A New Theory

1
A New Theory

• Eric Falkenstein

2
Problem

• There are a few areas where we see a risk premium
• Insurance against accidents
• Short end of the yield curve
• On average, no risk premium
• Negative risk premium to high risk
• People seem to be gambling, not investing, in
  practice, with the same expected return to
  gambling

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Simple Proof of no average risk-return

• Say utility is relative
• Y riskier than X in standard approach
• Y and X same risk in relative sense
• Risk is unnecessary choose ½(XY)
• Like idiosyncratic risk, unnecessary risk
  unpriced

4
Why Relative Risk?

• Easterlin Paradox
• Benchmarking, tracking error as risk
• Not a new idea
• Adam Smith, Karl Marx, Thorsten Veblen, Max
  Weber, all focused on status (didnt formalize)
• Pesendorfer (1995) and Rayo and Becker (2006)
  modeled status orientation formally

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Utility Proof

• Two assets, a risky security and a risk-free
  security, with returns RE and Rf
• Where
• (risk free return is certain)
• There are two identical agents, i and i, who
  have wealth in period 0 of k, and spends money,
  ?, on the two assets
• No consumption. In the next period, agent is
  wealth is thus

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Utility Proof

• Utility is relative
• Max utility subject to budget constraint
• Substituting for , because
  the budget constraint holds with equality.

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Utility Proof

• Taking the first order condition, we have
• Since each agent is identical, in equilibrium
  each agent holds the same amount
• So
• Or
• Which means, the expected return on risky assets
  is te risk free rate

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Utility Proof Benchmark

• Utility is absolute
• Max utility subject to budget constraint
• Substituting for , because
  the budget constraint holds with equality.

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Utility Proof Benchmark

• Taking the first order condition, we have
• This is the standard result

10
Arbitrage Argument

• Basically, if people are benchmarking against the
  market, ?1 has no risk
• Then,
• Through arbitrage
• Which means, trivially, that all assets have the
  same return

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Adjustment 1 Hope and Alpha

• Search for alpha like Optimal stopping problem
• Sample various investments xj where xjN(m,s2 )
• Each investment costs cgt0
• You get T draws (eg, 100)
• At any stage n, can stop and receive xn in until
  T
• Optimal to sample until xngtk(c,n, ?2 ,T), where k
  is the criterion for stopping

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Optimal to sample until xngtk(c,n)

• As the cost of sampling goes up, propensity to
  stop searching increases
• As the variance of the sampling information
  increases, the propensity to stop searching
  decreases
• As the time left in the draw goes down, the
  propensity to stop increases

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Risk taking increases return

• You are willing to pay to take risk because you
  get value from the extra sampling in many cases
  (cgt0)
• Sampling more than once, for most parameters,
  will be the optimal solution for most situations
• As time goes on, the ability to take such risk
  decreases because the benefits are not as great
• High variance increases value for search
• Forms our intuition

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What is 'sampling' in practice?

• Trying something to see if you have alpha
• Trying out for football
• Writing poetry
• Appearing on American Idol
• Most fail, miserably. Why it hurts to fail, it
  reflects on you.
• Finding your best fit has big payoffs
• Outside of organized sampling as in school,
  generally people will tell you, you have no chance

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Ignoring the Experts when searching for alpha

• That the automobile has practically reached the
  limit of its development
• Scientific American 1909
• Heavier-than-air flying machines are impossible
• Lord Kelvin 1895
• There is no reason anyone would want a computer
  in their home
• Ken Olson 1977
• We stand on the threshold of rocket mail
• US postmaster general Arthur Summerfield 1959
• Nuclear-powered vacuum cleaners will probably be
  a reality in 10 years
• Alex Lewyt President of Lewyt Vacuums, 1955

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So, ignore the odds

• People overconfident when they search for alpha
• Good meta-strategy , bad investment strategy
• Leads to excess demand for super risky assets

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Where there is no Alpha, no hope, and Cash is
Useful

• AAA-BBB spread, 3mo to 2 yr T-bills
• No alpha searching here
• Prescience too hard to prove
• Everyone needs some amount of safety assets, cash
• Cash is a medium of exchange, a property many
  investments do not have
• Repos (cash) have T-bills, AAA securities as
  collateral

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An Equilibrium Across All Assets
No Alpha Possible, no hope, no benchmarking
Alpha Possible, Benchmarking
Expected Return
Expensive Alpha Searching, Too much hope
Risk