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Janelle101 · Apr 11, 2010, 06:17 AM

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

ArcSine · Apr 11, 2010, 02:45 PM

You got it... nice job!

Janelle101 · Apr 13, 2010, 05:10 AM

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

ArcSine · Apr 13, 2010, 06:32 AM

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...

$PV\ =\ \text{pmt}\ \times\ \frac{1-(1+r)^{-n}}{r}$

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

$50,000\ +\ 50,000\ \times\ \frac{1-1.06^{-19}}{0.06}$

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

$50,000\ \times\ \frac{1-1.06^{-20}}{0.06}$ 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.

Janelle101 · Apr 13, 2010, 08:37 AM

Thank you so much.