// Numbas version: exam_results_page_options
{"name": "present value  - ordinary perpetuity", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "variables": {"P": {"description": "", "group": "Ungrouped variables", "definition": "C*period[1]/ipadec", "templateType": "anything", "name": "P"}, "C": {"description": "", "group": "Ungrouped variables", "definition": "if(period[1]=365,random(1..10),random(20..300#10))", "templateType": "anything", "name": "C"}, "ipadec": {"description": "", "group": "Ungrouped variables", "definition": "random(0.02..0.10#0.001)", "templateType": "anything", "name": "ipadec"}, "Prounded": {"description": "", "group": "Ungrouped variables", "definition": "precround(P,2)", "templateType": "anything", "name": "Prounded"}, "period": {"description": "", "group": "Ungrouped variables", "definition": "random([random('yearly','annually'),1,'year'],[random('half-yearly','semiannually'),2,'half-year'],['quarterly', 4, 'quarter'],['monthly',12,'month'],['daily', 365,'day'])", "templateType": "anything", "name": "period"}, "ipa": {"description": "", "group": "Ungrouped variables", "definition": "ipadec*100", "templateType": "anything", "name": "ipa"}}, "name": "present value  - ordinary perpetuity", "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"extendBaseMarkingAlgorithm": true, "gaps": [{"precisionType": "dp", "unitTests": [], "useCustomName": false, "variableReplacements": [], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "minValue": "P", "strictPrecision": true, "customName": "", "precisionMessage": "You have not given your answer to the nearest cent.", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "precision": "2", "mustBeReducedPC": 0, "marks": 1, "maxValue": "P", "type": "numberentry", "showPrecisionHint": false, "showCorrectAnswer": true, "scripts": {}, "correctAnswerFraction": false}], "useCustomName": false, "variableReplacements": [], "showFeedbackIcon": true, "sortAnswers": false, "unitTests": [], "customName": "", "marks": 0, "type": "gapfill", "prompt": "<p>Suppose you want to set up a scholarship or a donation that&nbsp;gives $\\$\\var{C}$ at the end of every {period[2]} forever. If you could invest at $\\var{ipa}\\%$ p.a. compounded&nbsp;{period[0]}, what amount would you need to invest at the start to ensure the cash flow lasts forever.</p>\n<p></p>\n<p>$\\$$&nbsp;[[0]] (to the nearest cent)</p>", "showCorrectAnswer": true, "scripts": {}, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst"}], "advice": "<p>You are asked to find the present value of an ordinary perpetuity&nbsp;(since the payments are at the end of each&nbsp;period and last forever). Therefore we will use the ordinary perpetuity&nbsp;formula</p>\n<p>$\\displaystyle P=\\frac{C}{i}$</p>\n<p>where $P$ is the present value, $C$ is the cash flow per period&nbsp;and&nbsp;$i$ is the interest rate per period.</p>\n<p>In our&nbsp;situation we have,</p>\n<p>$C=\\var{C}$,</p>\n<p><span data-jme-visible=\"period[1]&gt;1\">$i=\\frac{\\var{ipa}\\%}{\\var{period[1]}}=\\frac{\\var{ipadec}}{\\var{period[1]}}$,</span> <span data-jme-visible=\"period[1]=1\">$i=\\var{ipa}\\%=\\var{ipadec}$,</span></p>\n<p>and therefore we have</p>\n<p>$\\displaystyle P=\\frac{\\var{C}}{\\left(\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)}$</p>\n<p>Calculating this we find&nbsp;</p>\n<p>$\\begin{align}P&amp;\\approx \\var{P}\\\\&amp;=\\$\\var{Prounded}\\quad \\text{(to the nearest cent)}\\end{align}$</p>", "variable_groups": [], "metadata": {"description": "Financial maths. Present value of an ordinary perpetuity.", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "extensions": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["period", "ipadec", "ipa", "C", "P", "Prounded"], "rulesets": {}, "tags": [], "statement": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Reveal answers</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again. Do each question repeatedly to ensure you have mastered it.</p>", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}