[MUSIC]

In this video, we show how to calculate the present and

future values of an annuity due, where the cash flows occur at the beginning of

each period rather than at the end of each period for an ordinary annuity.

We will also discuss the relationship between the present and

future values of ordinary annuities and those annuity dues.

Remember that an annuity due is one where the cashflows occur at the beginning of

each time period.

What about calculating the present and future values of an annuity due.

When compared to an ordinary annuity, each payment or

cash flow occurs one period earlier in an annuity due.

Denoting the future value of an end payment annuity due as FVAd sub n,

we can write FVAd sub n equals FVA sub n times 1 plus r over m.

Using the formula for FVAb sub n, we get FVAd sub n equals

PMT time 1 plus r over m times 1 plus r over m raised to

the power of n minus 1, the whole thing divided by r over m.

Similarly, we can show that the present value today of an annuity

due denoted as PVAd sub 0 is PMT plus PMT divided by r over m times

1 minus 1 divided by 1 plus r over m, the whole raised to the power of n minus 1.

Let's look at an example, previously, we calculated the future value

of a four year $1,000 a year ordinary annuity to be $4,641.

The future value of a four year $1,000 a year annuity

due would be 4,641 times 1 plus 0.10 which equals to 5,105.10.

You can verify that the formula for the future value for

annuity due gives you the same answer.

Let's look at another example.

You have your eyes set on a new Audi A3, which sells for $34,000.

You decide to borrow the entire amount from a bank,

which will charge you an interest rate of 12% per year.

The loan is for five years and

the repayments to the bank will be on monthly basis.

The terms of the loan required you to make the payments at this beginning of each

period.

How much will you pay the bank each month.

Here you're given that the loan is annuity due.

So PVAd sub 0 equals34,000,

n is equal to 5 years times 12 months, which equals 60

months r equals 12%, m equals 12, you need to calculate PMT.

Let's plug in all the values in the formula.

PVAd sub 0 equals to PMT plus PMT divided by r over m

times 1 minus 1 divided by 1 plus r over m,

the whole ratio to the power of n minus 1.

Yeah, this gives 34,000 equals PMT plus PMT divided by 0.12 over 12

times 1 minus 1 divided 1 plus 0.12 over 12, the whole ratio to the power of 59.

Solving for PMT, we get PMT equals $748.82.

You will make 60 monthly payment of $748.82, with the first payment today.

What would happen to this present value calculation,

if the cash flow were occur forever.

In the next video, we will look at what perpetuities are,

how to calculate the present value and how they relate to stop valuation.

[MUSIC]