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that continues indefinitely which pays £{value} every {p} months at an AER of {interest}% if:Perpetuites are annuities that continue indefinitely, i.e $n \\to\\infty$.
\nAs $v={1 \\over 1+i}<1$, then as $n \\to\\infty$, the $\\lim_{n \\to\\infty}\\ v^n \\to 0$
\nThis is annuity immediate that continues indefinitely.
\nThe formula for the present value (one period before the first payment) for a standard annuity immediate is given by:
\n${a}_{n|}^{(p)}={1-v^n \\over i^{(p)}}$
\n(see Numbas- Annuities PV)
\nAs perpetuities are annuites that continue indefintely, the formula for the present value of a perpetuity immediate is:
\n$\\lim_{n \\to\\infty}\\ {a}_{n|}^{(p)}=\\lim_{n\\to\\infty}\\ {1-v^n \\over i^{(p)}}={1 \\over i^{(p)}}$
\nSo then
\n${a}_{\\infty}^{(p)}={1 \\over i^{(p)}}$
\nwhere:
\nIn order to get the present value of an annuity that pays a certain value per period, multiply ${a}_{\\infty|}^{(p)}$ by both the value and p. So the present value is given by:
\n$px{a}_{\\infty|}^{(p)}$
\nwhere:
\nThis is annuity due that continues indefinitely.
\nThe formula for the present value (at the time of the first payment) for a standard annuity is given by:
\n$\\ddot{a}_{n|}^{(p)}={1-v^n \\over d^{(p)}}$
\n(see Numbas- Annuities PV)
\nAs perpetuities are annuities that sontinue indefinitely, the formula for the present value of a perpetuity due is:
\n$lim_{n\\to\\infty} \\ddot{a}_{n|}^{(p)}=lim_{n\\to\\infty} {1-v^n \\over d^{(p)}}={1 \\over d^{(p)}}$
\nso then:
\n$\\ddot{a}_{\\infty|}^{(p)}={1 \\over d^{(p)}}$
\nwhere:
\nThe relationship between the effective and nominal discount rate is given by:
\n$1-d=(1-{d^{(p)} \\over p})^p$
\nRe-arranging this and making $d^{(p)}$ the subject of the formula gives the nominal discount rate.
\nIn order to get the present value of an annuity that pays a certain value per period, multiply $\\ddot{a}_{\\infty|}^{(p)}$ by both the value and p. So the present value is given by:
\n$px\\ddot{a}_{\\infty|}^{(p)}$
\nwhere:
\nIn this question:
\nSubstituting these numbers into the above formulas will give the present value for a perpetuity immediate and due.
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", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "2", "maxValue": "{value}/{int}", "minValue": "{value}/{int}", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "the annuity is annuity due?
", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "2", "maxValue": "{value}/{disc}", "minValue": "{value}/{disc}", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "What is the present value of an annuity that continues indefinitely which pays £{value} every {p} months at an AER of {interest}% if: