Quantitative Analysis, Risk Management, Modelling, Algo Trading, and Big Data Analysis

Modern Time Analysis of Black Swans

I decided to take the data analysis on Black Swan and Extreme Loss Modeling to the next level and examine the time distribution of extreme losses across the entire S&P 500 universe of traded stocks. Previously, we were interested, first, in finding the maximum loss among all trading days for a given stock in a specified time interval (stock life-time on the trading floor), secondly, in plotting the distribution of those extreme losses (found to be well fitted with the Gumbel distribution). The investigation when all those losses occurred in time can provide us with an extra vantage point, and if we are lucky researchers we can discover some new information on the population of Black Swans.

An excellent approach to data analysis is through time analysis. It is one of many, many data analytic techniques available but it is closest to my heart and I would like to provide you with a sophisticated taste of easy-to-digest mathematics applied to the financial time-series analysis. So what is a data analysis about in our case? Can we code it or be tempted by ready-to-use dedicated data analytic software? Both ways are accessible but let’s do it in more educational way taking computer programming class in Matlab.

By the time analysis we can understand the behaviour of a given quantity in time, e.g. the price of an asset traded in the exchange market. It is pretty straightforward that our eye looks for patterns and tries to classify them somehow. If we observe a repeating structure, or sequence, or shape $-$ that may help us to code its recognition within our trading platforms. The quest for hunting the most fundamental pattern of hidden volatility in the data remains obvious: a sine wave. Fairly painless in understanding, covered by maths teachers in high-schools, ‘up and down’ approach, good moods versus bad moods, bear markets followed by bull markets. A sinusoidal behaviour is everywhere around us. Therefore it is so important to know how to find it!

In quantitative finance and risk modeling, a periodicity or cyclicality constitutes an attractive and simplest model of volatility. To be able to figure out what spectrum of characteristic frequencies our given data set conceals we may need to use a proper tool. In time-series analysis this tool is known as a periodogram. However, before starting using it properly, it is essential to understand the theoretical background.

1. The Method of Period Detection

1.1. General Formulations and Model Orthogonality

In general, we learn from experiments by fitting data, $x$, with a model, $x_{\|}$. The data contain $n$ measurements, and the model, $n_{\|}$ free parameters. The consistency of the data with the model is measured by a function, $\Theta$, called a statistic. A given model, $x_{\|}$, using a given statistic (e.g., $\chi^2$), yields its particular value, $\Theta_1$. Various methods used in the analysis of time series differ both in their choice of the model and the statistic; hence are difficult to compare directly. To enable such a comparison and for determining the significance of results, $\Theta$ is converted into the false alarm probability,$P_1$. This is done considering a hypothetic situation, $H_1$, in which $x$ is pure white noise. Then each pair $(x_{\|},\Theta)$ corresponds to certain cumulative probability distribution of $\Theta$, namely $P(n_{\|},n;\Theta)$, with $P_1$ being the tail probability that under the hypothesis $H_1$ the experiment yields $\Theta>\Theta_1$, i.e., $P_1(\Theta>\Theta_1) = 1-P(n_{\|},n;\Theta_1)$.

Up to here, we have just outlined the classical Neyman-Pearson procedure of statistics. The specific method for analysis of time series used here differs from those commonly encountered in astronomy only in the choices of $x_{\|}$ and $\Theta$. Then, our accounting for variance, correlation and multiple frequencies in calculating $P$ is dictated by the laws of statistics. The probabilities derived by us from the data are the false alarm probabilities. However, we also call them below just probabilities or significance levels.

We note then that Fourier harmonics are not orthogonal in terms of the scalar product with weights at unevenly distributed observations. Certain statistical procedures employing classical probability distributions hold for orthogonal models only and fail in other cases. To avoid that, a popular variant of the power spectrum, Lomb (1976) and Scargle (1982, hereafter LS) Lomb-Scargle periodogram, $P_{\rm LS}(\nu)$, relies on a special choice of phase such that the sine and cosine functions become orthogonal:
$$P_{\rm LS}(\nu) = A_{\rm LS} \left|\hat{x}_{\rm LS}(\nu) \right|^2 .$$ Square of the Fourier amplitude, $\hat{x}(\nu)$, takes form:
$$\begin{eqnarray} \left|\hat{x}_{\rm LS}(\nu)\right|^2 & = \left[ \sum_{k=1}^{n} (x_k-\bar{x}) \cos (2\pi\nu (t_k-\tau)) \right]^2 + \nonumber \\ & \left[ \sum_{k=1}^{n} (x_k-\bar{x}) \sin (2\pi\nu (t_k-\tau)) \right]^2 . \end{eqnarray}$$ The phase $\tau$ is defined as:
$$\tan(4\pi\nu\tau) = \frac{\sum_{k=1}^{n} \sin(4\pi\nu x_k)} {\sum_{k=1}^{N_{\rm obs}} \cos(4\pi\nu x_k)}.$$ where, as usual, we consider a time-series $\{x_i\}\ (i=1,…,n)$; $\bar{x}$ denotes the subtracted mean value, and the discrete Fourier transform takes our signal from time to frequency domain. Orignally, normalization of LS periodogram was proposed as $A_{\rm LS}=1/2\sigma^2$ in order to account for normalization to the level of white noise but different variants are available as well.

In practice, some simplification to the full version of LS periodogram are applied where we are interested in understanding the power density spectrum of $|\hat{x}(\nu)|^2$ distribution. The following Matlab function allow us to obtain a modified LS solution for a time-series $[t,y]$:

 % Function calculates the power density spectrum (periodogram) for % a time-series [t,y] based on discrete Fourier transform. The % series is expected to be evenly distributed, but gaps are allowed. % % (c) 2013 QuantAtRisk.com, by Pawel Lachowicz % % Input: % t : time vector [1xn] % y : time-series [1xn] % dt : average time-series sampling time % Output: % freq : frequency vector [1xm] % psd : power spectral density [1xm]   function [freq,pds]=qarPDS(t,y,dt); n=length(t); % number of data points X=fft(y,n); Pyy=X.*conj(X); freq=[1:n/2]*(1/n/dt); % in physical units pds=(Pyy(2:(floor(n/2)+1))); pds=pds'; % pds end

We will use it later on to the time analysis of our S&P 500 data of extreme losses across the whole universe of stocks traded. The function computes the periodogram for any time-series so don’t be worried too much about initial data units.

Now, back to the first base. We may extend the approach used within LS method by employing Szego orthogonal trigonometric polynomials as model functions. A series of $n_{\|}=2N+1$ polynomials corresponds to the orthogonal combinations of the $N$ lowest Fourier harmonics (Schwarzenberg-Czerny 1996). Orthogonal series are optimal from the statistical point of view because, by virtue of the Fisher lemma (Fisz 1963; Schwarzenberg-Czerny 1998), they guarantee the minimum variance of the fit residuals for a given model complexity (given by $n_{\|}$). Szego polynomials are also convenient in computations since the least-square solution may be obtained using recurrence and orthogonal projections, resulting in high computational efficiency, with the number of steps $\propto N$ instead of $N^3$ for $N$ harmonics.

1.2. Variance, the AoV statistics, and model complexity

The LS method employs the sine as a model, and the quadratic norm, $$\Theta_{\chi^2}=\|x-x_{\|}\|^2 ,$$ as the statistic. The corresponding probability distribution is $\chi^2$ with 2 degrees of freedom. Prior to use of the $\chi^2$ distribution, $\Theta_{\chi^2}$ has to be divided by the signal variance, $V$. However,$V$ is usually not known and has to be estimated from the data themselves. Then, neither $\Theta_{\chi^2}$ and variance estimates are independent nor their ratio follows the $\chi^2$ distribution, which effect has to be accounted for. A simple way to do it is to apply the Fisher Analysis of Variance (AoV) statistic, $$\Theta\equiv (n-n_{\|}) \|x_{\|}\|^2/ (n_{\|}\|x – x_{\|}\|^2) .$$ Hence we call our method, involving Szego polynomials model and the AoV statistics, the multi-harmonic analysis of variance or mhAoV periodogram (Schwarzenberg-Czerny 1996). The probability distribution is then the Fisher-Snedecor distribution, $F$, rather then $\chi^2$, and $P_1= 1-F(n_{\|},n_{\perp};\Theta)$ where $n_{\perp}=n-n_{\|}$. For everything else fixed, replacing $\chi^2$ with $F$ for $n=100$ yields an increase of $P_1(\chi^2)=0.001$ to $P_1(F)=0.01$. Thus, accounting for the unknown variance yields the mhAoV detection less significant, but more trustworthy. In this work, $n$ usually is larger, for which $P_1(F)/ P_1(\chi^2)$ reduces to several.

Apart from the choice of the statistic, our method for $N=1$ differs from the LS one in the average flux being subtracted in the latter (thus yielding $n_\|=2$) whereas a constant term is fitted in the former (which can be often of significant advantage, see Foster 1995). If the periodic modulation in the data differs significantly from a sinusoid (e.g., due to dips, eclipses, etc.), then our $N>1$ models account for that more complex shape and perform considerably better then the LS one.

1.3. Multiple trials

Probability can be assigned to a period found in data according to one of two statistical hypotheses. Namely, (i) one knows in advance the trial frequency, $\nu_0$ (from other data), and would like to check whether it is also present in a given data set or (ii) one searches a whole range, $\Delta\nu$, of $N_{\rm eff}$ frequencies and finds the frequency, $\nu$, corresponding to the most significant modulation. The two cases correspond to the probabilities $P_1$ and $P_{N_{\rm eff}}$ to win in a lottery after 1 and $N_{\rm eff}$ trials, respectively, i.e., they represent the false alarm probabilities in single and multiple experiments, respectively. They are related by
$$P_{N_{\rm eff}}= 1-(1-P_1)^{N_{\rm eff}} .$$ Note that the hypothesis (ii) and the probability $P_{N_{\rm eff}}$ must be always employed in order to claim any new frequency in the object under study. The hypothesis (i) is rarely used. However, since $P_1\lt P_{N_{\rm eff}}$, it is the more sensitive one. For this reason, we advocate its use in the situations where the modulation frequency is already known, and we aim at checking for its manifestation in the same object but in a new band, new data set, etc. We stress that we do not use the hypothesis (i) to claim any new frequency.

An obstacle hampering use of the (ii) hypothesis is that no analytical method is known to calculate $N_{\rm eff}$. The number $N_{\rm eff}$ corresponds to independent trials, whereas values of periodograms at many frequencies are correlated because of the finite width of the peaks, $\delta\nu$, and because of aliasing. As no analytical method is known to determine $N_{\rm eff}$, Monte Carlo simulations have been used (e.g., Paltani 2004). Here, we use a simple conservative estimate, $N_{\rm eff}= \min(\Delta\nu/\delta\nu, N_{\rm calc},n)$, where $N_{\rm calc}$ is the number of the values at which the periodogram is calculated. The estimate is conservative in the sense that it corresponds to the upper limit on $P_{N_{\rm eff}}$, and thus the minimum significance of detection. This effect applies to all methods of period search (Horne & Baliunas 1986). In general, it may reduce significance of a new frequency detection for large $N_{\rm eff}$ as $P_{N_{\rm eff}}\gg P_1$. In practice, it underscores the role of any prior knowledge, in a way similar to the Bayesian statistics: with any prior knowledge of the given frequency we are able to use the hypothesis (i) to claim the detection with large significance (small $P_1$).

1.4. Correlation length

The $P_1$, and other common probability distributions used to set the detection criteria, are derived under the assumption of the noise being statistically
independent. Often this is not the case, as seen, e.g., in light curves of cataclysmic variables (CVs). The correlated noise, often termed red noise, obeys different probability distribution than the standard $P_1$, and hence may have a profound effect. For example, noise with a Gaussian autocorrelation function (ACF) correlated over a time interval, $\delta t$, yields a power spectrum with the Gaussian shape centered at $\nu=0$ and the width $\delta\nu=1/\delta t$. It may be demonstrated that the net effect of the correlation on $P_1$ in analysis of low frequency processes is to decimate the number of independent observations by a factor $n_{\rm corr}$, the average number of observations in the correlation interval $\delta t$ (Schwarzenberg-Czerny 1991). Effectively, one should use $n_{\perp}/n_{\rm corr}$ and $\Theta/n_{\rm corr}$ instead of $n_{\perp}$ and $\Theta$ in calculating $P_1$. This result holds generally, for both least squares and maximum likelihood analyses of time series.

For independent observations, $m=2$ consecutive residuals have the same sign on average (e.g., Fisz 1963). Thus, counting the average length, $m$, of series of residuals of the same sign provides an estimate of the number of consecutive observations being correlated, $n_{\rm corr}$. Note that $m=n/l$ where $l$ is the number of such series (both positive and negative). For correlated observations, the average length of series with the same sign is $m=2n_{\rm corr}$, which allows us to calculate $n_{\rm corr}$.

Let $\Theta$ denote the Fisher-Snedecor statistics from the mhAoV periodogram (i.e. from Fourier series fit) computed for $n_{\|}=2N+1$ parameters, $n$ observations and $n_{\perp}=n-n_{\|}$ degrees of freedom. To account for $n_{\rm corr}$, we calculate $P_1$ as follows,

P_1=1- F\left(n_{\|},\frac{n_{\perp}}{n_{\rm
corr}};\frac{\Theta}{n_{\rm corr}}\right)=
I_{z}\left(\frac{n_{\perp}}{2n_{\rm
corr}},\frac{n_{\|}}{2}\right)\label{P1},

where $z= n_{\perp}/(n_{\perp}+n_{\|}\Theta)$ and $I_z(a,b)$ is the incomplete (regularized) beta function (Abramowitz & Stegun 1971), see Schwarzenberg-Czerny 1998 and references therein. In the popular Mathematica (Wolfram 1996) that function is called BetaRegularized. In Matlab, the following function does the calculations for us:

 1 2 3 4 5 6 7 8 9 10 11 12 % Function computes the mhAoV periodogram peak significance % Usage: [P1,Pneff]=pneff(n,nh,ncorr,neff,theta)   function [P1,Pneff]=pneff(n,nh,ncorr,neff,theta); npar=2.0*nh+1; nper=n-npar; z=nper/(nper+npar*theta); a=nper/(2.0*ncorr); b=npar/2.0; P1=betainc(z,a,b); Pneff=1.0-(1.0-P1)^neff; end

In the following section we will apply both approaches of modified Lomb-Scargle (LS) and multi-harmonic AoV periodograms for financial data and we will discuss the potential consequences coming from time analysis of largest daily losses for stocks traded publicly within S&P500 universe. So buckle up as we are ready for take off!

2. The Annual Migration Routes of Black Swans

A theory can both beautiful and exhausting. So let’s do some work to capture the beauty. Our goal is to re-analyze the data of extreme losses extracted by us previously within Black Swan and Extreme Loss Modeling article. First, we extract the value of the maximum loss for each stock and store them in a matrix in Matlab data as follows:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 % Modern Time Analysis of Black Swans among % Traded Stocks in S&P 500 % % (c) 2013 QuantAtRisk.com, by Pawel Lachowicz     clear all; close all; clc; tic;   %% DATA READING AND VISUALIZATION   % read a list of stock names StockNames=dataread('file',['sp500u.lst'],'%s','delimiter', '\n'); K=length(StockNames); % the number of stocks in the universe % path to data files path=['./SP500_1984_2011'];   data=[]; fprintf('data reading and preprocessing..\n'); for si=1:K % --stock name stock=StockNames{si}; fprintf('%4.0f %7s\n',si,stock); % --load data n=[path,'/',stock,'.dat']; % check for NULL and change to NaN (using 'sed' command % in Unix/Linux/MacOS environment cmd=['sed -i ''s/NULL/NaN/g''',' ',n]; [status,result]=system(cmd); % construct FTS object for daily data FTS=ascii2fts(n,1,2); % fill any missing values denoted by NaNs FTS=fillts(FTS); % extract the close price of the stock cp=fts2mat(FTS.CLOSE,0); dd=fts2mat(FTS.CLOSE,1); % extract Matlab matrix containing value of maximum % loss per stock and corresponding day rtmp=cp(2:end)./cp(1:end-1)-1; % daily returns dtmp=dd(2:end,1); % time vector tmp{si}=[dtmp rtmp]; [tmp1,tmp2]=min(tmp{si}(:,2)); % maximum loss data=[data; dtmp(tmp2) tmp1]; % [time of maximum loss, loss value] end   data=sortrows(data,1); % sort data according to time of loss occurrence

where, again, the required data files can be downloaded here as sp500u.zip (23.8 MB) and sp500u.lst, respectively.

The visualization of collected data provides us with a new dimension on time distribution of the maximum losses (Black Swans events) across the S&P 500 universe as traded between 3-Jan-1984 and 8-Mar-2011:

 46 47 48 49 plot(data(:,1),data(:,2)*100,'.-','color',[0.7 0.7 0.7]) xlim([min(data(:,1)) max(data(:,1)) ]); xlabel('Time (days)'); ylabel('R_{\rm min} (%)');

All individual stock maximum losses have been depicted with dot markers. As we found within Gumbel distribution analysis, the expected value was -22.6% with a heavy tail extending up to nearly complete losses of $\sim$98%.

Changing the way how we look at our data, we allow to connect the dots and think about data as a new time-series $x_i\ (i=1,…,n=954)$. From this standpoint we can continue our analysis in various direction. Let’s have a look at one case in particular: annual average maximum losses in a function of time. Why? Such approach has been suggested as interesting by McNeil, Frey, and Embrechts in their book Quantitative Risk Management, section 7.1.4., making use of the block maxima method in order to find return levels for stress losses. We turn this idea in practice by rebinning our time-series $\{x_i\}$ with a new time step of 252 (trading) days utilizing the code published within my past post on Rebinning of Financial Time-Series as follows:

 51 52 [rx,ry]=rebin(data(:,1),data(:,2),[],252); hold on; plot(rx,ry*100,'or');

and allowing for inappropriate data profanity with a gentle data interpolation between the points:

 54 55 56 57 xi=rx(1):0.01:rx(end); rdatai=interp1(rx,ry,xi,'pchip'); rdatai=[xi; rdatai]'; hold on; plot(rdatai(:,1),rdatai(:,2)*100,'r-');

resulting in:

Next, based on non-interpolated data we compute the Fourier power spectrum (a modified LS periodogram) as follows:

 59 60 61 62 63 64 65 % Periodogram [freq,pds]=qarPDS(rx,ry,252);   figure(2); plot(freq,pds); xlabel('Frequency [1/d]'); ylabel('Power Spectrum');

which returns:

It is obvious that the periodogram is calculated based on a fixed frequency grid with a frequency step of $\Delta\nu = 1/T = 0.000104$ [1/d]. The peak of highest power corresponds to the sine modulation detected in the time-series which period is equal $1/0.001462$ or 684 days. The maximal allowed frequency is the Nyquist frequency of $1/(2\Delta t)$ or 0.00198 [1/d]. Honestly, the plot is terrible. To improve its quality it is allowed in spectral analysis of time-series to apply over-samling in frequency, i.e to adopt the frequency grid of computations with a step of $\Delta\nu = 1/(kT)$ where $k$ denotes the over-sampling (an integer) factor. Why do we need the over-sampling? One of the reasons is: to find the value of periodicity as accurately as possible.

Let’s see how mhAoV periodogram copes with this task in practice. The source codes for mhAoV can be downloaded directly from Alex’s webpage (available in Fortran 95 and Python), though I still make use of our old version executable directly in Unix/Linux environment: aov. Let’s first store rebinned data (with 252 d step) in an external file of rdata.dat:

 67 68 69 70 71 rdata=[rx ry]; fn=['./rdata.dat']; fid=fopen(fn,'wt'); fprintf(fid,'%f %f\n',rdata'); fclose(fid);

and, next, let’s compute aov periodogram:

 ./aov -f=0.000104,0.002 -nh=1 -fos=5 rdata.dat   mhAov periodogram, by Alex Schwarzenberg-Czerny ver. 27.9.2006 updated by Pawel Lachowicz   datname=rdata.dat trfname=rdata.trf maxname=rdata.max nobs=38 method=ORTAOV nh=1 nbf=20 fos=5.00 frstart=0.0001040 frend=0.0019809 frstep=0.0000107 nfr=176 frequency period theta quality 0.0014747 678.1132747 5.53729 0.743 0.0013152 760.3542866 1.39906 0.146 0.0016301 613.4435376 1.37416 0.138 0.0003351 2984.5922602 1.30742 0.116 0.0001733 5771.4262041 1.22450 0.088 0.0011538 866.7094426 1.12090 0.050

i.e. employing the model which contains a single sinusoid (nh=1) and adopting over-sampling in frequency with $k=5$ (fos). It occurs that the highest value of mhAoV $\Theta=5.537$ statistics corresponds to the period of 678 days.

Fitting the annual data with the model defined as:
$$f(t) = c + A\sin(2\pi t/P – g\pi)$$ we find for $P_1=678$ d the estimation for amplitude $A_1=0.12$ and $g_1=0.79$ and the best fit we over-plot as follows:

This model is not perfect but delivers us a good taste of concealed periodic pattern following a sinusoidal model in about 60% of time between 1984 to 2011. This is an interesting result though the computation of:

 >> [P1,Pneff]=pneff(38,1,1.3,7.1,5.53729)   P1 = 0.0138 Pneff = 0.0941

indicates at only 9% of significance of this periodicity. This can be understood as a poor fit of the model to the complicated and variable shape of annual changes in maximal losses for different traded stocks.

A sort of improvement of the model we could achieve by inclusion of variation of the amplitude in function of time, i.e. $A_1(t)$. This can be practically extracted from the wavelet analysis via computation of the continuous wavelet transform. If this is subject of your interest check a draft of this approach in the paper of Czerny et alii (2010) I co-authored.

3. Conclusions

Was Joseph Fourier a birdwatcher? We don’t know. But his approach to the time-series analysis allowed us to check whether any periodic patterns in annual migration (occurrences) of Black Swan events do exist? With a very low probability we found a cyclical trend repeating every 678 days. Will that allow us to forecast the future and next density of massive losses as incurred by individual stocks? Well, now, equipped with power tools of modern approach to time analysis, we can always come back and verify our hypotheses.

References

The theory on the methods of period detection based on publication of Lachowicz et alii (2006). For deeper references to source readings mentioned in the text, check the reference section inside the aforementioned publication.

Probability of Financial Ruin

Very often in the modeling of rare events in finance and insurance industry the analysts are interested in both the probability of events and their financial consequences. That revolves around the aspect of a probability of ruin, i.e. when one of two dealing parties becomes insolvent (loses all its assets).

The estimation of chances of going broke may be understood by considering the following gambler’s problem. Two parties X and Y start the game with $x$ and $y$ dollars. If X wins, and its probability of winning is $p$, Y pays $\$1$to X. The game ends when one of the parties amasses all money,$\$(a+b)$.

Let’s denote by $u_{n+1}$ the probability that X wins and holds $\$(n+1)$. If X wins again with$p$it will hold$\$(n+2)$ and probability of winning of $u_{n+2}$ or if Y wins with $q=1-p$, it will hold $\$n$and$u_n$: $$u_{n+1} = pu_{n+2} + qu_n \ \ \mbox{for}\ \ 0 < n+1 < x+y.$$ But because$u_{n+1} = pu_{n+1} + qu_{n+1}$we get: $$(u_{n+2}-u_{n+1}) = \frac{q}{p}(u_{n+1}-u_n).$$ Applying recurrence relation for the problem we may denote: $$(u_2-u_1) = \frac{q}{p}(u_1) \\ (u_3-u_2) = \frac{q}{p}(u_2-u1) = \left(\frac{q}{p}\right)^2 (u_1) \\ ... \\ (u_i-u_{i-1}) = \left(\frac{q}{p}\right)^{i-1}(u_1) .$$ Now, if we sum both sides we will find: $$(u_i-u_1) = \left[\sum_{j=1}^{i-1} \left(\frac{q}{p}\right)^{j}\right](u_1), \ \ \ \mbox{or} \\ u_i = \left[1+\frac{q}{p}+\left(\frac{q}{p}\right)^2+ ... + \left(\frac{q}{p}\right)^{i-1} \right](u_1)$$ Doing maths, one can prove that: $$\left(1-\frac{q}{p}\right) \left[1+\sum_{j=1}^{i-1} \left(\frac{q}{p}\right)^{j} \right] = 1-\left(\frac{q}{p}\right)^i$$ therefore for$p\neq q$we have $$u_i = \left[ \frac{1-\left(\frac{q}{p}\right)^i}{1-\left(\frac{q}{p}\right)} \right] u_1 .$$ Making a note that$u_{x+y}=1$, we evaluate it to: $$u_1 = \left[ \frac{1-\left(\frac{q}{p}\right)}{1-\left(\frac{q}{p}\right)^{x+y}} \right]u_{x+y}$$ what generalized for$i$gives $$u_i = \left[ \frac{1-\left(\frac{q}{p}\right)^i}{1-\left(\frac{q}{p}\right)^{x+y}} \right]$$ therefore we have: $$u_x = \left[ \frac{1-\left(\frac{q}{p}\right)^x}{1-\left(\frac{q}{p}\right)^{x+y}}\right] .$$ For the case of$p=q=\frac{1}{2}$we have: $$u_x = \frac{x}{x+y} .$$ So, what this$u_x$means for us? Let's illustrate it using a real-life example. Learn to count cards before going to Las Vegas If you start playing poker-like game having$y=\$1000$ in your pocket and casino’s total cash that you may win equals $x=\$$50,000,000 but your winning rate solely depends on your luck (favorably, say your$q=1-p=0.49$, i.e. only a bit lower that bank’s odds to win) then your chances of going broke are equal$u_x = 1$. However, if you can count carts and do it consistently over and over again (now your q=0.51) then$u_x \approx 4 \times 10^{-18}\$ what simply means you have a huge potential to be very rich over night using your skills and mathematics.