// Numbas version: finer_feedback_settings
{"name": "interest rate  - ordinary perpetuity", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ipadec": {"templateType": "anything", "definition": "ippdec*period[1]", "description": "", "group": "Ungrouped variables", "name": "ipadec"}, "P": {"templateType": "anything", "definition": "random(5000..100000#1000)", "description": "", "group": "Ungrouped variables", "name": "P"}, "C": {"templateType": "anything", "definition": "siground(seed*P/period[1],2)", "description": "", "group": "Ungrouped variables", "name": "C"}, "seed": {"templateType": "anything", "definition": "random(0.03..0.1#0.0001)", "description": "", "group": "Ungrouped variables", "name": "seed"}, "iparounded": {"templateType": "anything", "definition": "precround(ipa,2)", "description": "", "group": "Ungrouped variables", "name": "iparounded"}, "ipa": {"templateType": "anything", "definition": "ipadec*100", "description": "", "group": "Ungrouped variables", "name": "ipa"}, "ippdec": {"templateType": "anything", "definition": "C/P", "description": "", "group": "Ungrouped variables", "name": "ippdec"}, "period": {"templateType": "anything", "definition": "random([random('yearly','annually'),1,'year'],[random('half-yearly','semiannually'),2,'half-year'],['quarterly', 4, 'quarter'],['monthly',12,'month'],['daily', 365,'day'])", "description": "", "group": "Ungrouped variables", "name": "period"}}, "functions": {}, "name": "interest rate  - ordinary perpetuity", "rulesets": {}, "parts": [{"useCustomName": false, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "customName": "", "unitTests": [], "gaps": [{"useCustomName": false, "variableReplacements": [], "mustBeReduced": false, "marks": 1, "allowFractions": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "extendBaseMarkingAlgorithm": true, "showPrecisionHint": false, "showCorrectAnswer": true, "type": "numberentry", "scripts": {}, "minValue": "ipa", "customMarkingAlgorithm": "", "precision": "2", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "customName": "", "precisionMessage": "You have not given your answer to&nbsp;2 decimal places.", "precisionPartialCredit": 0, "maxValue": "ipa", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "strictPrecision": true, "unitTests": []}], "prompt": "<p>Suppose you want to set up a scholarship or a donation that&nbsp;gives $\\$\\var{C}$&nbsp;at the end of every {period[2]} forever. If you have $\\$\\var{P}$ to invest and the interest is compounded {period[0]} what&nbsp;would the nominal interest rate need to be?</p>\n<p>&nbsp;[[0]]&nbsp;$\\%$ p.a. (to 2&nbsp;decimal places)</p>"}], "extensions": [], "ungrouped_variables": ["period", "seed", "P", "C", "ippdec", "ipadec", "ipa", "iparounded"], "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "Financial maths. Cash flow amount of an ordinary perpetuity."}, "tags": [], "advice": "<p>You are asked to find the nominal interest rate&nbsp;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>$P=\\var{P}$,</p>\n<p>$C=\\var{C}$,</p>\n<p>and therefore&nbsp;we have</p>\n<p>$\\displaystyle \\var{P}=\\frac{\\var{C}}{i}$</p>\n<p>which we need to rearrange to solve for $i$.</p>\n<p>We multiply both sides by $i$ (so that $i$&nbsp;isn't in the denominator)</p>\n<p>$\\displaystyle \\var{P}i=\\var{C}$</p>\n<p>Then divide both sides by $\\var{P}$ (to remove the multiplication by it)</p>\n<p>$\\displaystyle i=\\frac{\\var{C}}{\\var{P}}$</p>\n<p>Recall that $i$ is&nbsp;the interest rate per time period but we were&nbsp;asked for&nbsp;the interest rate per annum so we multiply by $\\var{period[1]}$ to get&nbsp;</p>\n<p>$\\begin{array}\\displaystyle \\text{interest rate pa}&amp;=\\var{period[1]}\\times&nbsp;\\frac{\\var{C}}{\\var{P}}\\\\&amp;\\approx\\var{ipadec}\\\\&amp;=\\var{iparounded}\\%\\quad \\text{(2 decimal places)}\\end{array}$</p>", "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>", "type": "question", "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/"}]}