**Annuity Due in Terms of Ordinary Annuity**

**Annuity: Ordinary vs. Due**

The distinction is determined when the first payment is made:

Ordinary Annuity: the first payment is made at the end of the first period, with no payment made at the beginning (period 0) of the annuity.

Annuity Due: the first payment is made at the very beginning of the annuity, with no payment made at the end of the annuity.

For today's exercise, assume all payments in annuity are all equal. To simplify, I am going to set payments to 1 monetary unit (Dollars, Euros, Lira, Yen, etc).

**Present Value**

The present value of an annuity is today's value of the annuity, with future payments and cash flows are discounted back to today using the periodic interest rate. On financial calculators, the present value is symbolized by the PV key.

Let i be the interest rate of the annuity, in decimal format.

Example, for 5% periodic rate, i=0.05.

Let v = 1/(1 + i).

**Present Value of an Ordinary Annuity**

Given the payment of 1 monetary unit, an annuity of n periods, equally spaced out, and interest rate i, the present value of this ordinary annuity is the sum:

PVAF = v + v^2 + v^3 + ... + v^(n-1) + v^n = Σ(v^x for x=1 to n)

The above is known as the present value annuity factor, or PVAF.

A closed formula for PVAF is:

PVAF = (1 - (1 + i)^-n) / i

**Present Value of an Annuity Due**

Likewise, the present value of the annuity due is the sum:

PVAFD = 1 + v + v^2 + v^3 + ... + v^(n-1) = Σ(v^x for x=0 to n-1)

A closed formula for PVAFD is:

PVAFD = (1 - (1 + i)^-n) / (1 - 1/(1 + i))

We can express PVAFD in terms of PVAF by:

PVAFD = 1 + v + v^2 + v^3 + ... + v^(n-1)

PVAFD + v^n = 1 + v + v^2 + v^3 + ... + v^(n-1) + v^n

PAVFD + v^n = 1 + PVAF

PAVFD = 1 + PVAF - v^n

PAVFD = 1 + (1 - (1 + i)^-n) / i - (1 + i)^-n

Example: i = 0.04, n = 36

PAVFD = 1 + (1 - 1.04^-36) / 0.04 - 1.04^-36 ≈ 19.6646

This is the same as setting up TVM (time value of money) keys on most financial calculators or any calculator with a TVM solver as:

n = 36, I = 4, PMT = -1, FV = 0, (if needed, P/Y = 1), BEGIN mode

Solve for PV: 19.6646...

**Future Value**

The future value of an annuity is the final value of the annuity, including the accumulated interest of payments and cash flows from the date the payment to the end of the annuity. On financial calculators, the future value is symbolized by the FV key.

Let w = 1 + i

**Future Value of an Ordinary Annuity**

Given the payment of 1 monetary unit, an annuity of n periods, equally spaced out, and interest rate i, the future value of this ordinary annuity is the sum:

FVAF = 1 + w + ... + w^(n-3) + w^(n-2) + w^(n-1) = Σ(w^x for x=0 to n-1)

The above is known as the future value annuity factor, or FVAR.

A closed formula for FVAF is:

FVAF = ( (1 + i)^n - 1 ) / i

**Future Value of an Annuity Due**

Likewise, the future value of the annuity due is the sum:

FVAFD = w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n = Σ(w^x for x=1 to n)

A closed formula for FVAFD is:

FVAFD = (1 + i) * ( (1 + i)^n - 1 ) / i

We can express FVAFD in terms of FVAF by:

FAVFD = w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n

1 + FAVFD = 1 + w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n

1 + FAVFD = FAVD + w^n

FAVFD = FAVD + w^n - 1

FAVFD = ( (1 + i)^n - 1 ) / i + (1 + i)^n - 1

Example: i = 0.04, n = 36

FAVFD = (1.04^36 - 1) / 0.04 + 1.04^36 - 1 ≈ 80.7022

Using the TVM keys or solver:

n = 36, I = 4, PMT = -1, PV = 0, (if needed, P/Y = 1), BEGIN mode

Solve for FV: 80.7022...

On tomorrow's blog, January 24, I will cover two actuarial problems.

Source:

Finan, Marcel B. __A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2__ Arkansas Tech University, 2017.

Eddie

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