Trading Case CA1

Trading Case CA1

Case Objectives

To understand mean-variance 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 end-of-period 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 type-4 traders to every one type-1, -2 and -3 trader. That is, for every six traders there will be one type-1, one type-2, one type-3 and three type-4. 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, vis-a-vis 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 Cash²)

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 grade-cash 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 types-1, -2 and -3 have alpha equal to (2.105031/1000) and beta equal to 0.0000525 and trader type-4'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.