## Time Value of Money: Annuities Due

In Time Value of Money: Present & Future we looked at fixed amounts and expanded that understanding into Time Value of Money: Financial Tables and Time Value of Money: Ordinary Annuities. That brings us to Annuities Due.

### Future Value Interest Factor of an Annuity Due

Cash flows of an Annuity Due are realised at the beginning of each period. In contrast, the cash flows of an Ordinary Annuity are at the end of each period.

The formula to calculate the Future Value of an Annuity Due it is necessary to adjust the Future Value Interest Factor of an Annuity to include one extra period of accrued interest:

FVIFA_{i,n} (annuity due) = FVIFA_{i,n} * (1 + i)

Running with the previous example of $2,000 payments at five per cent interest over four years we can plug in the numbers and use Financial Tables to solve for the Future Value Interest Factor of an Annuity Due:

- FVIFA
_{0.05,4}(annuity due) = FVIFA_{0.05,4}* (1 + i) - FVIFA
_{0.05,4}(annuity due) = FVIFA_{0.05,4}* 1.05 - FVIFA
_{0.05,4}(annuity due) = 4.310 * 1.05 - FVIFA
_{0.05,4}(annuity due) = 4.5255

Having resolved the FVIFA_{i,n} our Future Value of the Annuity Due equation becomes:

- Future Value of the Annuity Due = PMT * FVIFA
_{i,n}(annuity due) - Future Value of the Annuity Due = $2,000 * FVIFA
_{0.05,4}(annuity due) - Future Value of the Annuity Due = $2,000 * 4.5255
- Future Value of the Annuity Due = $9,051

The Future Value of the Annuity Due is $9,051 compared to the Future Value of the Ordinary Annuity (in the previous article using the same interest and number of terms) of $8,620.

### Present Value Interest Factor of an Annuity Due

The formula to calculate the Present Value of an Annuity Due also requires an adjustment to the Present Value Interest Factor of an Annuity because of that one extra period of accrued interest:

PVIFA_{i,n} (annuity due) = PVIFA_{i,n} * (1 + i)

So to find the Present Value Interest Factor of an Annuity Due we follow through:

- PVIFA
_{0.05,4}(annuity due) = PVIFA_{0.05,4}* (1 + i) - PVIFA
_{0.05,4}(annuity due) = 3.546 * 1.05 - PVIFA
_{0.05,4}(annuity due) = 3.7233

Therefore the Present Value of the Annuity Due equation becomes:

- Present Value of the Annuity Due = PMT * PVIFA
_{i,n}(annuity due) - Present Value of the Annuity Due = $2,000 * PVIFA
_{i,n}(annuity due) - Present Value of the Annuity Due = $2,000 * 3.7233
- Present Value of the Annuity Due = $7,446.60

The Present Value of the Annuity Due is $7,446.60 compared to the Present Value of the Ordinary Annuity (in the previous article using the same interest and number of terms) of $7,092. Again, this shows that the value of an Annuity Due for any given interest factor and number of terms will be greater than the corresponding Ordinary Annuity.

### Present Value of a Perpetuity

When an annuity possesses infinite life the term we use is Perpetuity. The Perpetuity provides a continuous cash flow for at the end of each year. To find the Present Value of a Perpetuity you need to use the equation:

PVIFA_{i,infinity} = 1 / i

An example where the Present Value of a Perpetuity might be where you wanted to set up a Trust Fund that supplied your child with an annual $10,000 payment. Given an eight per cent interest factor the formula would follow through:

- PVIFA
_{0.08,infinity}= 1 / 0.08 - PVIFA
_{0.08,infinity}= 12.5

Therefore, to calculate the Present Value of a Perpetuity:

- Present Value of the Perpetuity = PVIFA
_{0.8,infinity} - Present Value of the Perpetuity = $10,000 x 12.5
- Present Value of the Perpetuity = $125,000

This means that given a rate or return of eight per cent and an initial investment of $125,000 the Trust Fund can pay out $10,000 indefinitely and not consume any of the $125,000 invested capital.

### The Next Step: Mixed Streams of Cash Flow

In the next instalment in this Time Value of Money series we will discuss the calculation for dealing with Mixed Streams of Cash Flow.

I would also advise you to pick up any decent copy of a managerial finance textbook to underpin these articles and to complete the exercises that will cement this understanding at the end of each chapter. The textbook will also provide more precise context important to your understanding.

### Time Value of Money 101 Series

- Time Value of Money: Present & Future
- Time Value of Money: Financial Tables
- Time Value of Money: Ordinary Annuities
- Time Value of Money: Annuities Due
- Time Value of Money: Mixed Streams
- Time Value of Money: Compound Interest (redux)
- Time Value of Money: Nominal versus Effective Interest Rates
- Time Value of Money: Accumulation of a Target Sum & Loan Amortisation
- Time Value of Money: Time Periods to Reach a Sum & Growth Rates