STOCK MARKETS

Usually organized exchange

(NASDAQ an exception)

Transaction costs: spread + commission

ADR’s (foreign stocks trading in US stock markets)

Buying on Margin and Short Sales are common

PREFERRED STOCK(nonvoting shares usually paying a fixed stream of dividends)

• Fixed dividends• Priority over common•Can be callable, convertible, adjustable rate- dividend tied to current interest rates

Uses

• Track average returns

• Serve as benchmarks to compare performance

Equally-weighted, value-weighted.

Price indexes vs. return indexes.

Stock Indexes

Buying on Margin

When purchasing securities, investors have easy access to a source of debt financing called broker’s call loans. The act of buying shares financed (at least) partly with broker’s call loans is called buying on margin.

Purchasing stocks on margin means the investor borrows part of the purchase of the stock from a broker. The brokers in turn borrow money from a bank at the call money rate to finance these purchases; they then charge their clients that rate plus a service charge for the loan.

All securities purchased on margin must be maintained with the brokerage firm as the securities are collateral for the loan.

The Board of Governors of the Federal Reserve System limits the extent to which stock purchases can be financed using margin loans.

The percentage margin = net worth / market value of position

Example: an investor initially pays $6,000 toward the purchase of $10,000 worth of stock (100 shares at $100 per share) borrowing the remaining $4,000 from a broker.The initial percentage margin is: $6,000/$10,000 = 60%

And investor’s balance sheet will look like:

Assets: Value of the stock: $10,000Liabilities and Owner’s Equity: Loan from the broker: $4,000Equity: $6,000

If the stock price declines to $70 then the percentage margin becomes $3,000/$7,000 =43%

And the balance sheet will look like:

Assets: Value of the stock: $7,000

Liabilities and Owner’s Equity: Loan from the broker: $4,000

Equity: $3,000

If the stock value were to fall below $4,000, investor’s equity would become negative, meaning that the value of the stock is no longer sufficient collateral to cover the loan from the broker.

To guard against this possibility, the broker sets a maintenance margin.

If the percentage margin falls below the maintenance level, the broker will issue a margin call which requires the investor to add new cash or securities to restore the percentage margin to an acceptable level.

Example:

Maintenance margin = 30%

P= price of stock

Value of the investor’s 100 shares is: P*100

The equity in the account is: 100P − $4,000

The percentage margin is (100P − $4,000) / 100P

Initial Margin

Maintenance Margin

Margin Call

Another example: Margin Trading

X Corp $70

50% Initial Margin

40% Maintenance Margin

1000 Shares Purchased

Initial Position

Stock $70,000 Borrowed $35,000

Equity 35,000

Stock price falls to $60 per shareNew PositionStock $60,000 Borrowed $35,000 Equity 25,000Margin% = $25,000/$60,000 = 41.67%

How far can the stock price fall before a margin call?

(1000P - $35,000)* / 1000P = 40%P = $58.33

* 1000P - Amt Borrowed = Equity

Short sale

The sale of shares not owned by the investor but borrowed through a broker and later purchased to replace the loan.

Normally, an investor would first buy a stock and later sell it. With a short sale the order is reversed.

Example: Short Sale

Z Corp 100 Shares

50% Initial Margin

30% Maintenance Margin

$100 Initial Price

Sale Proceeds $10,000 (100*$100)

Margin & Equity 5,000 (initial margin)

Value of Stock Owed 10,000 (100*$100)

Stock Price Rises to $110

Sale Proceeds $10,000 (as before)Initial Margin 5,000Value of Stock Owed 11,000 (100*$110)Net Equity 4,000 Margin % (4000/11000) 36%

How much can the stock price rise before a margin call?

($15,000* − 100P) / (100P) = 30%P = $115.38

* Initial margin plus sale proceeds

Short Sale - Example 2• Q: Initial Margin Requirement: 40% Maintenance Margin Requirement: 20%You borrow 1000 shares of IBM and sell short at $ 60 /

share.

• Proceeds from short sale: $ 60,000 Cash deposit: $ 24,000 so that: % margin = net equity / value of the position = (84000 − 60,000) / 60000 = 40%• Some time later, the stock price rises to $70.Value of the position = $ 70,000Net equity = 84,000 − 70,000 = 14000%margin = 14000 / 70000 = 20%

PORTFOLIO THEORY

• Two key concepts in finance theory: Return and Risk

• Risk is the variability (uncertainty) of Return.

• The risk of a security is measured by the variance or standard deviation of its returns.

• Economic Agents are risk averse (which implies that (i) Given equal return, they would prefer low-risk security (ii) To assume more risk, they require compensation in terms of higher expected returns).

Return – Risk Relationship

R = E(e) + time preference + risk premium

Hence: high risk → high return → low price

R

CFP

1

Stock Valuation

P0 = = r > g

As it is extremely difficult to estimate the correct discount rate r and to forecast the future growth rate g, this formula is quite useless in daily stock market practice.

Stock Analysts usually use ratio comparisons:

P/E Ratio = P / eps = Market Cap / Net Income

Book-to-Market ratio = Market Cap / Book value = P / shareholders’ equity per share

g-r

g)(1D0 gr

D

1

Diversification: Forming portfolios combining many different securities.

Unless the return correlation of securities included in the portfolio is +1 diversification reduces risk.

PORTFOLIO RETURN AND RISK CALCULATIONS:

Rp = Σ wi Ri σp = Σ Σ wi wj Covij

Covij = Corij σi σj

Two-Security Portfolio: Return

rrpp = w = w11rr1 1 ++ ww22rr22

ww11 = Proportion of funds in Security 1 = Proportion of funds in Security 1

ww22 = Proportion of funds in Security 2 = Proportion of funds in Security 2

rr11 = Expected return on Security 1 = Expected return on Security 1

rr22 = Expected return on Security 2 = Expected return on Security 2

ww iiii=1=1

nn

= 11

p2

= w121

2 + w222

2 + 2w1w2 Cov(r1r2)p2

= w121

2 + w222

2 + 2w1w2 Cov(r1r2)

12 = Variance of Security 112 = Variance of Security 1

22 = Variance of Security 222 = Variance of Security 2

Cov(r1r2) = Covariance of returns for Security 1 and Security 2Cov(r1r2) = Covariance of returns for Security 1 and Security 2

Two-Security Portfolio: Risk

Covariance

1,2 = Correlation coefficient of returns

1,2 = Correlation coefficient of returns

Cov(r1r2) = 12Cov(r1r2) = 12

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2

2p = w1

2122

p = w121

2 + w22

+ w22

+ 2w1w2+ 2w1w2

rp = w1r1 + w2r2 + w3r3rp = w1r1 + w2r2 + w3r3

Cov(r1r2)Cov(r1r2)

+ w323

2+ w323

2

Cov(r1r3)Cov(r1r3)+ 2w1w3+ 2w1w3

Cov(r2r3)Cov(r2r3)+ 2w2w3+ 2w2w3

Three-Security Portfolio

E(rp) = w1r1 + w2r2E(rp) = w1r1 + w2r2

Two-Security Portfolio (Return and Risk)

p2

= w121

2 + w222

2 + 2w1w2 Cov(r1r2)

p = [w1

212 + w2

222 + 2w1w2 Cov(r1r2)]1/2

= 0= 0

E(r)E(r)

= 1= 1 = -1= -1

= -1= -1

= .3= .3

13%13%

8%8%

12%12% 20%20% σp

TWO-SECURITY PORTFOLIOS WITH TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS COEF.DIFFERENT CORRELATIONS COEF.

E(r)E(r) n-security portfoliosn-security portfolios

EfficientEfficientfrontierfrontier

GlobalGlobalminimumminimumvariancevarianceportfolioportfolio MinimumMinimum

variancevariancefrontierfrontier

IndividualIndividualassetsassets

St. Dev.

Lowest risk

E(r)E(r)

E(rE(rMM))

rrff

MMCMLCML

mm

Capital Market Line

A Stock has two types of risk: Market Risk and Unique Risk.

Market Risk: results from economy-wide uncertainties, is non-diversifiable.

Unique Risk: results from firm-specific uncertainties, is diversifiable by forming portfolios.

By forming a well-diversified portfolio, it is possible to eliminate all unique risks; that is, to reduce portfolio risk to market risk.

( n: the number of stocks in the portfolio ) As n goes up, σp decreases and eventually converges to

market risk.

Diversification and Portfolio Risk

• Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns.

• This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another.

• However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion (market risk).

Portfolio Risk and Number of Stocks

Nondiversifiable risk; Systematic Risk; Market Risk

Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk

n

In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not.

Portfolio risk

Measuring Components of Risk

Run the regression: Rit = Rf + βi(Rm−Rf) + et

i2 = i

2 m2 + 2(ei)

where;

i2 = total variance

i2 m

2 = systematic variance

2(ei) = unsystematic variance

ASSET PRICING MODELS: CAPM

Because, unique risk can be eliminated via diversification, there should be no reward for it (i.e. no risk premium for unique risk).

Only, market risk rewarded.

The reward for 1 unit of market risk is:

E(RM) – Rf

where is M is the market portfolio (e.g. index)

Then, each stock’s E(R) should be proportional to how much market risk it has.

A stock’s degree of vulnerability to market risk is measured by its beta (βi).

βi is the responsiveness of stock i’s returns to market return. βi = CoviM / σ2

M

βM = 1

If βi > 1 , then i is called agressive stock.

If βi < 1 , then i is called defensive stock.

βi can also be estimated by regression

of Ri on Rm.

CAPM: E(Ri) = Rf + βi [ E(RM) – Rf ]

βi is the quantity of risk, [ E(RM) – Rf ] is the price of risk.

βi [ E(RM) – Rf ] is the risk premium.

The required rate r to be used in stock valuation should be determined by this formula. The discount rate r to be used in NPV computation should be determined accordingly.

Relationship Between Risk & Return

Exp

ecte

d re

turn

)(β FMiFi RRRR

FR

1.0

MR

Example

Exp

ecte

d re

turn

%3FR

%3

1.5

%5.13

5.1β i %10MR

%5.13%)3%10(5.1%3 iR

M = Market portfoliorf = Risk free rate

E(rM) - rf = Market risk premium (equilibrium)

E(rM) - rf = Market price of risk = Slope of the CML

Abnormal Return = Ri − E(R)

Excess Return = Ri − Rf

Slope and Market Risk Premium

MM

Expected Return and Risk on Individual Securities

• The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio

• Individual security’s risk premium is a function of the covariance of returns with the market portfolio

E(r)E(r)

E(rE(rMM))

rrff

SMLSML

MMßßßß = 1.0= 1.0

Security Market Line

SML Relationships

= [COV(ri,rm)] / m2

Slope SML = E(rm) - rf

= market risk premium

E(Ri) = rf + i [E(rm) - rf]

E(r)E(r)

15%15%

SMLSML

ßß1.01.0

RRmm=11%=11%

rrff=3%=3%

1.251.25

Disequilibrium Example

Disequilibrium Example• Suppose a security with a of 1.25 is

offering expected return of 15%

• According to SML, it should be 13%

• Underpriced: offering too high of a rate of return for its level of risk