Financial Engineering 4 Security Returns: The importance of volatility 4 Portfolio Returns: The case for diversification 4 Efficient Portfolios and mean variance analysis 4 The case for Index Funds 4 Futures and Options 4 Derivative Security Pricing: Black-Scholes 4 Using Options as a Hedging Tool 4 The Theory Police and Market Rationality 4 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 measures such 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 stakes were scaled up by a large factor?

The Importance of Volatility Volatility is also important because it can significantly affect 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%.

Probability of loss after n years approaches one half Ten Year Simulation of Returns

Portfolio Returns The case for diversification In finance, a portfolio is a group of securities owned by 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.

Probability of loss after n years becomes negligible Ten Year Simulation of Returns

Forming a Portfolio to Reduce Downside Risk

Forming a Portfolio to Reduce Downside Risk (continued)

Mean-Average Downside Risk

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-free T-bills and the portfolio of risky securities that provides the largest excess return per unit of risk.

T Bill 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? What is Passive Management?

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

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? 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 no value of their own. They derive their value from the value 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 he plants 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 100 shares 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

Option Pricing (continued)

Black Scholes Formula

Using Excel to Price Options

Call Price as a function of Strike Price

Using Options as a Hedging Tool

The Theory Police 4 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 Point A 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 what we 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 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.