### Drawdown (economics)

The drawdown is the measure of the decline from a historical peak in some variable (typically the cumulative profit or total open equity of a financial trading strategy).[1]

Somewhat more formally, if {\textstyle X(t),\;t\geq 0} is a stochastic process with {\textstyle X(0)=0}, the drawdown at time {\displaystyle T}, denoted {\textstyle D(T)}, is defined as:

{\displaystyle D(T)=\max \left[\max _{t\in (0,T)}X(t)-X(T),0\right]\equiv \left[\max _{t\in (0,T)}X(t)-X(T)\right]_{+}}

The average drawdown (AvDD) up to time {\displaystyle T} is the time average of drawdowns that have occurred up to time {\displaystyle T}:

{\displaystyle \operatorname {AvDD} (T)={1 \over T}\int _{0}^{T}D(t)\,dt}

The maximum drawdown (MDD) up to time {\displaystyle T} is the maximum of the drawdown over the history of the variable. More formally, the MDD is defined as:

{\displaystyle \operatorname {MDD} (T)=\max _{\tau \in (0,T)}D(\tau )=\max _{\tau \in (0,T)}\left[\max _{t\in (0,\tau )}X(t)-X(\tau )\right]}

## Pseudocode

The following pseudocode computes the Drawdown (“DD”) and Max Drawdown (“MDD”) of the variable “NAV”, the Net Asset Value of an investment. Drawdown and Max Drawdown are calculated as percentages:

MDD = 0
peak = -99999
for i = 1 to N step 1 do
# peak will be the maximum value seen so far (0 to i), only get updated when higher NAV is seen
if (NAV[i] > peak) then
peak = NAV[i]
end if
DD[i] = 100.0 × (peak - NAV[i]) / peak
# Same idea as peak variable, MDD keeps track of the maximum drawdown so far. Only get updated when higher DD is seen.
if (DD[i] > MDD) then
MDD = DD[i]
end if
end for


There are two main definitions of a drawdown:

### 1. How low it goes (the magnitude)

Putting it plainly, a drawdown is the “pain” period experienced by an investor between a peak (new highs) and subsequent valley (a low point before moving higher).[citation needed]
Next, the Maximum Drawdown, or more commonly referred to as Max DD. This is pretty much self explanatory, as the Max DD is the worst (the maximum) peak to valley loss since the investment’s inception.[citation needed]

In finance, the use of the maximum drawdown as an indicator of risk is particularly popular in the world of commodity trading advisors through the widespread use of three performance measures: the Calmar ratio, the Sterling ratio and the Burke ratio. These measures can be considered as a modification of the Sharpe ratio in the sense that the numerator is always the excess of mean returns over the risk-free rate while the standard deviation of returns in the denominator is replaced by some function of the drawdown.

### 2. How long it lasts (the duration)

The drawdown duration is the length of any peak to peak period, or the time between new equity highs.
The max drawdown duration is the worst (the maximum/longest) amount of time an investment has seen between peaks (equity highs).

Many assume Max DD Duration is the length of time between new highs during which the Max DD (magnitude) occurred. But that isn’t always the case. The Max DD duration is the longest time between peaks, period. So it could be the time when the program also had its biggest peak to valley loss (and usually is, because the program needs a long time to recover from the largest loss), but it doesn’t have to be.[citation needed]

When {\displaystyle X} is Brownian motion with drift, the expected behavior of the MDD as a function of time is known. If {\displaystyle X} is represented as:

{\displaystyle X(t)=\mu t+\sigma W(t)}

Where {\displaystyle W(t)} is a standard Wiener process, then there are three possible outcomes based on the behavior of the drift {\displaystyle \mu }:[2]

• {\displaystyle \mu >0} implies that the MDD grows logarithmically with time
• {\displaystyle \mu =0} implies that the MDD grows as the square root of time
• {\displaystyle \mu <0} implies that the MDD grows linearly with time

## Banking or other finance definitions

### Credit offered

Where an amount of credit is offered, a drawdown against the line of credit results in a debt (which may have associated interest terms if the debt is not cleared according to an agreement.)

### Funds offered

Where funds are made available, such as for a specific purpose, drawdowns occur if the funds – or a portion of the funds – are released when conditions are met.

## Optimization of drawdown

A passing glance at the mathematical definition of drawdown suggests significant difficulty in using an optimization framework to minimize the quantity, subject to other constraints; this is due to the non-convex nature of the problem. However, there is a way to turn the drawdown minimization problem into a linear program.[3][4]

The authors start by proposing an auxiliary function {\displaystyle \Delta _{\alpha }(x)}, where {\displaystyle x\in \mathbb {R} ^{p}} is a vector of portfolio returns, that is defined by:

{\displaystyle \Delta _{\alpha }(x)=\min _{\zeta }\left\{\zeta +{1 \over {(1-\alpha )T}}\int _{0}^{T}[D(x,t)-\zeta ]_{+}\,dt\right\}}

They call this the conditional drawdown-at-risk (CDaR); this is a nod to conditional value-at-risk (CVaR), which may also be optimized using linear programming. There are two limiting cases to be aware of:

• {\textstyle \lim _{\alpha \rightarrow 0}\Delta _{\alpha }(x)} is the average drawdown
• {\textstyle \lim _{\alpha \rightarrow 1}\Delta _{\alpha }(x)} is the maximum drawdown

• Linear programming
• Risk measure
• Risk return ratio

## References

1. ^ “What Is A Drawdown? – Fidelity”. www.fidelity.com. Retrieved 2019-08-04.
2. ^ Magdon-Ismail, Malik; Atiya, Amir F.; Pratap, Amrit; Abu-Mostafa, Yaser S. (2004). “On the Maximum Drawdown of a Brownian Motion” (PDF)Journal of Applied Probability41 (1): 147–161. doi:10.1239/jap/1077134674.
3. ^ Chekhlov, Alexei; Uryasev, Stanislav; Zabarankin, Michael (2003). “Portfolio Optimization with Drawdown Constraints” (PDF).
4. ^ Chekhlov, Alexei; Uryasev, Stanislav; Zabarankin, Michael (2005). “Drawdown Measure in Portfolio Optimization” (PDF)International Journal of Theoretical and Applied Finance8 (1): 13–58. doi:10.1142/S0219024905002767.

### Arrow–Debreu model

In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.[1]

The model is central to the theory of general (economic) equilibrium and it is often used as a general reference for other microeconomic models. It is named after Kenneth Arrow, Gérard Debreu,[2] and sometimes also Lionel W. McKenzie for his independent proof of equilibrium existence in 1954[3] as well as his later improvements in 1959.[4][5]

The A-D model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium (or Walrasian equilibrium) of an economy. In general, there may be many equilibria; however, with extra assumptions on consumer preferences, namely that their utility functions be strongly concave and twice continuously differentiable, a unique equilibrium exists. With weaker conditions, uniqueness can fail, according to the Sonnenschein–Mantel–Debreu theorem.

## Convex sets and fixed points

A quarter turn of the convex unit disk leaves the point (0,0) fixed but moves every point on the non–convex unit circle.

In 1954, McKenzie and the pair Arrow and Debreu independently proved the existence of general equilibria by invoking the Kakutani fixed-point theorem on the fixed points of a continuous function from a compact, convex set into itself. In the Arrow–Debreu approach, convexity is essential, because such fixed-point theorems are inapplicable to non-convex sets. For example, the rotation of the unit circle by 90 degrees lacks fixed points, although this rotation is a continuous transformation of a compact set into itself; although compact, the unit circle is non-convex. In contrast, the same rotation applied to the convex hull of the unit circle leaves the point (0,0) fixed. Notice that the Kakutani theorem does not assert that there exists exactly one fixed point. Reflecting the unit disk across the y-axis leaves a vertical segment fixed, so that this reflection has an infinite number of fixed points.

### Non-convexity in large economies

The assumption of convexity precluded many applications, which were discussed in the Journal of Political Economy from 1959 to 1961 by Francis M. Bator, M. J. Farrell, Tjalling Koopmans, and Thomas J. Rothenberg.[6] Ross M. Starr (1969) proved the existence of economic equilibria when some consumer preferences need not be convex.[6] In his paper, Starr proved that a “convexified” economy has general equilibria that are closely approximated by “quasi-equilbria” of the original economy; Starr’s proof used the Shapley–Folkman theorem.[7]

## Economics of uncertainty: insurance and finance

Compared to earlier models, the Arrow–Debreu model radically generalized the notion of a commodity, differentiating commodities by time and place of delivery. So, for example, “apples in New York in September” and “apples in Chicago in June” are regarded as distinct commodities. The Arrow–Debreu model applies to economies with maximally complete markets, in which there exists a market for every time period and forward prices for every commodity at all time periods and in all places.[citation needed]

The Arrow–Debreu model specifies the conditions of perfectly competitive markets.

In financial economics the term “Arrow–Debreu” is most commonly used with reference to an Arrow–Debreu security. A canonical Arrow–Debreu security is a security that pays one unit of numeraire if a particular state of the world is reached and zero otherwise (the price of such a security being a so-called “state price”). As such, any derivatives contract whose settlement value is a function on an underlying whose value is uncertain at contract date can be decomposed as linear combination of Arrow–Debreu securities.

Since the work of Breeden and Lizenberger in 1978,[8] a large number of researchers have used options to extract Arrow–Debreu prices for a variety of applications in financial economics.[9]

• Model (economics)
• Incomplete markets
• Fisher market – a simpler market model, in which the total quantity of each product is given, and each buyer comes only with a monetary budget.

## References

1. ^ Arrow, K. J.; Debreu, G. (1954). “Existence of an equilibrium for a competitive economy”. Econometrica22 (3): 265–290. doi:10.2307/1907353. JSTOR 1907353.
2. ^ EconomyProfessor.com Archived 2010-01-31 at the Wayback Machine, Retrieved 2010-05-23
3. ^ McKenzie, Lionel W. (1954). “On Equilibrium in Graham’s Model of World Trade and Other Competitive Systems”. Econometrica22(2): 147–161. doi:10.2307/1907539. JSTOR 1907539.
4. ^ McKenzie, Lionel W. (1959). “On the Existence of General Equilibrium for a Competitive Economy”. Econometrica27 (1): 54–71. doi:10.2307/1907777. JSTOR 1907777.
5. ^ For an exposition of the proof, see Takayama, Akira (1985). Mathematical Economics (2nd ed.). London: Cambridge University Press. pp. 265–274. ISBN 978-0-521-31498-5.
6. Jump up to:a b Starr, Ross M. (1969), “Quasi–equilibria in markets with non–convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37)”, Econometrica37 (1): 25–38, CiteSeerX 10.1.1.297.8498, doi:10.2307/1909201, JSTOR 1909201.
7. ^ Starr, Ross M. (2008). “Shapley–Folkman theorem”. In Durlauf, Steven N.; Blume, Lawrence E. (eds.). The New Palgrave Dictionary of Economics4 (Second ed.). Palgrave Macmillan. pp. 317–318. doi:10.1057/9780230226203.1518. ISBN 978-0-333-78676-5.
8. ^ Breeden, Douglas T.; Litzenberger, Robert H. (1978). “Prices of State-Contingent Claims Implicit in Option Prices”. Journal of Business51 (4): 621–651. doi:10.1086/296025. JSTOR 2352653.
9. ^ Almeida, Caio; Vicente, José (2008). “Are interest rate options important for the assessment of interest risk?” (PDF)Working Papers Series n. 179, Central Bank of Brazil.

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