Present value of an annuity due of 10 payments of 1000ドル at 6% effective interest:

Present value of that annuity due with specified initial and final payments:

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

Future value of that annuity due with specified initial and final payments:

Infinity may be used as the number of payment periods to specify a perpetuity due:

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

Future value of that annuity due with payments occurring twice per period:

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

Future value of a 10-period annuity due with a continuous payment flow at a rate such that the total payment flowing inward during one period is 100ドル:

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

AnnuityDue works with symbolic parameters. TimeValue [AnnuityDue[…],…] can find closed-form expressions:

Apart can be used to expose the discount factor coefficients on individual payments:

Apart may need to be applied multiple times to fully decompose the expression:

Solutions to equations involving AnnuityDue can be found in terms of symbolic parameters:

An integer can be used as a payment interval to specify payments occurring only once every several periods:

Value of a delayed annuity whose 7 payments start in 5 years:

At what annual effective interest is the present value of a series of payments of 1 every 6 months forever, with the first payment made immediately, equal to 10:

Find the accumulated value at the end of 10 years of an annuity in which payments are made at the beginning of each half-year for five years. The first payment is 2000,ドル and each payment is 98% of the prior payment. Interest is credited at 10% compounded quarterly:

TimeValue takes a reference point argument for cash flows. This argument can be used with Annuity to simulate an annuity due:

Annuity due is shifted Annuity :

An annuity due's payment growth pattern can be any Wolfram Language function or an arbitrary user-defined function:

Study the convergence of the value of high-frequency annuities due to a continuous annuity due: