{"id":47302,"date":"2020-04-24T19:42:19","date_gmt":"2020-04-24T18:42:19","guid":{"rendered":"https:\/\/blogs.bmj.com\/bmj\/?p=47302"},"modified":"2020-06-17T22:48:45","modified_gmt":"2020-06-17T21:48:45","slug":"jeffrey-aronson-when-i-use-a-word-exponential-finances-increasing-decreasing","status":"publish","type":"post","link":"https:\/\/blogs.bmj.com\/bmj\/2020\/04\/24\/jeffrey-aronson-when-i-use-a-word-exponential-finances-increasing-decreasing\/","title":{"rendered":"Jeffrey Aronson: When I Use a Word . . . Exponential finances\u2014increasing\/decreasing"},"content":{"rendered":"<p><a href=\"https:\/\/blogs.bmj.com\/bmj\/files\/2014\/12\/jeffrey_aronson.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-32935\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2014\/12\/jeffrey_aronson.jpg\" alt=\"\" width=\"112\" height=\"138\" \/><\/a><span style=\"font-weight: 400\">Last week I suggested that there is widespread misunderstanding about the meaning of \u201cexponential\u201d in mathematics. Colloquially, it means a huge increase, which it need not be. And when events change exponentially they can go down as well as going up. Considering how you invest your money introduces the way in which exponentials work.<\/span><\/p>\n<p><span style=\"font-weight: 400\">Suppose you invest \u00a3100 at a simple interest rate of 10 per cent per annum. Each year you earn \u00a310. The principal, what you have in the bank that has conned you into this deal, increases linearly (Table 1). Now you strike a better deal; instead of a fixed percentage of your <\/span><i><span style=\"font-weight: 400\">principal<\/span><\/i><span style=\"font-weight: 400\"> (\u00a3100) each year, the bank will give you a fixed percentage of the <\/span><i><span style=\"font-weight: 400\">cumulative<\/span><\/i><span style=\"font-weight: 400\"> amount. Now you do somewhat better (Table 1). Instead of increasing your principal by 120% after 20 years (6% per annum) you increase it by 220%, an annual average of 11%.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-47315 alignleft\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_again3.jpg\" alt=\"\" width=\"704\" height=\"146\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_again3.jpg 704w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_again3-300x62.jpg 300w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_again3-640x133.jpg 640w\" sizes=\"auto, (max-width: 704px) 100vw, 704px\" \/><\/p>\n<p><span style=\"font-weight: 400\">Let\u2019s call the principal P and the rate of interest k. In the case of simple interest, the amount, A(t), you have in the bank at any time, t, is given by the following equation: A(t) = P + (kP \u00d7 t). So, for example, when the rate of interest is 6% the amount you have after 5 years, A(5), can be calculated as A(5) = 100 + (0.06 \u00d7 100 \u00d7 5) = \u00a3130 (as in Table 1).<\/span><\/p>\n<p><span style=\"font-weight: 400\">However, when compound interest is calculated on the <\/span><i><span style=\"font-weight: 400\">cumulative<\/span><\/i><span style=\"font-weight: 400\"> amount, the equation becomes more complicated. In the first year the calculation is the same as for simple interest:<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400\">Year 1: A(1) = P + kP. In this case that\u2019s 100 + (0.06 \u00d7 100) = \u00a3106.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">But in the second year P becomes the new amount P + kP and so the equation becomes:<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400\">Year 2: A(2) = (P + kP) + k(P + kP). That\u2019s 106 + (0.06 \u00d7 106) = \u00a3112.36.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">And in the third year we have to use this new amount in the calculation:<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400\">Year 3: A(3) = [(P + kP) + k(P + kP)] + k[(P + kP) + k(P + kP)] = \u00a3119.10.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">We can keep on doing this and writing longer and longer equations. However, there is a simpler way of expressing it. If P increases by 6% each year it will grow at a rate of 1.06<sup>t<\/sup>, as shown in Table 1. After 20 years you will have \u00a3320.71 in the bank (100 \u00d7 1.06<sup>20<\/sup>). Before the days of electronic calculators you would have done this calculation using <\/span><a href=\"https:\/\/blogs.bmj.com\/bmj\/2020\/04\/17\/jeffrey-aronson-when-i-use-a-word-logarithmic-exponents\"><span style=\"font-weight: 400\">logarithms<\/span><\/a><span style=\"font-weight: 400\">.<\/span><\/p>\n<p><span style=\"font-weight: 400\">Now you negotiate a better deal with the bank. Why not get 3% every 6 months? Or 0.5% every month? Or even 6\/365 = 0.000164% every day? What would you have after 20 years?<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400\">Year 20: A(20) = 100 \u00d7 1.000164<sup>20 \u00d7365<\/sup> = \u00a3325.67.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">You might be surprised that this new amount, calculated day by day, isn\u2019t much more than it was when calculated year by year, with the same total interest. If you did it hour by hour the answer would be \u00a3332.01, not much more again.<\/span><\/p>\n<p><span style=\"font-weight: 400\">What we did was to calculate P(1 + k)<sup>t<\/sup>. Dividing k, the interest rate, by the number of days or hours or whatever, we calculated P(1 + k\/n)<sup>nt<\/sup>, where n is the number of days in a year (365) or the number of hours (8760). For example, 100(1 + 0.06\/8760)<sup>8760\u00d720<\/sup> = \u00a3332.01.<\/span><\/p>\n<p><span style=\"font-weight: 400\">Thus, the basic growth rate is given by (1 + 1\/n)<sup>n<\/sup>. When we increase n from one to very large numbers a surprising thing happens. The result doesn\u2019t go on increasing, or not by very much (Table 2). It reaches a limit of about 2.71828. And that limit has been given a symbol, the letter <\/span><i><span style=\"font-weight: 400\">e<\/span><\/i><span style=\"font-weight: 400\">. Enter a 1 into your calculator and press the key marked e<sup>x<\/sup>. You\u2019ll get 2.71828\u2026.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-47316 alignnone\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_integer_4.jpg\" alt=\"\" width=\"432\" height=\"343\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_integer_4.jpg 432w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_integer_4-300x238.jpg 300w\" sizes=\"auto, (max-width: 432px) 100vw, 432px\" \/><\/p>\n<p><span style=\"font-weight: 400\"><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-47313\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_again.jpg\" alt=\"\" width=\"241\" height=\"366\" \/>This number, which the Swiss mathematician Leonhard Euler called <\/span><i><span style=\"font-weight: 400\">e<\/span><\/i><span style=\"font-weight: 400\">, is the constant of natural growth. And instead of basing logarithms on 10<sup>x<\/sup>, where x is the logarithm of 10<sup>x<\/sup>, you can base it on e<sup>x<\/sup>, where x is the logarithm of e<sup>x<\/sup>, the natural logarithm (ln, as it is commonly called). So our principal increases by a factor of e<sup>kt<\/sup>, the expression of exponential change.<\/span><\/p>\n<p><span style=\"font-weight: 400\">What has all this to do with R<sub>0<\/sub> and R<sub>e<\/sub> the reproduction numbers of a virus? Well if R is 2, each person infects two people; so the first 100 people to be infected will infect 200 other people; those 200 infect another 400; those 400 infect 800. And so it goes, doubling all the while. That\u2019s an exponential increase. Even if R is only just above 1, say 1.1, the number of people will increase exponentially, but not as fast as when it was 2 (Figure 1). Suppose R falls to less than 1 during an epidemic, say 0.5. Then the first 100 people will infect another 50; they will infect 25, and so on; that\u2019s an exponential decrease (Figure 2) and the epidemic will die out.<\/span><\/p>\n<p><span style=\"font-weight: 400\">Remember: the value of your investment can go down as well as up.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-47310 alignnone\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_increase4.png\" alt=\"\" width=\"598\" height=\"1070\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_increase4.png 598w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_increase4-168x300.png 168w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_increase4-572x1024.png 572w\" sizes=\"auto, (max-width: 598px) 100vw, 598px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-47311 alignnone\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_increase5.jpg\" alt=\"\" width=\"635\" height=\"445\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_increase5.jpg 635w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_increase5-300x210.jpg 300w\" sizes=\"auto, (max-width: 635px) 100vw, 635px\" \/><\/p>\n<p style=\"text-align: left\"><span style=\"font-weight: 400\"><em><strong>Jeffrey Aronson<\/strong>\u00a0is a clinical pharmacologist, working in the Centre for Evidence Based Medicine in Oxford&#8217;s Nuffield Department of Primary Care Health Sciences. He is also president emeritus of the British Pharmacological Society.<\/em><\/span><\/p>\n<p style=\"text-align: left\"><strong>Competing interests:<\/strong> None declared.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-47816\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_271.png\" alt=\"\" width=\"702\" height=\"1590\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_271.png 702w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_271-132x300.png 132w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_271-452x1024.png 452w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_271-678x1536.png 678w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_271-640x1450.png 640w\" sizes=\"auto, (max-width: 702px) 100vw, 702px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-47323 alignleft\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_integeragain.png\" alt=\"\" width=\"694\" height=\"1106\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_integeragain.png 694w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_integeragain-188x300.png 188w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_integeragain-643x1024.png 643w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/04\/aronson_exponential_integeragain-640x1020.png 640w\" sizes=\"auto, (max-width: 694px) 100vw, 694px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Last week I suggested that there is widespread misunderstanding about the meaning of \u201cexponential\u201d in mathematics. Colloquially, it means a huge increase, which it need not be. And when events [&#8230;]<\/p>\n<p><a class=\"btn btn-secondary understrap-read-more-link\" href=\"https:\/\/blogs.bmj.com\/bmj\/2020\/04\/24\/jeffrey-aronson-when-i-use-a-word-exponential-finances-increasing-decreasing\/\">More&#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":38359,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5762],"tags":[],"class_list":["post-47302","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-jeff-aronsons-words"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Jeffrey Aronson: When I Use a Word . . . 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Exponential finances\u2014increasing\/decreasing - The BMJ","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/blogs.bmj.com\/bmj\/2020\/04\/24\/jeffrey-aronson-when-i-use-a-word-exponential-finances-increasing-decreasing\/","og_locale":"en_US","og_type":"article","og_title":"Jeffrey Aronson: When I Use a Word . . . Exponential finances\u2014increasing\/decreasing - The BMJ","og_description":"Last week I suggested that there is widespread misunderstanding about the meaning of \u201cexponential\u201d in mathematics. Colloquially, it means a huge increase, which it need not be. And when events [...]More...","og_url":"https:\/\/blogs.bmj.com\/bmj\/2020\/04\/24\/jeffrey-aronson-when-i-use-a-word-exponential-finances-increasing-decreasing\/","og_site_name":"The BMJ","article_publisher":"https:\/\/www.facebook.com\/bmjdotcom\/","article_published_time":"2020-04-24T18:42:19+00:00","article_modified_time":"2020-06-17T21:48:45+00:00","og_image":[{"width":540,"height":350,"url":"https:\/\/blogs.bmj.com\/bmj\/files\/2017\/02\/Jeffrey-Aronson.jpg","type":"image\/jpeg"}],"author":"BMJ","twitter_card":"summary_large_image","twitter_creator":"@bmj_latest","twitter_site":"@bmj_latest","twitter_misc":{"Written by":"BMJ","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/blogs.bmj.com\/bmj\/2020\/04\/24\/jeffrey-aronson-when-i-use-a-word-exponential-finances-increasing-decreasing\/#article","isPartOf":{"@id":"https:\/\/blogs.bmj.com\/bmj\/2020\/04\/24\/jeffrey-aronson-when-i-use-a-word-exponential-finances-increasing-decreasing\/"},"author":{"name":"BMJ","@id":"https:\/\/blogs.bmj.com\/bmj\/#\/schema\/person\/ba3da426ed20e8f1d933ca367d8216fe"},"headline":"Jeffrey Aronson: When I Use a Word . . . 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