For now, however, we return to our basic coin example, since it has enough dimensions to illustrate many concepts of risk management. We consider more complicated examples later.

 

Optimal Betting

 

In our coin toss example, we have constant luck at 50%, constant payoff at 2:1 and we always bet on heads. To find a risk management strategy, we have to find a way to manage the bet. This is similar to the problem confronting a risk manager in the business of trading stocks. Good managers realize that there is not much they can do about luck and payoff and that the essential problem is to determine how much to wager on the stock. We begin our game with $1,000.

 

Hunches and Systems

 

One way to determine a bet size is by HUNCH. We might have a hunch and and bet $100. 

 

Although hunch-centric betting is certainly popular and likely accounts for an enormous proportion of actual real world betting, it has several problems: the bets require the constant attention of an operator to generate hunches, and interpret them into bets, and the bets are likely to rely as much on moods and feelings as on science.

 

To improve on hunch-centric betting, we might come up with a betting SYSTEM. A system is a logical method that defines a series of bets. The advantages of a betting system, over a hunch method are (1) we don't need an operator, (2) the betting becomes regular, predictable and consistent and, very importantly, (3) we can perform a historical simulation, on a computer, to OPTIMIZE the betting system.

 

Despite almost universal agreement that a system offers clear advantages over hunches, very few risk managers actually have a definition of their own risk management systems that is clear enough to allow a computer to back-test it. 

Our coin-flip game, however is fairly simple and we can come up with some betting systems for it. Furthermore, we can test these systems and optimize the system parameters to find good risk management.

 

Fixed Bet and Fixed-Fraction Bet

 

Our betting system must define the bet. One way to define the bet is to make it a constant fixed amount, say $10 each time, no matter how much we win or lose. This is a FIXED BET system. In this case, as in fixed-betting systems in general, our $1,000 EQUITY might increase or decrease to the point where the $10 fixed bet becomes proportionately too large or small to be a good bet.

 

To remedy this problem of the equity drifting out of proportion to the fixed bet, we might define the bet as as FIXED-FRACTION of our equity. A 1% fixed-fraction bet would, on our original $1,000, also lead to a $10 bet. This time, however, as our equity rises and falls, our fixed-fraction bet stays in proportion to our equity.

 

One interesting artifact of fixed-fraction betting, is that, since the bet stays proportional to the equity, it is theoretically impossible to go entirely broke so the official risk of total ruin is zero. In actual practice, however the disintegration of an enterprise has more to do with the psychological UNCLE POINT; see below.

 

Simulations

 

In order to test our betting system, we can SIMULATE over a historical record of outcomes. Let's say we toss the coin ten times and we come up with five heads and five tails. We can arrange the simulation in a table such as figure 2.

 

 

Figure 2:  Simulation of Fixed-Bet and Fixed-Fraction Betting Systems.

 

 

 

Pyramiding and Martingale

 

In the case of a random process, such as coin tosses, streaks of heads or tails do occur, since it would be quite improbable to have a regular alternation of heads and tails. There is, however, no way to exploit this phenomenon, which is, itself random. In non-random processes, such as secular trends in stock prices, pyramiding and other trend-trading techniques may be effective.

 

Pyramiding is a method for increasing a position, as it becomes profitable. While this technique might be useful as a way for a trader to pyramid up to his optimal position, pyramiding on top of an already-optimal position is to invite the disasters of over-trading. In general, such micro-tinkering with executions is far less important than sticking to the system. To the extent that tinkering allows a window for further interpreting trading signals, it can invite hunch trading and weaken the fabric that supports sticking to the system.

 

The Martingale system is a method for doubling-up on losing bets. In case the doubled bet loses, the method re-doubles and so on. This method is like trying to take nickels from in front of a steam roller. Eventually, one losing streak flattens the account.

 

Optimizing - Using Simulation

 

Once we select a betting system, say the fixed-fraction betting system, we can then optimize the system by finding the PARAMETERS that yield the best EXPECTED VALUE. In the coin toss case, our only parameter is the fixed-fraction.  Again, we can get our answers by simulation. See figures 3 and 4.

 

Note: The coin-toss example intends to illuminate some of the elements of risk, and their inter-relationships. It specifically applies to a coin that pays 2:1 with a 50% chance of either heads or tails, in which an equal number of heads and tails appears. It does not consider the case in which the numbers of heads and tails are unequal or in which the heads and tails bunch up to create winning and losing streaks. It does not suggest any particular risk parameters for trading the markets.

 

 

Figure 3:  Simulation of equity from a fixed-fraction betting system.

Figure 4:  Expected value (ending equity) from ten tosses, versus bet fraction,

for a constant bet fraction system,  for a 2:1 payoff game,

from the first and last columns of figure 3.

 

 

 

Optimizing - Using Calculus

 

Since our coin flip game is relatively simple, we can also find the optimal bet fraction using calculus. Since we know that the best system becomes apparent after only one head-tail cycle, we can simplify the problem to solving for just one of the head-tail pairs.

 

The stake after one pair of flips:

S = (1 + b*P) * (1 - b) * S₀

_{S - the stake after one pair of flips}

_{b - the bet fraction}

_{P - the payoff from winning - 2:1}

S₀ - the stake before the pair of flips

(1 + b*P) - the effect of the winning flip

(1 - b) - the effect of the losing flip

So the effective return, R, of one pair of flips is:

R = S / S₀

R = (1 + bP) * (1 - b)

R = 1 - b + bP - b²P 

R = 1 + b(P-1) - b²P

 

Note how for small values of b, R increases with b(P-1) and how for large values of b, R decreases with b²P. These are the mathematical formulations of the timid and bold trader rules. 

 

We can plot R versus b to get a graph that looks similar to the one we get by simulation, above, and just pick out the maximum point by inspection. We can also notice that at the maximum, the slope is zero, so we can also solve for the maximum by taking the slope and setting it equal to zero.

 

Slope = dR/db = (P-1) - 2bP = 0,  therefore:

b = (P-1)/2P , and, for P = 2:1,

b = (2 - 1)/(2 * 2) = .25

 

So the optimal bet, as before, is 25% of equity.

 

 

Optimizing - Using The Kelly Formula

 

J. L. Kelly's seminal paper, A New Interpretation of Information Rate, 1956, examines ways to send data over telephone lines. One part of his work, The Kelly Formula, also applies to trading, to optimize bet size.

 

 

Figure 5:  The Kelly Formula

 

Note that the values of W and R are long-term average values,

so as time goes by, K might change a little.

 

Reference: http://www.racing.saratoga.ny.us/kelly.pdf