Update: Kelly pointed out to me that this hedging of betting on Nadal is nothing but canceling part of the bet put on Djokovic. (Weighted by the odds of course) Therefore it is actually trivial that in this case the expected return is proportional to risk, as both are proportional to the effective investment. In other words, this form of hedging does not change the inherent risk at all, which is just how certain Djokovic is going to win in this case. So the whole point is moot…
High risk, high reward is a very common slogan. I have never grasped why it is true though.
In the simplest set up, I have a random event which gives a few possible outcomes. As we all know, the expectation value (average) and the risk (standard deviation) for a distribution is not necessarily related in any way. So high risk does not imply high return in the long run.
In a realistic scenario, suppose I want to buy lottery. For $1, I can get one with maximum prize $2000, or another with maximum prize $50000. Is the one with higher prize more profitable in the long run? We don’t know. The merchant can set the average return independent of the prize spread. Again, higher risk does not imply a higher return. The same thing applies whether your average return is positive or negative.
(Strictly speaking, if you know the average return is positive, you should seek small risk to avoid lost; if it is negative then either you stop playing it, or you should maximize risk so that at some point you can swing to positive and leave at that point.)
If what the slogan implies is not the expected return but the highest possible return, than it is trivially true. But it is not useful at all.
However, it seems that I have just found a way to understand it. (No it’s not new at all, just I didn’t realize before.)
Suppose you want to bet in a tennis match, Djokovic vs. Nadal. For simplicity we assume there is no fee associated with the bet and the odds is set to 1/1 on both sides. However, we know better and assume from some scientific calculation we can say that actually Djokovic wins 3 to 1. How do we take advantage of our knowledge advantage?
Well, if we just bet on Djokovic, we can expect to have a return of +50%(1*0.75-1*0.25). However, there is 25% chance to lose money. To define the risk quantitatively let me just define it as the product of the chance to lose and the loss, where here it would be .25.
Note we can reduce the risk by hedging. Suppose we buy x Nadal and (1-x) Djokovic. Now our return would be (1-x)*.5-x*.5=(1-2x)*.5. The risk is -0.25*(x-(1-x))=0.25*(1-2x).
So now we see. In this simple model the risk is proportional to the expected return. Thus, higher risk, higher reward. We can see the proportionality constant is controlled by the discrepancy between the odds and the reality. (I have a suspicion that this proportionality constant is commonly called alpha.) For example, if the odds is 1/0.8 for Djokovic and 1/1.25 for Nadal, our return when we bet on Djoker will be 0.35(1-x)-0.4375*x=0.35(1-2.25x). The risk becomes 0.25*(1-2.25x), and the proportionality constant becomes smaller.
How does this relate to the simple example we first give? In the first example there is no set odds, so no inefficiency to begin with. The risk and the return are just set by the rules, and you can do nothing about it. With a set odds and if there is an inefficiency, you are in for a different game. Usually the odds is set by the market (so that the dealer will just collect the fee without risk) and your perspective separates you from the market and give you profitability. It’s interesting, however, to note that, at least in our simple example, you don’t really need to have that much perspective. basically you just need to know how the market is biased. If it is biased against Djokovic, you just buy Djokovic. Hedging only manages the risk, and the precise knowledge of the realistic probabilities only let you calculate the proportional constant, but does not change your strategy.
Last but not the least, the standard way to reduce risks still applies. If you can separate the event into individual mutually independent sub events, it is best to bet on all the sub events with maximum risk(the real hedging.) This way you will lower the overall risk and maintain the highest expected return. In the tennis example, this amounts to bet on individual sets. Although of course, in reality they are not really that independent.
One last job 