Please help me with the formula for this question.

Your father is about to retire, and he wants to buy an annuity that will provide him with $50,000 of income a year for 20 years, with the first payment coming immediately. The going rate on such annuities is 6%. How much would it cost him to buy the annuity today?

In excel: n = 20; I = 6; PMT = 50,000; FV = 0; = 607,905.82

You got it... nice job!

Sorry what I wanted to know was how to calculate it manually.

By "manually" I assume you mean with nothing more than the 3 Ps: pencil, paper, and pocket calculator.

Couple or three different ways you could hit it; at the end of the day they're mathematically equivalent. Here's one approach...

Pricing this annuity as described can be viewed as pricing two separate components: The first 50K which rolls in immediately, and then the other 19 payments taken together as a package which forms an 'ordinary' annuity (i.e. payments rolling in at the end of each year).

The present value of the immediately-received 50K is, well, 50K.

For the 19-payment ordinary annuity package you can use the familiar "PV-of-an-ordinary-annuity" shortcut formula...



For the thrilling finale you just put 'em together... the PV of your annuity is given by



Equivalently, you could price the whole 20-payment package as an ordinary annuity...

which would give you the value of the entire cash flow stream as of one year ago. Then multiply by 1.06 to arrive at today's value.

There is a shortcut formula for pricing 'beginning-of-year' type annuities, but the second method I just gave above is how that shortcut formula is derived.

Thank you so much.