Trading Case CA1
Case Objectives
To understand meanvariance portfolio efficiency; to
learn how prices of risky assets are determined in a market; to understand how
CAPM applies to price discovery.
Key Concepts
Capital market line and the market price of risk,
expected utility theory, diversification and price discovery.
Case
Description
The case has 3 companies: Company 1, 2 and 3. You will start with an initial position in
the three companies. You can trade (i.e., buy or sell) the stocks of these
companies for the first day of the year.
At the end of the trading day the remaining days of the calendar year
then “flash by” and you earn or pay interest on your net cash balance, that
resulted from your first day’s trading activities, at the annual rate of
12%. After interest is settled, one of
10 path of the economy is realized. Each
path determines a value for each company, and your portfolio is market to
market at these values. The set of
paths and associated values is shown in the table below, where each possible
path is equally likely. You are allowed
borrow cash (at 12%) and also short sell stocks. If you sell short a stock and don’t cover
your position, then you will have to pay the realized value at the end of the
period.
Prices in this case are determined by the traders,
so all trades will take place at bids and asks that either you or another
trader in the system puts in.
Case Data
The possible paths for the economy, and the
corresponding endofperiod realized values for the three companies are shown
here. The average value is also shown.
Path 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Average 
Co. 1 
5 
5 
5 
24 
25 
30 
32 
68 
75 
75 
34.4 
Co. 2 
4 
5 
10 
21 
66 
65 
65 
20 
20 
20 
29.6 
Co. 3 
55 
55 
49 
22 
22 
20 
10 
10 
10 
10 
26.3 
Trading Objective
Your
aim is to maximize your return at the same time minimizing your risk. You should work through the section titled Earning a Trading Bonus below to see exactly how your
trading bonus is computed in CA1.
Initial Trader Endowments
There are four types of
traders with the following initial endowment cash and shares:

Co. 1 
Co. 2 
Co. 3 
Market Cash 
Type 1 
316 
24 
52 
$8084 
Type 2 
78 
100 
52 
$3354 
Type 3 
78 
24 
210 
$5364 
Type 4 
0 
0 
0 
$4200 
Trader types have the following distribution in the market: There are approximately three type4 traders to every one type1, 2 and 3 trader. That is, for every six traders there will be one type1, one type2, one type3 and three type4. Over different market sessions your trader type can change.
Earning a Trading Bonus
The object is to earn as
much grade cash as possible by
managing both the risk and return of your portfolio. The market is open for one trading
period. In calendar time this is the
beginning of a year. At the end of the
year your position is marked to the market value associated with the realized
path for the economy. To ensure that
your performance, visavis
other traders, is not disadvantaged by an "unlucky path realization"
your position will be marked to market for a set of independently realized
paths for the economy. You will see
below how this provides an incentive for you to manage both risk and expected return in this trading case. The trading and the marking of your position is referred to as one trial. Trading
will continue over multiple independent trials where you start with a fresh
initial position at the beginning of each trial.
OPERATIONAL DETAILS FOR EARNING GRADE CASH
Suppose that the relevant
range of values is $0 to $10000, then the operational
details are provided in four steps.
Step
1: At the end of the trading period a
path is realized and the marked value of your portfolio is converted to market
cash.
Step 2: This total market cash is converted to a grade cash range using the following general functional form:
Grade Cash = a(Market Cash  b*Market Cash^{2})
where type I and type II's beta will vary. Alpha is just a scaling constant used to make grade cash a reasonable number.
Higher market cash corresponds to higher grade cash. However, higher market cash corresponds to higher gradecash using a conversion scheme that increases at a decreasing rate. An example is provided below to demonstrate that this latter property penalizes portfolio risk (i.e., volatility).
The example is a simple example that has selected alpha and beta so that the grade and market cash numbers range from 0 to 10000. In the example trader types1, 2 and 3 have alpha equal to (2.105031/1000) and beta equal to 0.0000525 and trader type4's alpha is (1.282015/1000) and beta is 0.000022.
EXAMPLE
Compare the expected
grade cash for the following two portfolios, A and B. Let portfolio A have
zero stocks and $5000 dollars of market cash.
At the end of the year,
before the stock values are realized, suppose portfolio B, for 5 paths of the
economy, realizes $1000 market cash, and for the 5
remaining paths of the economy, realizes $9000 market cash. Because each path is equally probable the
expected market cash value for each portfolio is $5000.
The expected grade cash
for portfolio A is 7.762, whereas the expected grade cash from portfolio B is
(1.995+9.994) * 0.5 = 5.994.
Observe that portfolio A
has a higher chance of earning grade cash, even though the two portfolios have
the same expected market cash value.
This is because portfolio A has zero
variance, whereas portfolio B has positive
variance. You can see that portfolio A has "cashed out" at
$5000 market cash, but portfolio B is worth either
$9000 or $1000 market cash.
The default CA1 trading case further places bounds so that the range of
grade cash is between 0 and 10. To make
sure that these bounds hold, at market cash
of 10000 or more, you earn 10 in grade cash; at zero or below, you earn zero
grade cash,
Trading is conducted over a number of independent
trials and a record of your cumulative grade cash is maintained.
© OS Financial Trading System 2001