5
$\begingroup$

If the returns of my strategy are distributed like 𝒩[μ,σ], what is the optimal fraction of capital to invest in each single trade, as a function μ and σ? Help!

PS. I know that normally distributed returns are an abstraction. But I'd like to grasp the concept in an ideal world, before exploring the implications of fat tails on the formula...

Improve this question
$\endgroup$
4
  • $\begingroup$ Do you assume you close each trade at the beginning of the next investment period ? $\endgroup$
    – Ezy
    Jan 4, 2019 at 19:09
  • $\begingroup$ yes, I do 1 trade per period. There are never 2 trades open at the same time. $\endgroup$
    – elemolotiv
    Jan 4, 2019 at 19:14
  • 1
    $\begingroup$ Have you perused Thorp’s papers, for instance, eecs.harvard.edu/cs286r/courses/fall12/papers/… , for one derivation? Another can be done via Itô’s lemma, iirc. $\endgroup$
    – Nap D. Lover
    Jan 4, 2019 at 22:59
  • $\begingroup$ @LoveTooNap29 thanks for the suggestion! $\endgroup$
    – elemolotiv
    Jan 5, 2019 at 7:24

2 Answers 2

Reset to default
10
$\begingroup$

This problem can be expressed as the original Merton's portfolio problem.

Consider wealth process defined by SDE

$$ d X _ { t } = \frac { X _ { t } \alpha _ { t } } { S _ { t } } d S _ { t } + \frac { X _ { t } \left( 1 - \alpha _ { t } \right) } { S _ { t } ^ { 0 } } d S _ { t } ^ { 0 } $$

where $\alpha_t$ is proportion of the investment in the risky asset $S_t$, and $S_t^0$ is the risk-free asset.

Optimality criterion may depend on the risk aversion of the investor, and the problem is to maximize expected utility of the investor for appropriate utility function $U$:

$$ E \left[ U \left( X _ { T } \right) \right] \rightarrow \max $$

Classical choice of the utility function is CRRA:

$$ u ( x ) = \frac { x ^ { 1 - \gamma } } { 1 - \gamma } $$

where $\gamma$ is constant and corresponds to the risk-aversion of the investor.

If the asset $S_t$ follows Black-Scholes dynamics (in conformance with your assumption of log-normal returns)

$$ \begin{aligned} d S _ { t } ^ { 0 } & = r S _ { t } ^ { 0 } d t \\ d S _ { t } & = \mu S _ { t } d t + \sigma S _ { t } d W _ { t } \end{aligned} $$

remarkably there is a closed-form solution which it is to invest a constant proportion of wealth in the risky asset

$$ \alpha_t = \frac { \mu - r } { \gamma \sigma ^ { 2 } } $$

Notice that the solution can be interpreted as the mean-variance trade-off.

Improve this answer
$\endgroup$
3
  • 3
    $\begingroup$ Note also that for log-utility (the Kelly case) $\gamma=1$ $\endgroup$
    – nbbo2
    Jan 5, 2019 at 20:50
  • $\begingroup$ @nakajuice thanks for the answer! I can't explain this case though. If we suppose γ=1 (kelly case), r=0, μ=0.01€ and σ=0.05€ we get ⍺=4. How can ⍺>1? $\endgroup$
    – elemolotiv
    Jan 6, 2019 at 8:31
  • 2
    $\begingroup$ We assume short-selling is allowed. $\alpha=4$ implies that for each 4 shares you hold, you need to short-sell 3 bonds (read: borrow money). $\endgroup$
    – starovoitovs
    Jan 6, 2019 at 12:29
2
$\begingroup$

ok I found it 🙂and this works for any distribution, not just the normal distribution

$f^*=\frac μ {σ^2 + μ^2} \approx \frac μ {σ^2} \space if μ\llσ$

here the steps: https://www.dropbox.com/s/4nqd5yfk2xcuag5/kelly.pdf

Improve this answer
$\endgroup$
5
  • $\begingroup$ This answer is close to, but not identical, to the other well-known one. It would be interesting to understand why the difference. $\endgroup$
    – nbbo2
    Jan 5, 2019 at 20:54
  • $\begingroup$ Why don't they just do $$ E(\prod_i (1 + f r_i)) = \prod_i E(1 + f r_i) = (1 + f \mu)^t$$ since the $$r_i$$ are IID? I am probably doing something dumb here. $\endgroup$
    – mathtick
    Apr 5, 2019 at 11:30
  • 1
    $\begingroup$ @mathtick I am trying to figure that out myself too. Can it be that: $\frac d {df} E\left[\prod_{i=1}^t(1+f\space r_i)\right] \ne \frac d {df}(1+f\space μ)^t$ that is, for some property of the $E[ ... ]$ operator, you must first differentiate and then apply the $E[ ... ]$ operator? $\endgroup$
    – elemolotiv
    Apr 13, 2019 at 20:55
  • $\begingroup$ @elemolotiv I read a lot more after that post ... so basically Kelly is exactly what I wrote above, and then then exp/log and end up maximizing the mean of the $\log(1 + f u)$ in the exponential. So basically, in the most general sense, "Kelly" just means use a log-utility when balancing risks. A key thing that they miss, is that the log is only defined if you survive. i.e. you don't go bust. So there is a term missing in most of this stuff to do with $P(1 + f r < 0)$ that one should do something with. $\endgroup$
    – mathtick
    Apr 14, 2019 at 9:04
  • 3
    $\begingroup$ Thanks! @mathtick Your comment helped me realise that maximising 𝐸[log(𝐺)] is not a mathematically convenient way to maximise 𝐸[𝐺]. That would be mathematically incorrect. 𝐸[log(𝐺)] is a whole different utility function that incorporates risk aversion. $\endgroup$
    – elemolotiv
    Apr 14, 2019 at 12:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.