// Numbas version: finer_feedback_settings {"name": "Annuities PV", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["time", "value", "p", "n", "interest", "intdec", "v", "int", "disc", "arrears", "advance"], "name": "Annuities PV", "tags": [], "advice": "

An annuity is a regular series of payments. There are two types of annuities :

Annuity Immediate where the payments are at the end of the period
Annuity Due where the payments are at the beginning of the period

The present value of an annuity is the discounted value at $t=0$ of the annuity paying a certain value every period for n years.

Arrears

This is when the payments are at the end of each period i.e. annuity immediate.

${a}_{n|}^{(p)}$ is the actuarial notation for the discounted value of an annuity paying p times a year for n years with payments of ${1 \\over p}$ at the end of each period.

The formula is given by:

${a}_{n|}^{(p)}={1-v^n \\over i^{(p)}}$

where:

$i^{(p)}$= nominal rate (see Numbas- Nominal Rates)
$p$= number of periods in a year
$n$= number of years
$v$=${1 \\over 1+i}$- the discount factor (see Numbas- Discounting)
$i$= annual effective rate

In order to get the discounted value of an annuity paying a certain value p times a year at the end of every period, multiply ${a}_{n|}^{(p)}$ by both the value and p.

So the present value is given by:

$px{a}_{n|}^{(p)}$

where:

$x$ is the amount paid every period.

Advance

This is when the payments are at the beginning of each period i.e. annuity due.

$\\ddot{a}_{n|}^{(p)}$ is the actuarial notation for the discounted value of an annuity paying ${1 \\over p}$ at the start of each period for n years.

The formula is given by:

$\\ddot{a}_{n|}^{(p)}={1-v^n \\over d^{(p)}}$

where:

$p$= number of periods in a year
$n$= number of years
$v$=${1 \\over 1+i}$- the discount factor
$i$= annual effective rate
$d^{(p)}$=nominal discount rate

The relationship between the nominal and effective discount rate is given by:

$1-d=(1-{d^{(p)} \\over p})^p$

Re-arranging this and making $d^{(p)}$ gives the nominal discount rate.

In order to get the present value of an annuity paying a certain value p times a year at the beginning of every period, multiply $\\ddot{a}_{n|}^{(p)}$ by both the value and p.

So the present value is given by:

$px\\ddot{a}_{n|}^{(p)}$

where:

$x$ is the amount paid every period.

Substituting in

$x$=£{value}
$p$={p}
$n$={n} years
$i$={interest}%

into the above formulas will give the present value of an annuity that is payed in arrears and in advance.

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

the payments are made in arrears?

", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "2", "maxValue": "{arrears}", "minValue": "{arrears}", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": "5", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "

the payments are made in advance?

", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "2", "maxValue": "{advance}", "minValue": "{advance}", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": "6", "type": "numberentry", "showPrecisionHint": false}], "statement": "

Compute the {time} value (to two decimal places) of an annuity paying £{value} {p} times a year for {n} years at an interest rate of {interest}% if:

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"advance": {"definition": "{value}*{p}*((1-({v}^{n}))/{disc})", "templateType": "anything", "group": "Ungrouped variables", "name": "advance", "description": ""}, "int": {"definition": "(((1+{intdec})^(1/{p}))-1)*{p}", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "v": {"definition": "1/(1+{intdec})", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "value": {"definition": "random(200..600#50)", "templateType": "randrange", "group": "Ungrouped variables", "name": "value", "description": ""}, "n": {"definition": "random(3..12#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}, "p": {"definition": "random([1,2,3,4,6,12,52,365])", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "disc": {"definition": "(1-({v}^(1/{p})))*{p}", "templateType": "anything", "group": "Ungrouped variables", "name": "disc", "description": ""}, "arrears": {"definition": "{value}*{p}*((1-({v}^{n}))/{int})", "templateType": "anything", "group": "Ungrouped variables", "name": "arrears", "description": ""}, "interest": {"definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "interest", "description": ""}, "time": {"definition": "\"present\"", "templateType": "string", "group": "Ungrouped variables", "name": "time", "description": ""}, "intdec": {"definition": "{interest}/100", "templateType": "anything", "group": "Ungrouped variables", "name": "intdec", "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/"}]}