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<h1 class="header-title">Welcome, and an algorithm</h1>
<p class="header-date"> <a href="https://bspeice.github.io/author/bradlee-speice.html">Bradlee Speice</a>, Thu 19 November 2015, Sat 05 December 2015, <a href="https://bspeice.github.io/category/blog.html">Blog</a></p>
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<span class="glyphicon glyphicon-tags mr5" aria-hidden="true"></span>
<a href="https://bspeice.github.io/tag/introduction.html">introduction</a>, <a href="https://bspeice.github.io/tag/trading.html">trading</a> </p>
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<p>Hello! Glad to meet you. I'm currently a student at Columbia University
studying Financial Engineering, and want to give an overview of the projects
I'm working on!</p>
<p>To start things off, Columbia has been hosting a trading competition that
myself and another partner are competing in. I'm including a notebook of the
algorithm that we're using, just to give a simple overview of a miniature
algorithm.</p>
<p>The competition is scored in 3 areas:</p>
<ul>
<li>Total return</li>
<li><a href="1">Sharpe ratio</a></li>
<li>Maximum drawdown</li>
</ul>
<p>Our algorithm uses a basic momentum strategy: in the given list of potential
portfolios, pick the stocks that have been performing well in the past 30
days. Then, optimize for return subject to the drawdown being below a specific
level. We didn't include the Sharpe ratio as a constraint, mostly because
we were a bit late entering the competition.</p>
<p>I'll be updating this post with the results of our algorithm as they come along!</p>
<hr />
<p><strong>UPDATE 12/5/2015</strong>: Now that the competition has ended, I wanted to update
how the algorithm performed. Unfortunately, it didn't do very well. I'm planning
to make some tweaks over the coming weeks, and do another forward test in January.</p>
<ul>
<li>After week 1: Down .1%</li>
<li>After week 2: Down 1.4%</li>
<li>After week 3: Flat</li>
</ul>
<p>And some statistics for all teams participating in the competition:</p>
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<td>Max Return</td>
<td>74.1%</td>
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<td>Min Return</td>
<td>-97.4%</td>
</tr>
<tr>
<td>Average Return</td>
<td>-.1%</td>
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<td>Std Dev of Returns</td>
<td>19.6%</td>
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<h1 id="Trading-Competition-Optimization">Trading Competition Optimization<a class="anchor-link" href="#Trading-Competition-Optimization">&#182;</a></h1><h3 id="Goal:-Max-return-given-maximum-Sharpe-and-Drawdown">Goal: Max return given maximum Sharpe and Drawdown<a class="anchor-link" href="#Goal:-Max-return-given-maximum-Sharpe-and-Drawdown">&#182;</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="k">import</span> <span class="n">display</span>
<span class="kn">import</span> <span class="nn">Quandl</span>
<span class="kn">from</span> <span class="nn">datetime</span> <span class="k">import</span> <span class="n">datetime</span><span class="p">,</span> <span class="n">timedelta</span>
<span class="n">tickers</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;XOM&#39;</span><span class="p">,</span> <span class="s1">&#39;CVX&#39;</span><span class="p">,</span> <span class="s1">&#39;CLB&#39;</span><span class="p">,</span> <span class="s1">&#39;OXY&#39;</span><span class="p">,</span> <span class="s1">&#39;SLB&#39;</span><span class="p">]</span>
<span class="n">market_ticker</span> <span class="o">=</span> <span class="s1">&#39;GOOG/NYSE_VOO&#39;</span>
<span class="n">lookback</span> <span class="o">=</span> <span class="mi">30</span>
<span class="n">d_col</span> <span class="o">=</span> <span class="s1">&#39;Close&#39;</span>
<span class="n">data</span> <span class="o">=</span> <span class="p">{</span><span class="n">tick</span><span class="p">:</span> <span class="n">Quandl</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;YAHOO/{}&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">tick</span><span class="p">))[</span><span class="o">-</span><span class="n">lookback</span><span class="p">:]</span> <span class="k">for</span> <span class="n">tick</span> <span class="ow">in</span> <span class="n">tickers</span><span class="p">}</span>
<span class="n">market</span> <span class="o">=</span> <span class="n">Quandl</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">market_ticker</span><span class="p">)</span>
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<h1 id="Calculating-the-Return">Calculating the Return<a class="anchor-link" href="#Calculating-the-Return">&#182;</a></h1><p>We first want to know how much each ticker returned over the prior period.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">returns</span> <span class="o">=</span> <span class="p">{</span><span class="n">tick</span><span class="p">:</span> <span class="n">data</span><span class="p">[</span><span class="n">tick</span><span class="p">][</span><span class="n">d_col</span><span class="p">]</span><span class="o">.</span><span class="n">pct_change</span><span class="p">()</span> <span class="k">for</span> <span class="n">tick</span> <span class="ow">in</span> <span class="n">tickers</span><span class="p">}</span>
<span class="n">display</span><span class="p">({</span><span class="n">tick</span><span class="p">:</span> <span class="n">returns</span><span class="p">[</span><span class="n">tick</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="k">for</span> <span class="n">tick</span> <span class="ow">in</span> <span class="n">tickers</span><span class="p">})</span>
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<pre>{&#39;CLB&#39;: -0.0016320202164526894,
&#39;CVX&#39;: 0.0010319531629488911,
&#39;OXY&#39;: 0.00093418904454400551,
&#39;SLB&#39;: 0.00098431254720448159,
&#39;XOM&#39;: 0.00044165797556096868}</pre>
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<h1 id="Calculating-the-Sharpe-ratio">Calculating the Sharpe ratio<a class="anchor-link" href="#Calculating-the-Sharpe-ratio">&#182;</a></h1><p>Sharpe: ${R - R_M \over \sigma}$</p>
<p>We use the average return over the lookback period, minus the market average return, over the ticker standard deviation to calculate the Sharpe. Shorting a stock turns a negative Sharpe positive.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">market_returns</span> <span class="o">=</span> <span class="n">market</span><span class="o">.</span><span class="n">pct_change</span><span class="p">()</span>
<span class="n">sharpe</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">ret</span><span class="p">:</span> <span class="p">(</span><span class="n">ret</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">-</span> <span class="n">market_returns</span><span class="p">[</span><span class="n">d_col</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="o">/</span> <span class="n">ret</span><span class="o">.</span><span class="n">std</span><span class="p">()</span>
<span class="n">sharpes</span> <span class="o">=</span> <span class="p">{</span><span class="n">tick</span><span class="p">:</span> <span class="n">sharpe</span><span class="p">(</span><span class="n">returns</span><span class="p">[</span><span class="n">tick</span><span class="p">])</span> <span class="k">for</span> <span class="n">tick</span> <span class="ow">in</span> <span class="n">tickers</span><span class="p">}</span>
<span class="n">display</span><span class="p">(</span><span class="n">sharpes</span><span class="p">)</span>
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<pre>{&#39;CLB&#39;: -0.10578734457846127,
&#39;CVX&#39;: 0.027303529817677398,
&#39;OXY&#39;: 0.022622210057414487,
&#39;SLB&#39;: 0.026950946344858676,
&#39;XOM&#39;: -0.0053519259698605499}</pre>
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<h1 id="Calculating-the-drawdown">Calculating the drawdown<a class="anchor-link" href="#Calculating-the-drawdown">&#182;</a></h1><p>This one is easy - what is the maximum daily change over the lookback period? That is, because we will allow short positions, we are not concerned strictly with maximum downturn, but in general, what is the largest 1-day change?</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">drawdown</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">ret</span><span class="p">:</span> <span class="n">ret</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="n">drawdowns</span> <span class="o">=</span> <span class="p">{</span><span class="n">tick</span><span class="p">:</span> <span class="n">drawdown</span><span class="p">(</span><span class="n">returns</span><span class="p">[</span><span class="n">tick</span><span class="p">])</span> <span class="k">for</span> <span class="n">tick</span> <span class="ow">in</span> <span class="n">tickers</span><span class="p">}</span>
<span class="n">display</span><span class="p">(</span><span class="n">drawdowns</span><span class="p">)</span>
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<pre>{&#39;CLB&#39;: 0.043551495607375035,
&#39;CVX&#39;: 0.044894389686214398,
&#39;OXY&#39;: 0.051424517867144637,
&#39;SLB&#39;: 0.034774627850375328,
&#39;XOM&#39;: 0.035851524605672758}</pre>
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<h1 id="Performing-the-optimization">Performing the optimization<a class="anchor-link" href="#Performing-the-optimization">&#182;</a></h1><p>$\begin{align}
max\ \ & \mu \cdot \omega\\
s.t.\ \ & \vec{1} \omega = 1\\
& \vec{S} \omega \ge s\\
& \vec{D} \cdot | \omega | \le d\\
& \left|\omega\right| \le l\\
\end{align}$</p>
<p>We want to maximize average return subject to having a full portfolio, Sharpe above a specific level, drawdown below a level, and leverage not too high - that is, don't have huge long/short positions.</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="k">import</span> <span class="n">minimize</span>
<span class="c1">#sharpe_limit = .1</span>
<span class="n">drawdown_limit</span> <span class="o">=</span> <span class="o">.</span><span class="mi">05</span>
<span class="n">leverage</span> <span class="o">=</span> <span class="mi">250</span>
<span class="c1"># Use the map so we can guarantee we maintain the correct order</span>
<span class="c1"># sharpe_a = np.array(list(map(lambda tick: sharpes[tick], tickers))) * -1 # So we can write as upper-bound</span>
<span class="n">dd_a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">tick</span><span class="p">:</span> <span class="n">drawdowns</span><span class="p">[</span><span class="n">tick</span><span class="p">],</span> <span class="n">tickers</span><span class="p">)))</span>
<span class="n">returns_a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">tick</span><span class="p">:</span> <span class="n">returns</span><span class="p">[</span><span class="n">tick</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">tickers</span><span class="p">)))</span> <span class="c1"># Because minimizing</span>
<span class="n">meets_sharpe</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">sharpe_a</span><span class="p">)</span> <span class="o">-</span> <span class="n">sharpe_limit</span>
<span class="k">def</span> <span class="nf">meets_dd</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">portfolio</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="k">if</span> <span class="n">portfolio</span> <span class="o">&lt;</span> <span class="o">.</span><span class="mi">1</span><span class="p">:</span>
<span class="c1"># If there are no stocks in the portfolio,</span>
<span class="c1"># we can accidentally induce division by 0,</span>
<span class="c1"># or division by something small enough to cause infinity</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">drawdown_limit</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">dd_a</span><span class="p">)</span> <span class="o">/</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="n">is_portfolio</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">sum</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">within_leverage</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">leverage</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="n">objective</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">sum</span><span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">returns_a</span><span class="p">)</span> <span class="o">*</span> <span class="o">-</span><span class="mi">1</span> <span class="c1"># Because we&#39;re minimizing</span>
<span class="n">bounds</span> <span class="o">=</span> <span class="p">((</span><span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">),)</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">tickers</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">tickers</span><span class="p">))</span>
<span class="n">constraints</span> <span class="o">=</span> <span class="p">[</span>
<span class="p">{</span>
<span class="s1">&#39;type&#39;</span><span class="p">:</span> <span class="s1">&#39;eq&#39;</span><span class="p">,</span>
<span class="s1">&#39;fun&#39;</span><span class="p">:</span> <span class="n">is_portfolio</span>
<span class="p">},</span> <span class="p">{</span>
<span class="s1">&#39;type&#39;</span><span class="p">:</span> <span class="s1">&#39;ineq&#39;</span><span class="p">,</span>
<span class="s1">&#39;fun&#39;</span><span class="p">:</span> <span class="n">within_leverage</span>
<span class="c1">#}, {</span>
<span class="c1"># &#39;type&#39;: &#39;ineq&#39;,</span>
<span class="c1"># &#39;fun&#39;: meets_sharpe</span>
<span class="p">},</span> <span class="p">{</span>
<span class="s1">&#39;type&#39;</span><span class="p">:</span> <span class="s1">&#39;ineq&#39;</span><span class="p">,</span>
<span class="s1">&#39;fun&#39;</span><span class="p">:</span> <span class="n">meets_dd</span>
<span class="p">}</span>
<span class="p">]</span>
<span class="n">optimal</span> <span class="o">=</span> <span class="n">minimize</span><span class="p">(</span><span class="n">objective</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">bounds</span><span class="o">=</span><span class="n">bounds</span><span class="p">,</span> <span class="n">constraints</span><span class="o">=</span><span class="n">constraints</span><span class="p">,</span>
<span class="n">options</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;maxiter&#39;</span><span class="p">:</span> <span class="mi">500</span><span class="p">})</span>
<span class="c1"># Optimization time!</span>
<span class="n">display</span><span class="p">(</span><span class="n">optimal</span><span class="o">.</span><span class="n">message</span><span class="p">)</span>
<span class="n">display</span><span class="p">(</span><span class="s2">&quot;Holdings: {}&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">tickers</span><span class="p">,</span> <span class="n">optimal</span><span class="o">.</span><span class="n">x</span><span class="p">))))</span>
<span class="n">expected_return</span> <span class="o">=</span> <span class="n">optimal</span><span class="o">.</span><span class="n">fun</span> <span class="o">*</span> <span class="o">-</span><span class="mi">100</span> <span class="c1"># multiply by -100 to scale, and compensate for minimizing</span>
<span class="n">display</span><span class="p">(</span><span class="s2">&quot;Expected Return: {:.3f}%&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">expected_return</span><span class="p">))</span>
<span class="n">expected_drawdown</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">optimal</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">dd_a</span><span class="p">)</span> <span class="o">/</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">optimal</span><span class="o">.</span><span class="n">x</span><span class="p">))</span> <span class="o">*</span> <span class="mi">100</span>
<span class="n">display</span><span class="p">(</span><span class="s2">&quot;Expected Max Drawdown: {0:.2f}%&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">expected_drawdown</span><span class="p">))</span>
<span class="c1"># TODO: Calculate expected Sharpe</span>
</pre></div>
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<pre>&#39;Optimization terminated successfully.&#39;</pre>
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<pre>&#34;Holdings: [(&#39;XOM&#39;, 5.8337945679814904), (&#39;CVX&#39;, 42.935064321851307), (&#39;CLB&#39;, -124.5), (&#39;OXY&#39;, 36.790387773552119), (&#39;SLB&#39;, 39.940753336615096)]&#34;</pre>
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<pre>&#39;Expected Return: 32.375%&#39;</pre>
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<pre>&#39;Expected Max Drawdown: 4.34%&#39;</pre>
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