**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 Cash^{2})

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.

© OS Financial Trading System 2001