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,{pinitial,pfinal}},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 s0 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,pfinal}},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,{pinitial,pfinal}},] 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 10-period 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, 30-year, 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 10-year, 7% semiannual coupon bond valued at $900:

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Future value at 5% interest of a 10-period annuity whose payments increase by 10%:

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Future value of a 10-period 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 very-high-frequency annuity with payments of $1 occurring 100 times per period:

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Introduced in 2010
(8.0)