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Financial Noise
And
Overconfidence
Ubiquitous News and Advice
 CNBC
 WSJ
 Jim Cramer
 Assorted talking heads
 How good can it be?
Thought Experiment
 Listen to CNBC every day
 Choose ten stocks from recommendations
 Buy stocks
 How good can this strategy be?
Thought Experiment
 10 stocks x 260 business days = 2,600 bets
 Assume the bets are independent
 How good does the information have to be such that the probability
of losing money is 1 in 10,000?
Detour: Gaussian Distribution
 Random Variable X depends upon two parameters:
‣ mean value: 
‣ uncertainty (standard deviation): 
1
 x    
Proba  X  b 
 exp 
 dx
2
2 

2
b
2
a
2
Detour: Gaussian Distribution
 Universal “Bell Curve” Distribution
 Prob(X < 0) depends only on the ratio of  / 
ProbX  0 
1
2
 /
 x2 
 exp   2  dx
The Central Limit Theorem
 {Xi}i = 1..N a set of r.v.:
‣ Identically distributed
‣ Independent
‣ All moments are finite
 If X = x1 + x2 + … + xN then:
‣
X has a Gaussian distribution with:
•  = N * average(x1)
•  = N * stddev(x1)
1
 x    
Proba  X  b 
 exp 
 dx
2
2 

2
b
2
a
2
The Central Limit Theorem
 X = x1 + x2 + … + xN is a Gaussian r.v. with:
‣ mean(X) =  = N * average(x1)
‣ stddev(X) =  = N * stddev(x1)
 For a Gaussian r.v. the probability that it is greater than zero is just
a function of the ratio of the mean to the stddev:


Prob(X 0)   
 
 In our experiment, our annual profit/loss will have a ratio of mean to
stddev sqrt(2,600)  50 greater than just one of our stock bets.
This is the advantage of aggregation.
How Good is the Information?
 Assume Prob(Xannual < 0)  1 / 10,000
 mean(Xannual)  3.7 * stddev(Xannual)
 mean(Xind)  (3.7 / 50) * stddev(Xind)  0.07 * stdev(Xind)
 Prob(Xind < 0)  47%
Probability That Advice Is Right?
 Any given piece of advice can’t be much more than 53% reliable,
compared to 50% for a random flip of a coin
 Otherwise, we could make a strategy that almost never loses
money by just aggregating all the different pieces of advice
 Contrast “my price target for this stock is 102 by mid-November”
with “I think it’s 53–47% that this goes up vs. down”
 Precision here greatly exceeds possible accuracy
 Overconfidence
 No error bars
Probability that Advice is Right?
 If we want Prob(Xannual < 0)  1 / 1,000,000 then:
‣ Prob(Xind < 0)  46%, instead of 47%
 A computer algorithm (or team of analysts) predicting something
everyday for all  2,000 U.S. stocks, instead of just 10 stocks, would
need only Prob(Xind < 0)  49.75% to get Prob(Xannual < 0) to 1 in
1,000,000!
 The existence of such a strategy seems unlikely
 Statements expressed with certainty about the market need to be
viewed with skepticism
 Low signal-to-noise ratio
Where Are Our Assumptions Wrong?
 Transaction costs matter
 Individual stock bets almost certainly not independent
 By placing bets on linear combinations of individual stocks one can
partially overcome correlations between bets, but you reduce the
effective number of bets.
 These aren’t just technical details. They can make the difference
between making or losing money.
Honesty Is The Best Policy
 Despite caveats, the big picture is correct
‣ Only small effects exist
‣ The data is very noisy
‣ One can find statistical significance only by aggregating lots of data
 As a result there is great value in:
‣ Clean data
‣ Quantitative sophistication
‣ Discipline
‣ Honesty
Overconfidence and Overfitting
 Humans are good at learning patterns when:
‣ Patterns are persistent
‣ Signal-to-noise ratio isn’t too bad
 Examples include: language acquisition, human motor control,
physics, chemistry, biology, etc.
 Humans are bad at finding weak patterns that come and go as
markets evolve
 They tend to overestimate their abilities to find patterns
Further Consequences of Aggregation
 If profit/loss distribution is Gaussian then only the mean and the
standard deviation matter
 Consider 3 strategies:
‣ Invest all your money in the S&P 500
‣ Invest all your money in the Nasdaq
‣ 50% in the Nasdaq and 50% in short-term U.S. Government
Bonds
 Over the past 40 years, these strategies returned approximately:
‣ 7% +/- 20%
‣ 8% +/- 35%
‣ 5.5% +/- 17.5%
Diversifying Risk
 Imagine we can lend or borrow at some risk-free rate r
 Suppose we have an amount of capital c, which we distribute:
‣ X in S&P 500
‣ Y in Nasdaq
‣ Z in “cash” (where c = X + Y + Z)
 Our investment returns are:
‣ rP = X*rSP + Y*rND + Z*r
‣ rP = X*(rSP – r) + Y*(rND – r) + c*r
‣ E(rP) = X*E(rSP – r) + Y*E(rND – r) + c*r
Consequences of Diversification
 As long as E(rSP) and/or E(rND) exceeds the risk-free rate, r, we
can target any desired return
 The price for high return is high volatility
 One measure of a strategy is the ratio of the return above the
risk-free rate divided by the volatility of the strategy
 Investing everything in Nasdaq gave the best return of the three
strategies in the original list
 Assuming the correlation between ND and SP is 0.85, the optimal
mixture of investments gives:
Y
 0.2
X
Consequences of Diversification
 For our example, we want Y  - 0.2*X, but we get to choose the
value of X. We can choose the level of risk!
 If we choose X  2.4 and Y  -0.5, we get the same risk as
investing everything in the Nasdaq but our return is  10.1%
rather than 8%
 Returns are meaningless if not quoted with:
‣ Volatility
‣ Correlations
 Why does everyone just quote absolute returns?
Past Results Do Not Guarantee Future Results
 If we only look back over the past 20 years, the numbers change:
‣ 3.5% +/- 20% for S&P
‣ 7% +/- 35% for Nasdaq
‣ 5.0% +/- 17.5% for 50% Nasdaq and 50 % Risk-free
 The same optimal weighting calculation gave X  -2.6 and Y  1.9
which gave the same risk as Nasdaq but with a return of 9.3%!
 40 years of data suggests that an investor should go long S&P and
short Nasdaq, but 20 years of data suggests to opposite. If we
used the 40 year weight on the last 20 years we’d end up making
2%.
Application to the Credit Crisis
 Securitization
‣ Bundles of loans meant to behave independently
‣ CDOs are slices of securitized bundles
‣ Rating agencies suggested 1 in 10,000 chance of investments
losing money
‣ All the loans defaulted together, rather than independently!
Credit Models Were Not Robust
 Even if mortgages were independent, the process of securitization
can be very unstable.
 Thought Experiment:
‣ Imagine a mortgage pays out $1 unless it defaults and then
pays $0.
‣ All mortgages are independent and have a default probability
of 5%.
‣ What happens to default probabilities when one bundles
mortgages?
Mortgage Bundling Primer: Tranches
 Combine the payout of 100 mortgages to make 100 new
instruments called “Tranches.”
 Tranche 1 pays $1 if no mortgage defaults.
 Tranche 2 pays $1 if only one mortgage defaults.
 Tranche i pays $1 if the number of defaulted mortgages < i.
 So far all we have done is transformed the cashflow.
 What are the default rates for the new Tranches?
Mortgage Bundling Primer: Tranches
 If each mortgage has default rate p=0.05 then the ith Tranche has
default rate:
100  j
 p (1  p)100 j
p T (i)   
j i  j 
 Where pT(i) is the default probability for the ith Tranche.
 For the first Tranche pT(1) = 99.4%, i.e., it’s very likely to default.
But for the 10th Tranche pT(10) = 2.8%.
 By the 10th Tranche we have created an instrument that is safer
than the original mortgages.
Securitizing Tranches: CDOs
 That was fun, let’s do it again!
 Take 100 type-10 Tranches and bundle them up together using the
same method.
 The default rate for the kth type-10 Tranche is then:
100 
 pT (10) j (1  pT (10))100 j
p CDO (k )   
j k  j 
 The default probability on the 10th type-10 Tranche is then
pCDO(k) = 0.05%. Only 1/100 the default probability of the original
mortgages!
Why Did We Do This: Risk Averse Investors
 After making these manipulations we still only have the same cash
flow from 10,000 mortgages, so why did we do it?
 Some investors will pay a premium for very low risk investments.
 They may even be required by their charter to only invest in
instruments rated very low risk (AAA) by rating agencies.
 They do not have direct access to the mortgage market, so they
can’t mimic securitization themselves.
Are They Really So Safe?
 These results are very sensitive to the assumptions.
 If the underlying mortgages actually have a default probability of
6% (a 20% increase) than the 10th Tranches have a default
probability of 7.8% (a 275% increase).
 Worse, the 10th type-10 Tranches will have a default probability of
25%, a 50,000% increase!
 These models are not robust to errors in the assumptions!
Application to the Credit Crisis
 Connections with thought experiment:
‣ Overconfidence (bad estimate of error bars)
‣ Insufficient independence between loans (bad use of Central
Limit Theorem)
‣ Illusory diversification (just one big bet houses wouldn’t lose
value)
About the D. E. Shaw Group
 Founded in 1988
 Quantitative and qualitative investment strategies
 Offices in North America, Europe, and Asia
 1,500 employees worldwide
 Managing approximately $22 billion (as of April 1, 2010)
About What I Do
 Quantitative Analyst (“Quant”)
 Most Quants have a background in math, physics, EE, CS, or
statistics
‣ Many hold a Ph.D. in one of these subjects
 We work on:
‣ Forecasting future prices
‣ Reducing transaction costs
‣ Modeling/managing risk