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{"name": "MATH1510 Financial Mathematics 1 Practical 4A", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["value1", "value2", "value3", "n", "m", "interest", "int", "intm", "intw", "intd", "force", "monthly", "weekly", "daily", "cont"], "name": "MATH1510 Financial Mathematics 1 Practical 4A", "tags": [], "advice": "<p>In order to obtain the NPV, you must first change the nominal rate per annum payable monthly/weekly/yearly/continuously into an effective rate per month.</p>\n<ul>\n<li><strong>{interest}%<i>&nbsp; </i>per annum&nbsp;payable monthly</strong></li>\n</ul>\n<p>As $i^{(12)}$={interest}%, divide this by 12 to obtain $i_{[p]}$ as $i_{[p]}={i^{(p)} \\over p}$ so then the interest rate payable per month is $i_{[12]}$={int}*$1 \\over 12$.</p>\n<ul>\n<li><strong>{interest}%<i>&nbsp; </i>per annum&nbsp;payable&nbsp;weekly</strong></li>\n</ul>\n<p>As we assume that each month has 4 weeks then we use:</p>\n<p style=\"text-align: center;\">$1+i=(1+{i^{(p)} \\over p})^p$</p>\n<p style=\"text-align: center;\">(see&nbsp;<a href=\"/numbas-previews/question-8449-nominal-rates/index.html\">Numbas- Nominal Rates</a>)</p>\n<p style=\"text-align: left;\">with:</p>\n<ul>\n<li>$p$= 4</li>\n<li>$i^{(p)}$=&nbsp;{intm}</li>\n</ul>\n<p>In this case, $i$={intw}</p>\n<ul>\n<li><strong>{interest}%<i>&nbsp; </i>per annum&nbsp;payable&nbsp;daily</strong></li>\n</ul>\n<p>As we assume that each month has 30 days then we use:</p>\n<p style=\"text-align: center;\">$1+i=(1+{i^{(p)} \\over p})^p$</p>\n<p style=\"text-align: center;\">(see&nbsp;<a href=\"/numbas-previews/question-8449-nominal-rates/index.html\">Numbas- Nominal Rates</a>)</p>\n<p style=\"text-align: left;\">with:</p>\n<ul>\n<li>$p$= 30</li>\n<li>$i^{(p)}$=&nbsp;{intm}</li>\n</ul>\n<p>In this case, $i$= {intd}</p>\n<p></p>\n<ul>\n<li><strong>Force of interest corresponds to a nominal rate of {interest}% per annum</strong></li>\n</ul>\n<p>The force of interest $\\delta$ is the nominal rate payable continuously.</p>\n<p style=\"text-align: center;\">$(1+{i^{(p)} \\over p})^p=e^\\delta$ and so $i_{[p]}=e^{\\delta \\over p}-1$</p>\n<p style=\"text-align: center;\">(see&nbsp;<a href=\"/numbas-previews/question-8466-force-of-interest/index.html\">Force of Interest</a>)</p>\n<p style=\"text-align: left;\">with:</p>\n<ul>\n<li>$p$= 12</li>\n<li>$\\delta$=&nbsp;{interest}%</li>\n</ul>\n<p>In this case, $i_{[12]}$= {force}</p>\n<ul>\n<li><strong>NPV</strong></li>\n</ul>\n<p>In order to work out the NPV, you must discount each value to time $t=0$:</p>\n<p style=\"text-align: center;\"><span>$NPV=x_1v+x_2v^2+...+x_nv^n$</span></p>\n<p style=\"text-align: center;\"><span>(see&nbsp;<a href=\"/numbas-previews/question-8456-net-present-value/index.html\">Numbas- Net Present Value</a>)</span></p>\n<p style=\"text-align: left;\"><span>In this case we have:</span></p>\n<p style=\"text-align: center;\"><span>$NPV=-x_1-x_2v^1-x_2v^2...-x_2v^{n-2}+x_3v^n$</span></p>\n<p style=\"text-align: left;\"><span>where:</span></p>\n<ul>\n<li>$x_1$=&pound;{value1} paid at $t=0$</li>\n<li>$x_2$=&pound;{value2} paid at $t=1,.....,t=n-2$</li>\n<li>$x_3$=&pound;{value3} received at $t=n$</li>\n<li>n= 12*{n}</li>\n</ul>\n<p>Substituting in the interest rates obtained above into the NPV expression gives the NPV if the nominal rate is {interest}% payable monthly/weekly/daily/continuously.</p>", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "<p>the nominal interest rate payable monthly is {interest}%?</p>", "precisionMessage": "<p><span>You have not given your answer to two decimal places.</span></p>", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{monthly}", "minValue": "{monthly}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "10", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>the nominal interest rate payable weekly&nbsp;{interest}%? (assume that there are 4 weeks per month)</p>", "precisionMessage": "<p><span>You have not given your answer to two decimal places.</span></p>", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{weekly}", "minValue": "{weekly}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "15", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>the nominal interest rate payable daily&nbsp;is {interest}%? (assume that there are 30 days per month)</p>", "precisionMessage": "<p><span>You have not given your answer to two decimal places.</span></p>", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{daily}", "minValue": "{daily}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "15", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>the force of interest, $\\delta$, corresponds to a nominal interest rate&nbsp;of {interest}% per annum?</p>", "precisionMessage": "<p>You have not given your answer to two decimal places.</p>", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{cont}", "minValue": "{cont}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "15", "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>At time t=0, a construction company pays &pound;{value1} to purchase a plot of land and materials to build a house. The company hires some workers that are paid a total of &pound;{value2} at the end of each month for {m}&nbsp;months. The company sells the house at the end of year {n} for &pound;{value3}.</p>\n<p>Construct an Excel spreadsheet that shows the cashflow of each month and which analytically calculates the net present value at different interest rates.</p>\n<p><em>(Either use the formula introduced in lecture to convert between interest rates or the Excel function <strong>=EFFECT</strong>)</em></p>\n<p><em></em></p>\n<p>What is the NPV (to two decimal places) if:</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"intd": {"definition": "((1+({intm}/30))^30)-1", "templateType": "anything", "group": "Ungrouped variables", "name": "intd", "description": ""}, "force": {"definition": "(e^{intm})-1", "templateType": "anything", "group": "Ungrouped variables", "name": "force", "description": ""}, "value1": {"definition": "random(30000..40000#1000)", "templateType": "randrange", "group": "Ungrouped variables", "name": "value1", "description": ""}, "int": {"definition": "random(0.02..0.05#0.001)", "templateType": "randrange", "group": "Ungrouped variables", "name": "int", "description": ""}, "m": {"definition": "(12*n)-2", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "daily": {"definition": "-{value1}-(({value2)*12*((1-((1+{intd})^-{m}))/(12*{intd})))+({value3}/((1+{intd})^(12*{n})))", "templateType": "anything", "group": "Ungrouped variables", "name": "daily", "description": ""}, "n": {"definition": "random(1..3#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}, "monthly": {"definition": "-{value1}-(({value2)*12*((1-((1+{intm})^-{m}))/{int}))+({value3}/((1+{intm})^(12*{n})))", "templateType": "anything", "group": "Ungrouped variables", "name": "monthly", "description": ""}, "intw": {"definition": "((1+({intm}/4))^4)-1", "templateType": "anything", "group": "Ungrouped variables", "name": "intw", "description": ""}, "value3": {"definition": "random(200000..300000#5000)", "templateType": "randrange", "group": "Ungrouped variables", "name": "value3", "description": ""}, "intm": {"definition": "{int}/12", "templateType": "anything", "group": "Ungrouped variables", "name": "intm", "description": ""}, "interest": {"definition": "{int}*100", "templateType": "anything", "group": "Ungrouped variables", "name": "interest", "description": ""}, "cont": {"definition": "-{value1}-(({value2)*12*((1-((1+{force})^-{m}))/(12*{force})))+({value3}/((1+{force})^(12*{n})))", "templateType": "anything", "group": "Ungrouped variables", "name": "cont", "description": ""}, "value2": {"definition": "random(3000..5500#500)", "templateType": "randrange", "group": "Ungrouped variables", "name": "value2", "description": ""}, "weekly": {"definition": "-{value1}-(({value2)*12*((1-((1+{intw})^-{m}))/(12*{intw})))+({value3}/((1+{intw})^(12*{n})))", "templateType": "anything", "group": "Ungrouped variables", "name": "weekly", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "James Fung", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/699/"}]}]}], "contributors": [{"name": "James Fung", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/699/"}]}