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{"name": "present value - annuity due", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["years", "period", "ipadec", "ipa", "C", "n", "P", "Prounded"], "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"variableReplacements": [], "marks": 0, "prompt": "<p>Suppose you are given $\\$\\var{C}$&nbsp;at the beginning&nbsp;of each&nbsp;{period[2]}&nbsp;for $\\var{years}$&nbsp;years. If the interest rate is $\\var{ipa}\\%$ per annum compounding {period[0]}, what is this cash flow worth at the beginning of the $\\var{years}$&nbsp;years?</p>\n<p></p>\n<p>$\\$$&nbsp;[[0]] (to the nearest cent)</p>", "useCustomName": false, "gaps": [{"variableReplacements": [], "allowFractions": false, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "minValue": "P", "useCustomName": false, "showCorrectAnswer": true, "precisionMessage": "You have not given your answer to the nearest cent.", "unitTests": [], "precisionType": "dp", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "P", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showPrecisionHint": false, "showFeedbackIcon": true, "precisionPartialCredit": 0, "customName": "", "strictPrecision": true, "precision": "2"}], "showFeedbackIcon": true, "showCorrectAnswer": true, "unitTests": [], "customName": "", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "scripts": {}, "sortAnswers": false}], "variable_groups": [], "extensions": [], "advice": "<p>You are asked to find the present value of an annuity due (since the payments are at the start&nbsp;of each&nbsp;period). Therefore we will use the present value of an annuity due formula</p>\n<p>$\\displaystyle P=\\frac{C}{i}\\left(1-\\frac{1}{(1+i)^n}\\right)(1+i)$</p>\n<p>where $P$ is the present value, $C$ is the cash flow per period, $i$ is the interest rate per period, and $n$ is the number of periods.</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><span data-jme-visible=\"period[1]&gt;1\">$n=\\var{years}\\times \\var{period[1]}=\\var{n}$,</span> <span data-jme-visible=\"period[1]=1\">$n=\\var{n}$,</span></p>\n<p>and therefore we have</p>\n<p>$\\displaystyle P=\\frac{\\var{C}}{\\left(\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)}\\left(1-\\frac{1}{\\left(1+\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)^\\var{n}}\\right)\\left(1+\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)$</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>", "name": "present value - annuity due", "variables": {"P": {"templateType": "anything", "definition": "(C*period[1]/ipadec)*(1-(1+ipadec/period[1])^(-n))(1+ipadec/period[1])", "description": "", "name": "P", "group": "Ungrouped variables"}, "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": "", "name": "period", "group": "Ungrouped variables"}, "ipadec": {"templateType": "anything", "definition": "random(0.02..0.10#0.001)", "description": "", "name": "ipadec", "group": "Ungrouped variables"}, "ipa": {"templateType": "anything", "definition": "ipadec*100", "description": "", "name": "ipa", "group": "Ungrouped variables"}, "n": {"templateType": "anything", "definition": "years*period[1]", "description": "", "name": "n", "group": "Ungrouped variables"}, "years": {"templateType": "anything", "definition": "random(3..20)", "description": "", "name": "years", "group": "Ungrouped variables"}, "C": {"templateType": "anything", "definition": "if(period[1]=365,random(1..10),random(20..300#10))", "description": "", "name": "C", "group": "Ungrouped variables"}, "Prounded": {"templateType": "anything", "definition": "precround(P,2)", "description": "", "name": "Prounded", "group": "Ungrouped variables"}}, "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>", "metadata": {"description": "Financial maths. Present value of an annuity due.", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "functions": {}, "rulesets": {}, "tags": [], "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/"}]}