Translating Bet Sizes into Portfolio Positions and Returns
Summary
The document examines how a sequence of bet sizes maps to share holdings, cash, and final profit. Using two strategies with identical price forecasts but different position adjustments, it shows that the path of exposure matters: the first strategy earns a gain, while the second loses despite the price ultimately rising. The stated arithmetic implies that bet sizes apply to the current position value rather than to uninvested budget.
The question then asks how to interpret a negative bet size, proposing a sequence that moves from long exposure through a short position. The document does not include an accepted answer to that part, so it does not establish a general convention for negative sizes or resolve how to convert them into orders. Its examples are illustrative and omit practical details such as fees, leverage constraints, and execution.
Key ideas
- Bet sizes applied over time can produce different returns even when forecasts are correct.
- The example's profit arithmetic treats each new size relative to the current position value.
- A position may be reduced as prices move against it, changing eventual performance.
- The document raises but does not resolve how negative bet sizes translate into short orders.
Tags
Full text
# Bet sizing to actual orders
# Bet sizing to actual orders
At chapter 10.2 in Advances in Financial Machine Learning it says:
> Suppose that one strategy produced a sequence of bet sizes $[m_{1,1}, m_{1,2}, m_{1,3}] = [.5, 1, 0]$, as the market price followed a sequence $[p_1, p_2, p_3] = [1, .5, 1.25]$, where $p_t$ is the price at time $t$. The other strategy produced a sequence $[m_{2,1} , m_{2,2} , m_{2,3} ] = [1, .5, 0]$, as it was forced to reduce its bet size once the market moved against the initial full position. Both strategies produced forecasts that turned out to be correct (the price increased by 25% between $p_1$ and $p_3$), however the first strategy made money (0.5) while the second strategy lost money (−.125).
I'm trying to figure out how this actually translates to number of shares to buy and sell given a budget as it appears to not be what intuition would dictate.
Taking the first example i.e. bet sizes .5, 1, 0 and a budget of \$1, we start first by wagering half of our budget at a price of \$1 so .5 shares. Then at $t_2$ we wager everything meaning the \$.5 we have left over so now we have 1.5 shares. And then at $t_3$ we sell everything. But this produces a final budget of \$1.875 not the \$1.5 stated ("however the first strategy made money (0.5)" i.e. \$1.50).
In order to arrive at \$1.50, at $t_1$ we again stake \$.5 so that's .5 shares. Then at $t_2$ we stake 1 (100%) of our current position (not the remaining budget as intuition might indicate) which is now worth \$.25 so that's another .5 shares, leaving us with a budget of \$.25. Then at $t_3$, we sell everything i.e. 1 share at 1.25 = 1.25 + .25 remaining budget = \$1.5. The same approach yields -.125 for the second strategy (at $t_2$ we sell half of our current position)
Is this interpretation correct? If so, how would a negative bet size be interpreted. For instance if the series was 1, -.2, 0 does this mean:
$t_1$: wager everything
$t_2$: sell everything and go short .2 the original long position size
$t_3$: cover the shortShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.