐸𝐸 𝑋𝑋 = 𝜇𝜇𝑋𝑋 = ∑𝑥𝑥 𝑥𝑥𝑥𝑥(𝑥𝑥) 𝐸𝐸 𝑋𝑋 = 𝜇𝜇𝑋𝑋 = ∫−∞ 𝑥𝑥 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑

2
𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 = 𝐸𝐸 𝑋𝑋 − 𝜇𝜇𝑋𝑋 = 𝐸𝐸 𝑋𝑋 2 − 𝐸𝐸 𝑋𝑋 2

Variance Variance
∞
𝜎𝜎𝑋𝑋2 = � 𝑥𝑥 − 𝜇𝜇𝑋𝑋 2 𝑃𝑃(𝑥𝑥) 𝜎𝜎𝑋𝑋2 = � 𝑥𝑥 − 𝜇𝜇𝑋𝑋 2
𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑
𝑥𝑥 −∞

Let g(𝑥𝑥) be some function of 𝑋𝑋 Let g(𝑥𝑥) be some function of 𝑋𝑋

𝐸𝐸 𝑔𝑔 𝑋𝑋 = � 𝑔𝑔 𝑥𝑥 𝑃𝑃(𝑥𝑥) 𝐸𝐸 𝑔𝑔 𝑋𝑋 = � 𝑔𝑔(𝑥𝑥)𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑

𝑥𝑥 𝑥𝑥

In general, 𝐸𝐸 𝑔𝑔 𝑋𝑋 ≠ 𝑔𝑔(𝐸𝐸[𝑋𝑋]) unless g(𝑥𝑥) is linear LOTUS: law of the

unconscious statistician
 Expectation
• The expected value (expectation, or mean), 𝐸𝐸[𝑋𝑋], of a continuous random variable 𝑋𝑋 is defined as
∞

𝐸𝐸 𝑋𝑋 = 𝜇𝜇𝑋𝑋 = � 𝑥𝑥 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑
−∞
– The probability of any specific value is zero for a continuous random variable, the expected values for
continuous random variables are computed using integral calculus.
– It is defined through an approximation of a discrete random variable
• The variance of 𝑋𝑋 is defined as the expectation of the squared deviation, 𝑋𝑋 − 𝜇𝜇𝑋𝑋 2 , of the random
variable from its mean
∞

𝜎𝜎𝑋𝑋2 = 𝐸𝐸 𝑋𝑋 − 𝜇𝜇𝑋𝑋 2
= � 𝑥𝑥 − 𝜇𝜇𝑋𝑋 2
𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑
−∞
or
∞

𝜎𝜎𝑋𝑋2 = 𝐸𝐸 𝑋𝑋 2 − 𝐸𝐸 𝑋𝑋 2 = � 𝑥𝑥 2 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 − 𝜇𝜇𝑋𝑋2

−∞
• The standard deviation of 𝑋𝑋, 𝜎𝜎𝑋𝑋 , is the positive square root of the variance
 Joint Distribution
• Let 𝑋𝑋 and 𝑌𝑌 be a pair of random variables (discrete or random) with mean 𝜇𝜇𝑋𝑋 and 𝜇𝜇𝑌𝑌 and
variances 𝜎𝜎𝑋𝑋2 and 𝜎𝜎𝑌𝑌2
𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 + 𝑌𝑌 = 𝜎𝜎𝑋𝑋2 + 𝜎𝜎𝑋𝑋2 + 2𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌)
– For any constant 𝑎𝑎 and 𝑏𝑏, 𝑊𝑊 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 has
 𝜇𝜇𝑊𝑊|𝑌𝑌=𝑦𝑦0 = 𝑎𝑎 + 𝑏𝑏𝜇𝜇𝑋𝑋|𝑌𝑌=𝑦𝑦0
2 2
 𝜎𝜎𝑊𝑊|𝑌𝑌=𝑦𝑦0
= 𝑏𝑏 2 𝜎𝜎𝑋𝑋|𝑌𝑌=𝑦𝑦0

• Let 𝑋𝑋1 , 𝑋𝑋2 , … , 𝑋𝑋𝑘𝑘 be 𝑘𝑘 random variables (discrete or random) with mean 𝜇𝜇1 , 𝜇𝜇2 , … , 𝜇𝜇𝑘𝑘 and
variances 𝜎𝜎12 , 𝜎𝜎22 , … , 𝜎𝜎𝑘𝑘2
𝑘𝑘−1 𝑘𝑘

𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋1 + 𝑋𝑋2 + ⋯ + 𝑋𝑋𝑘𝑘 = 𝜎𝜎12 + 𝜎𝜎22 + ⋯ + 𝜎𝜎𝑘𝑘2 + 2 � � 𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋𝑖𝑖 , 𝑋𝑋𝑗𝑗 )

𝑖𝑖=1 𝑗𝑗>𝑖𝑖
– For any constant 𝑎𝑎𝑖𝑖 , 𝑊𝑊 = 𝑎𝑎1 𝑋𝑋1 + 𝑎𝑎2 𝑋𝑋2 + ⋯ + 𝑎𝑎𝑘𝑘 𝑋𝑋𝑘𝑘 has
 𝜇𝜇𝑊𝑊 = ∑𝑘𝑘𝑖𝑖=1 𝑎𝑎𝑖𝑖 𝜇𝜇𝑖𝑖
2
 𝜎𝜎𝑊𝑊 = ∑𝑘𝑘𝑖𝑖=1 𝑎𝑎𝑖𝑖2 𝜎𝜎𝑖𝑖2 + 2 ∑𝑘𝑘−1 𝑘𝑘
𝑖𝑖=1 ∑𝑗𝑗>𝑖𝑖 𝑎𝑎𝑖𝑖 𝑎𝑎𝑗𝑗 𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋𝑖𝑖 , 𝑋𝑋𝑗𝑗 )
 Joint Distribution
Disregard market risk. As an investment consultant, you have a customer
named Amy who holds a portfolio consisting of 20 shares of Alpha and 30
shares of Beta. The price of Alpha follows a normal distribution with a mean
of $25 and a variance of 81, while the price of Beta follows a normal
distribution with a mean of 40 and a variance of 121.
1. Suppose their stock prices have a positive correlation 0.4. Find the
expected value and variance of the portfolio value.
2. Suppose their stock prices have a negative correlation −0.4. Find the
expected value and variance of the portfolio value.
3. If you are risk averse (you prefer outcomes with low uncertainty over
those outcomes with high uncertainty) and given options to choose from,
which portfolio would you prefer?
 The Normal Distribution
• Most often used continuous
probability distribution for
economics and business
applications
– bell shaped centered on the mean
– symmetric
– a single peak: peak corresponds to
the mean, median, and mode of the
distribution.
A RV 𝑋𝑋 follows a normal distribution (also known as Gaussian distribution) with parameters 𝜇𝜇 and 𝜎𝜎 2 ,
denoted as 𝑿𝑿~𝑵𝑵(𝝁𝝁, 𝝈𝝈𝟐𝟐 ), then for −∞ < 𝑋𝑋 < ∞
• The PDF
1 − 𝑥𝑥−𝜇𝜇 2
𝑓𝑓 𝑥𝑥 = 𝑒𝑒 2𝜎𝜎2 A normal distribution is
2𝜋𝜋𝜎𝜎 2
• The CDF determined completely
𝑥𝑥 by its mean and
𝐹𝐹 𝑥𝑥 = � 𝑓𝑓(𝑡𝑡)𝑑𝑑𝑑𝑑 variance
−∞
• The mean and variance
𝐸𝐸 𝑋𝑋 = 𝜇𝜇 𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 = 𝜎𝜎 2
where
 𝜇𝜇 and 𝜎𝜎 2 are any numbers such that −∞ < 𝜇𝜇 < ∞ and 0 < 𝜎𝜎 2 < ∞
 𝑒𝑒 = the mathematical constant approximated by 2.71828. It is the base for natural logarithms, called Euler’s
number
 𝜋𝜋 = the mathematical constant approximated by 3.14159, Archimedes’ constant
 same 𝜎𝜎 same 𝜇𝜇
with different 𝜇𝜇 with different 𝜎𝜎

location
𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 + 𝑐𝑐 = 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋)

𝑉𝑉𝑉𝑉𝑉𝑉 𝑐𝑐 ⋅ 𝑋𝑋 = 𝑐𝑐 2 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋) scale

 Appendix
 Separating commercial banking from investment banking
 Venture Capital and Private Equity
 Simulation Showing Bias in Sample Variance

 Not required for the exam

 Venture Capital and Private Equity
• While large firms can raise funds directly from the stock and bond markets with help from their investment
bankers, smaller and younger firms that have not yet issued securities to the public do not have that option.
Start-up companies rely instead on bank loans and investors who are willing to invest in them in return for an
ownership stake in the firm.
• The equity investment in these young companies is called venture capital (VC). Sources of venture capital are
dedicated venture capital funds, wealthy individuals known as angel investors, and institutions such as pension
funds.
• Most venture capital funds are set up as limited partnerships. A management company starts with its own
money and raises additional capital from limited partners such as pension funds. That capital may then be
invested in a variety of start-up companies.
• The management company usually sits on the start-up company’s board of directors, helps recruit senior
managers, and provides business advice. It charges a fee to the VC fund for overseeing the investments. After
some period of time, for example, 10 years, the fund is liquidated and proceeds are distributed to the investors.
• Venture capital investors commonly take an active role in the management of a start-up firm. Other active
investors may engage in similar hands-on management but focus instead on firms that are in distress or firms
that may be bought up, “improved,” and sold for a profit. Collectively, these investments in firms that do not
trade on public stock exchanges are known as private equity investments.
Simulation Showing Bias in Sample Variance