Statistics 701: CAPM from a statistical perspective

Announcement

• Homework 3 questions?
  • brushing
  • stepwise regression. how to. number of varaibles. breaking JMP IN.
  • which data.

Summary so far

• Ponzironi 1: Bonferonni p-value = n * regular p-value.
• Ponzironi 2: Unseen crashes can be delt with by adding 3 of them to your data.
• Ponzironi 3: Correct for risk since variance matters.
  • long run growth rate = mean - variance / 2
  • optimal amount of one investment: alpha/MSE
  • Figure of merit: alpha²/MSE = alpha²/RMSE²

Optimal investing

• What if you have more than one instrument?
  • Good life: means add AND variances add
  • Life is 1/2 good. means always add
  • Variances only add if uncorrelated

• How do you make something uncorrelated with the market?
  • Look at residInvestment = (Y-RF) - beta(M-RF)
  • sell beta of (M-R) buy (Y-RF)
  • Claim resid is uncorrelated with market if beta is choosen correct (otherwise residuals wouldn't be flat)
  • So use beta = beta of regression of Y on market
  • mean is now alpha of regression (just like CAPM)
  • SD = SD from regression

• How much to buy of each? (if uncorrelated)
  • optimize each separately
  • buy correct amount of market
  • buy a alpha/SD²
  • Key thing: is alpha significantly bigger than zero?

• If you are more risk adverse, buy more of the risk free and less of risky assets

CAPM

• Notice, if alpha > 0, everone wants to buy
• Notice, if alpha < 0, everone wants to sell
• Hence price is not in equilibrium unless alpha = 0

Putting it all together

• Figure of merit: R = alpha²/MSE
• Growth of log optimal portfolio increases by R/2
• It takes 4/R years to statistically prove this investment is profitable using out-of-sample data
  • Find an R of 1, and CAPM is dead in 4 years
  • Find an R of .01 and CAPM is indistinqushable from "better model" for next 400 years
  • For reference: T-bills vs. cash has R = (.03/.01)² = 9 (takes several months to prove better)
  • For reference: Market vs. t-bills has R = (.07/.2)² = .12 (takes 30 years to prove)

Example: The Quant-Jock

• Suppose you put a quant jock in a cage and ask him questions: Is XYZ corp a good investment? Five minutes later he gives an answer.
• A game the traders play is guess what the quant jock will say. No one can guess what he will say better than 50/50.
• To the world, the quant jock looks like a coin toss
• BUT, over the course of a year the quant jock's "buys" grow by 2 percent a year compared to his "sells".
• What is his figure of merit?
• Portfolio: Buy his "buys," short his "sells". Buy portfolio is correlated at least .99 with sell portfolio (same variance). Thus difference as has a variance of (1 - R²) * variance of market = .02 * (.2)².
• Figure of merit: (.02)²/(.02 * .2²) = .5
• Doubles every two years
• Takes 8 years to "prove" method is successful

Last modified: Tue Oct 30 08:11:31 2001