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Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus Chapter 13 Equity Valuation

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No Slide TitleFourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Basic Types of Models

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Intrinsic Value

Market Price

Trading Signal

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

No Growth Model

Stocks that have earnings and dividends that are expected to remain constant

Preferred Stock

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

V0 = 3.00 / (.15 - .08) = $42.86

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

ROE = Return on Equity for the firm

b = plowback or retention percentage rate

(1- dividend payout percentage rate)

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

D0 = $2.00 g1 = 20% g2 = 5%

k = 15% T = 3 D1 = 2.40

D2 = 2.88 D3 = 3.46 D4 = 3.63

V0 = D1/(1.15) + D2/(1.15)2 + D3/(1.15)3 +

D4 / (.15 - .05) ( (1.15)3

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Specified Holding Period Model

PN = the expected sales price for the stock at time N

N = the specified number of years the stock is expected to be held

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

PVGO = Present Value of Growth Opportunities

E1 = Earnings Per Share for period 1

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

g = .20 x .40 = .08 or 8%

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

PVGO = Present Value of Growth Opportunities

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Required Rates of Return (k)

Expected growth in Dividends

Fourth Edition

Irwin / McGraw-Hill

E1 - expected earnings for next year

E1 is equal to D1 under no growth

k - required rate of return

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

b = retention ration

Fourth Edition

Irwin / McGraw-Hill

P0 = D/k = $2.50/.125 = $20.00

PE = 1/k = 1/.125 = 8

Fourth Edition

Irwin / McGraw-Hill

E1 = $2.50 (1 + (.6)(.15)) = $2.73

D1 = $2.73 (1-.6) = $1.09

k = 12.5% g = 9%

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Coverage ratios

Leverage ratios

Liquidity ratios

Profitability ratios

Fourth Edition

Irwin / McGraw-Hill

T = number of periods to maturity

r = semi-annual discount rate or the semi-annual yield to maturity

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Ct = 40 (SA)

Fourth Edition

Irwin / McGraw-Hill

Bond Prices and Yields

Prices and Yields (required rates of return) have an inverse relationship

When yields get very high the value of the bond will be very low

When yields approach zero, the value of the bond approaches the sum of the cash flows

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Avg. Income = Int. +(Par-Price) / Yrs to maturity

Avg. Price = (Price + Par) / 2

Using the earlier example

Approx. YTM = 65.10/1074.50 = .0606 or 6.06%

Actual YTM = 6.00%

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Relationship between yields to maturity and maturity

Yield curve - a graph of the yields on bonds relative to the number of years to maturity

Usually Treasury Bonds

Have to be similar risk or other factors would be influencing yields

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Active strategy

Trade on market inefficiencies

Fourth Edition

Irwin / McGraw-Hill

Inverse relationship between price and yield

An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield

Long-term bonds tend to be more price sensitive than short-term bonds

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

As maturity increases, price sensitivity increases at a decreasing rate

Price sensitivity is inversely related to a bond’s coupon rate

Price sensitivity is inversely related to the yield to maturity at which the bond is selling

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

A measure of the effective maturity of a bond

The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment

Duration is shorter than maturity for all bonds except zero coupon bonds

Duration is equal to maturity for zero coupon bonds

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Price change is proportional to duration and not to maturity

DP/P = -D x [D(1+y) / (1+y)

D* = modified duration

Fourth Edition

Irwin / McGraw-Hill

Summary measure of length or effective maturity for a portfolio

Immunization of interest rate risk (passive management)

Net worth immunization

Target date immunization

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

In the Money - exercise of the option would be profitable

Call: market price>exercise price

Put: exercise price>market price

Out of the Money - exercise of the option would not be profitable

Call: market price>exercise price

Put: exercise price>market price

At the Money - exercise price and asset price are equal

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

American vs European Options

American - the option can be exercised at any time before expiration or maturity

European - the option can only be exercised on the expiration or maturity date

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Notation

Payoff to Call Holder

0 if ST < X

Profit to Call Holder

Fourth Edition

Irwin / McGraw-Hill

Payoff to Call Writer

0 if ST < X

Profit to Call Writer

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Payoffs to Put Holder

0 if ST > X

Profit to Put Holder

Fourth Edition

Irwin / McGraw-Hill

Payoffs to Put Writer

0 if ST > X

Profits to Put Writer

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Investment Strategy Investment

Leveraged Buy calls @ 10 100 options $1,000

equity Buy T-bills @ 2% $7,000

Yield

Fourth Edition

Irwin / McGraw-Hill

Microsoft Stock Price

$75 $80 $100

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Microsoft Stock Price

$75 $80 $100

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Payoff for

Total Payoff ST - X ST - X

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Long Call

Short Put

Fourth Edition

Irwin / McGraw-Hill

Arbitrage & Put Call Parity

Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal.

C - P = S0 - X / (1 + rf)T

If the prices are not equal arbitrage will be possible

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Stock Price = 110 Call Price = 17

Put Price = 5 Risk Free = 10.25%

Maturity = .5 yr X = 105

C - P > S0 - X / (1 + rf)T

17- 5 > 110 - (105/1.05)

12 > 10

Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Position Cashflow ST<105 ST> 105

Buy Stock -110 ST ST

Borrow

Sell Call +17 0 -(ST-105)

Buy Put -5 105-ST 0

Total 2 0 0

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Option Strategies

Spreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expiration

Vertical or money spread

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Option Values

Intrinsic value - profit that could be made if the option was immediately exercised

Call: stock price - exercise price

Put: exercise price - stock price

Time value - the difference between the option price and the intrinsic value

2

Fourth Edition

Irwin / McGraw-Hill

Option

value

X

Fourth Edition

Irwin / McGraw-Hill

Factor Effect on value

Time to expiration increases

Fourth Edition

Irwin / McGraw-Hill

d2 = d1 - (s T1/2)

So = Current stock price

N(d) = probability that a random draw from a normal dist. will be less than d.

9

Fourth Edition

Irwin / McGraw-Hill

e = 2.71828, the base of the nat. log.

r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option.

T = time to maturity of the option in years.

ln = Natural log function

s = Standard deviation of annualized cont. compounded rate of return on the stock

10

Fourth Edition

Irwin / McGraw-Hill

s = .50 d = 0

= .43

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Call Option Value

Co = Soe-dTN(d1) - Xe-rTN(d2)

Co = 100 X .6664 - 95 e- .10 X .25 X .5714

Co = 13.70

Implied Volatility

Using Black-Scholes and the actual price of the option, solve for volatility.

Is the implied volatility consistent with the stock?

14

Fourth Edition

Irwin / McGraw-Hill

Using the sample data

P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664)

Fourth Edition

Irwin / McGraw-Hill

P = C + PV (X) - So

= C + Xe-rT - So

r = .10 T = .25

P = 6.35

Fourth Edition

Irwin / McGraw-Hill

Hedging: Hedge ratio or delta

The number of stocks required to hedge against the price risk of holding one option

Call = N (d1)

Option Elasticity

Percentage change in the option’s value given a 1% change in the value of the underlying stock

16

Fourth Edition

Irwin / McGraw-Hill

Buying Puts - results in downside protection with unlimited upside potential

Limitations

Maturity of puts may be too short

Hedge ratios or deltas change as stock values change

17

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Futures and Forwards

Forward - an agreement calling for a future delivery of an asset at an agreed-upon price

Futures - similar to forward but feature formalized and standardized characteristics

Key difference in futures

Fourth Edition

Irwin / McGraw-Hill

Futures price - agreed-upon price at maturity

Long position - agree to purchase

Short position - agree to sell

Profits on positions at maturity

Long = spot minus original futures price

Short = original futures price minus spot

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Foreign currencies

Financial futures

Fourth Edition

Irwin / McGraw-Hill

Clearinghouse - acts as a party to all buyers and sellers.

Obligated to deliver or supply delivery

Closing out positions

Reversing the trade

Most trades are reversed and do not involve actual delivery

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Initial Margin - funds deposited to provide capital to absorb losses

Marking to Market - each day the profits or losses from the new futures price and reflected in the account.

Maintenance or variance margin - an established value below which a trader’s margin may not fall.

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Margin and Trading Arrangements

Margin call - when the maintenance margin is reached, broker will ask for additional margin funds

Convergence of Price - as maturity approaches the spot and futures price converge

Delivery - Actual commodity of a certain grade with a delivery location or for some contracts cash settlement

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Hedging -

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Basis and Basis Risk

Basis - the difference between the futures price and the spot price

over time the basis will likely change and will eventually converge

Basis Risk - the variability in the basis that will affect profits and/or hedging performance

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Futures Pricing

Spot-futures parity theorem - two ways to acquire an asset for some date in the future

Purchase it now and store it

Take a long position in futures

These two strategies must have the same market determined costs

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

no storage costs

no seasonal patterns in prices

Strategy 1: Buy the stock now and hold it until time T

Strategy 2: Put funds aside today to perform on a futures contract for delivery at time T that is acquired today

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Buy stock -So ST

Long futures 0 ST - FO

Invest in Bill

Total for B - FO(1+rf)T ST

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Since the strategies have the same flows at time T

FO / (1 + rf)T = SO

FO = SO (1 + rf)T

The futures price has to equal the carrying cost of the stock

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Implications for business and corporate finance

Implications for investment

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Random Walk - stock prices are random

Actually submartingale

Positive trend and random about the trend

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Prices react to information

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Once information becomes available, market participants analyze it

Competition assures prices reflect information

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Types of Stock Analysis

Technical Analysis - using prices and volume information to predict future prices

Weak form efficiency & technical analysis

Fundamental Analysis - using economic and accounting information to predict stock prices

Semi strong form efficiency & fundamental analysis

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Active Management

Security analysis

Fourth Edition

Irwin / McGraw-Hill

Market Efficiency and Portfolio Management

Even if the market is efficient a role exists for portfolio management

Appropriate risk level

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Basic Types of Models

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Intrinsic Value

Market Price

Trading Signal

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

No Growth Model

Stocks that have earnings and dividends that are expected to remain constant

Preferred Stock

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

V0 = 3.00 / (.15 - .08) = $42.86

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

ROE = Return on Equity for the firm

b = plowback or retention percentage rate

(1- dividend payout percentage rate)

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

D0 = $2.00 g1 = 20% g2 = 5%

k = 15% T = 3 D1 = 2.40

D2 = 2.88 D3 = 3.46 D4 = 3.63

V0 = D1/(1.15) + D2/(1.15)2 + D3/(1.15)3 +

D4 / (.15 - .05) ( (1.15)3

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Specified Holding Period Model

PN = the expected sales price for the stock at time N

N = the specified number of years the stock is expected to be held

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

PVGO = Present Value of Growth Opportunities

E1 = Earnings Per Share for period 1

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

g = .20 x .40 = .08 or 8%

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

PVGO = Present Value of Growth Opportunities

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Required Rates of Return (k)

Expected growth in Dividends

Fourth Edition

Irwin / McGraw-Hill

E1 - expected earnings for next year

E1 is equal to D1 under no growth

k - required rate of return

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

b = retention ration

Fourth Edition

Irwin / McGraw-Hill

P0 = D/k = $2.50/.125 = $20.00

PE = 1/k = 1/.125 = 8

Fourth Edition

Irwin / McGraw-Hill

E1 = $2.50 (1 + (.6)(.15)) = $2.73

D1 = $2.73 (1-.6) = $1.09

k = 12.5% g = 9%

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Coverage ratios

Leverage ratios

Liquidity ratios

Profitability ratios

Fourth Edition

Irwin / McGraw-Hill

T = number of periods to maturity

r = semi-annual discount rate or the semi-annual yield to maturity

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Ct = 40 (SA)

Fourth Edition

Irwin / McGraw-Hill

Bond Prices and Yields

Prices and Yields (required rates of return) have an inverse relationship

When yields get very high the value of the bond will be very low

When yields approach zero, the value of the bond approaches the sum of the cash flows

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Avg. Income = Int. +(Par-Price) / Yrs to maturity

Avg. Price = (Price + Par) / 2

Using the earlier example

Approx. YTM = 65.10/1074.50 = .0606 or 6.06%

Actual YTM = 6.00%

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Relationship between yields to maturity and maturity

Yield curve - a graph of the yields on bonds relative to the number of years to maturity

Usually Treasury Bonds

Have to be similar risk or other factors would be influencing yields

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Active strategy

Trade on market inefficiencies

Fourth Edition

Irwin / McGraw-Hill

Inverse relationship between price and yield

An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield

Long-term bonds tend to be more price sensitive than short-term bonds

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

As maturity increases, price sensitivity increases at a decreasing rate

Price sensitivity is inversely related to a bond’s coupon rate

Price sensitivity is inversely related to the yield to maturity at which the bond is selling

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

A measure of the effective maturity of a bond

The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment

Duration is shorter than maturity for all bonds except zero coupon bonds

Duration is equal to maturity for zero coupon bonds

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Price change is proportional to duration and not to maturity

DP/P = -D x [D(1+y) / (1+y)

D* = modified duration

Fourth Edition

Irwin / McGraw-Hill

Summary measure of length or effective maturity for a portfolio

Immunization of interest rate risk (passive management)

Net worth immunization

Target date immunization

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

In the Money - exercise of the option would be profitable

Call: market price>exercise price

Put: exercise price>market price

Out of the Money - exercise of the option would not be profitable

Call: market price>exercise price

Put: exercise price>market price

At the Money - exercise price and asset price are equal

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

American vs European Options

American - the option can be exercised at any time before expiration or maturity

European - the option can only be exercised on the expiration or maturity date

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Notation

Payoff to Call Holder

0 if ST < X

Profit to Call Holder

Fourth Edition

Irwin / McGraw-Hill

Payoff to Call Writer

0 if ST < X

Profit to Call Writer

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Payoffs to Put Holder

0 if ST > X

Profit to Put Holder

Fourth Edition

Irwin / McGraw-Hill

Payoffs to Put Writer

0 if ST > X

Profits to Put Writer

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Investment Strategy Investment

Leveraged Buy calls @ 10 100 options $1,000

equity Buy T-bills @ 2% $7,000

Yield

Fourth Edition

Irwin / McGraw-Hill

Microsoft Stock Price

$75 $80 $100

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Microsoft Stock Price

$75 $80 $100

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Payoff for

Total Payoff ST - X ST - X

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Long Call

Short Put

Fourth Edition

Irwin / McGraw-Hill

Arbitrage & Put Call Parity

Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal.

C - P = S0 - X / (1 + rf)T

If the prices are not equal arbitrage will be possible

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Stock Price = 110 Call Price = 17

Put Price = 5 Risk Free = 10.25%

Maturity = .5 yr X = 105

C - P > S0 - X / (1 + rf)T

17- 5 > 110 - (105/1.05)

12 > 10

Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Position Cashflow ST<105 ST> 105

Buy Stock -110 ST ST

Borrow

Sell Call +17 0 -(ST-105)

Buy Put -5 105-ST 0

Total 2 0 0

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Option Strategies

Spreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expiration

Vertical or money spread

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Option Values

Intrinsic value - profit that could be made if the option was immediately exercised

Call: stock price - exercise price

Put: exercise price - stock price

Time value - the difference between the option price and the intrinsic value

2

Fourth Edition

Irwin / McGraw-Hill

Option

value

X

Fourth Edition

Irwin / McGraw-Hill

Factor Effect on value

Time to expiration increases

Fourth Edition

Irwin / McGraw-Hill

d2 = d1 - (s T1/2)

So = Current stock price

N(d) = probability that a random draw from a normal dist. will be less than d.

9

Fourth Edition

Irwin / McGraw-Hill

e = 2.71828, the base of the nat. log.

r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option.

T = time to maturity of the option in years.

ln = Natural log function

s = Standard deviation of annualized cont. compounded rate of return on the stock

10

Fourth Edition

Irwin / McGraw-Hill

s = .50 d = 0

= .43

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Call Option Value

Co = Soe-dTN(d1) - Xe-rTN(d2)

Co = 100 X .6664 - 95 e- .10 X .25 X .5714

Co = 13.70

Implied Volatility

Using Black-Scholes and the actual price of the option, solve for volatility.

Is the implied volatility consistent with the stock?

14

Fourth Edition

Irwin / McGraw-Hill

Using the sample data

P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664)

Fourth Edition

Irwin / McGraw-Hill

P = C + PV (X) - So

= C + Xe-rT - So

r = .10 T = .25

P = 6.35

Fourth Edition

Irwin / McGraw-Hill

Hedging: Hedge ratio or delta

The number of stocks required to hedge against the price risk of holding one option

Call = N (d1)

Option Elasticity

Percentage change in the option’s value given a 1% change in the value of the underlying stock

16

Fourth Edition

Irwin / McGraw-Hill

Buying Puts - results in downside protection with unlimited upside potential

Limitations

Maturity of puts may be too short

Hedge ratios or deltas change as stock values change

17

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Futures and Forwards

Forward - an agreement calling for a future delivery of an asset at an agreed-upon price

Futures - similar to forward but feature formalized and standardized characteristics

Key difference in futures

Fourth Edition

Irwin / McGraw-Hill

Futures price - agreed-upon price at maturity

Long position - agree to purchase

Short position - agree to sell

Profits on positions at maturity

Long = spot minus original futures price

Short = original futures price minus spot

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Foreign currencies

Financial futures

Fourth Edition

Irwin / McGraw-Hill

Clearinghouse - acts as a party to all buyers and sellers.

Obligated to deliver or supply delivery

Closing out positions

Reversing the trade

Most trades are reversed and do not involve actual delivery

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Initial Margin - funds deposited to provide capital to absorb losses

Marking to Market - each day the profits or losses from the new futures price and reflected in the account.

Maintenance or variance margin - an established value below which a trader’s margin may not fall.

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Margin and Trading Arrangements

Margin call - when the maintenance margin is reached, broker will ask for additional margin funds

Convergence of Price - as maturity approaches the spot and futures price converge

Delivery - Actual commodity of a certain grade with a delivery location or for some contracts cash settlement

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Hedging -

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Basis and Basis Risk

Basis - the difference between the futures price and the spot price

over time the basis will likely change and will eventually converge

Basis Risk - the variability in the basis that will affect profits and/or hedging performance

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Futures Pricing

Spot-futures parity theorem - two ways to acquire an asset for some date in the future

Purchase it now and store it

Take a long position in futures

These two strategies must have the same market determined costs

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

no storage costs

no seasonal patterns in prices

Strategy 1: Buy the stock now and hold it until time T

Strategy 2: Put funds aside today to perform on a futures contract for delivery at time T that is acquired today

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Buy stock -So ST

Long futures 0 ST - FO

Invest in Bill

Total for B - FO(1+rf)T ST

Essentials of Investments

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Since the strategies have the same flows at time T

FO / (1 + rf)T = SO

FO = SO (1 + rf)T

The futures price has to equal the carrying cost of the stock

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Implications for business and corporate finance

Implications for investment

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Random Walk - stock prices are random

Actually submartingale

Positive trend and random about the trend

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Prices react to information

Essentials of Investments

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Once information becomes available, market participants analyze it

Competition assures prices reflect information

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Fourth Edition

Irwin / McGraw-Hill

Types of Stock Analysis

Technical Analysis - using prices and volume information to predict future prices

Weak form efficiency & technical analysis

Fundamental Analysis - using economic and accounting information to predict stock prices

Semi strong form efficiency & fundamental analysis

Essentials of Investments

Fourth Edition

Irwin / McGraw-Hill

Active Management

Security analysis

Fourth Edition

Irwin / McGraw-Hill

Market Efficiency and Portfolio Management

Even if the market is efficient a role exists for portfolio management

Appropriate risk level