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 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, 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:

In[1]:=
Click for copyable input
Out[1]=

Future value of an annuity of 5 payments of $1000 at 8% nominal interest compounded quarterly:

In[1]:=
Click for copyable input
Out[1]=

Future value of a 10-period annuity with payments occurring twice per period:

In[1]:=
Click for copyable input
Out[1]=

Find the monthly payments on a $200000 mortgage amortized over 30 years at 5.2% nominal interest:

In[1]:=
Click for copyable input
Out[1]=

Value of a $1000, 30-year, semiannual coupon bond with a 6% coupon and a yield of 5%:

In[1]:=
Click for copyable input
Out[1]=

Number of years it will take to pay off a $10000 loan with payments of $200 per month at 8% effective interest:

In[1]:=
Click for copyable input
Out[1]=

Monthly payment necessary to pay off a $5000 loan in 3 years:

In[1]:=
Click for copyable input
Out[1]=

Yield to maturity for a 10-year, 7% semiannual coupon bond valued at $900:

In[1]:=
Click for copyable input
Out[1]=

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

In[1]:=
Click for copyable input
Out[1]=

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:

In[1]:=
Click for copyable input
Out[1]=

Note the similarity in value to a very-high-frequency annuity with payments of $1 occurring 100 times per period:

In[2]:=
Click for copyable input
Out[2]=
New in 8
New to Mathematica? Find your learning path »
Have a question? Ask support »