Newsletter #168 Home
Newsletter Contributors
Previous Issues
BARRA Home




Developing and
Implementing
Risk Management
Systems

Fortis Group
Case Study



American Depository
Receipts in The
Global Equity Model



Analyzing the
Performance of
Crossing Networks
The Market
Impact Model™ — Part 4



The BARRA Brainteaser
 for Fall 1998
Solution to the Summer
1998 Brainteaser

 

 

Brainteaser Winners

Each winner will receive a BARRA Brainteaser T-shirt: Gary Sabot Sabot Associates, Inc. Larry Martin State Street Bank Alexandre Leal Bancu Icatu S.A. Honorable Mention: Geoff Ihle Wisconsin State Investment Board Chris Luck First Quadrant Ilan Ryfer Sagitta Fund Donald Tunnell Mellon Equity Associates, LLP

Solution to the Summer 1998 Brainteaser

Brainteaser from Summer 1998

Cab-driver-turned-stock-market-disciple Wade B. Cook offers "guaranteed" get-rich-quick courses to individual investors. His investment philosophy relies on technical analysis, high turnover, and leverage; and his courses cost $4,000 to $5,000 to attend.

This Brainteaser is concerned not with his philosophy or fees, but with Cook's money-back guarantee. He promises seminar attendees that within three months of their taking his course, at least three stocks listed on his stock-tip website will exhibit annualized returns of at least 300%—or he will return their course fees. Company spokesmen brag that no one has ever collected.

Assuming that all stock returns are independent, normally distributed, with expected annual returns of 12% and risk of 35%:

a) What is the probability that over one quarter at least three stocks out of 500 exhibit annualized returns of 300%?

b) How many stocks must his website include for that probability to be 50%?

c) Identify at least two real-world deviations from the above analysis and discuss how they would affect the calculated probabilities.

Brainteaser Solution

Wade B. Cook has guaranteed that at least three stocks listed on his website will exhibit annualized returns of 300% or more.

First, we'll convert the 300% annualized return into the associated quarterly (3-month) return hurdle:

(1)

 

(2)

So, the guarantee is that at least three stocks will be up by at least 41.42% over the quarter.

We are also told that each stock has an expected annual return of 12%, with 35% annual volatility. We can convert the expected annual return to an expected quarterly return of 2.87% using the methodology above. To convert annual to quarterly volatility, we simply divide by . The result is a quarterly volatility of 17.5%.

a) What is the probability that at least three stocks out of 500 exhibit the guaranteed return?

We are told that these stock returns are independent and normally distributed. The probability that a given stock will fall short of the hurdle is:

(3)

= 0.986

(4)

where N[] is the cumulative normal distribution function. The hurdle represents a 2.2 sigma event, so for any given stock the probability of a 300% annualized return is very small.

The probability that at least three out of 500 stocks exceed the hurdle is:

(5)

Using Equation 3, we see that the probability of exactly zero out of 500 stocks exceeding the hurdle is:

(6)

= 0.0009

(7)

The probability of exactly one stock exceeding the hurdle is:

(8)

where the factor of 500 arises because any one of the 500 stocks can exceed the hurdle. Therefore:

(9)

Similarly, the probability that exactly two stocks exceed the hurdle is:

(10)

= 0.0218

(11)

Hence, using Equation 5, we see that the probability of three or more stocks exceeding the hurdle is:

(12)

 

= 0.971

(13)

So, the probability that at least three out of 500 stocks will exceed the hurdle is extremely high.

b) How many stocks must the website include so that the answer to Question a) is 50% instead of 97.11%?

Clearly, the answer is less than 500. Equations 6, 8 and 10 show how our calculations depend on number of stocks. FIGURE 1 plots the probability that three or more stocks exceed the hurdle as a function of the number of stocks. We can see graphically, and verify in the underlying spreadsheet, that the probability is 50.12% with 191 stocks.

FIGURE 1: Probability of at least three Big Winners

c) What real world deviations would affect this analysis?

1. Non-Normal Distributions

Real world return distributions are non-normal, with fat tails. We calculated that the probability of exceeding the hurdle return, a 2.2 sigma event, was only 1.4%. The real world probability may be somewhat higher, which for a given number of stocks will increase the probability that at least three stocks exceed the hurdle.

2. Correlated Returns

We have assumed that all stocks are uncorrelated, whereas we know that in fact they are. The BARRA factor models capture this quite well.

A simplified model of correlation is Sharpe's market model where stocks' correlations are driven by their common exposure to the market. According to that model all residual returns are uncorrelated.

Even in this simple model the probability analysis becomes quite complex. We must analyze hurdle residual returns conditional on the market return, and integrate them over the distribution of market returns.

The net result should lower the unconditional probabilities that three or more stocks exceed the total return hurdle, since there is a reasonable probability that the quarterly market return will be negative, shifting all returns downward and dramatically decreasing the probability of individual stocks exceeding the hurdle. It would be interesting to see how many U.S. stocks were up at least 41.42% over the last quarter of 1987.

3. Distribution of Stock Volatilities

We have assumed that all stocks have a 35% annual volatility. Wade Cook can substantially improve his chances of meeting his guarantee by following the old stock picking contest strategy of choosing high volatility stocks.

At the end of July, the weighted average predicted volatility for S&P 500 stocks was 29%, but 152 stocks had volatilities above 35%, reaching a maximum of 73% for Parametric Technology.