layout: post title: "ML4T笔记 | 02-09 The Fundamental Law of active portfolio management" date: "2019-03-05 03:05:05" categories: 计算机科学 auth: conge
Quotes from Warren Buffett: Wide diversification is only necessary when investors do not know what they are doing.
Mr. Buffett is talking about two things, 1) Investor skill and 2) bre,adth (the number of investments).
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the Fundamental Law of Active Portfolio Management.
measure of skill and some measure of breadth.
information ratio refers to the Sharpe ratio of excess returns: the manner in which the portfolio manager is exceeding the market's performance.
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A Coin Flipping is analogous to a single trade in stock: you flip a coin, you get head or tail, you buy a stock and hold it, you make or lose money.
Here, the coin is biased: we have information about this coin
even money bet: bet n coins, If you win you get 2 x n coins; if you lose, your money is taken.
Which one of these is better?
Solution: 2 multiple coin-flips
In 1. there's a 49% chance you'll lose all your money In 2, there's very little chance you'll lose all your money. Option 2 has a lower risk.
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Reward: expected return.
winning_chance x bet times Winning change is 0.51 and losing chance is 0.49 or 49%. multiply this all out and add it up, then our expected return is 0.51 x 1000 - 0.49 * 1000 = $20.Solution: still $20.
So the reword of the single betting and multiple betting is the same, so why the multi-bet approach is better?
RISK
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Risk as the probability of lose it all: what's the chance you might lose it all?
single bet: there is a 49% chance to lose all in one bet. this is very high
multi-bet case: the chance of losing it all $p{loseAll} = \prod{i=1}^{1000}p_{i} =0.49^{1000}$
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Risk as the standard deviation of all those individual bets
for the multi-bet: Because for each table, the result is either -1 or 1, so that the standard deviation is easily calculated as 1.0.
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Risk adjusted reward
adjusted reward/ ratio = 0.63.__Solution: adjusted reward/ ratio = reword/risk = $20/1 =20
so, A thousand bets is better than a single bet.
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Performance is related to skill (how good are you at predicting the future return of the stock) and breadth (by the square root of your breadth).
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The Coin-Flip Casino experiment enabled us to allocate our 1,000 tokens to 1,000 tables.
We looked at two extreme cases: 1) a small bet on all 1,000 tables, 2) all our money on one table.
The expected return was the same in both cases, about $20, but the risk was substantially higher for the single bet case. True for both risk to lose everything and risk as standard deviation.
Three lessons.
1) higher alpha generates a higher Sharpe ratio. (reword) 2) more execution opportunities provides a higher Sharpe ratio. (breadth) 3) Sharpe ratio grows as the square root of breadth.
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alpha is about skill. the Sharpe ratio of this skill component is the information ratio.
$IC = mean({\alpha}) / std( \alpha) $ Note the risk and reword are calculated by looking back historically at the daily values of alpha.
The information ratio is essentially a Sharpe ratio of excess return, this part that's due to skill.
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the fundamental law as expressed by Richard Grinold.
IR = IC * sqrt(BR) or $IR = IC \times \sqrt{BR} = mean(\alpha) / std(\alpha) * \sqrt{BR}$Proof of the Fundamental Law
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Assuming that both Simons and Buffet have the same information ratio and that Simons' algorithms are only one 1000th as smart as Buffet and that Buffet trades 120 times per year.
Given that, how many trades must Simons execute in order to maintain the information ratio at the same level as Buffet?
Solution: 120,000,000
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Total Time: 00:25:45
2019-03-05 初稿