## 2017年7月18日 星期二

### 20170718 盈富基金的威力

(2017 年 (YTD) +15.93% 比我太太的「穩陣師奶」組合 (+21.80%) 大幅落後。原因可能是我半年來放了大部份盈富而換入了香港小輪(0050)，維他奶(0345)，置富(0778) 和粵海投資(0270)。

## 2017年7月1日 星期六

### 20170701 Kelly Criterion 在股票的應用

1. How to apply Kelly criterion to a portfolio made by a stock plus a option?

I know that by a portfolio made by only by one stock (and a risk free bond) I can use the formula:

f* = (R-Rf)/d^2

f* - Wealth fraction that maximize the log return
R - Asset Return
Rf - Risk-free return d - Standard Deviation

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Kelly is mostly based upon assets with zero correlation made independent of each other.
The way I approximate Kelly for multiple bets with correlation is:

Assume after your first bet the capital is gone.
Place a second bet based upon the Kelly of the remaining capital.
Factor in correlation..
Part 3 is the challenging part. I assume that with multiple bets at zero correlation placed simultaneously that I would bet the full Kelly per bet made. I assume that with multiple bets at a correlation of 1, I would divide the Kelly by the number of bets. So if for example I were to make 5 bets with a Kelly of 20%...
a correlation of 1 would be 20% divided by 5 or 4% per bet. A correlation of zero would be 1-(0.80^5)
to determine total capital at risk and then divide by 5 which is ~13.45% per bet.
A correlation of 50% is the average of the two or ~8.7% Anything else is a weighted average
but you have to be careful not to get the weightings backwards.
For example a correlation of 20% you take 80% of the Kelly amount 13.45 and 20% of 4% and sum them together.

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2.

### Quantitative Trading: Kelly vs. Markowitz Portfolio Optimization

In my book, I described a very simple and elegant formula for determining the optimal asset allocation among N assets:

F=C-1*M   (1)

where F is a Nx1 vector indicating the fraction of the equity to be allocated to each asset, C is the covariance matrix, and M is the mean vector for the excess returns of these assets. Note that these "assets" can in fact be "trading strategies" or "portfolios" themselves. If these are in fact real assets that incur a carry (financing) cost, then excess returns are returns minus the risk-free rate.

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3. Simple closed form solution for unconstrained Simultaneous bet Kelly staking
So given, for example, events A, B, C, D, and E, with corresponding single-bet Kelly stakes of κA, κB, κC, κD, and κE,
then the Kelly stake for the 1-team parlay consisting of only bet A would be:
κA * (1-κB) * (1-κC) * (1-κD) * (1-κE)
While the Kelly stake for the 3-team parlay consisting of bets A, B, and C would be:
κA * κB * κC * (1-κD) * (1-κE)
Much simpler, no?

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4. Algorithms for optimal allocation of bets on many simultaneous events
Chris Whitrow

Conclusions
When the number of bets is small, the optimal sizes of bet seem to be almost exactly proportional to the Kelly stakes on individual bets.

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(註：我的組合程式上的個別kelly是用以下的公式的：

Sharpe ratio S = (R-Rf)/d,
f = (R-Rf)/d^2 = S/d

We usually drop the risk-free rate, so we have g=S^2/2.