SOLUTIONS

Annuity
Annuity[p, t]
represents an annuity of fixed payments p made over t periods.
Annuity[p, t, q]
represents a series of payments occurring at time intervals q.
Annuity[{p, {p_{initial}, p_{final}}}, t, q]
represents an annuity with the specified initial and final payments.
Details and OptionsDetails and Options
 Annuity objects specify a class of financial instruments involving a series of payments. They can be used to represent loans or mortgages, loan amortizations, and bonds.
 TimeValue[Annuity[...], interest, t] computes the time value of an annuity as a single equivalent payment at time t.
 Annuity works with numeric or arbitrary symbolic expressions.
 In Annuity[p, t], payments are assumed to occur at times .
 In Annuity[p, t, q], payments occur at times .
 TimeValue[Annuity[p, t, q], r, s] gives the present value for s≤0 and the future value for .
 The present value of a common mortgage is given by TimeValue[Annuity[p, t, q], r, 0].
 The value of a typical bond is given by TimeValue[Annuity[{p, {0, p_{final}}}, t, q], r, 0].
 An annuity with payment interval q differing from the compounding interval d is given by TimeValue[Annuity[p, t, q], EffectiveInterest[r, d], s].
 Annuity[function, ...] represents an annuity in which payments are given as a function of time.
 In Annuity[function, t, 0], the payment rate is taken to be a continuous function of time integrated from 0 to t.
 Discrete payment functions are sometimes defined as recurrence relations. RSolve can be used to convert recurrence relations into functions of time alone for use in Annuity[function, t, q].
 Annuity[{function, {p_{initial}, p_{final}}}, ...] specifies payments as a function of time and also an initial and a final payment.
 Annuity[p, Infinity, ...] represents a perpetuity.
ExamplesExamplesopen allclose all
Basic Examples (10)Basic Examples (10)
Present value of an annuity of 10 payments of $1000 at 6% effective interest:
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Future value of an annuity of 5 payments of $1000 at 8% nominal interest compounded quarterly:
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Future value of a 10period annuity with payments occurring twice per period:
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Find the monthly payments on a $200000 mortgage amortized over 30 years at 5.2% nominal interest:
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Value of a $1000, 30year, semiannual coupon bond with a 6% coupon and a yield of 5%:
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Number of years it will take to pay off a $10000 loan with payments of $200 per month at 8% effective interest:
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Monthly payment necessary to pay off a $5000 loan in 3 years:
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Yield to maturity for a 10year, 7% semiannual coupon bond valued at $900:
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Future value at 5% interest of a 10period annuity whose payments increase by 10%:
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Future value of a 10period annuity with a continuous payment flow at a rate such that the total payment flowing inward during one period is $100:
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Note the similarity in value to a veryhighfrequency annuity with payments of $1 occurring 100 times per period:
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