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 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?
Utility
1200
1000
800
600
400
200
0
$0
$200,000
$400,000
$600,000
Wealth
$800,000
$1,000,000
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%.
Ten Year Simulation of Returns
year
0
1
2
3
4
5
6
7
8
9
10
return
100%
-50%
100%
-50%
100%
-50%
-50%
100%
-50%
100%
cum_wealth
$10,000
$20,000
$10,000
$20,000
$10,000
$20,000
$10,000
$5,000
$10,000
$5,000
$10,000
Probability of loss after n years approaches one half
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.
Ten Year Simulation of Returns
year
0
1
2
3
4
5
6
7
8
9
10
return sec_1 return sec_2
100%
-50%
-50%
100%
100%
100%
-50%
-50%
100%
-50%
-50%
100%
-50%
100%
100%
-50%
-50%
100%
100%
-50%
cum_wealth
$10,000
$12,500
$15,625
$31,250
$15,625
$19,531
$24,414
$30,518
$38,147
$47,684
$59,605
Probability of loss after n years becomes negligible
Forming a Portfolio to Reduce
Downside Risk
Assume each scenario is equally likely
Scenarios
1
2
3
4
Portfolio
Security Returns
1
2
5.51%
4.80%
-1.24%
0.61%
5.46%
3.60%
-1.70% -1.30%
100%
0%
Port_return Downside_risk
by scenario by scenario
3
2.56%
0.16%
-1.64%
0.30%
0%
5.5%
-1.2%
5.5%
-1.7%
Average Port_return
Average downside_risk
2.01%
0.74%
0.0%
1.2%
0.0%
1.7%
Forming a Portfolio to Reduce
Downside Risk
Assume each scenario is equally likely
Scenarios
1
2
3
4
Portfolio
Security Returns
2
1
4.80%
5.51%
0.61%
-1.24%
3.60%
5.46%
-1.70% -1.30%
73%
0%
Port_return Downside_risk
by scenario by scenario
3
2.56%
0.16%
-1.64%
0.30%
27%
4.2%
0.5%
2.2%
-0.9%
Average Port_return
Average downside_risk
1.50%
0.22%
0.0%
0.0%
0.0%
0.9%
Forming a Portfolio to Reduce
Downside Risk (continued)
Assume each scenario is equally likely
Scenarios
1
2
3
4
Portfolio
Security Returns
1
2
5.51% 4.80%
-1.24% 0.61%
5.46% 3.60%
-1.70% -1.30%
0%
89%
Port_return Downside_risk
by scenario by scenario
3
2.56%
0.16%
-1.64%
0.30%
11%
4.5%
0.6%
3.0%
-1.1%
Average Port_return
Average downside_risk
1.75%
0.28%
0.0%
0.0%
0.0%
1.1%
Forming a Portfolio to Reduce
Downside Risk (continued)
Assume each scenario is equally likely
Scenarios
1
2
3
4
Portfolio
Security Returns
1
2
5.51%
4.80%
-1.24%
0.61%
5.46%
3.60%
-1.70% -1.30%
91%
9%
Port_return Downside_risk
by scenario by scenario
3
2.56%
0.16%
-1.64%
0.30%
0%
5.4%
-1.1%
5.3%
-1.7%
Average Port_return
Average downside_risk
2.00%
0.68%
0.0%
1.1%
0.0%
1.7%
Mean-Average Downside Risk
Risk-return
Expected Return
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
0.0%
0.2%
0.4%
0.6%
Average Downside Risk
0.8%
Mean-Standard Deviation
Average Port_return
Risk Return
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
0.5%
1.5%
2.5%
3.5%
4.5%
Standard Deviation of Port_return
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.
The Capital Asset Line
Expected Port_return
2.5%
2.0%
Tangency Portfolio
1.5%
1.0%
0.5%
T Bill
0.0%
0%
1%
2%
3%
Standard Deviation of Port_return
4%
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
What is Passive Management?
Suppose that the market consists of three stocks
Company
A
B
C
Shares
Outstanding
150
300
150
Current
Price
$40
$20
$40
Market
Cap
$6,000
$6,000
$6,000
Question: How many shares of each stock should you buy
if you want to passively invest $1,200?
What is Passive Management?
Company
A
B
C
Shares
Outstanding
150
300
150
Current
Price
$40
$20
$40
Market
Cap
$6,000
$6,000
$6,000
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
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.
Company
A
B
C
Shares
Outstanding
150
300
150
Current Market
Price
Cap
$40 $6,000
$40 $12,000
$40 $6,000
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.
Company
A
B
C
Shares
Outstanding
150
300
150
Current Market
Price
Cap
$40 $6,000
$40 $12,000
$40 $6,000
Question: Do you need to rebalance your portfolio?
Answer: NO.
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?
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.
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
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.
•Exist in Europe since medieval times (lettres de faire)
•Used by Japanese feudal lords in the 1600s (cho-ai-mai)
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
Current Stock Price
Current Bond Price
Strike Price
$ 1.00
$ 1.00
$ 1.10
Values a Period Later:
Up
Down
Bond
$1.05
$1.05
Stock
$1.30
$0.80
Call
$0.20
$0.00
Question: What is the value of the call option?
Option Pricing (continued)
Replicating portfolio: 0.4 stocks and -0.3048 bonds
up
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
Let S0 be the current va lue of the stock.
Let K 0 be the present va lue of the strike price.
The price C of a call option is given by
C  S 0  ( d1 )  K 0  ( d 2 )
where
d1 
ln(
S0
1
)   2t
K0 2
 t
and
d2 
ln(
S0
1
)   2t
K0 2
 t
Using Excel to Price Options
Current Stock Price (in dollars)
Strike Price (in dollars)
Time to Expiration (in years)
Rate on T-Bills
Stock Volatility
$ 100.00
$ 100.00
0.25
5.00%
15%
Present Value of Strike Price
d_1
d_2
Phi(d_1)
Phi(d_2)
Price of Call Option
$
$
98.76
0.20
0.13
0.58
0.55
3.64
Call Price as a function of Strike Price
$12.00
Call Price
$10.00
$8.00
$6.00
$4.00
$2.00
$0.00
$90.00
$100.00
$110.00
Strike Price
$120.00
$130.00
Using Options as a Hedging Tool
OPTION_A.XLS
Call Option Strike Price
$
Can buy or sell at most 3000 calls
Initial budget is $10,000
Securities
Stock
Bond
Option bought
Option sold
Portfolio cost
Budget const
Initial budget
Cash leftover
Options Problem Spreadsheet, part (a)
15
Number
500
0
0
0
10000
<=
10000
0.00
Initial
Prices
20
900
10
-10
Stock Price Scenarios
1
2
40
20
1000
1000
25
5
-25
-5
20000
10000
Profit
10000
0
0.33
0.33
3
12
1000
0
0
6000
-4000
0.33
$ 12,000 <-Expected Portfolio Value
(including cash)
Using Options as a Hedging Tool
OPTION_A.XLS
Call Option Strike Price
$
Can buy or sell at most 3000 calls
Initial budget is $10,000
Securities
Stock
Bond
Option bought
Option sold
Portfolio cost
Budget const
Initial budget
Cash leftover
Options Problem Spreadsheet, part (a)
15
Number
2000
0
0
3000
10000
<=
10000
0.00
Initial
Prices
20
900
10
-10
Profit
Stock Price Scenarios
1
2
40
20
1000
1000
25
5
-25
-5
5000
25000
-5000
15000
0.33
0.33
3
12
1000
0
0
24000
14000
0.33
$ 18,000 <-Expected Portfolio Value
(including cash)
Using Options as a Hedging Tool
OPTION_B.XLS
Options Problem Spreadsheet, part (b)
Call Option Strike Price
$
15
Can buy or sell at most 3000 calls
Initial budget is $10,000
limit losses to $1,000
Initial
Stock Price Scenarios
Securities
Number
Prices
1
2
3
Stock
1600
20
40
20
12
Bond
0
900
1000
1000
1000
Option bought
0
10
25
5
0
Option sold
2200
-10
-25
-5
0
Portfolio cost
10000
9000
21000
19200
Budget const
<=
Profit
-1000
11000
9200
Initial budget
10000
0.33
0.33
0.33
Cash leftover
0
$ 16,400 <- Expected portfolio value
(including cash)
Using Options as a Hedging Tool
OPTION_C.XLS
Options Problem Spreadsheet, part (c)
Call Option Strike Price
Can buy or sell at most 3000 calls
Initial budget is $10,000
Maximizes minimum profit
Initial
Stock Price Scenarios
Securities
Number
Prices
1
2
3
Stock
1136
20
40
20
12
Bond
0
900
1000
1000
1000
Option bought
0
10
25
5
0
Option sold
1273
-10
-25
-5
0
Portfolio cost
10000
13636
16364
13636
Budget const
<=
Profit
3636
6364
3636
Initial budget
10000
Cash 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?
Select A or B
A. 80% Chance of losing $4M and 20%
chance of breaking even
B. 100% chance of losing $3M
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
Objective
5 Years to
March 1989
5 Years to
March 1994
International stocks
20.6%
14.3%
14.2%
13.3%
10.3%
8.9%
13.6%
9.4%
11.2%
11.9%
13.9%
15.9%
16.1%
13.1%
Income
Growth and Income
Growth
Small company
Aggressive growth
Average
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.