## Wednesday, September 7, 2011

### Present value makes the world go 'round...

... the financial world, at least.

If you were with me as we calculated the present value of future receipts, consider the fact that one can compute the present value of even an infinite stream of future receipts.  As explained in Chirelstein's appendix, at 5%, the present value of \$1 a year forever is \$20.

How can one calculate the value of an infinite annuity?  If we do it manually, it's going take a long time.  We can take the present values of \$1 every future year, and add them all up.  We'd start with year 1, then year 2, year 3, etc., and keep going until we drop or until we see that it's useless to continue.  Here are the first 10 years -- and let's use \$100 annual receipts:

PV of \$100, 1 year from now: \$95.24
PV of \$100, 2 years from now: \$90.70
PV of \$100, 3 years from now: \$86.38
PV of \$100, 4 years from now: \$82.27
PV of \$100, 5 years from now: \$78.35
PV of \$100, 6 years from now: \$74.62
PV of \$100, 7 years from now: \$71.07
PV of \$100, 8 years from now: \$67.68
PV of \$100, 9 years from now: \$64.46
PV of \$100, 10 years from now: \$61.39

Adding those all up, we get to \$772.16, with many more years to go.  But as you can see, the amounts get smaller as time marches on, and that trend will continue for as long as we want to play this game.  Here's year 15:

PV of \$100, 15 years from now: \$48.10

Here's year 30:    PV of \$100, 30 years from now: \$23.14

Here's year 50:    PV of \$100, 50 years from now: \$8.72

Here's year 100:    PV of \$100, 100 years from now: \$0.76

Eventually, the present values get so small that you can hardly see them, and as it turns out, they eventually disappear from the naked eye. For example, the present value of a \$100 payment to come in 250 years from now is \$0.0005.

In the end, the present value of the infinite stream is \$2,000.  In other words, if one deposits \$2,000 in a 5% account, one will get \$100 a year to spend, every year, forever, leaving the original \$2,000 in place to keep going.

When the annual payment is \$100 and the present value is \$2,000, it is sometimes said that the "multiplier" is 20.  That is, the present value is 20 times the annual payment.  The multiplier is just the inverse of the discount rate.  1 divided by .05 = 20.

Now here's where Wall Street comes into the picture.  When discussing the price of a stock that's traded on a national exchange, people often refer to the company's "price-earnings ratio."  And that's just another name for the multiplier.

For example, take a look at this entry on Yahoo! Finance for Dow Chemical:

Over on the right, do you see the line "P/E (ttm)"?  That stands for price/earnings (trailing 12 months).  What Yahoo! has done is compare the share price of the stock (P), \$27.07, with the earnings per share of the company for the most recent 12 months (E), \$2.18.  In this case, the stock price is 12.41 times annual earnings (\$27.07/\$2.18), and so the multiplier is 12.41.

Given that multiplier, the discount rate that the market is apparently applying to that stock -- the inverse of the multiplier -- is 1 divided by 12.41, or 8.06%.  The P/E ratio is a rough way of saying that the discount rate that investors are using in pricing that stock is 8.06% a year.

The P/E ratio shown there is crude in one respect -- it looks backward at the last 12 months of earnings, which is public knowledge because of the company's regular filings with the SEC.  In fact, a smart investor is looking to the future, not the past.  He or she is trying to predict, and then present-value, future receipts.  The past may be some indicator of future earnings, but there are certainly no guarantees.  And so it's not entirely certain what discount rate the market is actually using.  But P/E based on the last 12 months tells us something.

Most importantly, the fact that the P/E ratio is calculated and published for every public company, in real time, shows that investors are keenly interested in present value.  Present value is what that ratio is all about.