101 Drawdown and Its Linearity

Manu Rathi — Thu, 01 Dec 2022 00:00:00 GMT

In this article, we first discuss the conditional drawdown at risk (CDaR), its special cases — maximum drawdown and average drawdown — and then move to the optimization problem for the return–CDaR efficient frontier as discussed by Uryasev et al. ¹

Assumption

This approach concentrates on the portfolio equity curve over a particular sample path (historical, or the most probable future sample path) — i.e. we define a sample-path risk function rather than a risk measure over a set of sample paths. It makes no assumption about the underlying probability distribution, which unlocks various practical applications.

CDaR

For some value of the confidence parameter , the -CDaR is defined as the mean of the worst drawdowns experienced over some period of time on a sample path.

The CDaR risk function is based on the CVaR concept and can be viewed as a modification of CVaR to the case where the loss function is defined as a drawdown. Similarly to CVaR — the average of all losses greater than or equal to -VaR — CDaR is the average of all drawdowns greater than or equal to the drawdown-at-risk (), such that a low CVaR ensures a low VaR.

Risk functions are parameterized by the portfolio weight matrix ; is the total time period considered for the calculation — the sample-path time horizon.

where .

By definition, with a specified probability , the -DaR () of a portfolio is the lowest amount such that the probability the drawdown will not exceed is greater than or equal to .

We consider two risk functions that are special cases of CDaR — maximum drawdown (MaxDD) and average drawdown (AvDD) — which are limiting cases as and respectively:

Linear optimization

MaxDD

We consider a modification of the classic return–risk efficient-frontier optimization: maximize return while constraining maximum drawdown to a pre-defined level.

constrained on

where is the return matrix , is all constituents, and is the percentage of deployed capital allowed to be in drawdown.

Maximum drawdown is not linear in the parameter . It can be made linear using an auxiliary variable, as follows:

Converting the first maximum function in Equation 5, using the property of maximum, applies a constraint on each day to be less than the desired drawdown:

Converting the second max function:

where is an auxiliary variable.

Defining in Equation 7: is equal to a maximum function, hence it’s an increasing function.

Also, should be equal to or greater than :

So the optimization is converted to a convex optimization problem with a linear performance function and a piecewise-linear convex constraint, given by maximizing return subject to Equation 6 through Equation 10.

CDaR

constrained on

Rewriting Equation 1:

where the function is defined as .

The CDaR constraint function can be linearized as follows:

Defining :

Since is the positive part:

Using Equation 7, can be written in terms of :

And since is always positive:

Similarly to MaxDD, the optimization problem converts to maximizing return subject to Equation 14, Equation 15, and Equation 8 through Equation 10.

An important feature of this formulation is that it does not involve the threshold function directly. At the optimal solution, the variables and together give the optimal portfolio and the corresponding value of the threshold function — see ² for further discussion.

Footnotes

Alexei Chekhlov, Stanislav Uryasev, Michael Zabarankin. Portfolio Optimization With Drawdown Constraints. ise.ufl.edu/uryasev↩︎
R. Tyrrell Rockafellar, Stanislav Uryasev. Optimization of Conditional Value-at-Risk. ise.ufl.edu/uryasev↩︎

Four-Moment Simulation — Mean, Variance, Skew, Kurtosis

Manu Rathi — Thu, 14 Oct 2021 00:00:00 GMT

In the previous article, Two-Moment Simulation, we discussed how to simulate normally-distributed points for a given mean and standard deviation / covariance matrix.

Here, we discuss how to generate non-normal numbers with a unimodal distribution — i.e. simulate using all four moments (mean, standard deviation, skewness, and kurtosis), ignoring coskewness and cokurtosis.

We refer to the paper by Hao Luo ¹, which uses Fleishman’s power method ² to simulate non-normal data.

The idea is to first simulate a series using the power method:

where , are the skewness and excess kurtosis of the distribution , respectively.

Second, we use our understanding from the previous article to convert the first two moments of from to .

Algorithm

As per the power method, can be defined as a third-degree polynomial of :

where .

We solve for:

The following four equations follow from solving the above:

To solve for :

constrained on

A solution to the above exists only when

Finally, apply the transformation:

Results

Let’s simulate a beta distribution with and . The left chart shows the PDF of the beta distribution, the simulated beta using the methodology above, and a normal distribution with mean and standard deviation . The right chart shows the cumulative distribution of the same, excluding the normal.

The PDF and its cumulative distribution are very close to each other by the Kolmogorov–Smirnov statistic.

First four moment values for all three cases:

Stat	Beta	Normal	Simulation
Mean	0.82	0.82	0.81
Std	0.15	0.15	0.15
Skew	-1.13	0	-1.08
Kurt	1.02	0	0.88

Footnotes

Hao Luo (2011). Generation of Non-normal Data – A Study of Fleishman’s Power Method. diva-portal.org↩︎
Fleishman, A. I. (1978). A method for simulating non-normal distributions. Psychometrika. springer.com↩︎

Two-Moment Simulation — Mean and Variance

Manu Rathi — Tue, 12 Oct 2021 00:00:00 GMT

This article looks at how to simulate a series with a normal distribution — i.e. where the higher moments (skew, excess kurtosis) are zero. For simulation with four moments, see Four-Moment Simulation.

In the first part we look at the univariate case (1 dimension); in the second, we see how to incorporate covariance for more than one dimension.

Univariate simulation (1 dimension)

The idea is to simulate a series

given and .

The probability distribution for a random variable that is normally distributed: every point , where .

To generate we can use the Box–Muller transformation. In Excel:

NORM.S.INV(rand())

where rand() draws from a uniform distribution on (0,1), and NORM.S.INV is the inverse of the cumulative normal distribution.

The number of simulated data points is critical to the accuracy of the simulation — more data points lead to higher accuracy.

Higher accuracy for the expected mean and variance with fewer data points can be achieved using random matrix theory — a topic for a future post.

Multivariate simulation (>1 dimension)

The idea is to simulate a series

given (an array of means) and (a covariance matrix).

where is the Cholesky decomposition of .

Pinball Maze

Manu Rathi — Thu, 28 May 2020 00:00:00 GMT

Problem statement

Imagine a pinball maze like the one below. A ball is dropped into the maze times. A few assumptions: (1) no ball ever leaves the container placed below the maze, and (2) at each pin (each dot in the figure) the ball has only two options — left or right, each equally likely.

The question: as we do this for a large number of throws,

what does the distribution of balls across the containers (i.e. the frequency of balls in each slot) converge to?

Simulation

Let’s simulate the situation in Python.

# function for simulating one throw
def oneThrow(numOfLayers):
    # generate left(-1) / right(+1) using a binomial distribution, equal probability
    bernoulliNumbers = np.random.binomial(1, 0.5, numOfLayers)
    decideNumbers = (lambda arr: [-1 if x == 0 else 1 for x in arr])(bernoulliNumbers)
    # final container slot
    finalContainerSlot = 0
    for decide in decideNumbers:
        finalContainerSlot += decide
    return decideNumbers, finalContainerSlot

numberOfLayers = 100
numberOfThrows = 10000

finalContainerSlotArr = [oneThrow(numberOfLayers)[1] for i in range(numberOfThrows)]

Each throw is a sum of numberOfLayers independent steps — by the central limit theorem, as numberOfLayers grows, the distribution of finalContainerSlot across many throws converges to a normal distribution. The pinball maze is, in effect, a physical Galton board.