// Numbas version: exam_results_page_options {"name": "Variable Interest Rates", "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", "monthly", "weekly", "daily", "cont"], "name": "Variable Interest Rates", "tags": [], "advice": "

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.

3% per annum payable monthly

As $i^{(12)}$=3%, divide this by 12 to obtain $i_{[p]}$ as $i_{[p]}={i^{(p)} \\over p}$ so then the interest rate payable per month is $i_{[p]}={0.03 \\over 12}=0.0025$.

3% per annum payable weekly

As we have obtained above $i_{[12]}=0.0025$ and as we assume that each month has 4 weeks then we use:

$1+i=(1+{i^{(p)} \\over p})^p$

(see Numbas- Nominal Rates)

with:

$p$= 4
$i^{(p)}$= 0.0025

In this case, $i=0.0025023447$

3% per annum payable daily

As we have obtained above $i_{[12]}=0.0025$ and as we assume that each month has 30 days then we use:

$1+i=(1+{i^{(p)} \\over p})^p$

(see Numbas- Nominal Rates)

with:

$p$= 30
$i^{(p)}$= 0.0025

In this case, $i=0.0025030232$

Force of interest corresponds to a nominal rate of 3% per annum

The force of interest $\\delta$ is the nominal rate payable continuously.

$(1+{i^{(p)} \\over p})^p=e^\\delta$ and so $i_{[p]}=e^{\\delta \\over 12}-1$

(see Force of Interest)

with:

$p$= 12
$\\delta$= 0.03

In this case, $i_{[12]}=0.0025031276$

NPV

In order to work out the NPV, you must discount each value to time $t=0$:

$NPV=x_1v+x_2v^2+...+x_nv^n$

(see Numbas- Net Present Value)

In this case we have:

$NPV=-x_1-x_2v^1-x_2v^2...-x_2v^{n-2}+x_3v^n$

where:

$x_1$=£{value1} paid at $t=0$
$x_2$=£{value2} paid at $t=1,.....,t=n-2$
$x_3$=£{value3 received at $t=n$
n= 12

Substituting in the interest rates obtained above into the NPV expression gives the NPV if the nominal rate is 3% payable monthly/weekly/daily/continuously.

", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "

the nominal rate is 3% per annum payable monthly?

", "precisionMessage": "

You have not given your answer to two decimal places.

", "allowFractions": false, "scripts": {}, "precision": "2", "maxValue": "{monthly}", "minValue": "{monthly}", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "

the nominal rate is 3% per annum payable weekly? (Assume each month has 4 weeks)

", "precisionMessage": "

You have not given your answer to two decimal places.

", "allowFractions": false, "scripts": {}, "precision": "2", "maxValue": "{weekly}", "minValue": "{weekly}", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "

the nominal rate is 3% per annum payable daily? (Assume each month has 30 days)

", "precisionMessage": "

You have not given your answer to two decimal places.

", "allowFractions": false, "scripts": {}, "precision": "2", "maxValue": "{daily}", "minValue": "{daily}", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "

the force of interest $\\delta$ corresponds to a nominal interest rate of 3% per annum?

", "precisionMessage": "

You have not given your answer to two decimal places.

", "allowFractions": false, "scripts": {}, "precision": "2", "maxValue": "{cont}", "minValue": "{cont}", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "

At time t=0 a construction company pays £{value1} to purchase a plot of land and materials to build a house. The comapny hires some workers that are paid a total of £{value2} at the end of each month for {m} months. The company sells the house at the end of year {n} for £{value3}.

What is the NPV (to two decimal places) if:

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"cont": {"definition": "{value3}/(1.0025031276^(12*{n}))-{value1}-({value2}/(1.0025031276^1))-({value2}/(1.0025031276^2))-({value2}/(1.0025031276^3))-({value2}/(1.0025031276^4))-({value2}/(1.0025031276^5))-({value2}/(1.0025031276^6))-({value2}/(1.0025031276^7))-({value2}/(1.0025031276^8))-({value2}/(1.0025031276^9))-({value2}/(1.0025031276^10))", "templateType": "anything", "group": "Ungrouped variables", "name": "cont", "description": ""}, "monthly": {"definition": "{value3}/(1.0025^(12*{n}))-{value1}-({value2}/(1.0025^1))-({value2}/(1.0025^2))-({value2}/(1.0025^3))-({value2}/(1.0025^4))-({value2}/(1.0025^5))-({value2}/(1.0025^6))-({value2}/(1.0025^7))-({value2}/(1.0025^8))-({value2}/(1.0025^9))-({value2}/(1.0025^10))", "templateType": "anything", "group": "Ungrouped variables", "name": "monthly", "description": ""}, "m": {"definition": "(12*n)-2", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "daily": {"definition": "{value3}/(1.0025030232^(12*{n}))-{value1}-({value2}/(1.0025030232^1))-({value2}/(1.0025030232^2))-({value2}/(1.0025030232^3))-({value2}/(1.0025030232^4))-({value2}/(1.0025030232^5))-({value2}/(1.0025030232^6))-({value2}/(1.0025030232^7))-({value2}/(1.0025030232^8))-({value2}/(1.0025030232^9))-({value2}/(1.0025030232^10))", "templateType": "anything", "group": "Ungrouped variables", "name": "daily", "description": ""}, "n": {"definition": "random(1..1#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}, "value3": {"definition": "random(100000..200000#5000)", "templateType": "randrange", "group": "Ungrouped variables", "name": "value3", "description": ""}, "value2": {"definition": "random(4000..7500#500)", "templateType": "randrange", "group": "Ungrouped variables", "name": "value2", "description": ""}, "value1": {"definition": "random(40000..60000#1000)", "templateType": "randrange", "group": "Ungrouped variables", "name": "value1", "description": ""}, "weekly": {"definition": "{value3}/(1.0025023447^(12*{n}))-{value1}-({value2}/(1.0025023447^1))-({value2}/(1.0025023447^2))-({value2}/(1.0025023447^3))-({value2}/(1.0025023447^4))-({value2}/(1.0025023447^5))-({value2}/(1.0025023447^6))-({value2}/(1.0025023447^7))-({value2}/(1.0025023447^8))-({value2}/(1.0025023447^9))-({value2}/(1.0025023447^10))", "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": "Habiba Gora", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/494/"}]}]}], "contributors": [{"name": "Habiba Gora", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/494/"}]}