Trading Case CA3

Trading Case CA3

Case Objectives

To understand how risk preferences influences price discovery when fundamentals remain the same as in CA1.

Key Concepts

Capital market line and the market price of risk, expected utility theory.

Case Description

This trading case is the same as CA1 except for the way you earn your trading bonus.

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 above, is just a scaling constant used to make grade cash a reasonable number.

Higher market cash corresponds to a higher grade cash. However, higher market cash corresponds to higher grade-cash using a conversion scheme that increases, in CA3, at an increasing 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 (0.6557603/1000) and beta equal to -0.0000525 and trader type-4's alpha is (0.3125215/1000) and beta is -0.00022.

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 4.139, whereas the expected grade cash from portfolio B is (6.90+8.690) * 0.5 = 4.690.

Observe that portfolio B earns higher expected 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 for 1), portfolio A has "cashed out" at $5000 market cash, but portfolio B is worth either $9000 or $1000 market cash.

Important Note: The above functional form only works over a relevant trading range. That is, it becomes negative if market cash gets too high. To avoid this a further bound is placed on grade cash so that above 10000 market cash you earn 10000 grade cash for sure (and similarly below 0 market cash you earn 0 grade cash).

The default CA3 trading case imposes these bounds but your instructor may choose to change alpha and beta to permit a larger relevant range of market cash.

Trading is conducted over a number of independent trials and a record of your cumulative grade cash is maintained.