// Numbas version: exam_results_page_options
{"name": "MATH1510 Financial Mathematics 1 Practical 2B", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["value", "p", "n", "m", "r", "s", "t", "u", "x", "y", "interest", "intdec", "int", "intper", "v", "an", "repayment", "ob", "intcomp", "repaycomp", "sumintcomp", "sumobcomp"], "name": "MATH1510 Financial Mathematics 1 Practical 2B", "tags": [], "advice": "<p style=\"text-align: left;\">A loan&nbsp;scehdule can be used to determine the outstanding balance of a loan. A loan&nbsp;schedule consists of two components:</p>\n<ol>\n<li>interest component</li>\n<li>repayment component</li>\n</ol>\n<ul>\n<li><b>Payments</b></li>\n</ul>\n<p>The total number of payments is the number of payments in one year multiplied by the number of years.</p>\n<p style=\"text-align: center;\">total number of payments= $np$</p>\n<p style=\"text-align: left;\">where:</p>\n<ul>\n<li>$n$= the number of years of the loan</li>\n<li>$p$= number of periods in a year</li>\n</ul>\n<p>In the above question, substituting in $n$={n} and $p$={p} will give the total number of payments.</p>\n<ul>\n<li><strong>Payment every period</strong></li>\n</ul>\n<p>As we know the value of the loan&nbsp;and that the repayment each period is like an annuity, we know that the present value of the annuity&nbsp;multipled by the number of periods in a year will give the value of the loan&nbsp;(see Numbas- Annuities PV).</p>\n<p style=\"text-align: center;\">$px{a}_{n|}^{p}=loan$</p>\n<p style=\"text-align: left;\">where:</p>\n<ul>\n<li>$p$= number of periods in one year</li>\n<li>${a}_{n|}^{p}$= present value of a standard annuity&nbsp;paying p times a year for n years with payments of $1 \\over p$ each&nbsp;period</li>\n<li>$x$= payment every period</li>\n</ul>\n<p>Re-arranging the above:</p>\n<p style=\"text-align: center;\"><span>$x={loan \\over p{a}_{n|}^{p}}$</span></p>\n<p style=\"text-align: left;\"><span>In the above question, working out ${a}_{n|}^{p}$ with an&nbsp;effective interest rate&nbsp;of {interest}% and substituting in loan=&pound;{value} and $p$={p} will give the value of each payment.</span></p>\n<ul>\n<li><strong>Outstanding balance</strong></li>\n</ul>\n<p>The outstanding balance, OB, is the value of the loan after a number of payments have been made. This can be calculated by creating a table with&nbsp;the interest and repayment component of each payment and then reading off the table. Another way would be to use the <strong>prospective method</strong>. This is the value of the remaining payments after $m$ payments have already been made.</p>\n<p style=\"text-align: center;\">OB after the $m^{th}$ payment=$px{a}_{n-{m \\over p}|}^{(p)}$</p>\n<p style=\"text-align: left;\">where:</p>\n<ul>\n<li>$p$= number of periods in a year</li>\n<li>$x$= payment every period</li>\n<li>$n$= number of years of the loan</li>\n<li>${m \\over p}$= the number of payments that have already been made in terms of years</li>\n</ul>\n<p>Working out&nbsp;<span>${a}_{n-{m \\over p}|}^{(p)}$ with an&nbsp;effective interest rate&nbsp;of {interest}% and substituting in $p$={p}, $m$={m}, $n$={n} and $x$ from the previous part will give the OB after the {m}<sup>th</sup> payment.&nbsp;</span></p>\n<ul>\n<li><strong>Interest component</strong></li>\n</ul>\n<p>To work out the interest component of the $r^{th}$ payment, you first need to calculate the OB after the $(r-1)^{th}$ payment and then work out the interest on that value.</p>\n<p style=\"text-align: center;\"><span>OB after the $(r-1)^{th}$ payment=$px{a}_{n-{(r-1) \\over p}|}^{(p)}$</span></p>\n<p style=\"text-align: left;\"><span>As the payments are made every period, the interest component is found by multiplying&nbsp;$px{a}_{n-{(r-1) \\over p}|}^{(p)}$ by $i_{[p]}$.</span></p>\n<p style=\"text-align: left;\"><span>The interest component of the $r^{th}$ is:</span></p>\n<p style=\"text-align: center;\"><span>$i_{[p]}px{a}_{n-{(r-1) \\over p}|}^{(p)}$</span></p>\n<p style=\"text-align: left;\"><span>where:</span></p>\n<ul>\n<li>$p$= number of periods in a year</li>\n<li>$x$= payment every period</li>\n<li>$n$= number of years of the loan</li>\n<li>${(r-1) \\over p}$= the number of payments that have already been made in terms of years</li>\n<li>$i_{[p]}={i^{(p)} \\over p}$= the interest rate for each period</li>\n</ul>\n<p><span>Working out&nbsp;</span><span>${a}_{n-{(r-1) \\over p}|}^{(p)}$ with an&nbsp;effective interest rate&nbsp;of {interest}% and $i_{[p]}$ and substituting in&nbsp;$p$={p}, $r$={r}, $n$={n} and $x$ will give the &nbsp;interest component of the {r}<sup>th&nbsp;</sup>payment.</span></p>\n<ul>\n<li><strong>Repayment component</strong></li>\n</ul>\n<p>The repayment component of the $s^{th}$ payment can be foound by subtracting the interest component of the $s^{th}$ payment from the payment per period. The interest component of the $s^{th}$ payment is found using the same method as above.</p>\n<p>The repayment component of the $s^{th}$ payment is:</p>\n<p style=\"text-align: center;\"><span>$x-i_{[p]}px{a}_{n-{(s-1) \\over p}|}^{(p)}$</span></p>\n<p style=\"text-align: left;\"><span>where:</span></p>\n<ul>\n<li>$p$= number of periods in a year</li>\n<li>$x$= payment every period</li>\n<li>$n$= number of years of the loan</li>\n<li>${(s-1) \\over p}$= the number of payments that have already been made in terms of years</li>\n<li>$i_{[p]}={i^{(p)} \\over p}$= the interest rate for each period</li>\n</ul>\n<p><span>Working out&nbsp;</span><span>${a}_{n-{(s-1) \\over p}|}^{(p)}$ with an&nbsp;effective interest rate&nbsp;of {interest}% and $i_{[p]}$ and substituting in&nbsp;$p$={p}, $s$={s}, $n$={n} and $x$ will give the repayment&nbsp;component of the&nbsp;{s}<sup>th</sup><span> </span>payment.</span></p>", "rulesets": {}, "parts": [{"prompt": "<p>How many payments are there in total?</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "{p}*{n}", "minValue": "{p}*{n}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>What is the value (to two decimal places) of each payment?</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{repayment}", "minValue": "{repayment}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "5", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>What is the outstanding balance (to two decimal places) immediately after the {m}th payment?</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ob}", "minValue": "{ob}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "10", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>What is the interest component (to two decimal places) of the {r}th payment?</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{intcomp}", "minValue": "{intcomp}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "10", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>What is the repayment component (to two decimal places) of the {s}th payment?</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{repaycomp}", "minValue": "{repaycomp}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "10", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p><span>What is the sum (to two decimal places) of the outstanding balances&nbsp;from the {x}th to the {y}th payment?</span></p>", "precisionMessage": "<p>You have not given your answer to two decimal places.</p>", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{sumobcomp}", "minValue": "{sumobcomp}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "15", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>What is the sum (to two decimal places) of the interest components from the {t}th to the {u}th payment?</p>", "precisionMessage": "<p>You have not given your answer to two decimal places.</p>", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{sumintcomp}", "minValue": "{sumintcomp}", "variableReplacementStrategy": "originalfirst", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "15", "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>Suppose you borrow <strong>&pound;{value}</strong> to be repaid in equal&nbsp;installments <strong>{p}</strong> times a year in arrears over a period of <strong>{n}</strong> years at an AER of <strong>{interest}%</strong>.</p>\n<p>Construct an Excel spreadsheet (using either the formulas derived in lectures or the Excel financial functions) presenting a loan schedule showing the interest and repayment component of every payment and the outstanding balance.</p>\n<div id=\"showhide\" style=\"display: none;\">Do the question in Excel first and then input the numbers.</div>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"sumintcomp": {"definition": "(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}-1))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+1))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+2))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+3))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+4))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+5))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+6))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+7))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+8))/{p})))/{int}))*{intper})+(({repayment}*{p}*((1-({v}^((({p}*{n})-({t}+9))/{p})))/{int}))*{intper})", "templateType": "anything", "group": "Ungrouped variables", "name": "sumintcomp", "description": ""}, "int": {"definition": "(((1+{intdec})^(1/{p}))-1)*{p}", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "an": {"definition": "(1-({v}^{n}))/{int}", "templateType": "anything", "group": "Ungrouped variables", "name": "an", "description": ""}, "sumobcomp": {"definition": "{repayment}*{p}*((1-({v}^((({p}*{n})-{x})/{p})))/{int})+{repayment}*{p}*((1-({v}^((({p}*{n})-({x}+1))/{p})))/{int})+{repayment}*{p}*((1-({v}^((({p}*{n})-({x}+2))/{p})))/{int})+{repayment}*{p}*((1-({v}^((({p}*{n})-({x}+3))/{p})))/{int})+{repayment}*{p}*((1-({v}^((({p}*{n})-({x}+4))/{p})))/{int})+{repayment}*{p}*((1-({v}^((({p}*{n})-({x}+5))/{p})))/{int})", "templateType": "anything", "group": "Ungrouped variables", "name": "sumobcomp", "description": ""}, "intdec": {"definition": "{interest}/100", "templateType": "anything", "group": "Ungrouped variables", "name": "intdec", "description": ""}, "intcomp": {"definition": "({repayment}*{p}*((1-({v}^((({p}*{n})-({r}-1))/{p})))/{int}))*{intper}", "templateType": "anything", "group": "Ungrouped variables", "name": "intcomp", "description": ""}, "intper": {"definition": "{int}/{p}", "templateType": "anything", "group": "Ungrouped variables", "name": "intper", "description": ""}, "repaycomp": {"definition": "{repayment}-(({repayment}*{p}*((1-({v}^((({p}*{n})-({s}-1))/{p})))/{int}))*{intper})", "templateType": "anything", "group": "Ungrouped variables", "name": "repaycomp", "description": ""}, "interest": {"definition": "random(1..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "interest", "description": ""}, "m": {"definition": "({p}*{n})-17", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "ob": {"definition": "{repayment}*{p}*((1-({v}^((({p}*{n})-{m})/{p})))/{int})", "templateType": "anything", "group": "Ungrouped variables", "name": "ob", "description": ""}, "value": {"definition": "random(20000..50000#1000)", "templateType": "randrange", "group": "Ungrouped variables", "name": "value", "description": ""}, "n": {"definition": "random(5..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}, "p": {"definition": "random([4,12])", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "s": {"definition": "({p}*{n})-7", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "r": {"definition": "({p}*{n})-13", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "u": {"definition": "{t}+10", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "t": {"definition": "random([4,6,8,9])", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "repayment": {"definition": "{value}/({p}*{an})", "templateType": "anything", "group": "Ungrouped variables", "name": "repayment", "description": ""}, "v": {"definition": "1/(1+{intdec})", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "y": {"definition": "{x}+5", "templateType": "anything", "group": "Ungrouped variables", "name": "y", "description": ""}, "x": {"definition": "random([({p}*{n})-6,({p}*{n})-10])", "templateType": "anything", "group": "Ungrouped variables", "name": "x", "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/"}]}