{"id":33176,"date":"2025-01-04T13:00:12","date_gmt":"2025-01-04T13:00:12","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=33176"},"modified":"2025-06-02T00:58:21","modified_gmt":"2025-06-02T00:58:21","slug":"exemplum-simplex-mercatus-notiones-et-praesumptiones-elementariae","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/exemplum-simplex-mercatus-notiones-et-praesumptiones-elementariae\/","title":{"rendered":"Exemplum Simplex Mercatus: Notiones et Praesumptiones Elementariae"},"content":{"rendered":"<style>\np {\n    text-align: justify;\n}\n<\/style>\n<p><center><\/p>\n<h1><strong>Exemplar Simple Mercatus:<\/strong><br \/>Notiones et Praesumptiones Elementariae<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>Summarium:<\/strong><br \/>\nHaec lectio \u00abExemplar Simple Mercatus\u00bb introducit, rationem quae discendi notiones fundamentales de collocatione faciliorem reddit, coniungens bona sine periculo (vincula, cum reditu noto) et bona periculosa (actiones, cum reditu incerto). Videbimus quomodo haec bona in portifolio coniungi possint quod, recte gestum, reditus superiores quam usurae argentariae praebet, inter crescendum et securitatem aequilibrando. Praeterea, discemus quomodo reditum horum bonorum computare possimus in linea temporis simplicificata (praesens et futurum) et praesumptiones mercatus sicut aleatorietas pretii et solvendo considerare ad decisiones prudentes de collocatione et periculo capiendas.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Propositum Discendi:<\/strong><br \/>\nPost hanc lectionem, discipulus poterit<\/p>\n<ul>\n<li><strong>Agnoscere<\/strong> proprietates Exempli Simplicis Mercatus, bonorum periculorum et sine periculo in decisionibus collocandis.<\/li>\n<li><strong>Intelligere<\/strong> differentiam inter bona periculosa et sine periculo, intellegens quomodo unumquodque reditum et periculum in portifolio afficiat.<\/li>\n<li><strong>Applicare<\/strong> formulas ad reditum collocationis computandum in bonis periculosis et sine periculo, pretia initialia et finalia utens.<\/li>\n<li><strong>Analyzare<\/strong> constructionem portifoliorum quae bona periculosa et sine periculo coniungunt ad reditum optimandum periculo recte gesto in exemplo simplici mercatus.<\/li>\n<li><strong>Aestimare<\/strong> effectum condicionum mercatus in valore et reditu portifolii, variationes in pretio bonorum considerans.<\/li>\n<li><strong>Applicare<\/strong> probabilitatem ad reditum exspectatum computandum in condicionibus mercatus incertis, eventus financieros determinans.<\/li>\n<\/ul>\n<p style=\"text-align:center;\"><strong><u>INDEX CONTENTORUM<\/u><\/strong><br \/>\n<a href=\"#1\"><strong>Introductio<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Definitiones et Praesumptiones Theoricae<\/strong><\/a><br \/>\n<a href=\"#3\">Bona Periculosa et Bona Sine Periculo<\/a><br \/>\n<a href=\"#4\">Scala Temporis in Exemplo<\/a><br \/>\n<a href=\"#5\">Redditus Collocationis<\/a><br \/>\n<a href=\"#6\">Constructio et Aestimatio Portifolii<\/a><br \/>\n<a href=\"#7\">Praesumptiones Fundamentales Exempli<\/a><br \/>\n<a href=\"#8\"><strong>Problemata Soluta<\/strong><\/a><br \/>\n<a href=\"#9\"><strong>Exercitia Proposita<\/strong><\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/MK9owXS381U?si=7KmgnPbga5fihMVy\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><br \/>\n<center><\/p>\n<h2>Introductio<\/h2>\n<p><\/center><\/p>\n<p>Imaginare te modo praemium in opere tuo accepisse et quantitatem notabilem in argentaria servavisse. Attamen, cum spectas ad hodiernas usuras et ad effectum inflationis, sollicitus es ne potentia emendi pecuniae tuae per tempus minuatur. Vis ut pecunia tua non tantum conservetur, sed etiam crescat.<\/p>\n<p>Audivisti collocationem in actionibus et vinculis bonum modum esse ad pecuniam augendam. Scis aliqua bona, ut vincula, esse tuta, dum alia, ut actiones, reditus maiores offerunt sed maiorem periculum secum ferunt. Te interrogares utrum utrumque genus bonorum coniungere posses in consilio quod tibi plura quam usurae argentariae lucrari sineret, sine periculo nimio assumendo.<\/p>\n<p>Investigare statuis et rationem invenis nomine <strong>\u00abExemplum Simplex Mercatus\u00bb<\/strong>, quae discendi notiones fundamentales bonorum periculorum et sine periculo, redituum et portifoliorum constructionis faciliorem reddit. Hoc exemplar inceptivis aptum est, cum analysin financialem simplicet ad duos temporis punctos spectando: praesens et momentum futurum.<\/p>\n<p>Hac motivatione, plura discere vis quomodo reditum collocationis computare possis et quomodo portifolium aedificare quod reditus tuos maxime augeat. Dum procedimus, has notiones in profundum explorabimus ut decisiones prudentes capere possis et res tuas personales melius administrare.<\/p>\n<p>Nunc paratus es, immergamur in scientiam theoricam quam debes cognoscere ut hoc exemplar mercatus intellegas et ad tuas proprias decisiones collocandas applices.<\/p>\n<p><center><a name=\"2\"><\/a><\/p>\n<h2>Definitiones et Praesumptiones Theoricae<\/h2>\n<p><\/center><br \/>\n<a name=\"3\"><\/a><\/p>\n<h3>Bona Periculosa et Bona Sine Periculo<\/h3>\n<p>Ut exemplum simplex mercatus intellegere incipiamus, necesse est notiones <strong>bonorum periculorum<\/strong> et <strong>bonorum sine periculo<\/strong> cognoscere. Haec duo genera bonorum fundamentum constituunt plurimarum rationum collocationis.<\/p>\n<p><strong>Bonum sine periculo<\/strong> est genus collocationis cuius reditus notus est et certus. Exemplum classicum boni sine periculo est <em>vinculum<\/em> a gubernatione vel institutione nummaria stabili emissum, quod solutionem certam usurarum in fine temporis praestat. Haec vincula videri possunt tamquam depositiones in ratione argentaria vel instrumenta debiti quae reditum praedicibilem et stabilem offerunt.<\/p>\n<p>Contra, <strong>bonum periculosum<\/strong> est illud cuius pretium futurum incertum est et variari potest, sive in maius sive in minus. Commune exemplum boni periculosi sunt <em>actiones<\/em> societatibus quae in foro publico negotiantur. Actiones variabiles esse possunt et pretium earum a multis causis pendet, quod facit ut earum valor futurus praevideri non possit.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Scala Temporis in Exemplo<\/h3>\n<p>In exemplo simplici mercatus, analysin ad duos tantum temporis instantes restringimus: praesentem, quem <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span> appellamus, et momentum futurum, ut unum annum postea, quod <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> nominamus. Haec ratio simplicificata sinit ut mutationes in pretio bonorum sine nimia complexitate considerentur.<\/p>\n<p>Hoc exemplar duorum temporis punctorum inceptoribus valde utile est, cum intellectum praebeat quomodo pretia bonorum per tempus mutantur et quomodo hae mutationes valorem portifolii afficiant.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Redditus Collocationis<\/h3>\n<p>Redditus est mensura quanti valoris collocatio per tempus acquisiverit vel amiserit. Pro genere boni, computatio reditus incerta vel determinata esse potest.<\/p>\n<p>Pro bono periculoso, ut actio, reditus incertus est et computatur utens pretio initiali et pretio futuro boni. Si pretium actionis in tempore <span class=\"katex-eq\" data-katex-display=\"false\"> t <\/span> repraesentatur per <span class=\"katex-eq\" data-katex-display=\"false\"> S(t) <\/span>, reditus actionis inter <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> sic computatur:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S = \\dfrac{S(1) - S(0)}{S(0)} <\/span>\n<p>Hic reditus, per <span class=\"katex-eq\" data-katex-display=\"false\"> K_S <\/span> repraesentatus, est fractio valoris initialis actionis et potest esse positivus (si pretium actionis crevit), negativus (si cecidit) vel nullus (si pretium non mutatum est).<\/p>\n<p>Pro bono sine periculo, ut vinculum, reditus certo et antea notus est. Si pretium vinculi in tempore <span class=\"katex-eq\" data-katex-display=\"false\"> t <\/span> repraesentatur per <span class=\"katex-eq\" data-katex-display=\"false\"> A(t) <\/span>, reditus huius vinculi inter <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> sic computatur:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_A = \\dfrac{A(1) - A(0)}{A(0)} <\/span>\n<p>Hic reditus, <span class=\"katex-eq\" data-katex-display=\"false\"> K_A <\/span>, fixus est et ab emittente vinculi praestatur. Differentia principalis inter <span class=\"katex-eq\" data-katex-display=\"false\"> K_S <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> K_A <\/span> est certitudo: dum reditus actionis incertus est, reditus vinculi fixus et notus est.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Constructio et Aestimatio Portifolii<\/h3>\n<p>Nunc cum notionem reditus intellegimus, possumus bona periculosa et sine periculo coniungere ad formandum <strong>portifolium<\/strong>. Fingamus te velle portifolium constituere quod contineat <span class=\"katex-eq\" data-katex-display=\"false\"> x <\/span> actiones et <span class=\"katex-eq\" data-katex-display=\"false\"> y <\/span> vincula. Valor totalis portifolii in quolibet tempore <span class=\"katex-eq\" data-katex-display=\"false\"> t <\/span> est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(t) = xS(t) + yA(t) <\/span>\n<p>Hic, <span class=\"katex-eq\" data-katex-display=\"false\"> V(t) <\/span> repraesentat valorem totius portifolii, quod est summa valoris actionum (<span class=\"katex-eq\" data-katex-display=\"false\"> xS(t) <\/span>) et valoris vinculorum (<span class=\"katex-eq\" data-katex-display=\"false\"> yA(t) <\/span>).<\/p>\n<p>In tempore initiali (<span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span>), valor portifolii notus est si numerum actionum et vinculorum necnon pretia eorum praesentia novimus. Attamen, in tempore <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span>, valor actionum variari potest, quod facit ut valor portifolii incertus sit.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Praesumptiones Fundamentales Exempli<\/h3>\n<p>Ad exemplum simpliciorem reddendum, quasdam praesumptiones fundamentales statuimus quae nobis computationes et analysin faciliores reddunt:<\/p>\n<ul>\n<li><strong>Praesumptio Aleatorietatis:<\/strong> Pretium actionis in futuro (<span class=\"katex-eq\" data-katex-display=\"false\"> S(1) <\/span>) est <em>variabilis aleatoria<\/em>, quod significat eam varios valores accipere posse secundum factores mercatus praevideri non possibiles.<\/li>\n<li><strong>Positivitas Pretii:<\/strong> Omnia pretia actionum et vinculorum sunt stricte positiva, id est, <span class=\"katex-eq\" data-katex-display=\"false\"> S(t) &gt; 0 <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> A(t) &gt; 0 <\/span> pro <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0, 1 <\/span>. Haec praesumptio efficit ut bona valores reales habeant.<\/li>\n<li><strong>Divisibilitas, Liquiditas:<\/strong> Bona emi possunt in partibus fractis, quod sinit collocatores portifolia sua sine restrictionibus accommodare. Praeterea, praesumitur bona emptioni vel venditioni in quacumque quantitate subici posse.<\/li>\n<li><strong>Solventia:<\/strong> Divitiae totales collocatoris semper non-negativae esse debent, id est, <span class=\"katex-eq\" data-katex-display=\"false\"> V(t) \\geq 0 <\/span>. Hoc significat neminem perdere posse plus quam quod collocatum est.<\/li>\n<li><strong>Pretia Discreta:<\/strong> Pretium futurum <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) <\/span> actionis est variabilis aleatoria quae tantum finitam multitudinem valorum accipere potest. Hoc analysin et exemplificationem mercatus faciliorem reddit.<\/li>\n<\/ul>\n<p>His praesumptionibus positis, exemplar tractabilius fit, quod sinit nos reditus et valores portifoliorum sine difficultatibus additis examinare.<\/p>\n<p>Hactenus tractavimus notiones theoricas fundamentales ad exemplum simplex mercatus intellegendum. In sectione sequenti, hos intellectus adhibebimus in exercitiis practicis ut ostendamus quomodo valorem et reditum portifolii in variis casibus computare possimus.<\/p>\n<p><center><a name=\"8\"><\/a><\/p>\n<h2>Problemata Soluta<\/h2>\n<p><\/center><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/BdcylGfSgtA?si=QwVbLqRiAaULJ6v1\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<h3>Exercitatio 1: Computatio Redditus in Vinculis (Bonum Sine Periculo)<\/h3>\n<p>Fingamus te vinculum habere cuius pretium tempore initiali est <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> dollaria. In fine anni, valor vinculi crescit ad <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 110 <\/span> dollaria.<\/p>\n<p><strong>Quaestio:<\/strong> Quis est reditus huius collocationis in vinculis?<\/p>\n<p><strong>Solutio:<\/strong> Quia vinculum est bonum sine periculo, reditus certus est et computari potest formula reditus bonorum sine periculo utens:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_A = \\dfrac{A(1) - A(0)}{A(0)} <\/span>\n<p>Valores substituentes:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_A = \\dfrac{110 - 100}{100} = \\dfrac{10}{100} = 0.10 <\/span>\n<p>Redditus est 10%.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 2: Computatio Redditus in Actionibus (Bonum Periculosum)<\/h3>\n<p>Fingamus te actionem emere pretio <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span> dollaria. In fine anni, pretium actionis variari potest. Duo sunt eventus possibiles:<\/p>\n<ul>\n<li>Si mercatus crescit, pretium actionis erit <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 52 <\/span> dollaria, cum probabilitate <span class=\"katex-eq\" data-katex-display=\"false\"> p <\/span>.<\/li>\n<li>Si mercatus decrescit, pretium actionis erit <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 48 <\/span> dollaria, cum probabilitate <span class=\"katex-eq\" data-katex-display=\"false\"> 1 - p <\/span>.<\/li>\n<\/ul>\n<p><strong>Quaestio:<\/strong>In exemplo simplici mercatus, quis est reditus huius collocationis in unoquoque eventu?<\/p>\n<p><strong>Solutio:<\/strong> Redditus actionis, cum sit bonum periculosum, incertus est et computatur utens formula reditus bonorum periculorum:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S = \\dfrac{S(1) - S(0)}{S(0)} <\/span>\n<p>Computamus reditum in unoquoque casu:<\/p>\n<ul>\n<li><strong>Si pretium crescit ad 52 dollaria:<\/strong><\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S = \\dfrac{52 - 50}{50} = \\dfrac{2}{50} = 0.04 <\/span>\n<p>Redditus in hoc casu est 4%.<\/p>\n<li><strong>Si pretium decrescit ad 48 dollaria:<\/strong><\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S = \\dfrac{48 - 50}{50} = \\dfrac{-2}{50} = -0.04 <\/span>\n<p>Redditus in hoc casu est -4%.<\/p>\n<\/ul>\n<p>Ergo, secundum mercatus condicionem, reditus potest esse positivus (4%) vel negativus (-4%).<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 3: Valor Portifolii cum Bonis Periculosis et Sine Periculo<\/h3>\n<p>Supponas te constituere portifolium quod continet 20 actiones et 10 vincula. Novimus haec:<\/p>\n<ul>\n<li>Pretium unius actionis initio est <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span> dollaria.<\/li>\n<li>Pretium unius vinculi initio est <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> dollaria.<\/li>\n<\/ul>\n<p><strong>Quaestio:<\/strong> Quis est valor huius portifolii in tempore initiali <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span>?<\/p>\n<p><strong>Solutio:<\/strong> Valor portifolii in tempore <span class=\"katex-eq\" data-katex-display=\"false\"> t <\/span> computatur ut:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(t) = xS(t) + yA(t) <\/span>\n<p>Ubi <span class=\"katex-eq\" data-katex-display=\"false\"> x <\/span> est numerus actionum et <span class=\"katex-eq\" data-katex-display=\"false\"> y <\/span> est numerus vinculorum.<\/p>\n<p>Valores substituentes:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(0) = (20)(50) + (10)(100) <\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(0) = 1000 + 1000 = 2000 <\/span>\n<p>Valor portifolii in tempore initiali <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span> est 2000 dollaria.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 4: Computatio Redditus in Portifolio Mixto<\/h3>\n<p>Supponas pretia bonorum in portifolio Exercitationis 3 tempore <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> ita variari:<\/p>\n<ul>\n<li>Si mercatus crescit, pretium actionis erit <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 52 <\/span> et vinculi erit <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 110 <\/span>.<\/li>\n<li>Si mercatus decrescit, pretium actionis erit <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 48 <\/span> et vinculi erit <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 110 <\/span>.<\/li>\n<\/ul>\n<p><strong>Quaestio:<\/strong>In exemplo simplici mercatus, quis est valor et reditus portifolii in unoquoque casu?<\/p>\n<p><strong>Solutio:<\/strong><\/p>\n<p><strong>Casus 1: Mercatus crescit<\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = (20)(52) + (10)(110) <\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = 1040 + 1100 = 2140 <\/span>\n<p>Valor portifolii in hoc casu est 2140 dollaria.<\/p>\n<p>Redditus portifolii est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_V = \\dfrac{V(1) - V(0)}{V(0)} = \\dfrac{2140 - 2000}{2000} = \\dfrac{140}{2000} = 0.07 <\/span>\n<p>Redditus est 7%.<\/p>\n<p><strong>Casus 2: Mercatus decrescit<\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = (20)(48) + (10)(110) <\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = 960 + 1100 = 2060 <\/span>\n<p>Valor portifolii in hoc casu est 2060 dollaria.<\/p>\n<p>Redditus portifolii est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_V = \\dfrac{V(1) - V(0)}{V(0)} = \\dfrac{2060 - 2000}{2000} = \\dfrac{60}{2000} = 0.03 <\/span>\n<p>Redditus est 3%.<\/p>\n<p>Summatim, reditus portifolii a mercatus cursu pendet. Si mercatus crescit, reditus est 7%; si mercatus decrescit, reditus est 3%.<\/p>\n<h3>Exercitatio 5: Computatio Redditus Ponderati in Portifolio Mixto<\/h3>\n<p>Fingamus te constituere portifolium mixtum cum hac distributione initiali:<\/p>\n<ul>\n<li>50% collocationis tuae est in vinculorum sine periculo, cum pretio initiali <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> et pretio in fine anni <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 105 <\/span>.<\/li>\n<li>50% collocationis tuae est in actionibus periculosis, cum pretio initiali <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span>. Pretium actionis in <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> erit <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 55 <\/span> si mercatus crescit (probabilitas 0.7) vel <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 45 <\/span> si mercatus decrescit (probabilitas 0.3).<\/li>\n<\/ul>\n<p><strong>Quaestio:<\/strong> Qui est reditus exspectatus totius portifolii considerata probabilitate ascensus vel descensus mercatus?<\/p>\n<p><strong>Solutio:<\/strong><\/p>\n<p>1. Primum, computamus reditum cuiusque generis boni:<\/p>\n<ul>\n<li>Pro vinculorum sine periculo:<\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_A = \\dfrac{A(1) - A(0)}{A(0)} = \\dfrac{105 - 100}{100} = 0.05 <\/span> (5%)<\/p>\n<li>Pro actionibus in unoquoque casu:<\/li>\n<ul>\n<li>Si mercatus crescit:<\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S^{\\text{up}} = \\dfrac{55 - 50}{50} = 0.10 <\/span> (10%)<\/p>\n<li>Si mercatus decrescit:<\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S^{\\text{down}} = \\dfrac{45 - 50}{50} = -0.10 <\/span> (-10%)<\/p>\n<\/ul>\n<\/ul>\n<p>2. Computamus reditum exspectatum actionum secundum probabilitates:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\text{Redditus exspectatus actionum} = (0.7 \\times 0.10) + (0.3 \\times -0.10)  = 0.04 <\/span> (4%)<\/p>\n<p>3. Nunc computamus reditum ponderatum portifolii, dato quod 50% est in vinculorum et 50% in actionibus:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_{\\text{portafolio}} = (0.5 \\times 0.05) + (0.5 \\times 0.04) = 0.045 <\/span> (4.5%)<\/p>\n<p><strong>Responsum:<\/strong> Redditus exspectatus totius portifolii est 4.5%.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 6: Aestimatio Periculi et Redditus Portifolii cum Venditione Brevi in Exemplo Simplici Mercatus<\/h3>\n<p>Fingamus te habere consilium quo 2000 dollaria in vinculis sine periculo collocas, cum reditu 3% in fine anni certo. Praeterea, 1000 dollaria mutuas accipis ut actiones breviter vendas, sperans pretium earum decrescere posse ut lucrum obtineas. Nunc pretium actionum est <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span> dollaria pro singula actione, et in fine anni pretium esse potest:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 40 <\/span> dollaria si mercatus decrescit (probabilitas 0.6)<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 60 <\/span> dollaria si mercatus crescit (probabilitas 0.4)<\/li>\n<\/ul>\n<p><strong>Quaestio:<\/strong> Quis est reditus exspectatus portifolii et quod est periculum venditionis brevis, mensum per deviationem standardam redituum?<\/p>\n<p><strong>Solutio:<\/strong><\/p>\n<h4>Computatio Redditus Exspectati<\/h4>\n<p>Primum, computamus reditum vinculorum sine periculo:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_A = 0.03 <\/span> (3%)<\/p>\n<p>Pro venditione brevi, computamus lucrum vel damnum in unoquoque casu:<\/p>\n<ul>\n<li><strong>Si mercatus decrescit:<\/strong><\/li>\n<p>Venditio brevis facta est pretio 50 dollariorum pro actione, et pretium in fine anni est 40 dollaria. Lucrum pro singula actione est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 50 - 40 = 10 <\/span> dollaria<\/p>\n<p>Si 1000 dollaria mutuo sumpseris, hoc aequivalet venditioni brevi <span class=\"katex-eq\" data-katex-display=\"false\"> \\dfrac{1000}{50} = 20 <\/span> actionum. Lucrum totale est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 20 \\times 10 = 200 <\/span> dollaria<\/p>\n<li><strong>Si mercatus crescit:<\/strong><\/li>\n<p>Venditio brevis facta est pretio 50 dollariorum, et pretium in fine anni est 60 dollaria. Damnum per actionem est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 50 - 60 = -10 <\/span> dollaria<\/p>\n<p>Pro 20 actionibus, damnum totale est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 20 \\times -10 = -200 <\/span> dollaria<\/p>\n<\/ul>\n<p>Computamus reditum exspectatum venditionis brevis:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\text{Redditus exspectatus venditionis brevis} = (0.6 \\times 200) + (0.4 \\times -200) = 120 - 80 = 40 <\/span> dollaria<\/p>\n<h4>Computatio Variantiarum et Deviationis Standardae ad Mensurandum Periculum<\/h4>\n<p>Nunc, ad periculum mensurandum in exemplo simplici mercatus, computamus <strong>variantiam redituum<\/strong> venditionis brevis. Formula variantiae, secundum eventus possibiles et probabilitates, est:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\text{Variantia} = (0.6) \\times (200 - 40)^2 + (0.4) \\times (-200 - 40)^2 = 38400<\/span>\n<p>Postremo, computamus deviationem standardam ut radicem quadratam variantiae:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\text{Dev. standarda} = \\sqrt{38400} \\approx 196 <\/span>\n<h4>Interpretatio Deviationis Standardae in Contextu Distributionis Normae<\/h4>\n<p>Dev. standarda est mensura dispersionis valorum circa mediam. In contextu distributionis normae, dev. standarda magni momenti est ad periculi intellectum et ad probabilitatem quorundam redituum aestimandam.<\/p>\n<h4>Relatio inter Dev. Standardam et Distributionem Normae<\/h4>\n<p><strong>Distributio normalis<\/strong> (vel campana Gaussiana) est distributio probabilitatis symmetrica circa suam mediam, in qua plerique valores prope mediam congregantur. Multi reditus financieros, sicut reditus portifoliorum bene diversificatorum, saepe ad distributionem normalem appropinquant.<\/p>\n<p>In distributione normali:<\/p>\n<ul>\n<li>Circiter 68% valorum intra unam deviationem standardam ab media continentur.<\/li>\n<li>Circiter 95% valorum intra duas dev. standardas ab media continentur.<\/li>\n<li>Circiter 99.7% valorum intra tres dev. standardas ab media continentur.<\/li>\n<\/ul>\n<h4>Interpretatio in Contextu Periculi et Redituum Financialium<\/h4>\n<p>Si supponimus reditus venditionis brevis distributioni normali appropinquare, dev. standarda 196 dollariorum nobis permittit aestimare probabilitatem redituum circa mediam exspectatam. Exempli gratia:<\/p>\n<ul>\n<li>Cum dev. standarda 196 dollariorum et reditu exspectato 40 dollariorum, dicere possumus 68% eventuum in intervallo <span class=\"katex-eq\" data-katex-display=\"false\"> 40 \\pm 196 <\/span> dollariorum esse (inter -156 et 236 dollaria).<\/li>\n<li>Ad <strong>pericula extrema<\/strong> aestimanda, eventus ad duas vel tres dev. standardas a media considerare possumus. In distributione normali, tales eventus minus probabiles sunt (5% vel minus), sed magnum effectum in portifolio habere possunt.<\/li>\n<\/ul>\n<h4>Limitationes in Contextu Venditionis Brevis<\/h4>\n<p>Notandum est in venditione brevi reditus non semper distributionem perfecte normalem sequi propter <strong>asymmetriam<\/strong> consilii: pretium actionis in infinitum augeri potest, damna infinita generans, sed sub zero cadere non potest. Hoc in distributione inclinationem introducit, faciens ut pericula extremorum damnum maiore probabilitate eveniant quam distributio normalis suggereret.<\/p>\n<p><center><a name=\"9\"><\/a><\/p>\n<h2>Exercitia Proposita<\/h2>\n<p><\/center><\/p>\n<h3>Exercitatio 1: Computatio Redditus in Vinculo<\/h3>\n<p>Fingamus te emere vinculum sine periculo pretio initiali <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 200 <\/span> dollaria, et in fine anni pretium vinculi crescat ad <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 220 <\/span> dollaria.<\/p>\n<p><strong>Quaestio:<\/strong> Quis est reditus huius collocationis in vinculo?<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 2: Redditus in Actione Periculosa cum Casibus Probabilisticis<\/h3>\n<p>Emis actionem pretio initiali <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 100 <\/span> dollaria. In fine anni, pretium actionis esse potest <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 110 <\/span> cum probabilitate 0.5, vel <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 90 <\/span> cum probabilitate 0.5.<\/p>\n<p><strong>Quaestio:<\/strong> Computa reditum in unoquoque casu et reditum exspectatum huius collocationis in actione.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 3: Valor Portifolii Mixti<\/h3>\n<p>Constituis portifolium cum 15 actionibus et 5 vinculorum. Initio, pretium unius actionis est <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 30 <\/span> dollaria, et pretium unius vinculi est <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> dollaria.<\/p>\n<p><strong>Quaestio:<\/strong> Quis est valor totalis portifolii tui in tempore <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span>?<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 4: Redditus Portifolii in Diversis Cursibus Mercatus<\/h3>\n<p>Pro portifolio exercitationis praecedentis, in fine anni pretium actionis esse potest <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 35 <\/span> si mercatus crescit, vel <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 25 <\/span> si mercatus decrescit. Vinculum sine periculo habebit pretium <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 105 <\/span> in utroque casu.<\/p>\n<p><strong>Quaestio:<\/strong> Computa valorem et reditum portifolii in unoquoque casu mercatus.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 5: Effectus Mutationum Pretii Actionis in Portifolio<\/h3>\n<p>Fingamus te habere portifolium compositum ex 10 vinculorum et 40 actionibus. Pretium unius vinculi est <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 90 <\/span> dollaria, et unius actionis <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 20 <\/span> dollaria initio. In fine anni, pretium actionis crescit ad <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 30 <\/span> et pretium vinculi ad <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 95 <\/span>.<\/p>\n<p><strong>Quaestio:<\/strong> Computa valorem initialem et finalem portifolii, et determina reditum portifolii.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 6: Computatio Redditus Ponderati in Portifolio Diversificato in Exemplo Simplici Mercatus<\/h3>\n<p>Investis 60% portifolii tui in vinculis sine periculo et 40% in actionibus. Pretium initiale vinculorum est <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 200 <\/span> dollaria, et pretium finale <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 210 <\/span> dollaria. Pretium initiale actionum est <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span> et pretium finale dependet ex mercatus cursu: <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 55 <\/span> cum probabilitate 0.6, vel <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 45 <\/span> cum probabilitate 0.4.<\/p>\n<p><strong>Quaestio:<\/strong> Computa reditum exspectatum totalem portifolii.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 7: Aestimatio Periculi per Deviationem Standardam<\/h3>\n<p>In consilio venditionis brevis, 500 dollaria mutuo accipis ut actiones breviter vendas quae sunt pretio initiali <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 25 <\/span> dollaria. In fine anni, pretium actionum esse potest <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 20 <\/span> (probabilitas 0.7) vel <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 30 <\/span> (probabilitas 0.3).<\/p>\n<p><strong>Quaestio:<\/strong> Computa reditum exspectatum et deviationem standardam huius collocationis in venditione brevi.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 8: Constructio Portifolii quod Redditum Specificum Praestat<\/h3>\n<p>2000 dollaria habes et vis constituere portifolium cum vinculorum sine periculo (<span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> dollaria, cum reditu 5%) et actionum (<span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span> dollaria) quorum reditus exspectatus est 8%.<\/p>\n<p><strong>Quaestio:<\/strong> Quot vincula et actiones emere debes ut reditus exspectatus totalis portifolii sit 6%?<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 9: Analysis Effectus Diversificationis in Portifolio<\/h3>\n<p>Investis 3000 dollaria in portifolio composito ex vinculorum et actionibus. Dimidium collocationis tuae est in vinculorum (<span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 150 <\/span> dollaria, cum reditu garantito 4%) et reliquum dimidium in actionibus (<span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 75 <\/span> dollaria), quarum pretium in <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> esse potest 90 (probabilitas 0.5) vel 60 (probabilitas 0.5).<\/p>\n<p><strong>Quaestio:<\/strong> Computa reditum exspectatum et deviationem standardam portifolii.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercitatio 10: Effectus Mutationis Pretii in Portifolio et Solventia<\/h3>\n<p>Constituis portifolium cum 1000 dollariis, 300 in vinculis et 700 in actionibus collocans. Pro vinculorum parte, reditus fixus est 3%, dum pretium actionum (<span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 35 <\/span>) potest cadere ad 25 vel crescere ad 45 cum probabilitatibus aequis.<\/p>\n<p><strong>Quaestio:<\/strong> Quis est valor portifolii in unoquoque casu in exemplo simplici mercatus? Aestima an portifolium praesumptionem solventiae (<span class=\"katex-eq\" data-katex-display=\"false\"> V(t) \\geq 0 <\/span>) impleat.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Exemplar Simple Mercatus:Notiones et Praesumptiones Elementariae Summarium: Haec lectio \u00abExemplar Simple Mercatus\u00bb introducit, rationem quae discendi notiones fundamentales de collocatione faciliorem reddit, coniungens bona sine periculo (vincula, cum reditu noto) et bona periculosa (actiones, cum reditu incerto). Videbimus quomodo haec bona in portifolio coniungi possint quod, recte gestum, reditus superiores quam usurae argentariae praebet, inter [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29422,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":11,"footnotes":""},"categories":[1232,1220],"tags":[],"class_list":["post-33176","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematicae-financiariae","category-oeconomia-et-financiae"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Exemplum Simplex Mercatus: Notiones et Praesumptiones Elementariae<\/title>\n<meta name=\"description\" content=\"Fundamenta Exempli Simplicis Mercatus: disce quomodo bona cum periculo et sine periculo in portifolio coniungantur ad reditus et pericula computanda.\" \/>\n<meta name=\"robots\" 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