Financial Engineering

Security Returns: The importance of volatility Portfolio Returns: The case for diversification Efficient Portfolios and mean variance analysis The case for Index Funds Futures and Options Derivative Security Pricing: Black-Scholes Using Options as a Hedging Tool The Theory Police and Market Rationality System Risk

Security Returns: The Importance of Volatility

Volatility is a measure of uncertainty or risk. The volatility of security returns is often measured by the standard deviation of returns, but other measuressuch as average downside risk are also used.

The Importance of Volatility

Volatility of security returns is important because the utility (benefit) we derive from wealth is not linear.While more wealth is always preferred, the marginal utility of wealth tends to decrease as wealth increases.

Example: Most people would pay four dollar to enter into a game that pays $10 dollars if the outcome of tossing a fair coin is heads and nothing if it is tails. Question: Would you play the game if the stakeswere scaled up by a large factor?

Utility

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$0 $200,000 $400,000 $600,000 $800,000 $1,000,000

Wealth

The Importance of Volatility

Volatility is also important because it can significantlyaffect the way wealth accumulates.

Example: Suppose a security returns 100% with probability one half and -50% with probability one half.The expected return of this security is 25% but it has a standard deviation of 75%.

year return cum_wealth0 - $10,0001 100% $20,0002 -50% $10,0003 100% $20,0004 -50% $10,0005 100% $20,0006 -50% $10,0007 -50% $5,0008 100% $10,0009 -50% $5,000

10 100% $10,000

Probability of loss after n years approaches one half

Ten Year Simulation of Returns

Portfolio ReturnsThe case for diversification

•In finance, a portfolio is a group of securities ownedby an individual or corporation.

•A well diversified portfolio can greatly reduce risk.

The Case for Diversification (continued)

Example: Each year invest half your money on two securities. Assume that each security returns independently 100% with probability one half and -50% with probability one half.

year return sec_1 return sec_2 cum_wealth0 - - $10,0001 100% -50% $12,5002 -50% 100% $15,6253 100% 100% $31,2504 -50% -50% $15,6255 100% -50% $19,5316 -50% 100% $24,4147 -50% 100% $30,5188 100% -50% $38,1479 -50% 100% $47,684

10 100% -50% $59,605

Probability of loss after n years becomes negligible

Ten Year Simulation of Returns

Forming a Portfolio to Reduce Downside Risk

Assume each scenario is equally likely

Security ReturnsPort_return by scenario

Downside_risk by scenario

Scenarios 1 2 31 5.51% 4.80% 2.56% 5.5% 0.0%2 -1.24% 0.61% 0.16% -1.2% 1.2%3 5.46% 3.60% -1.64% 5.5% 0.0%4 -1.70% -1.30% 0.30% -1.7% 1.7%

Portfolio 100% 0% 0%

Average Port_return 2.01%Average downside_risk 0.74%

Forming a Portfolio to Reduce Downside Risk

Assume each scenario is equally likely

Security ReturnsPort_return by scenario

Downside_risk by scenario

Scenarios 1 2 31 5.51% 4.80% 2.56% 4.2% 0.0%2 -1.24% 0.61% 0.16% 0.5% 0.0%3 5.46% 3.60% -1.64% 2.2% 0.0%4 -1.70% -1.30% 0.30% -0.9% 0.9%

Portfolio 0% 73% 27%

Average Port_return 1.50%Average downside_risk 0.22%

Forming a Portfolio to Reduce Downside Risk (continued)

Assume each scenario is equally likely

Security ReturnsPort_return by scenario

Downside_risk by scenario

Scenarios 1 2 31 5.51% 4.80% 2.56% 4.5% 0.0%2 -1.24% 0.61% 0.16% 0.6% 0.0%3 5.46% 3.60% -1.64% 3.0% 0.0%4 -1.70% -1.30% 0.30% -1.1% 1.1%

Portfolio 0% 89% 11%

Average Port_return 1.75%Average downside_risk 0.28%

Forming a Portfolio to Reduce Downside Risk (continued)Assume each scenario is equally likely

Security ReturnsPort_return by scenario

Downside_risk by scenario

Scenarios 1 2 31 5.51% 4.80% 2.56% 5.4% 0.0%2 -1.24% 0.61% 0.16% -1.1% 1.1%3 5.46% 3.60% -1.64% 5.3% 0.0%4 -1.70% -1.30% 0.30% -1.7% 1.7%

Portfolio 91% 9% 0%

Average Port_return 2.00%Average downside_risk 0.68%

Risk-return

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Average Downside Risk

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Risk Return

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Standard Deviation of Port_return

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Mean-Standard Deviation

Taking Advantage of T-Bills

•The efficient frontier is the graph of the maximum expected return as a function of risk.

•The expected return, at any level of risk, can be improved by investing in a combination of risk-freeT-bills and the portfolio of risky securities that providesthe largest excess return per unit of risk.

The Capital Asset Line

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Standard Deviation of Port_return

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Tangency Portfolio

The Arithmetic of Active Management

If "active" and "passive" management styles are defined in sensible ways, it must be the case that

(1) before costs, the return on the average actively managed dollar will equal the return on the average passively managed dollar and

(2) after costs, the return on the average actively managed dollar will be less than the return on the average passively managed dollar

Problems with Active Management

•Past performances is a frail guide to the future.

•Winning strategies have a brief half-life

Advantages of Passive Management

•Low turnover resulting in low lower transaction costs and capital-gain taxes

•Fees charged by index funds run about 0.10%of assets. Active managers charges often exceed 1% of assets

Suppose that the market consists of three stocks

Question: How many shares of each stock should you buy if you want to passively invest $1,200?

CompanyShares

OutstandingCurrent

PriceMarket

CapA 150 $40 $6,000B 300 $20 $6,000C 150 $40 $6,000

What is Passive Management?

Question: How many shares of each stock should you buy if you want to passively invest $1,200?

CompanyShares

OutstandingCurrent

PriceMarket

CapA 150 $40 $6,000B 300 $20 $6,000C 150 $40 $6,000

Answer: 10 of A, 20 of B, 10 of C

What is Passive Management?

Suppose that a year later the prices of securities A, B, and C are respectively $40, $40, and $40 per share. No new shares are issued during the year.

Question: Do you need to rebalance your portfolio?

CompanyShares

OutstandingCurrent

PriceMarket

CapA 150 $40 $6,000B 300 $40 $12,000C 150 $40 $6,000

Suppose that a year later the prices of securities A, B, and C are respectively $40, $40, and $40 per share. No new shares are issued during the year.

Question: Do you need to rebalance your portfolio?

CompanyShares

OutstandingCurrent

PriceMarket

CapA 150 $40 $6,000B 300 $40 $12,000C 150 $40 $6,000

Answer: NO.

Suppose that Securities A, B, and C pay respectively $8, $0, and $8 per share in dividends. Immediately after paying dividends the prices of securities A, B, and C are respectively $40, $40, and $40 per share. No new shares are issued during the year.

Question: How would you reinvest the $160 in dividends to continue a passive investment strategy?

Passive Management (continued)

Suppose that Securities A, B, and C pay respectively $8, $0, and $8 per share in dividends. Immediately after paying dividends the prices of securities A, B, and C are respectively $40, $40, and $40 per share. No new shares are issued during the year.

Question: How would you reinvest the $160 in dividends to continue a passive investment strategy?

Answer: Buy 1 shares of A, 2 of B, and 1 of C.

Passive Management (continued)

Derivative Securities

Derivatives are financial instruments that have novalue of their own. They derive their value from thevalue of some other asset. They are use to hedge the risk of owning commodities, foreign currency, bonds, and common stocks. The product in derivative transactions is uncertainty itself.•Futures (contracts for future delivery at specified prices)•Options (give one side the opportunity to buy from or sell to the other side a prearranged price)

Futures

Contracts for future delivery at specified prices

•Exist in Europe since medieval times (lettres de faire)

•Used by Japanese feudal lords in the 1600s (cho-ai-mai)

Example: A farmer agrees to sell his crops before heplants it to protect himself from catastrophe if prices fall. The counter-party may be a food processor that wants protection against price increases.

Options

Give one side the opportunity to buy from or sell to the other side a prearranged price.•Aristotle (Politics) “a financial device which involves a principle of universal application.”•Used during the famous Dutch tulip bubble.

Options

Options are contingent claims on existing securities.

A call option gives the owner the right to buy a fixed number of shares of a stock at a fixed price, either before or at some fixed date. Example: Suppose you own an option to buy 100shares of IBM at $160 per share by May 1. If IBM trades at $168 on May 1 you would exercise the option and make $8 per share. If IBM trades under $160 the call option expires worthless.

Options (continued)

A put option gives the owner the right to sell a fixed number of shares of a stock at a fixed price, either before or at some fixed date.

Example: Suppose you own an option to sell 100 shares of IBM at $160 per share by May 1. If IBM trades at $140 on May 1 you would exercise the option and make $20 per share. If IBM trades over $160 the put option expires worthless.

Option Pricing

Current Stock Price 1.00$ Current Bond Price 1.00$ Strike Price 1.10$

Values a Period Later:Up Down

Bond $1.05 $1.05Stock $1.30 $0.80Call $0.20 $0.00

Question: What is the value of the call option?

Option Pricing (continued)

Replicating portfolio: 0.4 stocks and -0.3048 bondsup down

0.4 stocks 0.52$ 0.32$ -.3048 bonds (0.32)$ (0.32)$ payoff 0.20$ $0.00

cost of replicating portfolio 0.10$

value of call option 0.10$

Black Scholes Formula

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Using Excel to Price Options

Current Stock Price (in dollars) 100.00$ Strike Price (in dollars) 100.00$ Time to Expiration (in years) 0.25Rate on T-Bills 5.00%Stock Volatility 15%

Present Value of Strike Price 98.76$ d_1 0.20 d_2 0.13 Phi(d_1) 0.58 Phi(d_2) 0.55 Price of Call Option 3.64$

$0.00

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$90.00 $100.00 $110.00 $120.00 $130.00

Strike Price

Call P

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Call Price as a function of Strike Price

Using Options as a Hedging ToolOPTION_A.XLS Options Problem Spreadsheet, part (a)Call Option Strike Price 15$ Can buy or sell at most 3000 callsInitial budget is $10,000

Initial Stock Price ScenariosSecurities Number Prices 1 2 3Stock 500 20 40 20 12Bond 0 900 1000 1000 1000Option bought 0 10 25 5 0Option sold 0 -10 -25 -5 0Portfolio cost 10000 20000 10000 6000Budget const <= Profit 10000 0 -4000Initial budget 10000 0.33 0.33 0.33Cash leftover 0.00

12,000$ <-Expected Portfolio Value

(including cash)

Using Options as a Hedging ToolOPTION_A.XLS Options Problem Spreadsheet, part (a)Call Option Strike Price 15$ Can buy or sell at most 3000 callsInitial budget is $10,000

Initial Stock Price ScenariosSecurities Number Prices 1 2 3Stock 2000 20 40 20 12Bond 0 900 1000 1000 1000Option bought 0 10 25 5 0Option sold 3000 -10 -25 -5 0Portfolio cost 10000 5000 25000 24000Budget const <= Profit -5000 15000 14000Initial budget 10000 0.33 0.33 0.33Cash leftover 0.00

18,000$ <-Expected Portfolio Value

(including cash)

Using Options as a Hedging ToolOPTION_B.XLS Options Problem Spreadsheet, part (b)Call Option Strike Price 15$ Can buy or sell at most 3000 callsInitial budget is $10,000limit losses to $1,000 Initial Stock Price ScenariosSecurities Number Prices 1 2 3Stock 1600 20 40 20 12Bond 0 900 1000 1000 1000Option bought 0 10 25 5 0Option sold 2200 -10 -25 -5 0Portfolio cost 10000 9000 21000 19200Budget const <= Profit -1000 11000 9200Initial budget 10000 0.33 0.33 0.33Cash leftover 0

16,400$ <- Expected portfolio value

(including cash)

Using Options as a Hedging ToolOPTION_C.XLS Options Problem Spreadsheet, part (c)Call Option Strike PriceCan buy or sell at most 3000 callsInitial budget is $10,000Maximizes minimum profit Initial Stock Price ScenariosSecurities Number Prices 1 2 3Stock 1136 20 40 20 12Bond 0 900 1000 1000 1000Option bought 0 10 25 5 0Option sold 1273 -10 -25 -5 0Portfolio cost 10000 13636 16364 13636Budget const <= Profit 3636 6364 3636Initial budget 10000Cash leftover 0

14,545$ <- Expected portfolio value

(including cash)

The Theory Police

Prospect Theory. – Risk-averse or loss-averse?– Preferences can be manipulated by changes in

reference points.– Endowment effect – We extrapolate by overweighting new

information and forget about regression to the mean

Risk averse or loss averse?

A. 80% Chance of losing $4M and 20% chance of breaking even

B. 100% chance of losing $3M

Select A or B

Examples: Gibson Greetings vs Bankers Trust, and P&G vs Bankers Trust

Change in Reference PointA rare disease is expected to kill 600 people:

•Under plan A 200 people will be save.•Under plan B there is a 1/3 chance that everyone will be saved and a 2/3 chance that no one will be saved.

A rare disease is expected to kill 600 people:•Under plan C 400 people will die.•Under plan D there is a 1/3 chance that nobody will die, and a 2/3 chance that 600 will die.

Endowment Effect

We tend to set a higher selling price on whatwe own than what we would pay for the identical item if we did not own it.

Nationals of a country tend to overvalue domestic stocks and to undervalue foreign stocks.

Regression to the Mean

Objective 5 Years toMarch 1989

5 Years toMarch 1994

International stocks 20.6% 9.4%Income 14.3% 11.2%Growth and Income 14.2% 11.9%Growth 13.3% 13.9%Small company 10.3% 15.9%Aggressive growth 8.9% 16.1%Average 13.6% 13.1%

We extrapolate from the recent pass and forget about regression to the mean

How can you take advantage of

Quasi-Rational Behavior?•Even though investors are only quasi-rational, it is very hard to exploit lack of rationality.•Active portfolio managers have trouble keeping up with the market indexes they track.•Private partnerships managed by people with high performance quotients are accessible only to investors with at least $1M to invest.•Large institutional investors cannot allocate a significant portion of their assets to these partnerships.

System Risk

•The counterparties to most tailor-made derivatives are investment banks and insurance companies. The financial solvency of these institutions supports the solvency of the world economy. •The measurement of risk exposure in the system has become more comprehensive and sophisticated.