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rel="noopener noreferrer" class=authorSocialLink_owbf title=GitHub><svg viewBox="0 0 256 250" width=1em height=1em class="authorSocialLink_owbf githubSvg_Uu4N" style=--dark:#000;--light:#fff preserveAspectRatio=xMidYMid><path d="M128.001 0C57.317 0 0 57.307 0 128.001c0 56.554 36.676 104.535 87.535 121.46 6.397 1.185 8.746-2.777 8.746-6.158 0-3.052-.12-13.135-.174-23.83-35.61 7.742-43.124-15.103-43.124-15.103-5.823-14.795-14.213-18.73-14.213-18.73-11.613-7.944.876-7.78.876-7.78 12.853.902 19.621 13.19 19.621 13.19 11.417 19.568 29.945 13.911 37.249 10.64 1.149-8.272 4.466-13.92 8.127-17.116-28.431-3.236-58.318-14.212-58.318-63.258 0-13.975 5-25.394 13.188-34.358-1.329-3.224-5.71-16.242 1.24-33.874 0 0 10.749-3.44 35.21 13.121 10.21-2.836 21.16-4.258 32.038-4.307 10.878.049 21.837 1.47 32.066 4.307 24.431-16.56 35.165-13.12 35.165-13.12 6.967 17.63 2.584 30.65 1.255 33.873 8.207 8.964 13.173 20.383 13.173 34.358 0 49.163-29.944 59.988-58.447 63.157 4.591 3.972 8.682 11.762 8.682 23.704 0 17.126-.148 30.91-.148 35.126 0 3.407 2.304 7.398 8.792 6.14C219.37 232.5 256 184.537 256 128.002 256 57.307 198.691 0 128.001 0Zm-80.06 182.34c-.282.636-1.283.827-2.194.39-.929-.417-1.45-1.284-1.15-1.922.276-.655 1.279-.838 2.205-.399.93.418 1.46 1.293 1.139 1.931Zm6.296 5.618c-.61.566-1.804.303-2.614-.591-.837-.892-.994-2.086-.375-2.66.63-.566 1.787-.301 2.626.591.838.903 1 2.088.363 2.66Zm4.32 7.188c-.785.545-2.067.034-2.86-1.104-.784-1.138-.784-2.503.017-3.05.795-.547 2.058-.055 2.861 1.075.782 1.157.782 2.522-.019 3.08Zm7.304 8.325c-.701.774-2.196.566-3.29-.49-1.119-1.032-1.43-2.496-.726-3.27.71-.776 2.213-.558 3.315.49 1.11 1.03 1.45 2.505.701 3.27Zm9.442 2.81c-.31 1.003-1.75 1.459-3.199 1.033-1.448-.439-2.395-1.613-2.103-2.626.301-1.01 1.747-1.484 3.207-1.028 1.446.436 2.396 1.602 2.095 2.622Zm10.744 1.193c.036 1.055-1.193 1.93-2.715 1.95-1.53.034-2.769-.82-2.786-1.86 0-1.065 1.202-1.932 2.733-1.958 1.522-.03 2.768.818 2.768 1.868Zm10.555-.405c.182 1.03-.875 2.088-2.387 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<p>If you can see into the future, that is.</p>
<p>My previous class in Stochastic Calculus covered a lot of interesting topics, and the important one for today is the <a href="https://en.wikipedia.org/wiki/Gambler's_ruin" target=_blank rel="noopener noreferrer">Gambler's Ruin</a> problem. If you're interested in some of the theory behind it, also make sure to check out <a href=https://en.wikipedia.org/wiki/Random_walk target=_blank rel="noopener noreferrer">random walks</a>. The important bit is that we studied the <a href=https://en.wikipedia.org/wiki/Martingale_%28betting_system%29 target=_blank rel="noopener noreferrer">Martingale Betting Strategy</a>, which describes for us a <strong>guaranteed way</strong> to <small>eventually</small> make money.</p>
<p>The strategy goes like this: You are going to toss a fair coin with a friend. If you guess heads or tails correctly, you get back double the money you bet. If you guess incorrectly, you lose money. How should you bet?</p>
<p>The correct answer is that you should double your bet each time you lose. Then when you finally win, you'll be guaranteed to make back everything you lost and then $1 extra! Consider the scenario:</p>
<ol>
<li>You bet $1, and guess incorrectly. You're 1 dollar in the hole.</li>
<li>You bet $2, and guess incorrectly. You're 3 dollars in the hole now.</li>
<li>You bet $4, and guess incorrectly. You're 7 dollars in the hole.</li>
<li>You bet $8, and guess correctly! You now get back those 8 dollars you bet, plus 8 extra for winning, for a <strong>total profit of one dollar</strong>!</li>
</ol>
<p>Mathematically, we can prove that as long as you have unlimited money to bet, you are guaranteed to make money.</p>
<h2 class="anchor anchorWithStickyNavbar_LWe7" id=applying-the-martingale-strategy>Applying the Martingale Strategy<a href=#applying-the-martingale-strategy class=hash-link aria-label="Direct link to Applying the Martingale Strategy" title="Direct link to Applying the Martingale Strategy"></a></h2>
<p>But we're all realistic people, and once you start talking about "unlimited money" eyebrows should be raised. Even still, this is an interesting strategy to investigate, and I want to apply it to the stock market. As long as we can guarantee there's a single day in which the stock goes up, we should be able to make money right? The question is just how much we have to invest to guarantee this.</p>
<p>Now it's time for the math. We'll use the following definitions:</p>
<ul>
<li><span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>o</mi><mi>i</mi></msub></mrow><annotation encoding=application/x-tex>o_i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.5806em;vertical-align:-0.15em></span><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span> = the share price at the opening of day <span class=katex><span class=katex-mathml><math><semantics><mrow><mi>i</mi></mrow><annotation encoding=application/x-tex>i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.6595em></span><span class="mord mathnormal">i</span></span></span></span></li>
<li><span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub></mrow><annotation encoding=application/x-tex>c_i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.5806em;vertical-align:-0.15em></span><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span> = the share price at the close of day <span class=katex><span class=katex-mathml><math><semantics><mrow><mi>i</mi></mrow><annotation encoding=application/x-tex>i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.6595em></span><span class="mord mathnormal">i</span></span></span></span></li>
<li><span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub></mrow><annotation encoding=application/x-tex>d_i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.8444em;vertical-align:-0.15em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span> = the amount of money we want to invest at the beginning of day <span class=katex><span class=katex-mathml><math><semantics><mrow><mi>i</mi></mrow><annotation encoding=application/x-tex>i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.6595em></span><span class="mord mathnormal">i</span></span></span></span></li>
</ul>
<p>With those definitions in place, I'd like to present the formula that is <strong>guaranteed to make you money</strong>. I call it <em>Bradlee's Investment Formula</em>:</p>
<p><span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>c</mi><mi>n</mi></msub><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mfrac><msub><mi>d</mi><mi>i</mi></msub><msub><mi>o</mi><mi>i</mi></msub></mfrac><mo>></mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>d</mi><mi>i</mi></msub></mrow><annotation encoding=application/x-tex>c_n \sum_{i=1}^n \frac{d_i}{o_i} > \sum_{i=1}^{n} d_i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:1.3413em;vertical-align:-0.4451em></span><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mspace style=margin-right:0.1667em></span><span class=mop><span class="mop op-symbol small-op" style=position:relative;top:0em>∑</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.8043em><span style=top:-2.4003em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style=top:-3.2029em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.2997em><span></span></span></span></span></span></span><span class=mspace style=margin-right:0.1667em></span><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.8962em><span style=top:-2.655em><span class=pstrut style=height:3em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3281em><span style=top:-2.357em;margin-left:0em;margin-right:0.0714em><span class=pstrut style=height:2.5em></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.143em><span></span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.4101em><span class=pstrut style=height:3em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3281em><span style=top:-2.357em;margin-left:0em;margin-right:0.0714em><span class=pstrut style=height:2.5em></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.143em><span></span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.4451em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class=mspace style=margin-right:0.2778em></span><span class=mrel>></span><span class=mspace style=margin-right:0.2778em></span></span><span class=base><span class=strut style=height:1.104em;vertical-align:-0.2997em></span><span class=mop><span class="mop op-symbol small-op" style=position:relative;top:0em>∑</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.8043em><span style=top:-2.4003em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style=top:-3.2029em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.2997em><span></span></span></span></span></span></span><span class=mspace style=margin-right:0.1667em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span></p>
<p>It might not look like much, but if you can manage to make it so that this formula holds true, you will be guaranteed to make money. The intuition behind the formula is this: The closing share price times the number of shares you have purchased ends up greater than the amount of money you invested.</p>
<p>That is, on day <span class=katex><span class=katex-mathml><math><semantics><mrow><mi>n</mi></mrow><annotation encoding=application/x-tex>n</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.4306em></span><span class="mord mathnormal">n</span></span></span></span>, <small>if you know what the closing price will be</small> you can set up the amount of money you invest that day to <strong>guarantee you make money</strong>. I'll even teach you to figure out how much money that is! Take a look:</p>
<span class=katex-display><span class=katex><span class=katex-mathml><math display=block><semantics><mtable rowspacing=0.25em columnalign="right left" columnspacing=0em><mtr><mtd><mstyle scriptlevel=0 displaystyle=true><mrow><msub><mi>c</mi><mi>n</mi></msub><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><msub><mi>d</mi><mi>i</mi></msub><msub><mi>o</mi><mi>i</mi></msub></mfrac><mo>+</mo><mfrac><mrow><msub><mi>c</mi><mi>n</mi></msub><msub><mi>d</mi><mi>n</mi></msub></mrow><msub><mi>o</mi><mi>n</mi></msub></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel=0 displaystyle=true><mrow><mrow/><mo>></mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mi>d</mi><mi>i</mi></msub><mo>+</mo><msub><mi>d</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=0 displaystyle=true><mrow><mfrac><mrow><msub><mi>c</mi><mi>n</mi></msub><msub><mi>d</mi><mi>n</mi></msub></mrow><msub><mi>o</mi><mi>n</mi></msub></mfrac><mo>−</mo><msub><mi>d</mi><mi>n</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel=0 displaystyle=true><mrow><mrow/><mo>></mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy=false>(</mo><msub><mi>d</mi><mi>i</mi></msub><mo>−</mo><mfrac><mrow><msub><mi>c</mi><mi>n</mi></msub><msub><mi>d</mi><mi>i</mi></msub></mrow><msub><mi>o</mi><mi>i</mi></msub></mfrac><mo stretchy=false>)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=0 displaystyle=true><mrow><msub><mi>d</mi><mi>n</mi></msub><mo stretchy=false>(</mo><mfrac><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>−</mo><msub><mi>o</mi><mi>n</mi></msub></mrow><msub><mi>o</mi><mi>n</mi></msub></mfrac><mo stretchy=false>)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel=0 displaystyle=true><mrow><mrow/><mo>></mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mi>d</mi><mi>i</mi></msub><mo stretchy=false>(</mo><mn>1</mn><mo>−</mo><mfrac><msub><mi>c</mi><mi>n</mi></msub><msub><mi>o</mi><mi>i</mi></msub></mfrac><mo stretchy=false>)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=0 displaystyle=true><msub><mi>d</mi><mi>n</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel=0 displaystyle=true><mrow><mrow/><mo>></mo><mfrac><msub><mi>o</mi><mi>n</mi></msub><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>−</mo><msub><mi>o</mi><mi>n</mi></msub></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mi>d</mi><mi>i</mi></msub><mo stretchy=false>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><msub><mi>o</mi><mi>i</mi></msub></mfrac><mo stretchy=false>)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding=application/x-tex>\begin{align*}
c_n \sum_{i=1}^{n-1} \frac{d_i}{o_i} + \frac{c_nd_n}{o_n} &> \sum_{i=1}^{n-1}d_i + d_n\\
\frac{c_nd_n}{o_n} - d_n &> \sum_{i=1}^{n-1}(d_i - \frac{c_nd_i}{o_i})\\
d_n (\frac{c_n - o_n}{o_n}) &> \sum_{i=1}^{n-1} d_i(1 - \frac{c_n}{o_i})\\
d_n &> \frac{o_n}{c_n - o_n} \sum_{i=1}^{n-1} d_i(1 - \frac{1}{o_i})
\end{align*}</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:13.5151em;vertical-align:-6.5076em></span><span class=mord><span class=mtable><span class=col-align-r><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:7.0076em><span style=top:-9.0076em><span class=pstrut style=height:3.8011em></span><span class=mord><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mspace style=margin-right:0.1667em></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.8011em><span style=top:-1.8723em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style=top:-3.05em><span class=pstrut style=height:3.05em></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style=top:-4.3em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:1.2777em><span></span></span></span></span></span><span class=mspace style=margin-right:0.1667em></span><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.3714em><span style=top:-2.314em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.677em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.836em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>+</span><span class=mspace style=margin-right:0.2222em></span><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.3714em><span style=top:-2.314em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.677em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.836em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style=top:-5.6288em><span class=pstrut style=height:3.8011em></span><span class=mord><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.3714em><span style=top:-2.314em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.677em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.836em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>−</span><span class=mspace style=margin-right:0.2222em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-2.25em><span class=pstrut style=height:3.8011em></span><span class=mord><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mopen>(</span><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.2603em><span style=top:-2.314em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.677em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>−</span><span class=mspace style=margin-right:0.2222em></span><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.836em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class=mclose>)</span></span></span><span style=top:1.1288em><span class=pstrut style=height:3.8011em></span><span class=mord><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:6.5076em><span></span></span></span></span></span><span class=col-align-l><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:7.0076em><span style=top:-9.0076em><span class=pstrut style=height:3.8011em></span><span class=mord><span class=mord></span><span class=mspace style=margin-right:0.2778em></span><span class=mrel>></span><span class=mspace style=margin-right:0.2778em></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.8011em><span style=top:-1.8723em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style=top:-3.05em><span class=pstrut style=height:3.05em></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style=top:-4.3em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:1.2777em><span></span></span></span></span></span><span class=mspace style=margin-right:0.1667em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>+</span><span class=mspace style=margin-right:0.2222em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-5.6288em><span class=pstrut style=height:3.8011em></span><span class=mord><span class=mord></span><span class=mspace style=margin-right:0.2778em></span><span class=mrel>></span><span class=mspace style=margin-right:0.2778em></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.8011em><span style=top:-1.8723em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style=top:-3.05em><span class=pstrut style=height:3.05em></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style=top:-4.3em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:1.2777em><span></span></span></span></span></span><span class=mopen>(</span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>−</span><span class=mspace style=margin-right:0.2222em></span><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.3714em><span style=top:-2.314em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.677em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.836em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class=mclose>)</span></span></span><span style=top:-2.25em><span class=pstrut style=height:3.8011em></span><span class=mord><span class=mord></span><span class=mspace style=margin-right:0.2778em></span><span class=mrel>></span><span class=mspace style=margin-right:0.2778em></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.8011em><span style=top:-1.8723em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style=top:-3.05em><span class=pstrut style=height:3.05em></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style=top:-4.3em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:1.2777em><span></span></span></span></span></span><span class=mspace style=margin-right:0.1667em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mopen>(</span><span class=mord>1</span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>−</span><span class=mspace style=margin-right:0.2222em></span><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.1076em><span style=top:-2.314em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.677em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.836em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class=mclose>)</span></span></span><span style=top:1.1288em><span class=pstrut style=height:3.8011em></span><span class=mord><span class=mord></span><span class=mspace style=margin-right:0.2778em></span><span class=mrel>></span><span class=mspace style=margin-right:0.2778em></span><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.1076em><span style=top:-2.314em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">c</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>−</span><span class=mspace style=margin-right:0.2222em></span><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.677em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.836em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class=mspace style=margin-right:0.1667em></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.8011em><span style=top:-1.8723em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style=top:-3.05em><span class=pstrut style=height:3.05em></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style=top:-4.3em;margin-left:0em><span class=pstrut style=height:3.05em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:1.2777em><span></span></span></span></span></span><span class=mspace style=margin-right:0.1667em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span><span class=mopen>(</span><span class=mord>1</span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>−</span><span class=mspace style=margin-right:0.2222em></span><span class=mord><span class="mopen nulldelimiter"></span><span class=mfrac><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:1.3214em><span style=top:-2.314em><span class=pstrut style=height:3em></span><span class=mord><span class=mord><span class="mord mathnormal">o</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span><span style=top:-3.23em><span class=pstrut style=height:3em></span><span class=frac-line style=border-bottom-width:0.04em></span></span><span style=top:-3.677em><span class=pstrut style=height:3em></span><span class=mord><span class=mord>1</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.836em><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class=mclose>)</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:6.5076em><span></span></span></span></span></span></span></span></span></span></span></span>
<p>If you invest exactly <span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>d</mi><mi>n</mi></msub></mrow><annotation encoding=application/x-tex>d_n</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.8444em;vertical-align:-0.15em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span> that day, you'll break even. But if you can make sure the money you invest is greater than that quantity on the right <small>(which requires that you have a crystal ball tell you the stock's closing price)</small> you are <strong>guaranteed to make money!</strong></p>
<h2 class="anchor anchorWithStickyNavbar_LWe7" id=interesting-implications>Interesting Implications<a href=#interesting-implications class=hash-link aria-label="Direct link to Interesting Implications" title="Direct link to Interesting Implications"></a></h2>
<p>On a more serious note though, the formula above tells us a couple of interesting things:</p>
<ol>
<li>It's impossible to make money without the closing price at some point being greater than the opening price (or vice-versa if you are short selling) - there is no amount of money you can invest that will turn things in your favor.</li>
<li>Close prices of the past aren't important if you're concerned about the bottom line. While chart technicians use price history to make judgment calls, in the end, the closing price on anything other than the last day is irrelevant.</li>
<li>It's possible to make money as long as there is a single day where the closing price is greater than the opening price! You might have to invest a lot to do so, but it's possible.</li>
<li>You must make a prediction about where the stock will close at if you want to know how much to invest. That is, we can set up our investment for the day to make money if the stock goes up 1%, but if it only goes up .5% we'll still lose money.</li>
<li>It's possible the winning move is to scale back your position. Consider the scenario:<!-- -->
<ul>
<li>You invest money and the stock closes down the day .5%</li>
<li>You invest tomorrow expecting the stock to go up 1%</li>
<li>The winning investment to break even (assuming a 1% increase) is to scale back the position, since the shares you purchased at the beginning would then be profitable</li>
</ul>
</li>
</ol>
<h2 class="anchor anchorWithStickyNavbar_LWe7" id=running-the-simulation>Running the simulation<a href=#running-the-simulation class=hash-link aria-label="Direct link to Running the simulation" title="Direct link to Running the simulation"></a></h2>
<p>So now that we've defined our investment formula,we need to tweak a couple things in order to make an investment strategy we can actually work with. There are two issues we need to address:</p>
<ol>
<li>The formula only tells us how much to invest if we want to break even (<span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>d</mi><mi>n</mi></msub></mrow><annotation encoding=application/x-tex>d_n</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.8444em;vertical-align:-0.15em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span>). If we actually want to turn a profit, we need to invest more than that, which we will refer to as the <strong>bias</strong>.</li>
<li>The formula assumes we know what the closing price will be on any given day. If we don't know this, we can still invest assuming the stock price will close at a level we choose. If the price doesn't meet this objective, we try again tomorrow! This predetermined closing price will be referred to as the <strong>expectation</strong>.</li>
</ol>
<p>Now that we've defined our <em>bias</em> and <em>expectation</em>, we can actually build a strategy we can simulate. Much like the martingale strategy told you to bet twice your previous bet in order to make money, we've designed a system that tells us how much to bet in order to make money as well.</p>
<p>Now, let's get to the code!</p>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token keyword" style="color:hsl(301, 63%, 40%)">using</span><span class="token plain"> Quandl</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">api_key </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token string" style="color:hsl(119, 34%, 47%)">""</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">daily_investment </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">current_open</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> current_close</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> purchase_history</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> open_history</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># We're not going to safeguard against divide by 0 - that's the user's responsibility</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    t1 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> current_close </span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> current_open </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    t2 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> sum</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">purchase_history </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> purchase_history</span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain">current_close </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> open_history</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> t2 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> t1</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">end</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<p>And let's code a way to run simulations quickly:</p>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">is_profitable </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">current_price</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> purchase_history</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> open_history</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    shares </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> sum</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">purchase_history </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> open_history</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> current_price</span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain">shares </span><span class="token operator" style="color:hsl(221, 87%, 60%)">></span><span class="token plain"> sum</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">purchase_history</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">end</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">simulate </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">name</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> start</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> init</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> expected</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> bias</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    ticker_info </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> quandlget</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">name</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> from</span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain">start</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> api_key</span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain">api_key</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    open_vals </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> ticker_info</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token string" style="color:hsl(119, 34%, 47%)">"Open"</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">values</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    close_vals </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> ticker_info</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token string" style="color:hsl(119, 34%, 47%)">"Close"</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">values</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    invested </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">init</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># The simulation stops once we've made a profit</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    day </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    profitable </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> is_profitable</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">close_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">day</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">        is_profitable</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">day</span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">while</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">!</span><span class="token plain">profitable</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">        expected_close </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">day</span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> expected</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">        todays_purchase </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> daily_investment</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">day</span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> expected_close</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">day</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">        invested </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"> todays_purchase </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> bias</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">        </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># expected_profit = expected_close * sum(invested ./ open_vals[1:length(invested)]) - sum(invested)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">        day </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">        profitable </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> is_profitable</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">close_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">day</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">            is_profitable</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">day</span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">end</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    shares </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> sum</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">invested </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    max_profit </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> max</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">close_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">day</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> open_vals</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">day</span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    profit </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> shares </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> max_profit </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> sum</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">invested</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> profit</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">end</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">sim_summary </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">investments</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> profit</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    leverages </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">sum</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">investments</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">i</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">for</span><span class="token plain"> i</span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">investments</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    max_leverage </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> maximum</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">leverages</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> investments</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">println</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token string" style="color:hsl(119, 34%, 47%)">"Max leverage: $(max_leverage)"</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">println</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token string" style="color:hsl(119, 34%, 47%)">"Days invested: $(length(investments))"</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">println</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token string" style="color:hsl(119, 34%, 47%)">"Profit: $profit"</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">end</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<p>Now, let's get some data and run a simulation! Our first test:</p>
<ul>
<li>We'll invest 100 dollars in LMT, and expect that the stock will close up 1% every day. We'll invest <span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>d</mi><mi>n</mi></msub></mrow><annotation encoding=application/x-tex>d_n</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.8444em;vertical-align:-0.15em></span><span class=mord><span class="mord mathnormal">d</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.1514em><span style=top:-2.55em;margin-left:0em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class=vlist-s></span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span> + 10 dollars every day that we haven't turned a profit, and end the simulation once we've made a profit.</li>
</ul>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">investments</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> profit </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> simulate</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token string" style="color:hsl(119, 34%, 47%)">"YAHOO/LMT"</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> Date</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token number" style="color:hsl(35, 99%, 36%)">2015</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">11</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">29</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">100</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1.01</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">10</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">sim_summary</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">investments</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> profit</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    Max leverage: 5.590373200042106</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    Days invested: 5</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    Profit: 0.6894803101560001</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<p>The result: We need to invest 5.6x our initial position over a period of 5 days to make approximately .69¢</p>
<ul>
<li>Now let's try the same thing, but we'll assume the stock closes up 2% instead.</li>
</ul>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">investments</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> profit </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> simulate</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token string" style="color:hsl(119, 34%, 47%)">"YAHOO/LMT"</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> Date</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token number" style="color:hsl(35, 99%, 36%)">2015</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">11</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">29</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">100</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1.02</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">10</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">sim_summary</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">investments</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> profit</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    Max leverage: 1.854949900247809</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    Days invested: 25</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">    Profit: 0.08304813163696423</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<p>In this example, we only get up to a 1.85x leveraged position, but it takes 25 days to turn a profit of 8¢</p>
<h2 class="anchor anchorWithStickyNavbar_LWe7" id=summary>Summary<a href=#summary class=hash-link aria-label="Direct link to Summary" title="Direct link to Summary"></a></h2>
<p>We've defined an investment strategy that can tell us how much to invest when we know what the closing position of a stock will be. We can tweak the strategy to actually make money, but plenty of work needs to be done so that we can optimize the money invested.</p>
<p>In the next post I'm going to post more information about some backtests and strategy tests on this strategy (unless of course this experiment actually produces a significant profit potential, and then I'm keeping it for myself).</p>
<h3 class="anchor anchorWithStickyNavbar_LWe7" id=side-note-and-disclaimer>Side note and disclaimer<a href=#side-note-and-disclaimer class=hash-link aria-label="Direct link to Side note and disclaimer" title="Direct link to Side note and disclaimer"></a></h3>
<p>The claims made in this presentation about being able to guarantee making money are intended as a joke and do not constitute investment advice of any sort.</div></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Blog post page navigation"><a class="pagination-nav__link pagination-nav__link--prev" href=/2016/01/cloudy-in-seattle><div class=pagination-nav__sublabel>Older post</div><div class=pagination-nav__label>Cloudy in Seattle</div></a><a class="pagination-nav__link pagination-nav__link--next" href=/2016/02/profitability-using-the-investment-formula><div class=pagination-nav__sublabel>Newer post</div><div class=pagination-nav__label>Profitability using the investment formula</div></a></nav></main><div class="col col--2"><div class="tableOfContents_bqdL thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href=#applying-the-martingale-strategy class="table-of-contents__link toc-highlight">Applying the Martingale Strategy</a><li><a href=#interesting-implications class="table-of-contents__link toc-highlight">Interesting Implications</a><li><a href=#running-the-simulation class="table-of-contents__link toc-highlight">Running the simulation</a><li><a href=#summary class="table-of-contents__link toc-highlight">Summary</a><ul><li><a href=#side-note-and-disclaimer class="table-of-contents__link toc-highlight">Side note and disclaimer</a></ul></ul></div></div></div></div></div><footer class=footer><div class="container container-fluid"><div class="footer__bottom text--center"><div class=footer__copyright>Copyright © 2024 Bradlee Speice</div></div></div></footer></div>